Finite Group Databasebeta

← #34019237

Subgroups — #34019237

Order 18688.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626881C211yesmaximal, characteristic, complemented
2#22626791C211yesmaximal, characteristic, complemented
3#22625565C211yesmaximal, characteristic, complemented
4#22625562C211yesmaximal, characteristic, complemented
5#22625558C211yesmaximal, characteristic, complemented
6#22625552C211yesmaximal, characteristic, complemented
7#22625550C211yesmaximal, characteristic, complemented
8#22625548C211yesmaximal, characteristic, complemented
9SD16×#10596833C211yesmaximal, characteristic, complemented
10D8×#10596833C211yesmaximal, characteristic, complemented
11#22625505C211yesmaximal, characteristic, complemented
12#22625283C211yesmaximal, characteristic, complemented
13#22625217C211yesmaximal, characteristic, complemented
14#22625137C211yesmaximal, characteristic, complemented
15#22619977C211yesmaximal, characteristic
16#17989638C2211yescharacteristic, complemented
17Q8×#10596833C2211yescharacteristic, complemented
18#17989622C2211yescharacteristic, complemented
19#17989621C2211yescharacteristic
20#17989621C2211yescharacteristic
21#17989619—21no—
22#17989616—21no—
23#17989616—21no—
24#17989614—21no—
25D4×#10596833C2211yescharacteristic
26D4×#10596833C2211yescharacteristic
27D4×#10596833C2211yescharacteristic, complemented
28#17989611—21no—
29#17989609C2211yescharacteristic
30#17989606C2211yescharacteristic
31#17989593—21no—
32#17989588C2211yescharacteristic
33#17989558—21no—
34C2×#14556648—21no—
35C4×#11924036C2211yescharacteristic
36#17989451C2211yescharacteristic, complemented
37#17989450C2211yescharacteristic, complemented
38#17989449C2211yescharacteristic, complemented
39#17989448C2211yescharacteristic, complemented
40Q8⋊#10596833C2211yescharacteristic, complemented
41Q8⋊#10596833C2211yescharacteristic, complemented
42#17989435C2211yescharacteristic
43#17989435C2211yescharacteristic
44D4⋊#10596833C2211yescharacteristic
45D4⋊#10596833C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic, complemented
47D4⋊#10596833C2211yescharacteristic, complemented
48#17989429C2211yescharacteristic, complemented
49#17989429C2211yescharacteristic, complemented
50C8×#10596833C2211yescharacteristic, complemented
51C8×#10596833C2211yescharacteristic, complemented
52#17989116C2211yescharacteristic
53C4×D4×D73C2211yescharacteristic
54#17988921C411yescharacteristic
55#17988904C411yescharacteristic
56#17988888C411yescharacteristic
57#17988887C411yescharacteristic
58#17988799C411yescharacteristic
59#17988736C411yescharacteristic
60#17988719C411yescharacteristic
61SD16×Dic73C2211yescharacteristic
62#17988672C2211yescharacteristic
63D8×Dic73C2211yescharacteristic
64#17988667C2211yescharacteristic
65#17988639C2211yescharacteristic
66#463×C73C411yescharacteristic
67C2×#11924038—21no—
68C2×#11924038—21no—
69C2×#11924037—21no—
70C2×#11924036—21no—
71C2×#11924036—21no—
72C22×#10596833—21no—
73C22×#10596833—21no—
74C22×#10596833—21no—
75#14556658—21no—
76#14556658—21no—
77C4⋊#10596833—21no—
78C4⋊#10596833D411yescharacteristic
79C4⋊#10596833D411yescharacteristic
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84#14556648—21no—
85#14556648—21no—
86#14556648—21no—
87#14556648D411yescharacteristic
88#14556648D411yescharacteristic
89C22⋊#10596833—21no—
90C22⋊#10596833—21no—
91C22⋊#10596833—21no—
92C22⋊#10596833—21no—
93#14556646—21no—
94#14556646—21no—
95C4×#10596833—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C2×D4×D73C2×C411yescharacteristic
102C22×C4×D73—21no—
103C2×#11924015C2×C411yescharacteristic
104C2×#11924013C2×C411yescharacteristic
105#14556584—21no—
106#14556583C2×C411yescharacteristic
107#14556573—21no—
108#14556567C2×C411yescharacteristic
109#14556562C2×C411yescharacteristic
110Q8×Dic73C2311yescharacteristic
111C4×D292C2×C411yescharacteristic
112D4×Dic73C2311yescharacteristic
113D4×Dic73C2311yescharacteristic
114#14556553C2311yescharacteristic
115C4×C22⋊D73—21no—
116#14556551—21no—
117#14556551—21no—
118#14556547—21no—
119C4⋊C4×D73C2×C411yescharacteristic
120C22⋊C4×D73—21no—
121C42×D73C2311yescharacteristic
122#14556525C2×C411yescharacteristic
123#14556524C2×C411yescharacteristic
124#14556523C2×C411yescharacteristic
125#14556522C2×C411yescharacteristic
126C4⋊#10596831C2×C411yescharacteristic
127#14556515C2×C411yescharacteristic
128#14556513C2×C411yescharacteristic
129#14556513C2×C411yescharacteristic
130#14556512C2×C411yescharacteristic
131#14556512C2×C411yescharacteristic
132#14556512C2×C411yescharacteristic
133#14556503C2×C411yescharacteristic
134#14556503C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137C2×SD16×C73C2×C411yescharacteristic
138C2×D8×C73C2×C411yescharacteristic
139#124×C73C2×C411yescharacteristic
140C4×D4×C73C2×C411yescharacteristic
141C4⋊C8×C73C2×C411yescharacteristic
142#103×C73C2×C411yescharacteristic
143#102×C73C2×C411yescharacteristic
144#11924038—81no—
145#11924038—81no—
146#11924037—81no—
147#11924036—24no—
148#11924036—81no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—21no—
154C2×#10596833—21no—
155C2×#10596833—81no—
156C2×#10596833—81no—
157C2×#10596833—81no—
158C2×#10596833C2×D411yescharacteristic
159C2×#10596833C2×D411yescharacteristic
160C2×#10596833C2×D411yescharacteristic
161C2×#10596833C2×D411yescharacteristic
162C23×D73—21no—
163C2×D292C22×C411yescharacteristic
164C2×C22⋊D73—21no—
165C2×C22⋊D73—21no—
166D4×D73—22no—
167D4×D73—41no—
168C22×Dic73—21no—
169C22×Dic73—21no—
170C2×C4×D73—21no—
171C2×C4×D73—21no—
172C2×C4×D73—21no—
173C2×C4×D73—81no—
174C2×C4×D73—81no—
175C2×C4×D73C22×C411yescharacteristic
176C2×C4×D73C2×D411yescharacteristic
177C2×C4×D73#4111yescharacteristic
178#11924015—41no—
179#11924013—41no—
180C2×#10596831C22×C411yescharacteristic
181C2×#10596831C22×C411yescharacteristic
182C4⋊Dic73—21no—
183C4⋊Dic73C22×C411yescharacteristic
184C4⋊Dic73C22×C411yescharacteristic
185C4⋊Dic73C22×C411yescharacteristic
186#11924007—21no—
187#11924007—21no—
188#11924007—21no—
189#11924007—21no—
190C22⋊Dic73—21no—
191C22⋊Dic73—21no—
192C22⋊Dic73—21no—
193#11924005—21no—
194#11924005—21no—
195#11924005—21no—
196C4×Dic73—21no—
197C4×Dic73C2×D411yescharacteristic
198C4×Dic73#4111yescharacteristic
199C4×Dic73C2411yescharacteristic
200C2×Q8×C73C22×C411yescharacteristic
201C2×D4×C73C22×C411yescharacteristic
202C2×D4×C73C22×C411yescharacteristic
203C22×C4×C73—21no—
204SD16×C73—22no—
205D8×C73—22no—
206C2×C8×C73C22×C411yescharacteristic
207C2×C8×C73C22×C411yescharacteristic
208C4⋊C4×C73C22×C411yescharacteristic
209C22⋊C4×C73—21no—
210C22⋊C4×C73—21no—
211C22⋊C4×C73—21no—
212C42×C73C22×C411yescharacteristic
213C23×C73—21no—
214C23×C73—21no—
215C22×D73—21no—
216C22×D73—42no—
217C22×D73—42no—
218C22×D73—81no—
219C22×D73C4×D411yescharacteristic
220Q8×C73—21no—
221Q8×C73—21no—
222D4×C73—21no—
223D4×C73—21no—
224D4×C73—21no—
225D4×C73#13112yes—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—81no—
229C2×C4×C73C4×D411yescharacteristic
230C2×C4×C73C4×D411yescharacteristic
231C2×C4×C73C23×C411yescharacteristic
232C22⋊D73—42no—
233C22⋊D73—42no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73C4×D411yescharacteristic
242C2×Dic73C22×D411yescharacteristic
243C2×Dic73C2×#4111yescharacteristic
244D292—21no—
245D292—21no—
246C4×D73—22no—
247C4×D73—21no—
248C4×D73—21no—
249C4×D73—21no—
250C4×D73—21no—
251C4×D73—41no—
252C4×D73—41no—
253C4×D73—81no—
254C4×D73—81no—
255#10596833—22no—
256#10596833—22no—
257#10596833—21no—
258#10596833—21no—
259#10596833—21no—
260#10596833—21no—
261#10596833—41no—
262#10596833—41no—
263#10596833—41no—
264#10596833—41no—
265C8×C73—21no—
266C8×C73—21no—
267#10596831—21no—
268#10596831—21no—
269C22×C73—22no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73C2×C4×D411yescharacteristic
276C2×D73—21no—
277C2×D73—21no—
278C2×D73—41no—
279C2×D73—41no—
280C2×D73—41no—
281C2×D73—84no—
282C2×D73#44312yes—
283C4×C73—21no—
284C4×C73—21no—
285C4×C73—41no—
286C4×C73—41no—
287C4×C73C2×#13111yescharacteristic
288C4×C73#56811yescharacteristic
289Dic73—21no—
290Dic73—21no—
291Dic73—41no—
292Dic73—41no—
293Dic73—41no—
294Dic73#57611yescharacteristic
295Dic73#57711yescharacteristic
296#18014—731nomaximal, Sylow
297C2×C73—42no—
298C2×C73—81no—
299C2×C73#293311yescharacteristic
300C2×C73#294611yescharacteristic
301C2×C73C2×#44311yescharacteristic
302D73—22no—
303D73—81no—
304#3105—731no—
305#2994—731no—
306#2974—731no—
307#1590—731no—
308#1589—731no—
309#1588—731no—
310#1586—731no—
311#1586—731no—
312#1585—731no—
313C8×SD16—731no—
314C8×D8—731no—
315#1573—731no—
316#1494—731no—
317C2×D4⋊C8—731no—
318#1450—731no—
319C73#1801411yescharacteristic, complemented, Sylow
320#529—731no—
321C2×C4×D4—731no—
322#484—731no—
323#481—731no—
324#478—731no—
325#478—731no—
326#463—731no—
327#463#6440811yescharacteristic
328#450—731no—
329#447—731no—
330#446—731no—
331C8×Q8—731no—
332C4×SD16—731no—
333C4×SD16—731no—
334C4×D8—731no—
335C4×D8—731no—
336#436—731no—
337#436—731no—
338#436—731no—
339#435—1461no—
340#435—1461no—
341#435—1461no—
342C8×D4—731no—
343C8×D4—731no—
344C8×D4—731no—
345C8×D4—1461no—
346#433—1461no—
347#432—731no—
348#431—731no—
349#431—731no—
350#424—1461no—
351C2×C4⋊C8—731no—
352#407—1461no—
353C2×C22⋊C8—1461no—
354C2×C4×C8—731no—
355#335—731no—
356#335—731no—
357#334—731no—
358#334—731no—
359Q8⋊C8—731no—
360Q8⋊C8—731no—
361D4⋊C8—731no—
362D4⋊C8—731no—
363D4⋊C8—731no—
364D4⋊C8—731no—
365D4⋊C8—731no—
366D4⋊C8—731no—
367#322—731no—
368#322—731no—
369C82—731no—
370C82—731no—
371C22×D4—731no—
372C23×C4—1461no—
373C2×SD16—731no—
374C2×SD16C2×#6440811yescharacteristic
375C2×D8—731no—
376C2×D8C2×#6440811yescharacteristic
377C2×M16—1461no—
378C2×M16—1461no—
379C2×M16—1461no—
380C22×C8—1461no—
381C22×C8—1461no—
382C22×C8—1461no—
383C22×C8—1461no—
384C22×C8—1461no—
385#126—1461no—
386#125—731no—
387#124C2×#6440811yescharacteristic
388#123—1461no—
389C22⋊Q8—731no—
390#121—731no—
391C4×Q8—731no—
392C4×D4—731no—
393C4×D4—731no—
394C4×D4—731no—
395C4×D4—731no—
396C4×D4—1461no—
397C4×D4—1461no—
398C4×D4—1461no—
399C4×D4C2×#6440811yescharacteristic
400#117—1461no—
401C2×C4⋊C4—731no—
402C2×C22⋊C4—1461no—
403C2×C42—731no—
404#107—731no—
405#107—731no—
406#106—731no—
407#106—731no—
408C4⋊C8—731no—
409C4⋊C8—731no—
410C4⋊C8—731no—
411C4⋊C8—731no—
412C4⋊C8—731no—
413C4⋊C8—731no—
414C4⋊C8—731no—
415C4⋊C8—1461no—
416C4⋊C8—1461no—
417C4⋊C8—1461no—
418C4⋊C8C2×#6440811yescharacteristic
419#103—731no—
420#103C2×#6440811yescharacteristic
421#102—731no—
422#102—731no—
423#102—731no—
424#102—731no—
425#102—731no—
426#102C2×#6440811yescharacteristic
427C22⋊C8—731no—
428C22⋊C8—731no—
429C22⋊C8—731no—
430C22⋊C8—731no—
431C22⋊C8—1461no—
432C22⋊C8—1461no—
433C22⋊C8—1461no—
434C22⋊C8—1461no—
435C22⋊C8—1461no—
436C22⋊C8—1461no—
437C22⋊C8—1461no—
438#97—1461no—
439#97—1461no—
440C4×C8—731no—
441C4×C8—731no—
442C4×C8—731no—
443C4×C8—731no—
444C4×C8—731no—
445C4×C8—731no—
446C4×C8—1461no—
447C4×C8—1461no—
448C24—1461no—
449C2×Q8C22×#6440811yescharacteristic
450C2×D4—2921no—
451C2×D4—731no—
452C2×D4—1461no—
453C2×D4—1461no—
454C2×D4—1462no—
455C2×D4C22×#6440811yescharacteristic
456C2×D4C22×#6440811yescharacteristic
457C22×C4—21no—
458C22×C4—5841no—
459C22×C4—5841no—
460C22×C4—731no—
461C22×C4—731no—
462C22×C4—731no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1461no—
468SD16—22no—
469SD16—2921no—
470D8—22no—
471D8—2921no—
472M16—5841no—
473M16—5841no—
474M16—5841no—
475C2×C8—5841no—
476C2×C8—5841no—
477C2×C8—5841no—
478C2×C8—5841no—
479C2×C8—731no—
480C2×C8—731no—
481C2×C8—731no—
482C2×C8—731no—
483C2×C8—731no—
484C2×C8—731no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C2×C8—1464no—
492C2×C8C22×#6440811yescharacteristic
493C2×C8C22×#6440811yescharacteristic
494C4⋊C4—731no—
495C4⋊C4—731no—
496C4⋊C4—731no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4C22×#6440811yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—1461no—
507C22⋊C4—1461no—
508C22⋊C4—1461no—
509C22⋊C4—1461no—
510C22⋊C4—1461no—
511C22⋊C4—1461no—
512C42—731no—
513C42—731no—
514C42—731no—
515C42—1461no—
516C42C22×#6440811yescharacteristic
517C23—21no—
518C23—21no—
519C23—2922no—
520C23—2922no—
521C23—5841no—
522C23—731no—
523C23—1461no—
524Q8—21no—
525Q8—21no—
526D4—21no—
527D4—21no—
528D4—21no—
529D4—2922no—
530D4—2922no—
531D4—1461no—
532D4—1461no—
533D4#1455667812yes—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—81no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—5841no—
544C2×C4—5841no—
545C2×C4—731no—
546C2×C4—731no—
547C2×C4—731no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1462no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4D4×#6440811yescharacteristic
557C2×C4D4×#6440811yescharacteristic
558C2×C4C23×#6440811yescharacteristic
559C8—21no—
560C8—21no—
561C8—2921no—
562C8—2921no—
563C8—2921no—
564C8—2921no—
565C8—1461no—
566C8—1461no—
567C8—1461no—
568C8—1461no—
569C8—1462no—
570C8—1462no—
571C8—1461no—
572C8—1461no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—2921no—
580C22—2921no—
581C22—2921no—
582C22—5844no—
583C22—732no—
584C22—1461no—
585C22—1461no—
586C22C2×D4×#6440811yescharacteristic
587C4—21no—
588C4—21no—
589C4—41no—
590C4—41no—
591C4—2921no—
592C4—2921no—
593C4—2921no—
594C4—731no—
595C4—731no—
596C4—1461no—
597C4—1461no—
598C4C2×#1455667811yescharacteristic
599C4#1798966611yescharacteristic
600C2—42no—
601C2—81no—
602C2—5841no—
603C2—1462no—
604C2#2262666111yescharacteristic
605C2#2262668811yescharacteristic
606C2D73⋊#44311yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#463×C73C4
socleC22×C73C2×C4×D4
prime radical#17988719C4
cosocleC4×Dic73C24
hypercenter#463#64408
hypocenterC73#18014
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #22626881C2semidirect
2. #22626791C2semidirect
3. #22625565C2semidirect
4. #22625562C2semidirect
5. #22625558C2semidirect
6. #22625552C2semidirect
7. #22625550C2semidirect
8. #22625548C2semidirect
9. SD16×#10596833C2semidirect
10. D8×#10596833C2semidirect
11. #22625505C2semidirect
12. #22625283C2semidirect
13. #22625217C2semidirect
14. #22625137C2semidirect
16. #17989638C22semidirect
17. Q8×#10596833C22semidirect
18. #17989622C22semidirect
27. D4×#10596833C22semidirect
36. #17989451C22semidirect
37. #17989450C22semidirect
38. #17989449C22semidirect
39. #17989448C22semidirect
40. Q8⋊#10596833C22semidirect
41. Q8⋊#10596833C22semidirect
46. D4⋊#10596833C22semidirect
47. D4⋊#10596833C22semidirect
48. #17989429C22semidirect
49. #17989429C22semidirect
50. C8×#10596833C22semidirect
51. C8×#10596833C22semidirect
319. C73#18014semidirect

Maximal subgroups

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

subgroupindex
#226268812
#226267912
#226255652
#226255622
#226255582
#226255522
#226255502
#226255482
SD16×#105968332
D8×#105968332
#226255052
#226252832
#226252172
#226251372
#226199772
#1801473