Finite Group Databasebeta

← #26219528

Subgroups — #26219528

Order 12032.

764 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252156C211yesmaximal, characteristic
2#20252038C211yesmaximal, characteristic, complemented
3#20251814C211yesmaximal, characteristic, complemented
4#20244037C211yesmaximal, characteristic, complemented
5#20244035C211yesmaximal, characteristic, complemented
6#20243990C211yesmaximal, characteristic, complemented
7#20243984C211yesmaximal, characteristic, complemented
8#20243715C211yesmaximal, characteristic
9#20243713C211yesmaximal, characteristic
10#20243665C211yesmaximal, characteristic, complemented
11#108×Dic47C211yesmaximal, characteristic, complemented
12#20243200C211yesmaximal, characteristic, complemented
13#20243197C211yesmaximal, characteristic, complemented
14#20243138C211yesmaximal, characteristic, complemented
15C47×#1866C211yesmaximal, characteristic, complemented
16C2×#13094044C2211yescharacteristic
17C2×#13094031C2211yescharacteristic
18#41×Dic47C2211yescharacteristic
19#15689591C2211yescharacteristic
20#15689215C411yescharacteristic, complemented
21#15689215C411yescharacteristic, complemented
22#15689214C411yescharacteristic, complemented
23#15689214C411yescharacteristic, complemented
24#15689208C411yescharacteristic, complemented
25#15689208C411yescharacteristic, complemented
26#15689203C411yescharacteristic, complemented
27#15689203C411yescharacteristic, complemented
28#15689183—21no—
29#15689183—21no—
30#15689179—21no—
31#15689179—21no—
32C2×#13093960C2211yescharacteristic
33#15689159C2211yescharacteristic
34#15689150C2211yescharacteristic
35C2×#13093952C2211yescharacteristic
36C2×#13093951C2211yescharacteristic
37C2×#13093950C2211yescharacteristic
38#15689120C2211yescharacteristic
39#15689091C2211yescharacteristic
40#15689089C2211yescharacteristic
41#15689081—21no—
42#15689081C2211yescharacteristic
43#15689078—21no—
44#15689078C2211yescharacteristic
45M16×Dic47—21no—
46M16×Dic47C2211yescharacteristic
47M16×Dic47C2211yescharacteristic
48C2×#13093934—21no—
49C2×#13093934C2211yescharacteristic
50C2×C8×Dic47C2211yescharacteristic
51#15688969C2211yescharacteristic
52#15688887C2211yescharacteristic
53#15688887C2211yescharacteristic
54#15688884C2211yescharacteristic
55#15688883C2211yescharacteristic
56#15688882C2211yescharacteristic
57#15688878C2211yescharacteristic
58#15688877C2211yescharacteristic
59#15688877C2211yescharacteristic
60C2×#131×C47C2211yescharacteristic
61C2×#108×C47C2211yescharacteristic
62C47×#424C2211yescharacteristic
63C2×#104×C47C2211yescharacteristic
64C47×#416C2211yescharacteristic
65C47×#339C2211yescharacteristic
66C47×#336C2211yescharacteristic
67C2×#10625350C2311yescharacteristic
68#13094044—22no—
69#13094044—22no—
70C2×#10625337C2×C411yescharacteristic
71C2×#10625335C2×C411yescharacteristic
72C2×#10625336C2×C411yescharacteristic
73C2×#10625334C2×C411yescharacteristic
74#13094031—22no—
75#13094031—22no—
76C2×#10625329—21no—
77C2×#10625329C2311yescharacteristic
78C2×#10625329C2311yescharacteristic
79C22×C47⋊C8—21no—
80C22×C47⋊C8C2311yescharacteristic
81C22×C47⋊C8C2311yescharacteristic
82Q8×Dic47C2×C411yescharacteristic
83Q8×Dic47C2×C411yescharacteristic
84#13093989C2×C411yescharacteristic
85#13093989C2×C411yescharacteristic
86D4×Dic47—21no—
87D4×Dic47—21no—
88D4×Dic47C2×C411yescharacteristic
89D4×Dic47C2×C411yescharacteristic
90#13093986C2×C411yescharacteristic
91#13093986C2×C411yescharacteristic
92(C22⋊D47)⋊C4—21no—
93(C22⋊D47)⋊C4—21no—
94#13093983—21no—
95#13093983C2311yescharacteristic
96#13093981—21no—
97C2×C4×Dic47—21no—
98C2×C4×Dic47C2311yescharacteristic
99C2×C4×Dic47C2311yescharacteristic
100C42×D47—21no—
101#13093960—22no—
102#13093957C2×C411yescharacteristic
103#13093957C2×C411yescharacteristic
104#13093955C2×C411yescharacteristic
105#13093955C2×C411yescharacteristic
106C4⋊(C47⋊C8)—21no—
107C4⋊(C47⋊C8)—21no—
108#13093952—22no—
109#13093951—22no—
110#13093950—22no—
111#13093949C2×C411yescharacteristic
112#13093949C2×C411yescharacteristic
113#13093946C2×C411yescharacteristic
114#13093946C2×C411yescharacteristic
115(C2×D47)⋊C8—21no—
116(C2×D47)⋊C8—21no—
117(C2×D47)⋊C8—21no—
118(C2×D47)⋊C8—21no—
119#13093934—21no—
120#13093934—21no—
121#13093934—21no—
122#13093934—21no—
123#13093934D411yescharacteristic
124#13093934D411yescharacteristic
125#13093934D411yescharacteristic
126#13093934D411yescharacteristic
127C8×Dic47—21no—
128C8×Dic47—21no—
129C8×Dic47—21no—
130C8×Dic47—21no—
131C8×Dic47C2×C411yescharacteristic
132C8×Dic47C2×C411yescharacteristic
133C8×Dic47C2×C411yescharacteristic
134C8×Dic47C2×C411yescharacteristic
135C47⋊#95—21no—
136C47⋊#95—21no—
137C2×#41×C47C2311yescharacteristic
138#131×C47—22no—
139#131×C47—22no—
140C2×M16×C47—21no—
141C2×M16×C47C2311yescharacteristic
142C2×M16×C47C2311yescharacteristic
143C22×C8×C47—21no—
144C22×C8×C47C2311yescharacteristic
145C22×C8×C47C2311yescharacteristic
146#117×C47C2311yescharacteristic
147C2×C42×C47C2311yescharacteristic
148#108×C47—22no—
149C4⋊C8×C47—21no—
150C4⋊C8×C47—21no—
151#104×C47—22no—
152#103×C47C2×C411yescharacteristic
153#103×C47C2×C411yescharacteristic
154#102×C47C2×C411yescharacteristic
155#102×C47C2×C411yescharacteristic
156#41×C47—21no—
157#41×C47—21no—
158#41×C47#4112yes—
159#41×C47#4112yes—
160C2×Q8×C47C22×C411yescharacteristic
161C2×D4×C47—21no—
162C2×D4×C47C22×C411yescharacteristic
163C22×C4×C47—21no—
164C22×C4×C47—21no—
165C22×C4×C47C2411yescharacteristic
166C2×C22⋊D47—21no—
167C22×Dic47—21no—
168C22×Dic47—21no—
169C22×Dic47—21no—
170C22×Dic47#4111yescharacteristic
171C22×Dic47#4111yescharacteristic
172#10625350—21no—
173#10625350—21no—
174#10625350—21no—
175#10625350—21no—
176C2×D188C22×C411yescharacteristic
177C2×C4×D47—21no—
178C2×C4×D47—21no—
179C2×C4×D47—21no—
180C2×Dic94C22×C411yescharacteristic
181M16×C47—24no—
182M16×C47—21no—
183M16×C47—21no—
184M16×C47—21no—
185M16×C47#4112yes—
186C2×C8×C47—24no—
187C2×C8×C47—21no—
188C2×C8×C47—21no—
189C2×C8×C47—21no—
190C2×C8×C47—21no—
191C2×C8×C47—21no—
192C2×C8×C47—21no—
193C2×C8×C47C22×C411yescharacteristic
194C2×C8×C47C22×C411yescharacteristic
195C2×C8×C47C2×D411yescharacteristic
196C2×C8×C47C2×D411yescharacteristic
197C4⋊C4×C47C22×C411yescharacteristic
198C4⋊C4×C47C22×C411yescharacteristic
199C22⋊C4×C47—21no—
200C42×C47—21no—
201C42×C47—21no—
202C42×C47—21no—
203C22⋊Dic47—21no—
204C22⋊Dic47—21no—
205C22⋊Dic47—21no—
206#10625337—22no—
207#10625336—22no—
208#10625335—22no—
209#10625334—22no—
210(C2×D47)⋊C4—21no—
211(C2×D47)⋊C4—21no—
212C4⋊Dic47—21no—
213C4⋊Dic47—21no—
214C4⋊Dic47C22×C411yescharacteristic
215C4⋊Dic47C22×C411yescharacteristic
216Dic47⋊C4—21no—
217Dic47⋊C4—21no—
218C4×Dic47—21no—
219C4×Dic47—21no—
220C4×Dic47—21no—
221C4×Dic47—21no—
222C4×Dic47—21no—
223C4×Dic47—21no—
224C4×Dic47—21no—
225C4×Dic47C22×C411yescharacteristic
226C4×Dic47C22×C411yescharacteristic
227C4×Dic47C2×D411yescharacteristic
228C4×Dic47C2×D411yescharacteristic
229#10625329—24no—
230#10625329—21no—
231#10625329—21no—
232#10625329—21no—
233#10625329#4112yes—
234C2×C47⋊C8—24no—
235C2×C47⋊C8—21no—
236C2×C47⋊C8—21no—
237C2×C47⋊C8—21no—
238C2×C47⋊C8—21no—
239C2×C47⋊C8—21no—
240C2×C47⋊C8—21no—
241C2×C47⋊C8C22×C411yescharacteristic
242C2×C47⋊C8C22×C411yescharacteristic
243C2×C47⋊C8C2×D411yescharacteristic
244C2×C47⋊C8C2×D411yescharacteristic
245C23×C47—21no—
246C23×C47C2×#4111yescharacteristic
247C22×D47—21no—
248Q8×C47—21no—
249Q8×C47#11712yes—
250D4×C47—22no—
251D4×C47—22no—
252D4×C47—21no—
253D4×C47#11712yes—
254C2×C4×C47—21no—
255C2×C4×C47—22no—
256C2×C4×C47—22no—
257C2×C4×C47—21no—
258C2×C4×C47—21no—
259C2×C4×C47—41no—
260C2×C4×C47—41no—
261C2×C4×C47—41no—
262C2×C4×C47C23×C411yescharacteristic
263C2×C4×C47C22×D411yescharacteristic
264C2×C4×C47C2×#4111yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C2×C4×C47C2×#4111yescharacteristic
267C2×C4×C47C2×#4111yescharacteristic
268C22⋊D47—41no—
269C22⋊D47—41no—
270C2×Dic47—21no—
271C2×Dic47—21no—
272C2×Dic47—21no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—81no—
279C2×Dic47—81no—
280C2×Dic47C4×D411yescharacteristic
281C2×Dic47C4×D411yescharacteristic
282C2×Dic47C4×D411yescharacteristic
283C2×Dic47C4×D411yescharacteristic
284D188—21no—
285D188—21no—
286C4×D47—41no—
287C4×D47—41no—
288C4×D47—81no—
289C4×D47—81no—
290Dic94—21no—
291Dic94—21no—
292C8×C47—21no—
293C8×C47—21no—
294C8×C47—21no—
295C8×C47—22no—
296C8×C47—21no—
297C8×C47—41no—
298C47⋊C8—21no—
299C47⋊C8—21no—
300C47⋊C8—21no—
301C47⋊C8—22no—
302C47⋊C8—21no—
303C47⋊C8—41no—
304#21140—471nomaximal, Sylow
305C22×C47—21no—
306C22×C47—21no—
307C22×C47—22no—
308C22×C47—41no—
309C22×C47—41no—
310C22×C47C2×C4×D411yescharacteristic
311C22×C47#52911yescharacteristic
312C22×C47#52911yescharacteristic
313C2×D47—41no—
314C2×D47—41no—
315C2×D47—41no—
316C4×C47—22no—
317C4×C47—41no—
318C4×C47—41no—
319C4×C47—81no—
320C4×C47C2×#11711yescharacteristic
321C4×C47C4×#4111yescharacteristic
322C4×C47#52511yescharacteristic
323C4×C47#52511yescharacteristic
324Dic47—21no—
325Dic47—21no—
326Dic47—21no—
327Dic47—21no—
328Dic47—41no—
329Dic47—41no—
330Dic47—41no—
331Dic47—81no—
332C4×#135—471no—
333#2923—471no—
334#2875—471no—
335#1929—471no—
336#1929—471no—
337#1922—471no—
338#1922—471no—
339#1876—471no—
340#1876—471no—
341#1866—471no—
342#1866D4711yescharacteristic, complemented
343C4×#108—471no—
344#1759—471no—
345#1759—471no—
346#1746—471no—
347C2×C47—21no—
348C2×C47—21no—
349C2×C47—42no—
350C2×C47#229011yescharacteristic
351C2×C47#295711yescharacteristic
352C2×C47#295711yescharacteristic
353D47—81no—
354C2×#135—471no—
355C2×#131—471no—
356C2×#131C2×D4711yescharacteristic
357C4×#41—471no—
358C4×#41—471no—
359C4×Q16—471no—
360C4×Q16—471no—
361C4×SD16—471no—
362C4×SD16—471no—
363C4×SD16—471no—
364C4×SD16—471no—
365C4×D8—471no—
366C4×D8—471no—
367#433—941no—
368#433—941no—
369#432—941no—
370#432—941no—
371C2×#108—471no—
372C2×#108C2×D4711yescharacteristic
373#427—471no—
374#424—471no—
375#424C2×D4711yescharacteristic
376C2×#104—471no—
377C2×#104—471no—
378C2×#104—471no—
379C2×#104C2×D4711yescharacteristic
380#416—471no—
381#416C2×D4711yescharacteristic
382#408—471no—
383#408—471no—
384#405—471no—
385#405—471no—
386#405—941no—
387#405—941no—
388C4×M16—471no—
389C4×M16—471no—
390C4×M16—941no—
391C2×#97—471no—
392C2×#97—941no—
393C2×C4×C8—471no—
394#376—471no—
395#341—471no—
396#341—471no—
397#339—471no—
398#339—471no—
399#339—471no—
400#339C2×D4711yescharacteristic
401#336—471no—
402#336—471no—
403#336—471no—
404#336C2×D4711yescharacteristic
405C47#2114011yescharacteristic, complemented, Sylow
406C2×#41—471no—
407C2×#41C22×D4711yescharacteristic
408#135—942no—
409#135—942no—
410C2×Q16—471no—
411C2×SD16—471no—
412C2×SD16—471no—
413C2×D8—471no—
414#131—22no—
415#131—22no—
416#131—942no—
417#131—942no—
418C2×M16—21no—
419C2×M16—471no—
420C2×M16—471no—
421C2×M16—941no—
422C2×M16C22×D4711yescharacteristic
423C2×M16C22×D4711yescharacteristic
424C22×C8—21no—
425C22×C8—471no—
426C22×C8—471no—
427C22×C8—941no—
428C22×C8C22×D4711yescharacteristic
429C22×C8C22×D4711yescharacteristic
430C4×Q8—471no—
431C4×Q8—471no—
432C4×Q8—471no—
433C4×Q8—471no—
434C4×D4—471no—
435C4×D4—471no—
436C4×D4—471no—
437C4×D4—471no—
438C4×D4—941no—
439C4×D4—941no—
440C4×D4—941no—
441C4×D4—941no—
442#117—471no—
443#117—941no—
444#117—941no—
445#117C22×D4711yescharacteristic
446C2×C42—471no—
447C2×C42—471no—
448C2×C42—941no—
449C2×C42—941no—
450C2×C42C22×D4711yescharacteristic
451#108—22no—
452#108—942no—
453#107—471no—
454#107—471no—
455#106—471no—
456#106—471no—
457C4⋊C8—21no—
458C4⋊C8—21no—
459C4⋊C8—941no—
460C4⋊C8—941no—
461#104—22no—
462#104—942no—
463#104—942no—
464#104—942no—
465#103—471no—
466#103—471no—
467#103C4×D4711yescharacteristic
468#103C4×D4711yescharacteristic
469#102—471no—
470#102—471no—
471#102C4×D4711yescharacteristic
472#102C4×D4711yescharacteristic
473C22⋊C8—941no—
474C22⋊C8—941no—
475C22⋊C8—941no—
476C22⋊C8—941no—
477#97—471no—
478#97—471no—
479#97—471no—
480#97—471no—
481#97—941no—
482#97—941no—
483#97—941no—
484#97—941no—
485C4×C8—471no—
486C4×C8—471no—
487C4×C8—471no—
488C4×C8—471no—
489C4×C8—941no—
490C4×C8—941no—
491C4×C8—941no—
492C4×C8—941no—
493#95—941no—
494#95—941no—
495#41—21no—
496#41—21no—
497#41—941no—
498#41—941no—
499#41—941no—
500#41—941no—
501#41#1062535212yes—
502#41#1062535412yes—
503C2×Q8—471no—
504C2×Q8C2×C4×D4711yescharacteristic
505C2×D4—21no—
506C2×D4—471no—
507C2×D4—941no—
508C2×D4C2×C4×D4711yescharacteristic
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—471no—
512C22×C4—471no—
513C22×C4—941no—
514C22×C4—941no—
515C22×C4—941no—
516C22×C4—941no—
517C22×C4—941no—
518C22×C4—941no—
519C22×C4C23×D4711yescharacteristic
520Q16—942no—
521SD16—942no—
522SD16—942no—
523D8—942no—
524M16—24no—
525M16—21no—
526M16—21no—
527M16—21no—
528M16—472no—
529M16—944no—
530M16—941no—
531M16—941no—
532M16—941no—
533M16#1062535212yes—
534C2×C8—24no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—471no—
542C2×C8—471no—
543C2×C8—471no—
544C2×C8—471no—
545C2×C8—944no—
546C2×C8—941no—
547C2×C8—941no—
548C2×C8—941no—
549C2×C8—941no—
550C2×C8—941no—
551C2×C8—941no—
552C2×C8C2×C4×D4711yescharacteristic
553C2×C8C2×C4×D4711yescharacteristic
554C2×C8D4×D4711yescharacteristic
555C2×C8D4×D4711yescharacteristic
556C4⋊C4—471no—
557C4⋊C4—471no—
558C4⋊C4—941no—
559C4⋊C4—941no—
560C4⋊C4—941no—
561C4⋊C4—941no—
562C4⋊C4C2×C4×D4711yescharacteristic
563C4⋊C4C2×C4×D4711yescharacteristic
564C22⋊C4—21no—
565C22⋊C4—941no—
566C22⋊C4—941no—
567C22⋊C4—941no—
568C22⋊C4—941no—
569C22⋊C4—941no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42—471no—
574C42—471no—
575C42—471no—
576C42—471no—
577C42—941no—
578C42—941no—
579C42—941no—
580C42—941no—
581C42—941no—
582C42—941no—
583C42—941no—
584C23—21no—
585C23—941no—
586C23#41×D4711yescharacteristic
587Q8—21no—
588Q8—941no—
589Q8—941no—
590Q8#1309398212yes—
591D4—22no—
592D4—22no—
593D4—21no—
594D4—941no—
595D4—941no—
596D4—1881no—
597D4—1881no—
598D4#1309398212yes—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—471no—
608C2×C4—471no—
609C2×C4—471no—
610C2×C4—471no—
611C2×C4—941no—
612C2×C4—941no—
613C2×C4—941no—
614C2×C4—3761no—
615C2×C4—3761no—
616C2×C4—3761no—
617C2×C4—3761no—
618C2×C4—1881no—
619C2×C4—1881no—
620C2×C4—1881no—
621C2×C4—1881no—
622C2×C4—1881no—
623C2×C4—1881no—
624C2×C4—1881no—
625C2×C4C22×C4×D4711yescharacteristic
626C2×C4C2×D4×D4711yescharacteristic
627C2×C4#41×D4711yescharacteristic
628C2×C4C2×#1062535211yescharacteristic
629C2×C4C2×#1062535211yescharacteristic
630C2×C4C2×#1062535411yescharacteristic
631C8—21no—
632C8—21no—
633C8—21no—
634C8—22no—
635C8—21no—
636C8—41no—
637C8—941no—
638C8—941no—
639C8—941no—
640C8—941no—
641C8—942no—
642C8—1881no—
643C22—21no—
644C22—21no—
645C22—22no—
646C22—41no—
647C22—41no—
648C22—1881no—
649C22—1881no—
650C22—1881no—
651C22C4×D4×D4711yescharacteristic
652C22#1568965011yescharacteristic
653C22#1568965911yescharacteristic
654C4—22no—
655C4—41no—
656C4—41no—
657C4—81no—
658C4—941no—
659C4—941no—
660C4—941no—
661C4—941no—
662C4—3761no—
663C4—1881no—
664C4—1881no—
665C4—1881no—
666C4C2×#1309398211yescharacteristic
667C4C4×#1062535211yescharacteristic
668C4#1568962911yescharacteristic
669C4#1568963611yescharacteristic
670C2—21no—
671C2—21no—
672C2—42no—
673C2—3761no—
674C2#2024582411yescharacteristic
675C2#2025230011yescharacteristic
676C2#2025230211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1866C2
socleC22×C47C2×C4×D4
prime radicalC2×#13094044C22
cosocleC22×C4×C47C24
hypercenter#1866D47
hypocenterC47#21140
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1866

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20252038C2semidirect
3. #20251814C2semidirect
4. #20244037C2semidirect
5. #20244035C2semidirect
6. #20243990C2semidirect
7. #20243984C2semidirect
10. #20243665C2semidirect
11. #108×Dic47C2semidirect
12. #20243200C2semidirect
13. #20243197C2semidirect
14. #20243138C2semidirect
15. C47×#1866C2semidirect
20. #15689215C4semidirect
21. #15689215C4semidirect
22. #15689214C4semidirect
23. #15689214C4semidirect
24. #15689208C4semidirect
25. #15689208C4semidirect
26. #15689203C4semidirect
27. #15689203C4semidirect
342. #1866D47semidirect
405. C47#21140semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202521562
#202520382
#202518142
#202440372
#202440352
#202439902
#202439842
#202437152
#202437132
#202436652
#108×Dic472
#202432002
#202431972
#202431382
C47×#18662
#2114047