Finite Group Databasebeta

← #26222820

Subgroups — #26222820

Order 12032.

954 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689363C211yesmaximal, characteristic
2C2×C4×#10625337C211yesmaximal, characteristic, complemented
3C2×#15689080C211yesmaximal, characteristic
4#20246071C211yesmaximal, characteristic
5#20246064C211yesmaximal, characteristic
6#20244246C211yesmaximal, characteristic, complemented
7#20244120C211yesmaximal, characteristic, complemented
8#20244080C211yesmaximal, characteristic, complemented
9#20244076C211yesmaximal, characteristic, complemented
10#20243820C211yesmaximal, characteristic
11#20243818C211yesmaximal, characteristic
12#20243416C211yesmaximal, characteristic
13C4×#13093957C211yesmaximal, characteristic, complemented
14#20243219C211yesmaximal, characteristic, complemented
15C47×#1763C211yesmaximal, characteristic, complemented
16C22×#10625337C2211yescharacteristic
17C22×#10625324C2211yescharacteristic
18C22×C8×D47C2211yescharacteristic
19C2×#13094004C2211yescharacteristic
20C2×#13094000C2211yescharacteristic
21C2×C4×Dic94C2211yescharacteristic
22C2×#13093981C2211yescharacteristic
23#15689363C418yescomplemented
24C4×#10625337—24no—
25C2×#13093957C2211yescharacteristic
26C2×#13093957C2211yescharacteristic
27C2×#13093957C2211yescharacteristic
28C2×#13093949C2211yescharacteristic
29C2×#13093949C2211yescharacteristic
30C2×#13093948C2211yescharacteristic
31C2×#13093947C2211yescharacteristic
32#15689080—24no—
33C2×#13093934C2211yescharacteristic
34C2×C4×C47⋊C8C2211yescharacteristic
35C2×C8×Dic47C2211yescharacteristic
36#15689030C2211yescharacteristic
37#15689028C2211yescharacteristic
38#15689022C2211yescharacteristic
39#15689021—21no—
40#15689019C2211yescharacteristic
41#15689018—21no—
42#15689015—21no—
43#15689014—21no—
44#15689014—21no—
45#15689013—21no—
46#15689008—21no—
47#15689007—21no—
48#15688980C2211yescharacteristic
49#15688980C2211yescharacteristic
50C4×C4⋊Dic47C2211yescharacteristic
51#15688886C2211yescharacteristic
52#15688886C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C2×C4×Q8×C47C2211yescharacteristic
56C2×#103×C47C2211yescharacteristic
57C2×#103×C47C2211yescharacteristic
58C2×#97×C47C2211yescharacteristic
59C4×C4⋊C4×C47C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×Dic94C2311yescharacteristic
63C22×C4×D47C2311yescharacteristic
64C2×#10625337—22no—
65C2×#10625337—22no—
66C2×#10625337C2×C414yes—
67C2×#10625324—22no—
68C2×#10625324—22no—
69C2×#10625324—22no—
70C22×C47⋊C8C2311yescharacteristic
71C22×C47⋊C8C2311yescharacteristic
72C2×C8×D47—22no—
73C2×C8×D47—22no—
74C2×C8×D47C2×C414yes—
75#13094004—22no—
76#13094004C2×C414yes—
77#13094000—22no—
78#13094000C2×C414yes—
79C4×Dic94—22no—
80C4×Dic94—22no—
81#13093981—22no—
82#13093981—22no—
83C2×C4⋊Dic47—21no—
84C2×C4⋊Dic47—21no—
85C2×C4⋊Dic47C2311yescharacteristic
86C2×C4⋊Dic47C2311yescharacteristic
87C2×C4⋊Dic47C2311yescharacteristic
88C2×Dic47⋊C4—21no—
89C2×Dic47⋊C4—21no—
90C2×Dic47⋊C4—21no—
91C2×Dic47⋊C4D411yescharacteristic
92C2×Dic47⋊C4D411yescharacteristic
93C2×(C2×D47)⋊C4—21no—
94C2×(C2×D47)⋊C4—21no—
95C2×(C2×D47)⋊C4—21no—
96C2×(C2×D47)⋊C4—21no—
97C2×(C2×D47)⋊C4D411yescharacteristic
98C2×(C2×D47)⋊C4D411yescharacteristic
99C2×C4×Dic47—21no—
100C2×C4×Dic47—21no—
101C2×C4×Dic47C2311yescharacteristic
102#13093957—22no—
103#13093957—22no—
104#13093957C2×C414yes—
105#13093949—22no—
106#13093949C2×C414yes—
107#13093948—22no—
108#13093947—22no—
109#13093934—22no—
110C4×C47⋊C8D414yes—
111C8×Dic47—22no—
112C47⋊#95—21no—
113C47⋊#95—21no—
114C47⋊#95—21no—
115C47⋊#95—21no—
116C47⋊#95—21no—
117C47⋊#95—21no—
118C47⋊#95—21no—
119C47⋊#95—21no—
120C47⋊#95—21no—
121C22×Q8×C47C2311yescharacteristic
122C22×C8×C47C2311yescharacteristic
123C22×C8×C47C2311yescharacteristic
124C4×Q8×C47—22no—
125C4×Q8×C47—22no—
126C2×C4⋊C4×C47—21no—
127C2×C4⋊C4×C47C2311yescharacteristic
128C2×C4⋊C4×C47C2311yescharacteristic
129C2×C4⋊C4×C47C2311yescharacteristic
130C2×C42×C47—21no—
131C2×C42×C47—21no—
132C2×C42×C47C2311yescharacteristic
133#103×C47—22no—
134#103×C47C2×C414yes—
135#97×C47—22no—
136#95×C47—21no—
137C23×D47C2×D411yescharacteristic
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47—21no—
142C2×Q8×C47—21no—
143C2×Q8×C47C22×C412yes—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—21no—
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2411yescharacteristic
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—21no—
154C22×Dic47—21no—
155C22×Dic47C2×D411yescharacteristic
156C22×Dic47C2×D411yescharacteristic
157C22×Dic47C2×D411yescharacteristic
158C2×C4×D47—21no—
159C2×C4×D47—21no—
160C2×C4×D47—21no—
161C2×C4×D47—21no—
162C2×C4×D47C22×C412yes—
163C2×C4×D47#4112yes—
164C2×Dic94—21no—
165C2×Dic94—21no—
166C2×Dic94—21no—
167C2×Dic94—21no—
168C2×Dic94—21no—
169C2×Dic94C22×C412yes—
170C2×C8×C47—21no—
171C2×C8×C47—21no—
172C2×C8×C47—21no—
173C2×C8×C47—21no—
174C2×C8×C47C22×C412yes—
175C2×C8×C47#4112yes—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—42no—
182C4⋊C4×C47C22×C412yes—
183C42×C47—42no—
184C42×C47—81no—
185C42×C47C2×D412yes—
186C42×C47#4112yes—
187#10625337—28no—
188(C2×D47)⋊C4—24no—
189(C2×D47)⋊C4—24no—
190(C2×D47)⋊C4—22no—
191(C2×D47)⋊C4—22no—
192(C2×D47)⋊C4—81no—
193(C2×D47)⋊C4—81no—
194C4⋊Dic47—21no—
195C4⋊Dic47—24no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47—81no—
201C4⋊Dic47C22×C412yes—
202Dic47⋊C4—22no—
203Dic47⋊C4—22no—
204Dic47⋊C4—24no—
205Dic47⋊C4—42no—
206Dic47⋊C4—81no—
207C4×Dic47—21no—
208C4×Dic47—21no—
209C4×Dic47—42no—
210C4×Dic47—81no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8C22×C412yes—
214C2×C47⋊C8C2×D412yes—
215C2×C47⋊C8C2×D412yes—
216C2×C47⋊C8#4112yes—
217#10625324—28no—
218C8×D47—28no—
219C23×C47C22×D411yescharacteristic
220C22×D47—21no—
221C22×D47—21no—
222C22×D47—21no—
223C22×D47—21no—
224C22×D47—21no—
225C22×D47—21no—
226C22×D47C4×D412yes—
227Q8×C47—22no—
228Q8×C47—22no—
229Q8×C47—41no—
230Q8×C47—41no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47#12112yes—
248C2×C4×C47#12112yes—
249C2×C4×C47C23×C411yescharacteristic
250C2×C4×C47C22×D411yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C2×C4×C47C2×#4111yescharacteristic
253C2×C4×C47C2×#4111yescharacteristic
254C2×C4×C47C2×#4111yescharacteristic
255C2×Dic47—21no—
256C2×Dic47—21no—
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—21no—
264C2×Dic47—21no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47C4×D412yes—
276C4×D47—22no—
277C4×D47—22no—
278C4×D47—22no—
279C4×D47—22no—
280Dic94—22no—
281Dic94—22no—
282Dic94—41no—
283Dic94—41no—
284C8×C47—22no—
285C8×C47—22no—
286C47⋊C8—22no—
287C47⋊C8C4×D414yes—
288#21525—471nomaximal, Sylow
289C22×C47C2×C4×D411yescharacteristic
290C22×C47C2×#12111yescharacteristic
291C22×C47#53211yescharacteristic
292C22×C47#57611yescharacteristic
293C22×C47#57811yescharacteristic
294C22×C47#57811yescharacteristic
295C22×C47#57811yescharacteristic
296C2×D47—22no—
297C2×D47—22no—
298C2×D47—22no—
299C2×D47—41no—
300C2×D47—41no—
301C2×D47—41no—
302C2×D47—41no—
303C4×C47—22no—
304C4×C47—22no—
305C4×C47—42no—
306C4×C47—42no—
307C4×C47—81no—
308C4×C47—81no—
309C4×C47C2×C4×D411yescharacteristic
310C4×C47C4×#4111yescharacteristic
311C4×C47#52511yescharacteristic
312C4×C47#52911yescharacteristic
313Dic47—22no—
314Dic47—42no—
315Dic47—41no—
316Dic47—41no—
317Dic47—42no—
318Dic47—81no—
319Dic47—81no—
320C2×#467—471no—
321C2×C4×Q16—471no—
322C2×#405—471no—
323#2323—471no—
324#2323—471no—
325#1969—471no—
326#1942—471no—
327#1935—471no—
328#1935—471no—
329#1894—471no—
330#1894—471no—
331#1808—471no—
332C4×#107—471no—
333#1763—471no—
334#1763D4711yescharacteristic, complemented
335C2×C47C4×#12111yescharacteristic
336C2×C47#295011yescharacteristic
337C2×C47#295111yescharacteristic
338C2×C47#305911yescharacteristic
339C2×C47#306111yescharacteristic
340C2×C47#317511yescharacteristic
341C2×C47#317711yescharacteristic
342D47—42no—
343C22×Q16—471no—
344C22×M16—471no—
345C23×C8—471no—
346C2×C22⋊Q8—471no—
347C2×C22⋊Q8—471no—
348C2×C4×Q8—471no—
349C2×C4×Q8C2×D4711yescharacteristic
350C2×#117—471no—
351#467—478no—
352C4×Q16—944no—
353C2×#107—471no—
354C2×#107—471no—
355C2×#107—471no—
356C2×#103—471no—
357C2×#103—471no—
358C2×#103—471no—
359C2×#103—471no—
360C2×#103C2×D4711yescharacteristic
361C2×#103C2×D4711yescharacteristic
362#405—944no—
363C2×#97—471no—
364C2×#97C2×D4711yescharacteristic
365C2×C4×C8—471no—
366C2×C4×C8—471no—
367#391—471no—
368#391—471no—
369#389—471no—
370#389—471no—
371#389—941no—
372#389—941no—
373#388—941no—
374#388—941no—
375#388—941no—
376#388—941no—
377#387—941no—
378#387—941no—
379#380—471no—
380#380—471no—
381C4×C4⋊C4—471no—
382C4×C4⋊C4C2×D4711yescharacteristic
383#340—471no—
384#340—471no—
385#340—471no—
386#340—471no—
387#340C2×D4711yescharacteristic
388#340C2×D4711yescharacteristic
389C47#2152511yescharacteristic, complemented, Sylow
390C22×Q8—471no—
391C22×Q8C22×D4711yescharacteristic
392C23×C4—471no—
393C2×Q16—474no—
394C2×Q16—942no—
395C2×Q16—942no—
396C2×M16—942no—
397C2×M16—942no—
398C2×M16—942no—
399C22×C8—471no—
400C22×C8—471no—
401C22×C8—474no—
402C22×C8—942no—
403C22×C8—942no—
404C22×C8C22×D4711yescharacteristic
405C22×C8C22×D4711yescharacteristic
406C22⋊Q8—474no—
407C22⋊Q8—474no—
408C22⋊Q8—942no—
409C22⋊Q8—942no—
410C4×Q8—22no—
411C4×Q8—22no—
412C4×Q8—942no—
413C4×Q8—942no—
414#117—942no—
415#117—942no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—471no—
418C2×C4⋊C4—471no—
419C2×C4⋊C4—471no—
420C2×C4⋊C4—471no—
421C2×C4⋊C4—471no—
422C2×C4⋊C4—941no—
423C2×C4⋊C4—941no—
424C2×C4⋊C4—941no—
425C2×C4⋊C4—941no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4C22×D4711yescharacteristic
428C2×C4⋊C4C22×D4711yescharacteristic
429C2×C4⋊C4C22×D4711yescharacteristic
430C2×C22⋊C4—471no—
431C2×C22⋊C4—471no—
432C2×C22⋊C4—941no—
433C2×C22⋊C4—941no—
434C2×C22⋊C4—941no—
435C2×C22⋊C4—941no—
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—471no—
439C2×C42—941no—
440C2×C42—941no—
441C2×C42C22×D4711yescharacteristic
442#107—474no—
443#107—942no—
444#107—942no—
445#103—22no—
446#103—474no—
447#103—942no—
448#103—942no—
449#103—942no—
450#103C4×D4714yes—
451#97—22no—
452#97—942no—
453C4×C8—474no—
454C4×C8—942no—
455#95—21no—
456#95—941no—
457#95—941no—
458#95—941no—
459#95—941no—
460#95—941no—
461#95—941no—
462#95—941no—
463#95—941no—
464#95—941no—
465C24—471no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—472no—
472C2×Q8—941no—
473C2×Q8—941no—
474C2×Q8—941no—
475C2×Q8—941no—
476C2×Q8—941no—
477C2×Q8C2×C4×D4712yes—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—471no—
483C22×C4—472no—
484C22×C4—472no—
485C22×C4—471no—
486C22×C4—471no—
487C22×C4—941no—
488C22×C4—941no—
489C22×C4—941no—
490C22×C4—941no—
491C22×C4—941no—
492C22×C4—941no—
493C22×C4—941no—
494C22×C4—941no—
495C22×C4D4×D4711yescharacteristic
496C22×C4D4×D4711yescharacteristic
497C22×C4C23×D4711yescharacteristic
498Q16—948no—
499M16—948no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—472no—
505C2×C8—472no—
506C2×C8—472no—
507C2×C8—472no—
508C2×C8—948no—
509C2×C8—941no—
510C2×C8—941no—
511C2×C8C2×C4×D4712yes—
512C2×C8#1062535012yes—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—42no—
519C4⋊C4—472no—
520C4⋊C4—944no—
521C4⋊C4—944no—
522C4⋊C4—942no—
523C4⋊C4—942no—
524C4⋊C4—941no—
525C4⋊C4—941no—
526C4⋊C4—941no—
527C4⋊C4—941no—
528C4⋊C4—941no—
529C4⋊C4—3761no—
530C4⋊C4—3761no—
531C4⋊C4—1882no—
532C4⋊C4C2×C4×D4712yes—
533C22⋊C4—944no—
534C22⋊C4—944no—
535C22⋊C4—942no—
536C22⋊C4—942no—
537C22⋊C4—3761no—
538C22⋊C4—3761no—
539C42—42no—
540C42—81no—
541C42—941no—
542C42—941no—
543C42—3761no—
544C42—1882no—
545C42#1062535012yes—
546C42D4×D4712yes—
547C23—472no—
548C23—941no—
549C23—941no—
550C23—941no—
551C23—941no—
552C23—941no—
553C23—941no—
554C23C2×D4×D4711yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559Q8—942no—
560Q8—942no—
561Q8—1881no—
562Q8—1881no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—472no—
580C2×C4—941no—
581C2×C4—941no—
582C2×C4—942no—
583C2×C4—942no—
584C2×C4—942no—
585C2×C4—942no—
586C2×C4—942no—
587C2×C4—942no—
588C2×C4—941no—
589C2×C4—941no—
590C2×C4—941no—
591C2×C4—941no—
592C2×C4—941no—
593C2×C4—941no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4—1881no—
598C2×C4—1881no—
599C2×C4—1881no—
600C2×C4—1881no—
601C2×C4—1881no—
602C2×C4—1881no—
603C2×C4—1881no—
604C2×C4#1309399712yes—
605C2×C4#1309399712yes—
606C2×C4C22×C4×D4711yescharacteristic
607C2×C4C2×D4×D4711yescharacteristic
608C2×C4#41×D4711yescharacteristic
609C2×C4#41×D4711yescharacteristic
610C2×C4C2×#1062535011yescharacteristic
611C2×C4C2×#1062535011yescharacteristic
612C8—22no—
613C8—22no—
614C8—474no—
615C8—942no—
616C22—942no—
617C22—942no—
618C22—942no—
619C22—1881no—
620C22—1881no—
621C22—1881no—
622C22—1881no—
623C22C4×D4×D4711yescharacteristic
624C22C2×#1309399711yescharacteristic
625C22#1568967011yescharacteristic
626C22#1568992711yescharacteristic
627C22#1568993911yescharacteristic
628C22#1568993911yescharacteristic
629C22#1568993911yescharacteristic
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—81no—
635C4—81no—
636C4—942no—
637C4—3761no—
638C4—3761no—
639C4—1882no—
640C4—1881no—
641C4—1881no—
642C4—1882no—
643C4C4×D4×D4711yescharacteristic
644C4C4×#1062535011yescharacteristic
645C4#1568963211yescharacteristic
646C4#1568965111yescharacteristic
647C2—1882no—
648C2C4×#1309399711yescharacteristic
649C2#2025225311yescharacteristic
650C2#2025226011yescharacteristic
651C2#2025304211yescharacteristic
652C2#2025305711yescharacteristic
653C2#2025402511yescharacteristic
654C2#2025404511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1763C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1763D47
hypocenterC47#21525
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
2. C2×C4×#10625337C2semidirect
6. #20244246C2semidirect
7. #20244120C2semidirect
8. #20244080C2semidirect
9. #20244076C2semidirect
13. C4×#13093957C2semidirect
14. #20243219C2semidirect
15. C47×#1763C2semidirect
23. #15689363C4semidirect
334. #1763D47semidirect
389. C47#21525semidirect

Maximal subgroups

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

subgroupindex
C2×#156893632
C2×C4×#106253372
C2×#156890802
#202460712
#202460642
#202442462
#202441202
#202440802
#202440762
#202438202
#202438182
#202434162
C4×#130939572
#202432192
C47×#17632
#2152547