Finite Group Databasebeta

← #34021824

Subgroups — #34021824

Order 18688.

898 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626412C211yesmaximal, characteristic
2#22626368C211yesmaximal, characteristic
3#22625995C212yesmaximal, complemented
4#22625994C212yesmaximal, complemented
5#22625993C212yesmaximal, complemented
6#22625992C212yesmaximal, complemented
7#22625620C211yesmaximal, characteristic
8#22625620C211yesmaximal, characteristic
9C2×#17989461C211yesmaximal, characteristic
10#22625596C211yesmaximal, characteristic
11#427×D73C211yesmaximal, characteristic
12C2×D73⋊M16C2211yescharacteristic
13C2×#14556670C2211yescharacteristic
14C22×C4⋊#64408C2211yescharacteristic
15C22×C4⋊#64408C2211yescharacteristic
16#17989538C2212yes—
17#17989538C2212yes—
18#17989534—22no—
19#17989533—21no—
20#17989531—21no—
21C4⋊C4×#64408C2212yes—
22C4⋊C4×#64408C2212yes—
23C4×C4⋊#64408—21no—
24C4×C4⋊#64408—21no—
25C22⋊C4×#64408—22no—
26#17989461C2212yes—
27#17989461C2212yes—
28#17989460C2212yes—
29#17989460C2212yes—
30#17989460C2212yes—
31#17989460C2212yes—
32#17989452C2212yes—
33#17989452C2212yes—
34C22×C8×D73C2211yescharacteristic
35#117×D73C2211yescharacteristic
36#117×D73C2211yescharacteristic
37#17988625C411yescharacteristic, complemented
38#17988624C411yescharacteristic, complemented
39#17988624C411yescharacteristic, complemented
40#17988623C411yescharacteristic, complemented
41#17988623C411yescharacteristic, complemented
42#17988622C411yescharacteristic, complemented
43#17988622C411yescharacteristic, complemented
44#107×D73C2211yescharacteristic
45#107×D73C2211yescharacteristic
46#106×D73C2211yescharacteristic
47#106×D73C2211yescharacteristic
48#427×C73C411yescharacteristic, complemented
49C23×#64408—21no—
50C23×#64408—21no—
51C2×#11924038—21no—
52D73⋊M16—22no—
53D73⋊M16—22no—
54C2×#11924037—21no—
55C2×#11924036C2312yes—
56#14556670—22no—
57#14556670—22no—
58C2×C4⋊#64408—22no—
59C2×C4⋊#64408—22no—
60C2×C4⋊#64408—21no—
61C2×C4⋊#64408—21no—
62C2×C4⋊#64408—22no—
63C2×C4⋊#64408—22no—
64C2×C4⋊#64408—22no—
65C2×C4⋊#64408D411yescharacteristic
66C2×C4⋊#64408D411yescharacteristic
67C2×C4⋊#64408Q814yes—
68C2×C4⋊#64408C2312yes—
69C2×C4⋊#64408C2312yes—
70C2×C22⋊#64408D412yes—
71C2×C4×#64408—22no—
72C2×C4×#64408—22no—
73C2×C4×#64408C2312yes—
74(C2×#64408)⋊C4—22no—
75(C2×#64408)⋊C4—22no—
76(C2×#64408)⋊C4—22no—
77(C2×#64408)⋊C4—22no—
78C22×C4×D73C2311yescharacteristic
79C22×#10596831C2×C411yescharacteristic
80C2×C8×D73—22no—
81C2×C8×D73—22no—
82C2×C8×D73C2×C411yescharacteristic
83C2×C8×D73C2×C411yescharacteristic
84C2×C8×D73C2311yescharacteristic
85C2×C8×D73C2311yescharacteristic
86#14556550C2×C411yescharacteristic
87#14556550C2×C411yescharacteristic
88#14556549C2×C411yescharacteristic
89#14556549C2×C411yescharacteristic
90#14556549C2×C411yescharacteristic
91#14556549C2×C411yescharacteristic
92#14556548—21no—
93#14556548—21no—
94#14556547—21no—
95#14556547—21no—
96C4⋊C4×D73C2311yescharacteristic
97C4⋊C4×D73C2311yescharacteristic
98C4⋊C4×D73C2311yescharacteristic
99C4⋊C4×D73C2311yescharacteristic
100C22⋊C4×D73—21no—
101C22⋊C4×D73—21no—
102C42×D73—21no—
103C42×D73—21no—
104#14556525C2×C411yescharacteristic, complemented
105#14556525C2×C411yescharacteristic, complemented
106#14556524C2×C411yescharacteristic, complemented
107#14556524C2×C411yescharacteristic, complemented
108#14556524C2×C411yescharacteristic, complemented
109#14556524C2×C411yescharacteristic, complemented
110#14556523C2×C411yescharacteristic, complemented
111#14556523C2×C411yescharacteristic, complemented
112#14556522C2×C411yescharacteristic, complemented
113#14556522C2×C411yescharacteristic, complemented
114#14556522C2×C411yescharacteristic, complemented
115#14556522C2×C411yescharacteristic, complemented
116C22×C8×C73C2×C411yescharacteristic
117#117×C73C2×C411yescharacteristic
118#117×C73C2×C411yescharacteristic
119#107×C73C2×C411yescharacteristic, complemented
120#107×C73C2×C411yescharacteristic, complemented
121#106×C73C2×C411yescharacteristic, complemented
122#106×C73C2×C411yescharacteristic, complemented
123C22×#64408—22no—
124C22×#64408—24no—
125C22×#64408—22no—
126C22×#64408—42no—
127C22×#64408—81no—
128C22×#64408—81no—
129C22×#64408—81no—
130C22×#64408—81no—
131C22×#64408C2×D412yes—
132C22×#64408C2×D412yes—
133#11924038—42no—
134#11924037—42no—
135#11924036—22no—
136#11924036—22no—
137C2×#10596833—22no—
138C4⋊#64408—24no—
139C4⋊#64408—22no—
140C4⋊#64408—22no—
141C4⋊#64408—22no—
142C4⋊#64408—22no—
143C4⋊#64408—22no—
144C4⋊#64408—42no—
145C4⋊#64408—82no—
146C4⋊#64408—82no—
147C4⋊#64408C2×Q814yes—
148C22⋊#64408—24no—
149C4×#64408—22no—
150C4×#64408—22no—
151C4×#64408—82no—
152C4×#64408—82no—
153C23×D73C2×D411yescharacteristic
154C22×Dic73#4111yescharacteristic
155C2×C4×D73—21no—
156C2×C4×D73—21no—
157C2×C4×D73—21no—
158C2×C4×D73—21no—
159C2×C4×D73C22×C411yescharacteristic
160C2×C4×D73C2×D411yescharacteristic
161C2×C4×D73C2×Q812yes—
162C2×C4×D73#4112yes—
163C2×C4×D73#4112yes—
164C2×C4×D73#4112yes—
165C2×C4×D73#4111yescharacteristic
166C2×C4×D73C2411yescharacteristic
167C2×#10596831—22no—
168C2×#10596831C22×C411yescharacteristic
169C2×#10596831C22×C411yescharacteristic
170C8×D73—22no—
171C8×D73—22no—
172C8×D73—21no—
173C8×D73—21no—
174C8×D73—21no—
175C8×D73—21no—
176C4⋊Dic73C22×C411yescharacteristic
177C4⋊Dic73C22×C411yescharacteristic
178C4⋊Dic73C22×C411yescharacteristic
179C4⋊Dic73C22×C411yescharacteristic
180#11924007—21no—
181#11924007—21no—
182#11924007—21no—
183#11924007—21no—
184C22⋊Dic73—21no—
185C22⋊Dic73—21no—
186#11924005—21no—
187#11924005—21no—
188#11924005—21no—
189#11924005—21no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C4×Dic73—21no—
193C4×Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196C22×C4×C73C22×C411yescharacteristic
197C2×C8×C73—22no—
198C2×C8×C73C22×C411yescharacteristic
199C2×C8×C73C22×C411yescharacteristic
200C4⋊C4×C73C22×C411yescharacteristic
201C4⋊C4×C73C22×C411yescharacteristic
202C4⋊C4×C73C22×C411yescharacteristic
203C4⋊C4×C73C22×C411yescharacteristic
204C22⋊C4×C73—21no—
205C22⋊C4×C73—21no—
206C42×C73—21no—
207C42×C73—21no—
208C23×C73C4×D411yescharacteristic
209C22×D73—21no—
210C22×D73—21no—
211C22×D73—21no—
212C22×D73—21no—
213C22×D73C4×D411yescharacteristic
214C22×D73C22⋊Q812yes—
215C22×D73#12312yes—
216C22×D73C22×D411yescharacteristic
217C2×#64408—24no—
218C2×#64408—22no—
219C2×#64408—22no—
220C2×#64408—22no—
221C2×#64408—22no—
222C2×#64408—22no—
223C2×#64408—22no—
224C2×#64408—42no—
225C2×#64408—42no—
226C2×#64408—42no—
227C2×#64408—42no—
228C2×#64408—42no—
229C2×#64408—84no—
230C2×#64408—84no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73#11712yes—
236C2×C4×C73C4×D411yescharacteristic
237C2×C4×C73C4×Q812yes—
238C2×C4×C73C23×C411yescharacteristic
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73C4×D411yescharacteristic
244C2×Dic73#12512yes—
245C2×Dic73#12612yes—
246C2×Dic73C2×#4111yescharacteristic
247C4×D73—41no—
248C4×D73—41no—
249C4×D73—41no—
250C4×D73—41no—
251C4×D73—41no—
252C4×D73—41no—
253C4×D73—41no—
254C4×D73—41no—
255C4×D73#11712yes—
256C4×D73C4×Q812yes—
257C4×D73C22⋊Q812yes—
258C4×D73#12312yes—
259C4×D73#12512yes—
260C4×D73#12612yes—
261C4×D73C22×Q811yescharacteristic
262C4×D73C2×#4111yescharacteristic
263C4×D73C2×#4111yescharacteristic
264C4×D73C2×#4111yescharacteristic
265#10596833—42no—
266C8×C73—21no—
267C8×C73—21no—
268#10596831—21no—
269#10596831—21no—
270C22×C73—21no—
271C22×C73—21no—
272C22×C73#38712yes—
273C22×C73C2×C4×D411yescharacteristic
274C2×D73—21no—
275C2×D73—21no—
276C2×D73—22no—
277C2×D73—22no—
278C2×D73—22no—
279C2×D73—22no—
280C2×D73#38712yes—
281C2×D73C2×C22⋊Q811yescharacteristic
282C2×D73C2×#12311yescharacteristic
283C2×D73#57611yescharacteristic
284C2×D73#57711yescharacteristic
285C2×D73#57711yescharacteristic
286C2×D73#57811yescharacteristic
287#64408—44no—
288#64408—42no—
289#64408—42no—
290#64408—82no—
291C4×C73—41no—
292C4×C73—41no—
293C4×C73—41no—
294C4×C73—41no—
295C4×C73#38712yes—
296C4×C73C2×#11711yescharacteristic
297C4×C73C2×C4×Q811yescharacteristic
298Dic73—41no—
299Dic73—41no—
300Dic73—41no—
301Dic73—41no—
302Dic73#38712yes—
303Dic73C2×#12511yescharacteristic
304Dic73C2×#12611yescharacteristic
305#21039—731nomaximal, Sylow
306C2×C73—22no—
307C2×C73C2×#38711yescharacteristic
308C2×C73#295111yescharacteristic
309C2×C73#295211yescharacteristic
310D73—22no—
311D73#308111yescharacteristic
312D73#308211yescharacteristic
313D73#309611yescharacteristic
314D73#309711yescharacteristic
315C2×#427—731no—
316#2317—731no—
317#2274—731no—
318#1921—732no—
319#1920—732no—
320#1919—732no—
321#1918—732no—
322#1736—731no—
323#1736—731no—
324C2×#340—731no—
325#1729—731no—
326C73#2103911yescharacteristic, complemented, Sylow
327C22×M16—731no—
328C23×C8—731no—
329C2×#117—731no—
330C2×#117—731no—
331C2×#117—731no—
332C22×C4⋊C4—731no—
333C22×C4⋊C4—731no—
334#427—731no—
335#427—731no—
336#427—731no—
337#427—731no—
338#427—731no—
339#427—731no—
340#427—731no—
341#427#6440811yescharacteristic, complemented
342C2×#107—731no—
343C2×#107—731no—
344C2×#106—731no—
345C2×#106—731no—
346#389—732no—
347#389—732no—
348#385—1462no—
349#384—1461no—
350#382—1461no—
351C4×C4⋊C4—732no—
352C4×C4⋊C4—732no—
353C4×C4⋊C4—1461no—
354C4×C4⋊C4—1461no—
355C4×C22⋊C4—1462no—
356#340—732no—
357#340—732no—
358#340—732no—
359#340—732no—
360#340—732no—
361#340—732no—
362#336—732no—
363#336—732no—
364C23×C4—731no—
365C23×C4—1461no—
366C23×C4—1461no—
367C2×M16—1461no—
368C2×M16—1462no—
369C2×M16—1462no—
370C2×M16—1461no—
371C22×C8—731no—
372C22×C8—731no—
373C22×C8—731no—
374C22×C8—731no—
375C22×C8—731no—
376C22×C8—732no—
377C22×C8—1462no—
378C22×C8—1462no—
379C22×C8C2×#6440811yescharacteristic
380#117—731no—
381#117—731no—
382#117—731no—
383#117—731no—
384#117—731no—
385#117—731no—
386#117—1461no—
387#117—1461no—
388#117—1461no—
389#117—1461no—
390#117—1462no—
391#117—1462no—
392#117C2×#6440811yescharacteristic
393#117C2×#6440811yescharacteristic
394C2×C4⋊C4—732no—
395C2×C4⋊C4—734no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—732no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1462no—
405C2×C4⋊C4—1462no—
406C2×C4⋊C4—1462no—
407C2×C4⋊C4—1462no—
408C2×C4⋊C4—1461no—
409C2×C4⋊C4—1462no—
410C2×C22⋊C4—732no—
411C2×C22⋊C4—1461no—
412C2×C22⋊C4—1461no—
413C2×C42—732no—
414C2×C42—1461no—
415C2×C42—1461no—
416C2×C42—1462no—
417C2×C42—1462no—
418#107—731no—
419#107—731no—
420#107—731no—
421#107—731no—
422#107—731no—
423#107—731no—
424#107C2×#6440811yescharacteristic, complemented
425#107C2×#6440811yescharacteristic, complemented
426#106—731no—
427#106—731no—
428#106—731no—
429#106—731no—
430#106—731no—
431#106—731no—
432#106C2×#6440811yescharacteristic, complemented
433#106C2×#6440811yescharacteristic, complemented
434#95—1462no—
435#95—1462no—
436#95—1462no—
437#95—1462no—
438C24—731no—
439C22×C4—2922no—
440C22×C4—5841no—
441C22×C4—5841no—
442C22×C4—5841no—
443C22×C4—5841no—
444C22×C4—731no—
445C22×C4—732no—
446C22×C4—732no—
447C22×C4—731no—
448C22×C4—731no—
449C22×C4—732no—
450C22×C4—732no—
451C22×C4—731no—
452C22×C4—731no—
453C22×C4—732no—
454C22×C4—732no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4—1462no—
460C22×C4—1462no—
461C22×C4—1464no—
462C22×C4C22×#6440811yescharacteristic
463M16—2922no—
464M16—2922no—
465C2×C8—22no—
466C2×C8—731no—
467C2×C8—731no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1462no—
473C2×C8—1462no—
474C2×C8—1462no—
475C2×C8—1462no—
476C2×C8—1462no—
477C2×C8—1462no—
478C2×C8C22×#6440811yescharacteristic
479C2×C8C22×#6440811yescharacteristic
480C4⋊C4—2922no—
481C4⋊C4—5842no—
482C4⋊C4—5842no—
483C4⋊C4—734no—
484C4⋊C4—731no—
485C4⋊C4—731no—
486C4⋊C4—731no—
487C4⋊C4—731no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1462no—
493C4⋊C4—1462no—
494C4⋊C4—1462no—
495C4⋊C4—1464no—
496C4⋊C4—1462no—
497C4⋊C4—1462no—
498C4⋊C4C22×#6440811yescharacteristic
499C4⋊C4C22×#6440811yescharacteristic
500C4⋊C4C22×#6440811yescharacteristic
501C4⋊C4C22×#6440811yescharacteristic
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—1461no—
505C22⋊C4—1461no—
506C22⋊C4—1461no—
507C22⋊C4—1461no—
508C22⋊C4—1464no—
509C22⋊C4—1461no—
510C22⋊C4—1461no—
511C42—21no—
512C42—21no—
513C42—5842no—
514C42—5842no—
515C42—1461no—
516C42—1461no—
517C42—1461no—
518C42—1461no—
519C42—1462no—
520C42—1462no—
521C42—1461no—
522C42—1461no—
523C23—731no—
524C23—731no—
525C23—732no—
526C23—732no—
527C23—1461no—
528C23—1461no—
529C23—1461no—
530C23—1461no—
531C23D4×#6440811yescharacteristic
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2922no—
545C2×C4—2922no—
546C2×C4—2922no—
547C2×C4—2922no—
548C2×C4—2922no—
549C2×C4—5844no—
550C2×C4—5844no—
551C2×C4—731no—
552C2×C4—731no—
553C2×C4—732no—
554C2×C4—732no—
555C2×C4—731no—
556C2×C4—731no—
557C2×C4—732no—
558C2×C4—731no—
559C2×C4—731no—
560C2×C4—732no—
561C2×C4—732no—
562C2×C4—732no—
563C2×C4—732no—
564C2×C4—732no—
565C2×C4—1462no—
566C2×C4—1462no—
567C2×C4—1462no—
568C2×C4—1462no—
569C2×C4—1462no—
570C2×C4—1464no—
571C2×C4—1462no—
572C2×C4—1461no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4—1461no—
576C2×C4#1455667012yes—
577C2×C4D4×#6440811yescharacteristic
578C2×C4Q8×#6440812yes—
579C2×C4C23×#6440811yescharacteristic
580C8—21no—
581C8—21no—
582C8—2922no—
583C8—1461no—
584C8—1461no—
585C22—21no—
586C22—21no—
587C22—731no—
588C22—731no—
589C22—731no—
590C22—731no—
591C22—731no—
592C22—731no—
593C22—732no—
594C22—1462no—
595C22—1462no—
596C22—1462no—
597C22—1462no—
598C22—1461no—
599C22—1461no—
600C22#1798953612yes—
601C22C2×D4×#6440811yescharacteristic
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—2924no—
607C4—2922no—
608C4—2922no—
609C4—2921no—
610C4—2921no—
611C4—2921no—
612C4—2921no—
613C4—5842no—
614C4—731no—
615C4—731no—
616C4—732no—
617C4#1798953612yes—
618C4C2×#1455667011yescharacteristic
619C4C2×Q8×#6440811yescharacteristic
620C2—22no—
621C2—731no—
622C2—731no—
623C2—731no—
624C2—731no—
625C2—1462no—
626C2C2×#1798953611yescharacteristic
627C2#2262669311yescharacteristic
628C2#2262669411yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#427×C73C4
socleC22×C73C2×C4×D4
prime radicalC23×D73C2×D4
cosocleC2×C4×D73C24
hypercenter#427#64408
hypocenterC73#21039
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
3. #22625995C2semidirect
4. #22625994C2semidirect
5. #22625993C2semidirect
6. #22625992C2semidirect
37. #17988625C4semidirect
38. #17988624C4semidirect
39. #17988624C4semidirect
40. #17988623C4semidirect
41. #17988623C4semidirect
42. #17988622C4semidirect
43. #17988622C4semidirect
48. #427×C73C4semidirect
104. #14556525C2×C4semidirect
105. #14556525C2×C4semidirect
106. #14556524C2×C4semidirect
107. #14556524C2×C4semidirect
108. #14556524C2×C4semidirect
109. #14556524C2×C4semidirect
110. #14556523C2×C4semidirect
111. #14556523C2×C4semidirect
112. #14556522C2×C4semidirect
113. #14556522C2×C4semidirect
114. #14556522C2×C4semidirect
115. #14556522C2×C4semidirect
119. #107×C73C2×C4semidirect
120. #107×C73C2×C4semidirect
121. #106×C73C2×C4semidirect
122. #106×C73C2×C4semidirect
326. C73#21039semidirect
341. #427#64408semidirect
424. #107C2×#64408semidirect
425. #107C2×#64408semidirect
432. #106C2×#64408semidirect
433. #106C2×#64408semidirect

Maximal subgroups

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

subgroupindex
#226264122
#226263682
#226259952
#226259942
#226259932
#226259922
#226256202
C2×#179894612
#226255962
#427×D732
#2103973