Finite Group Databasebeta

← #10786185

Subgroups — #10786185

Order 768.

572 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3380C211yesmaximal, characteristic
2#83626C211yesmaximal, characteristic, complemented
3#83623C211yesmaximal, characteristic, complemented
4#79568C211yesmaximal, characteristic
5#71925C211yesmaximal, characteristic, complemented
6#71924C211yesmaximal, characteristic, complemented
7#71378C211yesmaximal, characteristic, complemented
8#71368C211yesmaximal, characteristic, complemented
9#71147C211yesmaximal, characteristic, complemented
10#71144C211yesmaximal, characteristic, complemented
11#71137C211yesmaximal, characteristic, complemented
12#71132C211yesmaximal, characteristic, complemented
13#70947C211yesmaximal, characteristic, complemented
14#70932C211yesmaximal, characteristic, complemented
15#70879C211yesmaximal, characteristic
16#19865—31nomaximal, Sylow
17C3×#554C2211yescharacteristic
18C3×#554C2211yescharacteristic
19#5543C2211yescharacteristic
20#5542C2211yescharacteristic
21C3×#500C2211yescharacteristic
22C3×#499—21no—
23C3×#478—21no—
24C3×#478—21no—
25C3×#476C2211yescharacteristic
26C3×#476C2211yescharacteristic
27C3×C8×Q8C2211yescharacteristic
28C3×C4×D8—21no—
29C3×C4×D8C2211yescharacteristic
30#5153—21no—
31#5092C2211yescharacteristic
32#5091C2211yescharacteristic
33#5091C2211yescharacteristic
34#5080C2211yescharacteristic
35#5079C2211yescharacteristic
36#5078C2211yescharacteristic
37#5052—21no—
38#5051—21no—
39#5045—21no—
40#5044—21no—
41C4×#216C2211yescharacteristic
42#5021C2211yescharacteristic
43#5020C2211yescharacteristic
44#5019C2211yescharacteristic
45#5012C2211yescharacteristic
46D4×C3⋊C8C2211yescharacteristic
47#5008C2211yescharacteristic
48#4971—21no—
49#4970—21no—
50#4964—21no—
51C2×#807—21no—
52#4877—21no—
53#4768—21no—
54#4758—21no—
55#4681C2211yescharacteristic
56C8×Dic6C2211yescharacteristic
57#4546C2211yescharacteristic
58#4546C2211yescharacteristic
59#4535C2211yescharacteristic
60D4⋊(C3⋊C8)C2211yescharacteristic
61D4⋊(C3⋊C8)C2211yescharacteristic
62#4484C2211yescharacteristic
63#4482C2211yescharacteristic
64#4481C2211yescharacteristic
65#4480C2211yescharacteristic
66#4458C2211yescharacteristic
67#4454C2211yescharacteristic
68#3380D311yescharacteristic
69#3360—31no—
70#3356—31no—
71#3338—31no—
72#1723—31no—
73#1695—31no—
74#1684—31no—
75#1676—31no—
76#1671—31no—
77#1643—31no—
78#1638—31no—
79#1632—31no—
80#1632—31no—
81#1581—31no—
82#1579—31no—
83C2×C3×D8D411yescharacteristic
84C3×C4⋊Q8C2311yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×#125—21no—
87C3×#125—21no—
88C3×C22⋊Q8—21no—
89C3×C22⋊Q8—21no—
90C3×C22⋊Q8—41no—
91C3×C22⋊Q8—41no—
92C3×C4×Q8C2311yescharacteristic
93C3×C4×D4—21no—
94C3×C4×D4—21no—
95C3×C4×D4C2311yescharacteristic
96C3×C4×D4C2311yescharacteristic
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×#216—41no—
100C2×#216D411yescharacteristic
101#936—21no—
102D4×Dic3—21no—
103C2×#214—21no—
104#926—21no—
105C2×C4⋊Dic3—21no—
106#924—21no—
107#924—21no—
108C2×#208—21no—
109C22×C3⋊C8—21no—
110#890—21no—
111#887—21no—
112#879—41no—
113#878—41no—
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116C4×Dic6C2311yescharacteristic
117C3×#107—21no—
118C3×#107—21no—
119C3×#107D411yescharacteristic
120C3×#106—21no—
121C3×#106—21no—
122C3×C4⋊C8—21no—
123C3×C4⋊C8C2311yescharacteristic
124C3×#102—21no—
125C3×#102—21no—
126C3×#102D411yescharacteristic
127C3×#102D411yescharacteristic
128C3×C4×C8—21no—
129C3×C4×C8C2311yescharacteristic
130#835—21no—
131#835—21no—
132#835—41no—
133#835D411yescharacteristic
134#832—21no—
135#830—21no—
136#830—21no—
137#816—21no—
138#816—21no—
139Dic3⋊C8—21no—
140C8×Dic3—21no—
141#810—21no—
142#810—21no—
143#810—21no—
144#810D411yescharacteristic
145#808—21no—
146#808—21no—
147#808—21no—
148#808—21no—
149#807—21no—
150#807—21no—
151#807—21no—
152#807—21no—
153#807—21no—
154#807D411yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#556—31no—
161#554—31no—
162#554C2×D311yescharacteristic
163#554C2×D311yescharacteristic
164#500—31no—
165#500C2×D311yescharacteristic
166#499—21no—
167#498—31no—
168#494—31no—
169#488—61no—
170#487—31no—
171#487—31no—
172#485—61no—
173#484—61no—
174#484—61no—
175#483—61no—
176#479—61no—
177#478—21no—
178#478—21no—
179#477—31no—
180#477—31no—
181#477—31no—
182#476—31no—
183#476C2×D311yescharacteristic
184#476C2×D311yescharacteristic
185#475—31no—
186#470—61no—
187#467—61no—
188#463—61no—
189#461—31no—
190C8×Q8—31no—
191C8×Q8C2×D311yescharacteristic
192C4×Q16—31no—
193C4×SD16—61no—
194C4×D8—21no—
195C4×D8C2×D311yescharacteristic
196#436—31no—
197C8×D4—31no—
198#428—61no—
199C2×#107—61no—
200#335—31no—
201#334—31no—
202#333—31no—
203#333—31no—
204#333—31no—
205#332—31no—
206Q8⋊C8—31no—
207Q8⋊C8—31no—
208D4⋊C8—31no—
209D4⋊C8—31no—
210#322—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8C2×D411yescharacteristic
214C2×C3×D4C2×D411yescharacteristic
215C2×C3×D4C2×D411yescharacteristic
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—41no—
219C22×C3×C4—41no—
220C22×Dic3—41no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6C2×D411yescharacteristic
224C3×D8—22no—
225C2×C3×C8—21no—
226C2×C3×C8C2×D411yescharacteristic
227C2×C3×C8C2×D411yescharacteristic
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—41no—
234C3×C4⋊C4—41no—
235C3×C4⋊C4—41no—
236C3×C4⋊C4—41no—
237C3×C4⋊C4C2×D411yescharacteristic
238C3×C4⋊C4C2×D411yescharacteristic
239C3×C4⋊C4C2×D411yescharacteristic
240C3×C22⋊C4—21no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—41no—
243C3×C22⋊C4—41no—
244C3×C42—21no—
245C3×C42C2411yescharacteristic
246#217—41no—
247#217—41no—
248#217—41no—
249#216—22no—
250#216—82no—
251#214—42no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—41no—
257C4⋊Dic3C2×D411yescharacteristic
258Dic3⋊C4—21no—
259Dic3⋊C4—41no—
260Dic3⋊C4—41no—
261C4×Dic3—21no—
262C4×Dic3—41no—
263#208—42no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—42no—
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271C2×Q16—31no—
272C2×Q16—121no—
273C2×SD16—61no—
274C2×D8#6711yescharacteristic
275C2×M16—61no—
276C22×C8—61no—
277C4⋊Q8—31no—
278C4⋊Q8—31no—
279C4⋊Q8C22×D311yescharacteristic
280C4⋊Q8C22×D311yescharacteristic
281#126—121no—
282#125—21no—
283#125—21no—
284#125—61no—
285#124—61no—
286C22⋊Q8—21no—
287C22⋊Q8—21no—
288C22⋊Q8—41no—
289C22⋊Q8—41no—
290C22⋊Q8—61no—
291C22⋊Q8—61no—
292C22⋊Q8—121no—
293C4×Q8—31no—
294C4×Q8—61no—
295C4×Q8C22×D311yescharacteristic
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—61no—
299C4×D4C22×D311yescharacteristic
300C4×D4C22×D311yescharacteristic
301#117—61no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—61no—
305#107—21no—
306#107—21no—
307#107—31no—
308#107—61no—
309#107—61no—
310#107—61no—
311#107—61no—
312#107—61no—
313#107#6711yescharacteristic
314#106—21no—
315#106—21no—
316#106—61no—
317#106—61no—
318#106—61no—
319#106—61no—
320C4⋊C8—21no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—61no—
325C4⋊C8C22×D311yescharacteristic
326#103—31no—
327#103—31no—
328#103—61no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—121no—
336#102—21no—
337#102—21no—
338#102—61no—
339#102#6711yescharacteristic
340#102#6711yescharacteristic
341C22⋊C8—61no—
342C22⋊C8—61no—
343C4×C8—21no—
344C4×C8—31no—
345C4×C8—31no—
346C4×C8—61no—
347C4×C8C22×D311yescharacteristic
348C23×C3—21no—
349C23×C3—21no—
350C3×Q8—21no—
351C3×Q8—21no—
352C3×Q8—81no—
353C3×Q8—81no—
354C3×D4—21no—
355C3×D4—21no—
356C3×D4—21no—
357C3×D4#13712yes—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—161no—
369C2×C3×C4—161no—
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4H(5,𝔽2)11yescharacteristic
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—161no—
378Dic6—21no—
379Dic6—21no—
380Dic6—41no—
381Dic6—41no—
382Dic6—81no—
383C3×C8—21no—
384C3×C8—21no—
385C3×C8—41no—
386C3⋊C8—21no—
387C3⋊C8—41no—
388C3⋊C8—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C3⋊C8#13512yes—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—31no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8D3×D411yescharacteristic
398C2×D4C2×#6711yescharacteristic
399C2×D4C2×#6711yescharacteristic
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—121no—
405Q16—62no—
406Q16—242no—
407SD16—122no—
408D8—22no—
409M16—122no—
410C2×C8—21no—
411C2×C8—31no—
412C2×C8—31no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—122no—
418C2×C8C2×#6711yescharacteristic
419C2×C8C2×#6711yescharacteristic
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—31no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—121no—
436C4⋊C4—121no—
437C4⋊C4—121no—
438C4⋊C4D3×D411yescharacteristic
439C4⋊C4C2×#6711yescharacteristic
440C4⋊C4C2×#6711yescharacteristic
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—121no—
446C22⋊C4—121no—
447C22⋊C4—121no—
448C42—21no—
449C42—61no—
450C42—121no—
451C42C23×D311yescharacteristic
452C22×C3—22no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3D4211yescharacteristic
459C3×C4—21no—
460C3×C4—41no—
461C3×C4—41no—
462C3×C4—41no—
463C3×C4—41no—
464C3×C4—81no—
465C3×C4—81no—
466C3×C4C2×#13511yescharacteristic
467C3×C4C2×#13711yescharacteristic
468C3×C4#57811yescharacteristic
469C3×C4#57811yescharacteristic
470Dic3—41no—
471Dic3—41no—
472Dic3—81no—
473Dic3—81no—
474C23—21no—
475C23—21no—
476Q8—21no—
477Q8—21no—
478Q8—61no—
479Q8—61no—
480Q8—81no—
481Q8—81no—
482Q8—121no—
483Q8—121no—
484Q8—241no—
485D4—21no—
486D4—21no—
487D4—21no—
488D4#94212yes—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—161no—
504C2×C4—161no—
505C2×C4—481no—
506C2×C4C2×D3×D411yescharacteristic
507C2×C4#100411yescharacteristic
508C2×C4C22×#6711yescharacteristic
509C8—21no—
510C8—21no—
511C8—32no—
512C8—41no—
513C8—61no—
514C8—121no—
515C8—121no—
516C8—121no—
517C8—121no—
518C2×C3—42no—
519C2×C3—81no—
520C2×C3#327011yescharacteristic
521C2×C3#328311yescharacteristic
522C2×C3#328811yescharacteristic
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22D4×#6711yescharacteristic
530C4—21no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—81no—
536C4—81no—
537C4—121no—
538C4—121no—
539C4—241no—
540C4—241no—
541C4D3×#13511yescharacteristic
542C4#577111yescharacteristic
543C4C2×#94211yescharacteristic
544C4#583811yescharacteristic
545C3#1986511yescharacteristic, complemented, Sylow
546C2—42no—
547C2—81no—
548C2#8356511yescharacteristic
549C2#8392611yescharacteristic
550C2#8400911yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3380C2
socleC22×C3D42
prime radicalC2×C3×D8D4
hypercenter#3380D3
hypocenterC3#19865
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #3380

Complements

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

NH ≅ G/Ntype
2. #83626C2semidirect
3. #83623C2semidirect
5. #71925C2semidirect
6. #71924C2semidirect
7. #71378C2semidirect
8. #71368C2semidirect
9. #71147C2semidirect
10. #71144C2semidirect
11. #71137C2semidirect
12. #71132C2semidirect
13. #70947C2semidirect
14. #70932C2semidirect
545. C3#19865semidirect

Maximal subgroups

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

subgroupindex
C3×#33802
#836262
#836232
#795682
#719252
#719242
#713782
#713682
#711472
#711442
#711372
#711322
#709472
#709322
#708792
#198653