Finite Group Databasebeta

← #26238372

Subgroups — #26238372

Order 12032.

692 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253174C211yesmaximal, characteristic
2#20252332C211yesmaximal, characteristic, complemented
3#20251955C211yesmaximal, characteristic
4#20244367C211yesmaximal, characteristic, complemented
5#20244162C211yesmaximal, characteristic, complemented
6#20244142C211yesmaximal, characteristic, complemented
7#20244133C211yesmaximal, characteristic, complemented
8#20243990C211yesmaximal, characteristic, complemented
9#20243666C211yesmaximal, characteristic
10#20243664C211yesmaximal, characteristic
11#20243244C211yesmaximal, characteristic, complemented
12#20243231C211yesmaximal, characteristic
13#20243227C211yesmaximal, characteristic
14#20243197C211yesmaximal, characteristic
15C47×#1922C211yesmaximal, characteristic, complemented
16C2×#13094032C2211yescharacteristic
17C2×#13094030C2211yescharacteristic
18#15689676C2211yescharacteristic
19#15689591C2211yescharacteristic
20#15689433C411yescharacteristic, complemented
21#15689429C411yescharacteristic, complemented
22#15689426C411yescharacteristic, complemented
23#15689422C411yescharacteristic, complemented
24#15689417C411yescharacteristic, complemented
25#15689412C411yescharacteristic, complemented
26#15689407C411yescharacteristic, complemented
27#15689402C411yescharacteristic, complemented
28#15689253—21no—
29#15689251—21no—
30#15689251—21no—
31C8×Dic94—21no—
32C2×#13093960C2211yescharacteristic
33C2×#13093959C2211yescharacteristic
34#15689159C2211yescharacteristic
35#15689150C2211yescharacteristic
36#15689150C2211yescharacteristic
37#15689149C2211yescharacteristic
38#15689148—21no—
39#15689148—21no—
40#15689146C2211yescharacteristic
41#15689145—21no—
42C2×Dic47⋊C8C2211yescharacteristic
43C2×Dic47⋊C8C2211yescharacteristic
44C4⋊C8×D47—21no—
45C2×#13093952C2211yescharacteristic
46C2×#13093950C2211yescharacteristic
47#15689120C2211yescharacteristic
48#15689117C2211yescharacteristic
49#15689081C2211yescharacteristic
50M16×Dic47C2211yescharacteristic
51C2×#13093934C2211yescharacteristic
52C2×C8×Dic47C2211yescharacteristic
53#15688969C2211yescharacteristic
54#15688887C2211yescharacteristic
55#15688884C2211yescharacteristic
56#15688883C2211yescharacteristic
57#15688882C2211yescharacteristic
58#15688882C2211yescharacteristic
59#15688878C2211yescharacteristic
60C47×#427C2211yescharacteristic
61C47×#424C2211yescharacteristic
62C47×#405C2211yescharacteristic
63C47×#341C2211yescharacteristic
64C47×#339C2211yescharacteristic
65C47×#339C2211yescharacteristic
66C47×#336C2211yescharacteristic
67C2×#10625350C2311yescharacteristic
68#13094032—22no—
69#13094032—41no—
70#13094030—22no—
71#13094030—41no—
72C2×#10625329C2311yescharacteristic
73C2×#10625329C2311yescharacteristic
74C2×#10625324—21no—
75C2×#10625324—21no—
76C22×C47⋊C8C2311yescharacteristic
77C22×C47⋊C8C2311yescharacteristic
78C2×C8×D47—21no—
79C2×C8×D47—21no—
80#13094023C2×C411yescharacteristic
81#13094023C2×C411yescharacteristic
82#13094015C2×C411yescharacteristic
83#13094015C2×C411yescharacteristic
84#13094001—21no—
85#13094001—21no—
86C4×Dic94—21no—
87C4×Dic94—21no—
88#13093989C2×C411yescharacteristic
89#13093989C2×C411yescharacteristic
90#13093986C2×C411yescharacteristic
91#13093986C2×C411yescharacteristic
92(C22⋊D47)⋊C4—21no—
93(C22⋊D47)⋊C4—21no—
94#13093983C2311yescharacteristic
95#13093981—21no—
96C2×C4×Dic47C2311yescharacteristic
97C2×C4×Dic47C2311yescharacteristic
98C42×D47—21no—
99#13093960—41no—
100#13093959—41no—
101#13093957C2×C411yescharacteristic
102#13093957C2×C411yescharacteristic
103#13093955C2×C411yescharacteristic
104#13093955C2×C411yescharacteristic
105C4⋊(C47⋊C8)—21no—
106C4⋊(C47⋊C8)—21no—
107C4⋊(C47⋊C8)—21no—
108C4⋊(C47⋊C8)—21no—
109Dic47⋊C8—21no—
110Dic47⋊C8—21no—
111Dic47⋊C8—21no—
112Dic47⋊C8—21no—
113Dic47⋊C8C2×C411yescharacteristic
114Dic47⋊C8C2×C411yescharacteristic
115Dic47⋊C8C2×C411yescharacteristic
116Dic47⋊C8C2×C411yescharacteristic
117Dic47⋊C8D411yescharacteristic
118Dic47⋊C8D411yescharacteristic
119Dic47⋊C8D411yescharacteristic
120Dic47⋊C8D411yescharacteristic
121#13093952—22no—
122#13093950Q814yescomplemented
123#13093949C2×C411yescharacteristic
124#13093949C2×C411yescharacteristic
125#13093947C2×C411yescharacteristic
126#13093947C2×C411yescharacteristic
127#13093946C2×C411yescharacteristic
128#13093946C2×C411yescharacteristic
129#13093944C2×C411yescharacteristic
130#13093944C2×C411yescharacteristic
131#13093934—21no—
132#13093934—21no—
133#13093934—41no—
134C8×Dic47—21no—
135C8×Dic47—21no—
136C8×Dic47—41no—
137C47⋊#95—21no—
138C47⋊#95—21no—
139C2×M16×C47C2311yescharacteristic
140C2×M16×C47C2311yescharacteristic
141C22×C8×C47C2311yescharacteristic
142C22×C8×C47C2311yescharacteristic
143#117×C47C2311yescharacteristic
144#117×C47C2311yescharacteristic
145C2×C42×C47C2311yescharacteristic
146#107×C47C2×C411yescharacteristic
147#107×C47C2×C411yescharacteristic
148#106×C47C2×C411yescharacteristic
149#106×C47C2×C411yescharacteristic
150C4⋊C8×C47—21no—
151C4⋊C8×C47—21no—
152#97×C47—21no—
153C4×C8×C47—21no—
154C22×C4×C47—21no—
155C22×C4×C47C2411yescharacteristic
156C2×C22⋊D47—21no—
157C22×Dic47—21no—
158C22×Dic47#4111yescharacteristic
159C22×Dic47#4111yescharacteristic
160#10625350—21no—
161#10625350—21no—
162#10625350C2×Q812yes—
163#10625350#4112yes—
164C2×D188C22×C411yescharacteristic
165C2×C4×D47—21no—
166C2×C4×D47—21no—
167C2×C4×D47—21no—
168C2×Dic94—21no—
169C2×Dic94—21no—
170C2×Dic94C22×C411yescharacteristic
171M16×C47—21no—
172M16×C47—21no—
173M16×C47—21no—
174M16×C47C2×Q812yes—
175C2×C8×C47—21no—
176C2×C8×C47—21no—
177C2×C8×C47—21no—
178C2×C8×C47—21no—
179C2×C8×C47—21no—
180C2×C8×C47—21no—
181C2×C8×C47C22×C411yescharacteristic
182C2×C8×C47C22×C411yescharacteristic
183C2×C8×C47C2×D411yescharacteristic
184C2×C8×C47C2×D411yescharacteristic
185C4⋊C4×C47C22×C411yescharacteristic
186C4⋊C4×C47C22×C411yescharacteristic
187C4⋊C4×C47C22×C411yescharacteristic
188C4⋊C4×C47C22×C411yescharacteristic
189C22⋊C4×C47—21no—
190C22⋊C4×C47—21no—
191C42×C47—21no—
192C42×C47—21no—
193C42×C47—21no—
194C42×C47—21no—
195C22⋊Dic47—21no—
196(C2×D47)⋊C4—21no—
197(C2×D47)⋊C4—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47C22×C411yescharacteristic
201C4⋊Dic47C22×C411yescharacteristic
202Dic47⋊C4—21no—
203Dic47⋊C4—21no—
204Dic47⋊C4—21no—
205Dic47⋊C4—21no—
206Dic47⋊C4—21no—
207Dic47⋊C4—21no—
208C4×Dic47—21no—
209C4×Dic47—21no—
210C4×Dic47—21no—
211C4×Dic47—21no—
212C4×Dic47C22×C411yescharacteristic
213C4×Dic47C22×C411yescharacteristic
214C4×Dic47C2×D411yescharacteristic
215C4×Dic47C2×D411yescharacteristic
216C4×Dic47C2×Q812yes—
217#10625329—21no—
218#10625329—21no—
219#10625329—21no—
220#10625329—21no—
221C2×C47⋊C8—21no—
222C2×C47⋊C8—21no—
223C2×C47⋊C8—21no—
224C2×C47⋊C8—21no—
225C2×C47⋊C8—21no—
226C2×C47⋊C8—21no—
227C2×C47⋊C8C22×C411yescharacteristic
228C2×C47⋊C8C22×C411yescharacteristic
229C2×C47⋊C8C2×D411yescharacteristic
230C2×C47⋊C8C2×D411yescharacteristic
231#10625324—42no—
232#10625324—42no—
233C8×D47—42no—
234C8×D47—42no—
235C23×C47C2×#4111yescharacteristic
236C22×D47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47C23×C411yescharacteristic
245C2×C4×C47C22×D411yescharacteristic
246C2×C4×C47C22×Q811yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47H(5,𝔽2)11yescharacteristic
249C2×C4×C472−1+411yescharacteristic
250C22⋊D47—22no—
251C22⋊D47—22no—
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—21no—
255C2×Dic47—22no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47C4×D411yescharacteristic
261C2×Dic47C4×D411yescharacteristic
262C2×Dic47C4×D411yescharacteristic
263C2×Dic47C4×D411yescharacteristic
264D188—21no—
265D188C4×Q812yes—
266C4×D47—22no—
267C4×D47—22no—
268C4×D47—42no—
269C4×D47—81no—
270Dic94—21no—
271Dic94—42no—
272Dic94—81no—
273Dic94C4×Q812yes—
274C8×C47—21no—
275C8×C47—21no—
276C8×C47—21no—
277C8×C47—21no—
278C8×C47—22no—
279C8×C47—41no—
280C47⋊C8—21no—
281C47⋊C8—21no—
282C47⋊C8—21no—
283C47⋊C8—21no—
284C47⋊C8—41no—
285C47⋊C8—41no—
286#22984—471nomaximal, Sylow
287C22×C47—21no—
288C22×C47—21no—
289C22×C47C2×C4×D411yescharacteristic
290C22×C47#55211yescharacteristic
291C22×C47#55411yescharacteristic
292C2×D47—22no—
293C2×D47—41no—
294C2×D47—41no—
295C4×C47—41no—
296C4×C47—41no—
297C4×C47—41no—
298C4×C47—41no—
299C4×C47—81no—
300C4×C47C2×C4×Q811yescharacteristic
301C4×C47#52011yescharacteristic
302C4×C47#54611yescharacteristic
303C4×C47D4×Q811yescharacteristic
304Dic47—22no—
305Dic47—21no—
306Dic47—21no—
307Dic47—21no—
308Dic47—21no—
309Dic47—42no—
310Dic47—41no—
311Dic47—41no—
312#3076—471no—
313#2962—471no—
314#2906—471no—
315#1995—471no—
316#1950—471no—
317#1946—471no—
318#1945—471no—
319#1922—471no—
320#1922D4711yescharacteristic, complemented
321#1866—471no—
322#1866—471no—
323#1768—471no—
324#1765—471no—
325#1764—471no—
326#1759—471no—
327C2×C47—21no—
328C2×C47—21no—
329C2×C47#236011yescharacteristic
330C2×C47#295511yescharacteristic
331C2×C47#295711yescharacteristic
332D47—42no—
333C2×#131—471no—
334C2×#131—471no—
335#533—471no—
336C4×#41—471no—
337#479—471no—
338#479—471no—
339#478—471no—
340#478—471no—
341#477—471no—
342#476—471no—
343#475—471no—
344#474—471no—
345#446—941no—
346#446—941no—
347#446—941no—
348C8×Q8—941no—
349C2×#108—471no—
350C2×#108—471no—
351#427—471no—
352#427C2×D4711yescharacteristic
353#424—471no—
354#424—471no—
355#424—471no—
356#424—941no—
357#424—941no—
358#424C2×D4711yescharacteristic
359#423—471no—
360#423—941no—
361C2×C4⋊C8—471no—
362C2×C4⋊C8—471no—
363C2×C4⋊C8—941no—
364C2×#104—471no—
365C2×#104—471no—
366#416—471no—
367#416—471no—
368#405—471no—
369#405C2×D4711yescharacteristic
370C4×M16—471no—
371C2×#97—471no—
372C2×C4×C8—471no—
373#376—471no—
374#341—471no—
375#341C2×D4711yescharacteristic
376#339—471no—
377#339—471no—
378#339—471no—
379#339—471no—
380#339C2×D4711yescharacteristic
381#339C2×D4711yescharacteristic
382#336—471no—
383#336C2×D4711yescharacteristic
384C47#2298411yescharacteristic, complemented, Sylow
385C2×#41—471no—
386#131—942no—
387#131—942no—
388#131—1881no—
389#131—1881no—
390C2×M16—471no—
391C2×M16—471no—
392C2×M16—941no—
393C2×M16—941no—
394C2×M16C22×D4711yescharacteristic
395C2×M16C22×D4711yescharacteristic
396C22×C8—471no—
397C22×C8—471no—
398C22×C8—941no—
399C22×C8—941no—
400C22×C8C22×D4711yescharacteristic
401C22×C8C22×D4711yescharacteristic
402C4⋊Q8—471no—
403C4⋊Q8—471no—
404#125—471no—
405#125—471no—
406C22⋊Q8—941no—
407C22⋊Q8—941no—
408C4×Q8—471no—
409C4×Q8—471no—
410C4×Q8—941no—
411C4×Q8—941no—
412C4×D4—471no—
413C4×D4—471no—
414C4×D4—941no—
415C4×D4—941no—
416#117—471no—
417#117—941no—
418#117C22×D4711yescharacteristic
419#117C22×D4711yescharacteristic
420C2×C42—471no—
421C2×C42—471no—
422C2×C42—941no—
423C2×C42C22×D4711yescharacteristic
424#108—1881no—
425#108—1881no—
426#107—471no—
427#107—471no—
428#107C4×D4711yescharacteristic
429#107C4×D4711yescharacteristic
430#106—471no—
431#106—471no—
432#106C4×D4711yescharacteristic
433#106C4×D4711yescharacteristic
434C4⋊C8—21no—
435C4⋊C8—21no—
436C4⋊C8—471no—
437C4⋊C8—471no—
438C4⋊C8—471no—
439C4⋊C8—471no—
440C4⋊C8—471no—
441C4⋊C8—471no—
442C4⋊C8—471no—
443C4⋊C8—471no—
444C4⋊C8—941no—
445C4⋊C8—941no—
446C4⋊C8—941no—
447C4⋊C8—941no—
448C4⋊C8—941no—
449C4⋊C8—941no—
450C4⋊C8—941no—
451C4⋊C8—941no—
452#104—474no—
453#104—942no—
454#103—471no—
455#103—471no—
456#103—471no—
457#103—471no—
458#102—471no—
459#102—471no—
460#102—471no—
461#102—471no—
462#97—21no—
463#97—941no—
464#97—941no—
465#97—1881no—
466C4×C8—21no—
467C4×C8—941no—
468C4×C8—941no—
469C4×C8—1881no—
470#95—941no—
471#95—941no—
472#41—472no—
473#41—472no—
474#41—941no—
475#41—941no—
476C2×Q8—471no—
477C2×Q8—941no—
478C2×Q8—941no—
479C2×D4—471no—
480C2×D4—941no—
481C22×C4—21no—
482C22×C4—471no—
483C22×C4—471no—
484C22×C4—941no—
485C22×C4—941no—
486C22×C4—941no—
487C22×C4—941no—
488C22×C4C23×D4711yescharacteristic
489M16—21no—
490M16—21no—
491M16—21no—
492M16—941no—
493M16—941no—
494M16—941no—
495M16—941no—
496M16—1882no—
497M16—1882no—
498M16Q8×D4712yes—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—471no—
506C2×C8—471no—
507C2×C8—471no—
508C2×C8—471no—
509C2×C8—941no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8—941no—
514C2×C8—941no—
515C2×C8—1882no—
516C2×C8—1882no—
517C2×C8C2×C4×D4711yescharacteristic
518C2×C8C2×C4×D4711yescharacteristic
519C2×C8D4×D4711yescharacteristic
520C2×C8D4×D4711yescharacteristic
521C4⋊C4—471no—
522C4⋊C4—471no—
523C4⋊C4—941no—
524C4⋊C4—941no—
525C4⋊C4—941no—
526C4⋊C4—941no—
527C4⋊C4—941no—
528C4⋊C4—941no—
529C4⋊C4—941no—
530C4⋊C4—941no—
531C4⋊C4C2×C4×D4711yescharacteristic
532C4⋊C4C2×C4×D4711yescharacteristic
533C4⋊C4C2×C4×D4711yescharacteristic
534C4⋊C4C2×C4×D4711yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—941no—
538C22⋊C4—941no—
539C22⋊C4—941no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—471no—
545C42—471no—
546C42—471no—
547C42—471no—
548C42—472no—
549C42—941no—
550C42—941no—
551C42—941no—
552C42—941no—
553C23—941no—
554C23#41×D4711yescharacteristic
555Q8—472no—
556Q8—941no—
557Q8—3761no—
558Q8—1882no—
559D4—472no—
560D4—942no—
561D4—942no—
562D4—941no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—471no—
571C2×C4—471no—
572C2×C4—471no—
573C2×C4—471no—
574C2×C4—941no—
575C2×C4—942no—
576C2×C4—942no—
577C2×C4—942no—
578C2×C4—941no—
579C2×C4—941no—
580C2×C4—3761no—
581C2×C4—1882no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4—1881no—
585C2×C4—1881no—
586C2×C4C22×C4×D4711yescharacteristic
587C2×C4C2×D4×D4711yescharacteristic
588C2×C4C2×Q8×D4711yescharacteristic
589C2×C4#41×D4711yescharacteristic
590C2×C4#1309407011yescharacteristic
591C2×C4#1309407111yescharacteristic
592C8—21no—
593C8—21no—
594C8—22no—
595C8—21no—
596C8—21no—
597C8—41no—
598C8—941no—
599C8—941no—
600C8—941no—
601C8—941no—
602C8—1881no—
603C8—1881no—
604C22—21no—
605C22—21no—
606C22—942no—
607C22—1881no—
608C22—1881no—
609C22C4×D4×D4711yescharacteristic
610C22#1568980911yescharacteristic
611C22#1568982911yescharacteristic
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—81no—
617C4—942no—
618C4—941no—
619C4—941no—
620C4—941no—
621C4—941no—
622C4—1882no—
623C4—1881no—
624C4—1881no—
625C4C4×Q8×D4711yescharacteristic
626C4#1568960311yescharacteristic
627C4#1568976511yescharacteristic
628C4#1568978611yescharacteristic
629C2—21no—
630C2—21no—
631C2—1882no—
632C2#2024639611yescharacteristic
633C2#2025228611yescharacteristic
634C2#2025230211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1922C2
socleC22×C47C2×C4×D4
prime radicalC2×#10625350C23
cosocleC22×C4×C47C24
hypercenter#1922D47
hypocenterC47#22984
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1922

Complements

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

NH ≅ G/Ntype
2. #20252332C2semidirect
4. #20244367C2semidirect
5. #20244162C2semidirect
6. #20244142C2semidirect
7. #20244133C2semidirect
8. #20243990C2semidirect
11. #20243244C2semidirect
15. C47×#1922C2semidirect
20. #15689433C4semidirect
21. #15689429C4semidirect
22. #15689426C4semidirect
23. #15689422C4semidirect
24. #15689417C4semidirect
25. #15689412C4semidirect
26. #15689407C4semidirect
27. #15689402C4semidirect
122. #13093950Q8semidirect
320. #1922D47semidirect
384. C47#22984semidirect

Maximal subgroups

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

subgroupindex
#202531742
#202523322
#202519552
#202443672
#202441622
#202441422
#202441332
#202439902
#202436662
#202436642
#202432442
#202432312
#202432272
#202431972
C47×#19222
#2298447