Finite Group Databasebeta

← #33966693

Subgroups — #33966693

Order 18688.

1706 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623752C211yesmaximal, characteristic, complemented
2#22622876C214yesmaximal, complemented
3#22622870C214yesmaximal, complemented
4#22622818C214yesmaximal, complemented
5#22622794C214yesmaximal, complemented
6#22620465C212yesmaximal, complemented
7C2×#17988986C211yesmaximal, characteristic, complemented
8#22619793C212yesmaximal, complemented
9#22619789C212yesmaximal, complemented
10#22619753C211yesmaximal, characteristic, complemented
11#22619749C211yesmaximal, characteristic, complemented
12#22618866C211yesmaximal, characteristic, complemented
13#22618804C212yesmaximal
14C4×#14556611C211yesmaximal, characteristic
15C73×#3504C211yesmaximal, characteristic, complemented
16C2×#14556611C2211yescharacteristic
17#17989315C2214yes—
18#17989297C2214yes—
19#17989291—24no—
20#17989215C2212yes—
21#17989214—22no—
22#17989143—21no—
23C2×#14556583C2211yescharacteristic
24#125×D73—21no—
25#17989067C2212yes—
26C4×#11924028C2211yescharacteristic
27#17988997C2214yes—
28#17988993C2214yes—
29#17988990C2214yes—
30#17988986C2214yes—
31#17988986C2214yes—
32#17988986C2214yes—
33#17988894C2214yes—
34#17988894C2214yes—
35#17988891C2214yes—
36#17988891C2214yes—
37#17988887C2214yes—
38#17988887C2214yes—
39#17988884C2214yes—
40#17988884C2214yes—
41#17988877C2214yes—
42#17988874C2214yes—
43#17988872C2214yes—
44#17988869C2214yes—
45#17988718C2214yes—
46Q8×#10596831C2214yes—
47#17988687C2214yes—
48#17988685C2214yes—
49C4×#11924018C2212yes—
50C4×#11924018C2214yes—
51C4×#11924015C2212yes—
52C4×#11924016C2212yes—
53C4×#11924013C2212yes—
54C4×#11924013C2214yes—
55#17988630C2212yes—
56#17988627C2212yes—
57#17988623C2211yescharacteristic
58C2×#14556524C2212yes—
59C2×#14556522C2212yes—
60#17988614C2212yes—
61#17988611C2211yescharacteristic
62#17988588C2212yes—
63#17988586C2212yes—
64#17988583C2211yescharacteristic
65#17988581C2211yescharacteristic
66#17988543C2212yes—
67C2×C4×#10596831C2211yescharacteristic
68#557×C73C2214yes—
69#554×C73—24no—
70#554×C73C2214yes—
71#544×C73—22no—
72#544×C73C2212yes—
73#533×C73—21no—
74#533×C73C2211yescharacteristic
75C2×#125×C73—21no—
76C2×#125×C73C2211yescharacteristic
77#520×C73C2212yes—
78C4×#41×C73C2211yescharacteristic
79C2×#11924028C2311yescharacteristic
80#14556611Q818yescomplemented
81C2×#11924018C2311yescharacteristic
82C2×#11924016C2311yescharacteristic
83C2×#11924015C2311yescharacteristic
84C2×#11924013C2311yescharacteristic
85C2×#11924012C2312yes—
86C22×#10596831C2311yescharacteristic
87C22×#10596831C2311yescharacteristic
88#14556591C2314yes—
89#14556590—22no—
90#14556589—24no—
91#14556583—21no—
92#14556583—21no—
93#14556583C2314yes—
94#14556583C2314yes—
95#14556582—22no—
96#14556581—24no—
97#14556581—24no—
98#14556569—24no—
99#14556569—24no—
100#14556568—24no—
101#14556567—24no—
102C4×Dic146C2314yes—
103C4×Dic146C2312yes—
104#14556556—24no—
105C4×D292C2314yes—
106C4×D292C2312yes—
107C4×C22⋊D73—22no—
108#14556551—24no—
109#14556550C2311yescharacteristic
110#14556549—22no—
111#14556548—21no—
112C2×C4⋊Dic73C2312yes—
113C2×#11924007—22no—
114C4⋊C4×D73—22no—
115C4⋊C4×D73—24no—
116C42×D73—21no—
117#14556524C2314yes—
118#14556524C2314yes—
119#14556524C2312yes—
120#14556524C2314yes—
121#14556524C2314yes—
122#14556522C2314yes—
123#14556522C2314yes—
124#14556522C2314yes—
125#14556522C2314yes—
126#14556522C2312yes—
127C4⋊#10596831C2314yes—
128C4⋊#10596831C2314yes—
129C4⋊#10596831C2314yes—
130#14556515C2312yes—
131#14556515C2314yes—
132#14556514C2312yes—
133#14556514C2314yes—
134#14556512C2312yes—
135#14556512C2314yes—
136#14556511C2312yes—
137#14556511C2314yes—
138#14556502C2314yes—
139C4×#10596831C2312yes—
140C4×#10596831C2314yes—
141C4×#10596831C2312yes—
142C2×#41×C73C2311yescharacteristic
143C4⋊Q8×C73—24no—
144C4⋊Q8×C73—22no—
145C4⋊Q8×C73D412yes—
146C4⋊Q8×C73C2314yes—
147#125×C73—22no—
148#125×C73—24no—
149#125×C73—24no—
150#125×C73—21no—
151#125×C73—21no—
152#125×C73D411yescharacteristic
153#125×C73D411yescharacteristic
154#125×C73C2314yes—
155#125×C73C2314yes—
156C22⋊Q8×C73—24no—
157C22⋊Q8×C73—24no—
158C22⋊Q8×C73—24no—
159C22⋊Q8×C73—24no—
160C22⋊Q8×C73D414yes—
161C4×Q8×C73—24no—
162C4×Q8×C73C2314yes—
163C4×Q8×C73C2312yes—
164C4×Q8×C73C2314yes—
165C4×D4×C73—22no—
166C4×D4×C73—24no—
167C4×D4×C73C2314yes—
168C4×D4×C73C2312yes—
169#117×C73—22no—
170#117×C73—21no—
171#117×C73C2311yescharacteristic
172#117×C73C2312yes—
173C2×C4⋊C4×C73—22no—
174C2×C4⋊C4×C73—22no—
175C2×C4⋊C4×C73—24no—
176C2×C4⋊C4×C73C2312yes—
177C2×C42×C73—21no—
178C2×C42×C73C2311yescharacteristic
179#11924028—22no—
180#11924028C2×Q814yes—
181C2×Dic146—24no—
182C2×Dic146C2411yescharacteristic
183C2×D292C2411yescharacteristic
184C2×C22⋊D73—21no—
185C22×Dic73—22no—
186C2×C4×D73—21no—
187C2×C4×D73—22no—
188C2×C4×D73—24no—
189#11924018C2×Q814yes—
190#11924016C2×Q814yes—
191#11924015C2×Q814yes—
192#11924013C2×Q814yes—
193#11924012—24no—
194C2×#10596831—22no—
195C2×#10596831C2×Q814yes—
196C2×#10596831C2412yes—
197C2×#10596831C2414yes—
198C2×#10596831C2411yescharacteristic
199C2×#10596831C2411yescharacteristic
200C4⋊Dic73—22no—
201C4⋊Dic73—24no—
202C4⋊Dic73C2414yes—
203C4⋊Dic73C2412yes—
204#11924007—24no—
205#11924007—24no—
206#11924007—24no—
207#11924007—24no—
208#11924007—22no—
209#11924007—24no—
210C22⋊Dic73—21no—
211#11924005—22no—
212#11924005—24no—
213C4×Dic73—21no—
214C4×Dic73—22no—
215C4×Dic73—24no—
216#41×C73—22no—
217#41×C73C2×Q814yes—
218C2×Q8×C73—24no—
219C2×Q8×C73C2×D414yes—
220C2×Q8×C73C2411yescharacteristic
221C2×D4×C73—21no—
222C2×D4×C73C2411yescharacteristic
223C22×C4×C73—21no—
224C22×C4×C73—22no—
225C22×C4×C73—22no—
226C22×C4×C73—24no—
227C22×C4×C73C2×D412yes—
228C22×C4×C73C2411yescharacteristic
229C4⋊C4×C73—24no—
230C4⋊C4×C73—22no—
231C4⋊C4×C73—24no—
232C4⋊C4×C73—24no—
233C4⋊C4×C73—24no—
234C4⋊C4×C73—24no—
235C4⋊C4×C73—22no—
236C4⋊C4×C73—24no—
237C4⋊C4×C73C2×D412yes—
238C4⋊C4×C73C2×D414yes—
239C4⋊C4×C73C2×D412yes—
240C4⋊C4×C73C2×D414yes—
241C4⋊C4×C73C2×D414yes—
242C4⋊C4×C73C2414yes—
243C4⋊C4×C73C2412yes—
244C22⋊C4×C73—21no—
245C22⋊C4×C73—22no—
246C22⋊C4×C73—24no—
247C22⋊C4×C73C2×D414yes—
248C42×C73—21no—
249C42×C73—22no—
250C42×C73—24no—
251C42×C73C2×D411yescharacteristic
252C42×C73C2×D411yescharacteristic
253C42×C73C2412yes—
254C42×C73C2414yes—
255C23×C73—21no—
256C23×C73C22×D411yescharacteristic
257C22×D73—21no—
258Q8×C73—21no—
259Q8×C73—44no—
260Q8×C73—84no—
261Q8×C73C22×Q812yes—
262D4×C73—24no—
263D4×C73—21no—
264D4×C73C22×Q812yes—
265C2×C4×C73—21no—
266C2×C4×C73—24no—
267C2×C4×C73—22no—
268C2×C4×C73—24no—
269C2×C4×C73—42no—
270C2×C4×C73—82no—
271C2×C4×C73—84no—
272C2×C4×C73—82no—
273C2×C4×C73C22×D412yes—
274C2×C4×C73C22×D414yes—
275C2×C4×C73C22×D412yes—
276C2×C4×C73C22×D414yes—
277C2×C4×C73C22×D411yescharacteristic
278C2×C4×C73C22×Q812yes—
279C2×C4×C732−1+412yes—
280C2×C4×C73C2511yescharacteristic
281C22⋊D73—24no—
282C2×Dic73—21no—
283C2×Dic73—22no—
284C2×Dic73—24no—
285C2×Dic73—82no—
286D292—21no—
287D292C22×Q812yes—
288C4×D73—24no—
289C4×D73—82no—
290C4×D73—84no—
291Dic146—21no—
292Dic146—84no—
293Dic146C22×Q812yes—
294#10596831—22no—
295#10596831—24no—
296#10596831C22×Q812yes—
297#10596831C22×Q812yes—
298C22×C73—22no—
299C22×C73—22no—
300C22×C73—42no—
301C22×C73D4×Q812yes—
302C22×C73C23×D411yescharacteristic
303C2×D73—22no—
304C2×D73—42no—
305C4×C73—24no—
306C4×C73—22no—
307C4×C73—24no—
308C4×C73—22no—
309C4×C73—22no—
310C4×C73—44no—
311C4×C73—42no—
312C4×C73D4×Q812yes—
313C4×C73C23×Q811yescharacteristic
314C4×C73C2×2−1+411yescharacteristic
315Dic73—22no—
316Dic73—42no—
317Dic73—44no—
318#61376—731nomaximal, Sylow
319C2×C73—22no—
320C2×C73—42no—
321C2×C73C2×D4×Q811yescharacteristic
322C2×C73#358412yes—
323D73—42no—
324#3504—731no—
325#3504D7311yescharacteristic, complemented
326#3388—734no—
327#3388—734no—
328#3383—734no—
329#3380—734no—
330#3164—732no—
331C2×#499—731no—
332#3084—732no—
333#3084—732no—
334#3080—731no—
335#3080—731no—
336#2960—731no—
337#2951—732no—
338C4×#135—731no—
339C73#6137611yescharacteristic, complemented, Sylow
340C2×#135—731no—
341#557—734no—
342#557C2×D7314yes—
343#554—24no—
344#554—734no—
345#554—1464no—
346#554C2×D7314yes—
347#544—22no—
348#544—732no—
349#544—1462no—
350#544C2×D7312yes—
351#533—21no—
352#533—1461no—
353#533C2×D7311yescharacteristic
354C2×#125—21no—
355C2×#125—731no—
356C2×#125—1461no—
357C2×#125C2×D7311yescharacteristic
358#520—732no—
359#520C2×D7312yes—
360C4×#41—731no—
361C4×#41C2×D7311yescharacteristic
362#501—734no—
363#501—734no—
364#500—734no—
365#499—734no—
366#499—734no—
367#499—734no—
368#479—734no—
369#479—734no—
370#479—734no—
371#479—734no—
372#478—734no—
373#478—734no—
374#478—734no—
375#478—734no—
376#476—734no—
377#476—734no—
378#475—734no—
379#475—734no—
380#446—734no—
381C8×Q8—734no—
382#440—734no—
383#440—734no—
384C4×Q16—732no—
385C4×Q16—734no—
386C4×SD16—732no—
387C4×SD16—732no—
388C4×D8—732no—
389C4×D8—734no—
390#428—732no—
391#428—732no—
392#427—731no—
393C2×#107—732no—
394C2×#106—732no—
395#424—732no—
396#423—731no—
397#417—732no—
398#417—732no—
399#416—731no—
400#416—731no—
401#405—732no—
402C2×C4×C8—731no—
403C2×#41—731no—
404C2×#41C22×D7311yescharacteristic
405#135—738no—
406C2×Q16—731no—
407C2×SD16—731no—
408C2×SD16—731no—
409C2×D8—731no—
410C2×M16—732no—
411C22×C8—731no—
412C22×C8—731no—
413C4⋊Q8—24no—
414C4⋊Q8—22no—
415C4⋊Q8—734no—
416C4⋊Q8—1464no—
417C4⋊Q8—1462no—
418C4⋊Q8C22⋊D7312yes—
419C4⋊Q8C22×D7314yes—
420#125—22no—
421#125—24no—
422#125—24no—
423#125—21no—
424#125—21no—
425#125—734no—
426#125—734no—
427#125—1462no—
428#125—1464no—
429#125—1464no—
430#125—1461no—
431#125—1461no—
432#125C22⋊D7311yescharacteristic
433#125C22⋊D7311yescharacteristic
434#125C22×D7314yes—
435#125C22×D7314yes—
436C22⋊Q8—24no—
437C22⋊Q8—24no—
438C22⋊Q8—24no—
439C22⋊Q8—24no—
440C22⋊Q8—1464no—
441C22⋊Q8—1464no—
442C22⋊Q8—1464no—
443C22⋊Q8—1464no—
444C22⋊Q8C22⋊D7314yes—
445C4×Q8—24no—
446C4×Q8—732no—
447C4×Q8—734no—
448C4×Q8—1464no—
449C4×Q8C22×D7312yes—
450C4×Q8C22×D7314yes—
451C4×Q8C22×D7314yes—
452C4×D4—22no—
453C4×D4—24no—
454C4×D4—732no—
455C4×D4—734no—
456C4×D4—1462no—
457C4×D4—1464no—
458C4×D4C22×D7312yes—
459C4×D4C22×D7314yes—
460#117—22no—
461#117—21no—
462#117—731no—
463#117—1462no—
464#117—1461no—
465#117C22×D7311yescharacteristic
466#117C22×D7312yes—
467C2×C4⋊C4—22no—
468C2×C4⋊C4—22no—
469C2×C4⋊C4—24no—
470C2×C4⋊C4—732no—
471C2×C4⋊C4—1462no—
472C2×C4⋊C4—1462no—
473C2×C4⋊C4—1464no—
474C2×C4⋊C4C22×D7312yes—
475C2×C42—21no—
476C2×C42—1461no—
477C2×C42C22×D7311yescharacteristic
478#107—734no—
479#107—734no—
480#107—732no—
481#107—734no—
482#107—734no—
483#106—734no—
484#106—734no—
485#106—732no—
486#106—734no—
487#106—734no—
488C4⋊C8—734no—
489C4⋊C8—734no—
490C4⋊C8—734no—
491#103—732no—
492#103—734no—
493#103—732no—
494#103—734no—
495#102—732no—
496#102—734no—
497#102—732no—
498#102—734no—
499#97—734no—
500C4×C8—732no—
501C4×C8—732no—
502C4×C8—734no—
503#41—22no—
504#41—734no—
505#41—1462no—
506#41Q8×D7314yes—
507C2×Q8—24no—
508C2×Q8—731no—
509C2×Q8—1464no—
510C2×Q8C2×C22⋊D7314yes—
511C2×Q8C23×D7311yescharacteristic
512C2×D4—21no—
513C2×D4—731no—
514C2×D4—1461no—
515C2×D4C23×D7311yescharacteristic
516C22×C4—21no—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—24no—
520C22×C4—1462no—
521C22×C4—1462no—
522C22×C4—1464no—
523C22×C4—1461no—
524C22×C4C2×C22⋊D7312yes—
525C22×C4C23×D7311yescharacteristic
526Q16—734no—
527SD16—734no—
528SD16—734no—
529D8—734no—
530M16—1464no—
531C2×C8—734no—
532C2×C8—731no—
533C2×C8—731no—
534C2×C8—732no—
535C2×C8—734no—
536C2×C8—1462no—
537C4⋊C4—24no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—22no—
544C4⋊C4—24no—
545C4⋊C4—734no—
546C4⋊C4—732no—
547C4⋊C4—1464no—
548C4⋊C4—1464no—
549C4⋊C4—1464no—
550C4⋊C4—1464no—
551C4⋊C4—1462no—
552C4⋊C4—1464no—
553C4⋊C4—1462no—
554C4⋊C4—1464no—
555C4⋊C4C2×C22⋊D7312yes—
556C4⋊C4C2×C22⋊D7314yes—
557C4⋊C4C2×C22⋊D7312yes—
558C4⋊C4C2×C22⋊D7314yes—
559C4⋊C4C2×C22⋊D7314yes—
560C4⋊C4C23×D7314yes—
561C4⋊C4C23×D7312yes—
562C22⋊C4—21no—
563C22⋊C4—22no—
564C22⋊C4—24no—
565C22⋊C4—1461no—
566C22⋊C4—1462no—
567C22⋊C4—1464no—
568C22⋊C4C2×C22⋊D7314yes—
569C42—21no—
570C42—22no—
571C42—24no—
572C42—1461no—
573C42—1462no—
574C42—1464no—
575C42C2×C22⋊D7311yescharacteristic
576C42C2×C22⋊D7311yescharacteristic
577C42C23×D7312yes—
578C42C23×D7314yes—
579C23—21no—
580C23—1461no—
581C23C22×C22⋊D7311yescharacteristic
582Q8—21no—
583Q8—44no—
584Q8—84no—
585Q8—5844no—
586Q8—732no—
587Q8—1461no—
588Q8C2×Q8×D7312yes—
589D4—24no—
590D4—21no—
591D4—732no—
592D4—1464no—
593D4—1461no—
594D4C2×Q8×D7312yes—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—42no—
600C2×C4—82no—
601C2×C4—84no—
602C2×C4—82no—
603C2×C4—5842no—
604C2×C4—5844no—
605C2×C4—5842no—
606C2×C4—1462no—
607C2×C4—1464no—
608C2×C4—1461no—
609C2×C4—1464no—
610C2×C4C22×C22⋊D7312yes—
611C2×C4C22×C22⋊D7314yes—
612C2×C4C22×C22⋊D7312yes—
613C2×C4C22×C22⋊D7314yes—
614C2×C4C22×C22⋊D7311yescharacteristic
615C2×C4C2×Q8×D7312yes—
616C2×C4#1455663912yes—
617C2×C4C24×D7311yescharacteristic
618C8—732no—
619C8—732no—
620C8—1462no—
621C8—1464no—
622C22—22no—
623C22—22no—
624C22—42no—
625C22—2922no—
626C22—1462no—
627C22Q8×C22⋊D7312yes—
628C22C23×C22⋊D7311yescharacteristic
629C4—22no—
630C4—22no—
631C4—22no—
632C4—24no—
633C4—24no—
634C4—44no—
635C4—42no—
636C4—2924no—
637C4—2922no—
638C4—1462no—
639C4Q8×C22⋊D7312yes—
640C4C22×Q8×D7311yescharacteristic
641C4C2×#1455663911yescharacteristic
642C2—22no—
643C2—42no—
644C2—2922no—
645C2C2×Q8×C22⋊D7311yescharacteristic
646C2#2262470212yes—
subgroupG/H
centerC22C23×C22⋊D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC73×#3504C2
socleC22×C73C23×D4
prime radicalC2×#14556611C22
hypercenter#3504D73
hypocenterC73#61376
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ #533 ◁ #3504

Complements

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

NH ≅ G/Ntype
1. #22623752C2semidirect
2. #22622876C2semidirect
3. #22622870C2semidirect
4. #22622818C2semidirect
5. #22622794C2semidirect
6. #22620465C2semidirect
7. C2×#17988986C2semidirect
8. #22619793C2semidirect
9. #22619789C2semidirect
10. #22619753C2semidirect
11. #22619749C2semidirect
12. #22618866C2semidirect
15. C73×#3504C2semidirect
80. #14556611Q8semidirect
325. #3504D73semidirect
339. C73#61376semidirect

Maximal subgroups

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

subgroupindex
#226237522
#226228762
#226228702
#226228182
#226227942
#226204652
C2×#179889862
#226197932
#226197892
#226197532
#226197492
#226188662
#226188042
C4×#145566112
C73×#35042
#6137673