Finite Group Databasebeta

← #32024894

Subgroups — #32024894

Order 18176.

1164 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22601745C211yesmaximal, characteristic, complemented
2C2×#17985973C211yesmaximal, characteristic
3C2×#17985973C211yesmaximal, characteristic
4#22589517C218yesmaximal, complemented
5#22588073C211yesmaximal, characteristic, complemented
6#22588072C211yesmaximal, characteristic, complemented
7#22588072C211yesmaximal, characteristic, complemented
8C2×#410×C71C211yesmaximal, characteristic
9#17986630—22no—
10#17986627—22no—
11#17986627—22no—
12#17986625C2214yes—
13#17986556—22no—
14#17986552—22no—
15#17986552—22no—
16C2×#14556087C2211yescharacteristic
17C2×#14556087C2211yescharacteristic
18C2×#14556087C2211yescharacteristic
19C2×#14556087C2211yescharacteristic
20#17985973—22no—
21#17985973—22no—
22#17985973—22no—
23#17985973—22no—
24C2×#14556043C2214yes—
25C2×#14556043C2214yes—
26#17985766C2214yes—
27#17985765C2214yes—
28#17985765C2214yes—
29C22×#41×C71C2211yescharacteristic
30C2×#117×C71C2211yescharacteristic
31C2×#117×C71C2211yescharacteristic
32#410×C71—22no—
33#410×C71—22no—
34C2×#99×C71C2214yes—
35#14556128—24no—
36#14556127—24no—
37#14556127—24no—
38#14556120—24no—
39#14556119—24no—
40#14556119—24no—
41#14556107—28no—
42#14556105—28no—
43#14556105—28no—
44C4×Dic142—28no—
45#14556087—21no—
46#14556087—21no—
47#14556087—21no—
48#14556087—21no—
49#14556087—21no—
50#14556087—21no—
51#14556087—21no—
52#14556087—21no—
53#14556087—21no—
54#14556087—21no—
55#14556087—21no—
56#14556087—21no—
57#14556087D412yes—
58#14556087D412yes—
59#14556087D412yes—
60#14556087D412yes—
61C2×C4⋊Dic71—21no—
62C2×C4⋊Dic71—21no—
63C2×C4⋊Dic71—21no—
64C2×C4⋊Dic71—21no—
65C2×C22⋊Dic71C2312yes—
66C2×C22⋊Dic71C2312yes—
67C2×C22⋊Dic71C2312yes—
68C2×C22⋊Dic71C2312yes—
69C2×C4×Dic71—21no—
70C2×C4×Dic71—21no—
71C2×C4×Dic71—21no—
72C2×C4×Dic71—21no—
73#14556043—28no—
74#14556043—28no—
75C2×#41×C71—21no—
76C2×#41×C71—21no—
77C2×#41×C71—21no—
78C2×#41×C71—21no—
79C2×#41×C71—21no—
80C2×#41×C71—21no—
81C2×#41×C71—21no—
82C2×#41×C71—21no—
83C2×#41×C71—21no—
84C2×#41×C71—21no—
85C2×#41×C71—21no—
86C2×#41×C71—21no—
87C22×Q8×C71Q811yescharacteristic, complemented
88C22×D4×C71Q811yescharacteristic, complemented
89C22×D4×C71C2312yes—
90C23×C4×C71Q812yescomplemented
91C23×C4×C71C2311yescharacteristic
92#117×C71—21no—
93#117×C71—21no—
94#117×C71—21no—
95#117×C71—21no—
96#117×C71—21no—
97#117×C71—21no—
98#117×C71D412yes—
99#117×C71D412yes—
100C2×C4⋊C4×C71—21no—
101C2×C4⋊C4×C71—21no—
102C2×C22⋊C4×C71C2312yes—
103C2×C22⋊C4×C71C2312yes—
104C2×C42×C71—21no—
105C2×C42×C71—21no—
106#99×C71—28no—
107C2×Dic142—28no—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—22no—
111C22×Dic71—22no—
112C4⋊Dic71—28no—
113C4⋊Dic71—22no—
114C4⋊Dic71—22no—
115C4⋊Dic71—22no—
116C4⋊Dic71—22no—
117C4⋊Dic71—41no—
118C4⋊Dic71—41no—
119C4⋊Dic71—41no—
120C4⋊Dic71—41no—
121Dic71⋊C4—28no—
122Dic71⋊C4—28no—
123C22⋊Dic71—22no—
124C22⋊Dic71—22no—
125C22⋊Dic71—22no—
126C22⋊Dic71—22no—
127C22⋊Dic71—22no—
128C22⋊Dic71—22no—
129C22⋊Dic71C2×D414yes—
130C22⋊Dic71C2×D414yes—
131C4×Dic71—22no—
132C4×Dic71—22no—
133C4×Dic71—22no—
134C4×Dic71—22no—
135C4×Dic71—41no—
136C4×Dic71—41no—
137C4×Dic71—41no—
138C4×Dic71—41no—
139C24×C71C2×Q812yes—
140C24×C71C2411yescharacteristic
141#41×C71—41no—
142#41×C71—41no—
143#41×C71—41no—
144#41×C71—41no—
145#41×C71—41no—
146#41×C71—41no—
147#41×C71—41no—
148#41×C71—41no—
149#41×C71—41no—
150#41×C71—41no—
151#41×C71—41no—
152#41×C71—41no—
153#41×C71—41no—
154#41×C71—41no—
155#41×C71—41no—
156#41×C71—41no—
157C2×Q8×C71—21no—
158C2×Q8×C71—21no—
159C2×Q8×C71—21no—
160C2×Q8×C71—21no—
161C2×Q8×C71—21no—
162C2×Q8×C71—21no—
163C2×D4×C71—22no—
164C2×D4×C71—22no—
165C2×D4×C71—22no—
166C2×D4×C71—22no—
167C2×D4×C71—22no—
168C2×D4×C71—22no—
169C2×D4×C71—21no—
170C2×D4×C71—21no—
171C2×D4×C71—21no—
172C2×D4×C71—21no—
173C2×D4×C71—21no—
174C2×D4×C71—21no—
175C22×C4×C71—22no—
176C22×C4×C71—22no—
177C22×C4×C71—22no—
178C22×C4×C71—22no—
179C22×C4×C71—22no—
180C22×C4×C71—22no—
181C22×C4×C71—22no—
182C22×C4×C71—22no—
183C22×C4×C71C2×D411yescharacteristic
184C22×C4×C71C2×D411yescharacteristic
185C22×C4×C71C2×D411yescharacteristic
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2×D411yescharacteristic
188C22×C4×C71C2×D411yescharacteristic
189C22×C4×C71C2×Q812yes—
190C22×C4×C71C2×Q811yescharacteristic
191C22×C4×C71C2×Q811yescharacteristic
192C22×C4×C71#4111yescharacteristic
193C22×C4×C71#4111yescharacteristic
194C22×C4×C71#4111yescharacteristic
195C22×C4×C71#4111yescharacteristic
196C22×C4×C71#4111yescharacteristic
197C22×C4×C71#4111yescharacteristic
198C4⋊C4×C71—22no—
199C4⋊C4×C71—22no—
200C4⋊C4×C71—41no—
201C4⋊C4×C71—41no—
202C22⋊C4×C71—22no—
203C22⋊C4×C71—22no—
204C22⋊C4×C71—22no—
205C22⋊C4×C71C2×D414yes—
206C42×C71—22no—
207C42×C71—22no—
208C42×C71—41no—
209C42×C71—41no—
210C23×C71—22no—
211C23×C71—22no—
212C23×C71—22no—
213C23×C71—81no—
214C23×C71—82no—
215C23×C71C22×D411yescharacteristic
216C23×C71C22×D411yescharacteristic
217C23×C71C22×D411yescharacteristic
218C23×C71C22×Q811yescharacteristic
219C23×C71C2×#4111yescharacteristic
220C23×C71C2×#4111yescharacteristic
221C23×C71C2×#4111yescharacteristic
222Q8×C71—41no—
223Q8×C71—41no—
224Q8×C71—41no—
225Q8×C71—41no—
226D4×C71—42no—
227D4×C71—42no—
228D4×C71—42no—
229D4×C71—42no—
230D4×C71—41no—
231D4×C71—41no—
232D4×C71—41no—
233D4×C71—41no—
234C2×C4×C71—24no—
235C2×C4×C71—22no—
236C2×C4×C71—22no—
237C2×C4×C71—22no—
238C2×C4×C71—42no—
239C2×C4×C71—42no—
240C2×C4×C71—42no—
241C2×C4×C71—42no—
242C2×C4×C71—42no—
243C2×C4×C71—82no—
244C2×C4×C71—82no—
245C2×C4×C71#12011yescharacteristic
246C2×C4×C71#12011yescharacteristic
247C2×C4×C71#12011yescharacteristic
248C2×C4×C71#12011yescharacteristic
249C2×C4×C71C22⋊Q811yescharacteristic
250C2×C4×C71C22⋊Q811yescharacteristic
251C2×C4×C71C22⋊Q811yescharacteristic
252C2×C4×C71C22⋊Q811yescharacteristic
253C2×C4×C71C22⋊Q811yescharacteristic
254C2×C4×C71C22⋊Q811yescharacteristic
255C2×C4×C71C22⋊Q811yescharacteristic
256C2×C4×C71C22⋊Q811yescharacteristic
257C2×C4×C71C22⋊Q811yescharacteristic
258C2×C4×C71C22⋊Q811yescharacteristic
259C2×C4×C71C22⋊Q811yescharacteristic
260C2×C4×C71C22⋊Q811yescharacteristic
261C2×C4×C71#12411yescharacteristic
262C2×C4×C71#12411yescharacteristic
263C2×C4×C71#12411yescharacteristic
264C2×C4×C71#12411yescharacteristic
265C2×C4×C71#12411yescharacteristic
266C2×C4×C71#12411yescharacteristic
267C2×C4×C71#12411yescharacteristic
268C2×C4×C71#12411yescharacteristic
269C2×C4×C71#12411yescharacteristic
270C2×C4×C71#12411yescharacteristic
271C2×C4×C71#12411yescharacteristic
272C2×C4×C71#12411yescharacteristic
273C2×Dic71—24no—
274C2×Dic71—24no—
275C2×Dic71—42no—
276C2×Dic71—42no—
277C2×Dic71—82no—
278C2×Dic71—82no—
279C2×Dic71—82no—
280C2×Dic71—82no—
281Dic142—88no—
282C22×C71—41no—
283C22×C71—41no—
284C22×C71—41no—
285C22×C71—41no—
286C22×C71—41no—
287C22×C71—41no—
288C22×C71—41no—
289C22×C71—42no—
290C22×C71—82no—
291C22×C71—82no—
292C22×C71—82no—
293C22×C71C2×#12011yescharacteristic
294C22×C71C2×C22⋊Q811yescharacteristic
295C22×C71C2×C22⋊Q811yescharacteristic
296C22×C71C2×C22⋊Q811yescharacteristic
297C22×C71C2×#12411yescharacteristic
298C22×C71C2×#12411yescharacteristic
299C22×C71C2×#12411yescharacteristic
300C4×C71—44no—
301C4×C71—42no—
302C4×C71—82no—
303C4×C71#39311yescharacteristic
304C4×C71#39311yescharacteristic
305C4×C71#39311yescharacteristic
306C4×C71#39311yescharacteristic
307C4×C71#39311yescharacteristic
308C4×C71#39311yescharacteristic
309C4×C71#39311yescharacteristic
310C4×C71#39311yescharacteristic
311Dic71—44no—
312Dic71—44no—
313Dic71—82no—
314Dic71—82no—
315#23474—711nomaximal, Sylow
316C2×C71—21no—
317C2×C71—21no—
318C2×C71—21no—
319C2×C71—21no—
320C2×C71—21no—
321C2×C71—21no—
322C2×C71—21no—
323C2×C71—82no—
324C2×C71C2×#39311yescharacteristic
325#3458—711no—
326C2×#410—711no—
327C2×#410—711no—
328C2×#410D7111yescharacteristic
329#2032—718no—
330#1740—711no—
331#1740—711no—
332#1740—711no—
333C71#2347411yescharacteristic, complemented, Sylow
334C22×#41C2×D7111yescharacteristic
335#544—1422no—
336#544—1422no—
337#544—1422no—
338#543—714no—
339#533—1422no—
340#533—1422no—
341#533—1422no—
342C2×#117—711no—
343C2×#117—711no—
344C2×#117—711no—
345C2×#117—711no—
346C2×#117C2×D7111yescharacteristic
347C2×#117C2×D7111yescharacteristic
348#410—22no—
349#410—22no—
350#410—1422no—
351#410—1422no—
352#410—1422no—
353#410—1422no—
354C2×#99—714no—
355C2×#99—714no—
356C2×#99C2×D7114yes—
357#342—714no—
358#342—714no—
359#342—714no—
360C2×#41—21no—
361C2×#41—21no—
362C2×#41—21no—
363C2×#41—21no—
364C2×#41—21no—
365C2×#41—21no—
366C2×#41—21no—
367C2×#41—21no—
368C2×#41—21no—
369C2×#41—21no—
370C2×#41—21no—
371C2×#41—21no—
372C22×Q8Dic14211yescharacteristic, complemented
373C22×D4Dic14211yescharacteristic, complemented
374C22×D4C22×D7112yes—
375C23×C4Dic14212yescomplemented
376C23×C4C22×D7111yescharacteristic
377C4⋊Q8—1424no—
378C4⋊Q8—1424no—
379C4⋊Q8—1424no—
380#125—1424no—
381#125—1424no—
382#125—1424no—
383C22⋊Q8—1428no—
384C22⋊Q8—1428no—
385C22⋊Q8—1428no—
386C4×Q8—1428no—
387#117—21no—
388#117—21no—
389#117—21no—
390#117—21no—
391#117—21no—
392#117—21no—
393#117—712no—
394#117—712no—
395#117—712no—
396#117—712no—
397#117—1421no—
398#117—1421no—
399#117—1421no—
400#117—1421no—
401#117—1421no—
402#117—1421no—
403#117—1421no—
404#117—1421no—
405#117—1421no—
406#117—1421no—
407#117—1421no—
408#117—1421no—
409#117C22⋊D7112yes—
410#117C22⋊D7112yes—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—1421no—
414C2×C4⋊C4—1421no—
415C2×C4⋊C4—1421no—
416C2×C4⋊C4—1421no—
417C2×C22⋊C4—712no—
418C2×C22⋊C4—712no—
419C2×C22⋊C4—712no—
420C2×C22⋊C4—712no—
421C2×C22⋊C4C22×D7112yes—
422C2×C22⋊C4C22×D7112yes—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—1421no—
426C2×C42—1421no—
427C2×C42—1421no—
428C2×C42—1421no—
429#99—28no—
430#99—1428no—
431#99—1428no—
432C24C2×Dic14212yes—
433C24C23×D7111yescharacteristic
434#41—41no—
435#41—41no—
436#41—41no—
437#41—41no—
438#41—41no—
439#41—41no—
440#41—41no—
441#41—41no—
442#41—41no—
443#41—41no—
444#41—41no—
445#41—41no—
446#41—41no—
447#41—41no—
448#41—41no—
449#41—41no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—1428no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—1422no—
478C22×C4—1422no—
479C22×C4—1422no—
480C22×C4—1422no—
481C22×C4D4×D7111yescharacteristic
482C22×C4D4×D7111yescharacteristic
483C22×C4D4×D7111yescharacteristic
484C22×C4D4×D7111yescharacteristic
485C22×C4C2×C22⋊D7111yescharacteristic
486C22×C4C2×C22⋊D7111yescharacteristic
487C22×C4C2×Dic14212yes—
488C22×C4C2×Dic14211yescharacteristic
489C22×C4C2×Dic14211yescharacteristic
490C22×C4#1176586411yescharacteristic
491C22×C4#1176586411yescharacteristic
492C22×C4#1176586411yescharacteristic
493C22×C4#1176586411yescharacteristic
494C22×C4#1176586511yescharacteristic
495C22×C4#1176586511yescharacteristic
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—2841no—
501C4⋊C4—2841no—
502C4⋊C4—2841no—
503C4⋊C4—2841no—
504C4⋊C4—1428no—
505C4⋊C4—1428no—
506C4⋊C4—1428no—
507C4⋊C4—1422no—
508C4⋊C4—1422no—
509C4⋊C4—1422no—
510C4⋊C4—1422no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—714no—
515C22⋊C4—714no—
516C22⋊C4—1422no—
517C22⋊C4—1422no—
518C22⋊C4—1422no—
519C22⋊C4—1422no—
520C22⋊C4—1422no—
521C22⋊C4—1422no—
522C22⋊C4C2×C22⋊D7114yes—
523C42—22no—
524C42—22no—
525C42—41no—
526C42—41no—
527C42—2841no—
528C42—2841no—
529C42—2841no—
530C42—2841no—
531C42—1422no—
532C42—1422no—
533C42—1422no—
534C42—1422no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—81no—
539C23—82no—
540C23C2×D4×D7111yescharacteristic
541C23C2×D4×D7111yescharacteristic
542C23C22×C22⋊D7111yescharacteristic
543C23C22×Dic14211yescharacteristic
544C23C2×#1176586411yescharacteristic
545C23C2×#1176586411yescharacteristic
546C23C2×#1176586511yescharacteristic
547Q8—41no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551Q8—5688no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—41no—
557D4—41no—
558D4—41no—
559D4—41no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—2842no—
572C2×C4—2842no—
573C2×C4—5682no—
574C2×C4—5682no—
575C2×C4—5682no—
576C2×C4—5682no—
577C2×C4—1424no—
578C2×C4—1424no—
579C2×C4#1455609611yescharacteristic
580C2×C4#1455609611yescharacteristic
581C2×C4#1455609611yescharacteristic
582C2×C4#1455609611yescharacteristic
583C2×C4#1455610511yescharacteristic
584C2×C4#1455610511yescharacteristic
585C2×C4#1455610511yescharacteristic
586C2×C4#1455610511yescharacteristic
587C2×C4#1455610511yescharacteristic
588C2×C4#1455610511yescharacteristic
589C2×C4#1455610511yescharacteristic
590C2×C4#1455610511yescharacteristic
591C2×C4#1455610711yescharacteristic
592C2×C4#1455610711yescharacteristic
593C2×C4#1455610711yescharacteristic
594C2×C4#1455610711yescharacteristic
595C2×C4#1455611411yescharacteristic
596C2×C4#1455611411yescharacteristic
597C2×C4#1455611411yescharacteristic
598C2×C4#1455611411yescharacteristic
599C2×C4#1455611411yescharacteristic
600C2×C4#1455611411yescharacteristic
601C2×C4#1455611411yescharacteristic
602C2×C4#1455611411yescharacteristic
603C2×C4#1455611511yescharacteristic
604C2×C4#1455611511yescharacteristic
605C2×C4#1455611511yescharacteristic
606C2×C4#1455611511yescharacteristic
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22C2×#1455609611yescharacteristic
619C22C2×#1455610511yescharacteristic
620C22C2×#1455610511yescharacteristic
621C22C2×#1455610711yescharacteristic
622C22C2×#1455611411yescharacteristic
623C22C2×#1455611411yescharacteristic
624C22C2×#1455611511yescharacteristic
625C4—44no—
626C4—42no—
627C4—82no—
628C4—2844no—
629C4—2844no—
630C4—5682no—
631C4—5682no—
632C4#1798591111yescharacteristic
633C4#1798591111yescharacteristic
634C4#1798591111yescharacteristic
635C4#1798591111yescharacteristic
636C4#1798591111yescharacteristic
637C4#1798591111yescharacteristic
638C4#1798591111yescharacteristic
639C4#1798591111yescharacteristic
640C2—21no—
641C2—21no—
642C2—21no—
643C2—21no—
644C2—21no—
645C2—21no—
646C2—21no—
647C2—82no—
648C2C2×#1798591111yescharacteristic
subgroupG/H
centerC4#17985911
commutator, cosocleC24×C71C24
FrattiniC24C23×D71
FittingC2×#410×C71C2
socleC2×C71C2×#393
prime radicalC22×#41×C71C22
hypercenterC2×#410D71
hypocenterC71#23474
LCSG ▷ C24×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
1. #22601745C2semidirect
4. #22589517C2semidirect
5. #22588073C2semidirect
6. #22588072C2semidirect
7. #22588072C2semidirect
87. C22×Q8×C71Q8semidirect
88. C22×D4×C71Q8semidirect
90. C23×C4×C71Q8semidirect
333. C71#23474semidirect
372. C22×Q8Dic142semidirect
373. C22×D4Dic142semidirect
375. C23×C4Dic142semidirect

Maximal subgroups

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

subgroupindex
#226017452
C2×#179859732
#225895172
#225880732
#225880722
C2×#410×C712
#2347471