Finite Group Databasebeta

← #10777924

Subgroups — #10777924

Order 768.

620 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83352C211yesmaximal, characteristic, complemented
2#83343C211yesmaximal, characteristic, complemented
3#79301C211yesmaximal, characteristic
4#79277C211yesmaximal, characteristic, complemented
5C3×#1544C211yesmaximal, characteristic, complemented
6#70773C211yesmaximal, characteristic, complemented
7#70745C211yesmaximal, characteristic, complemented
8#68479C211yesmaximal, characteristic, complemented
9#68478C211yesmaximal, characteristic, complemented
10#68474C211yesmaximal, characteristic, complemented
11#68472C211yesmaximal, characteristic, complemented
12#68455C211yesmaximal, characteristic, complemented
13#68454C211yesmaximal, characteristic, complemented
14#68450C211yesmaximal, characteristic, complemented
15#68448C211yesmaximal, characteristic, complemented
16#19159—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18#5477C2211yescharacteristic
19C4×#235C2211yescharacteristic
20C3×#431C2211yescharacteristic
21C3×#431C2211yescharacteristic
22#5094C2211yescharacteristic
23#5090C2211yescharacteristic
24#5088C2211yescharacteristic
25#5087C2211yescharacteristic
26#5085C2211yescharacteristic
27#5083C2211yescharacteristic
28#5080C2211yescharacteristic
29#5079C2211yescharacteristic
30#5048—21no—
31#5048—21no—
32#5047—21no—
33#5045—21no—
34C4×#216—21no—
35C4×#215—21no—
36C4×#215—21no—
37C4×#215—21no—
38#4925C2211yescharacteristic
39C2×C4×C3⋊C8C2211yescharacteristic
40#4850C2211yescharacteristic
41#4848C2211yescharacteristic
42#4843C2211yescharacteristic
43#4842C2211yescharacteristic
44#4838C2211yescharacteristic
45#4838C2211yescharacteristic
46#4831C2211yescharacteristic
47#4831C2211yescharacteristic
48#4737—21no—
49#4734—21no—
50#4730—21no—
51#4730—21no—
52C4×D24—21no—
53C4×#204—21no—
54C4×ΓL(1,𝔽25)—21no—
55#4685—21no—
56C3×#332C2211yescharacteristic
57C3×#332C2211yescharacteristic
58C3×#332C2211yescharacteristic
59C3×#332C2211yescharacteristic
60#4485C2211yescharacteristic
61#4485C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3167—31no—
69#3107—31no—
70#3106—31no—
71#2965—31no—
72#1704—31no—
73#1703—31no—
74#1702—31no—
75#1701—31no—
76#1692—31no—
77#1691—31no—
78#1686—31no—
79#1685—31no—
80#1564—31no—
81#1544D311yescharacteristic, complemented
82#1543—31no—
83C2×#235—21no—
84C2×#235D411yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×C4⋊Q8C2311yescharacteristic
87C3×#125—21no—
88C3×C22⋊Q8—21no—
89C3×C22⋊Q8—21no—
90C3×C4×Q8—21no—
91C3×C4×Q8—21no—
92C3×#117—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#216—21no—
95C2×#215—21no—
96C2×#215—21no—
97C2×#215—21no—
98#930—21no—
99#930—21no—
100C4×#67—21no—
101C4×#67—21no—
102#926D411yescharacteristic
103C22×C3⋊C8D411yescharacteristic
104C22×C3⋊C8D411yescharacteristic
105C2×D24—21no—
106C2×#204—21no—
107C2×ΓL(1,𝔽25)—21no—
108C2×ΓL(1,𝔽25)—21no—
109#875—21no—
110#874C2311yescharacteristic
111C4×D12—21no—
112C4×D12—21no—
113C4×D12D411yescharacteristic
114C4×D12C2311yescharacteristic
115#872—21no—
116C42×D3—21no—
117#869C2311yescharacteristic
118C4×Dic6D411yescharacteristic
119C4×Dic6C2311yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×C22⋊C8—21no—
125C3×C22⋊C8—21no—
126C3×C4×C8—21no—
127C3×C4×C8—21no—
128#835—21no—
129#835—21no—
130#835—21no—
131#835—21no—
132#830—21no—
133#821—21no—
134#821—21no—
135#821—21no—
136(C2×D3)⋊C8—21no—
137(C2×D3)⋊C8—21no—
138#818—21no—
139#817—21no—
140#816—21no—
141#815—21no—
142#815—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145#810—21no—
146#809—21no—
147#809—21no—
148#809—21no—
149#808—21no—
150#808—21no—
151#808—21no—
152#807—21no—
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4×C3⋊C8—21no—
156C4×C3⋊C8D411yescharacteristic
157C4×C3⋊C8D411yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533C2×D311yescharacteristic
161#532—31no—
162C4×#41—31no—
163#498—31no—
164#492—31no—
165#487—31no—
166#486—31no—
167#485—61no—
168#482—61no—
169#477—31no—
170#476—31no—
171#475—31no—
172#474—31no—
173#465—61no—
174#465—61no—
175#462—31no—
176#461—31no—
177#460—31no—
178#459—31no—
179#452—61no—
180#449—61no—
181#446—31no—
182#446—31no—
183C4×Q16—61no—
184C4×SD16—61no—
185C4×SD16—61no—
186C4×SD16—61no—
187C4×SD16—61no—
188C4×D8—61no—
189#436—31no—
190#436—31no—
191#435—61no—
192#435—61no—
193#431—31no—
194#431—61no—
195#431C2×D311yescharacteristic
196#431C2×D311yescharacteristic
197C4×M16—61no—
198C2×C4×C8—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#332C2×D311yescharacteristic
206#332C2×D311yescharacteristic
207#332C2×D311yescharacteristic
208#332C2×D311yescharacteristic
209#331—31no—
210#331—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4#4111yescharacteristic
216C2×#67—21no—
217C2×#67—41no—
218#235—41no—
219#235—41no—
220#235—82no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12C2×D411yescharacteristic
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—41no—
229C2×Dic6—21no—
230C2×Dic6C2×D411yescharacteristic
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—41no—
240C3×C4⋊C4—41no—
241C3×C22⋊C4—41no—
242C3×C42—41no—
243C3×C42C2×D411yescharacteristic
244C3×C42C2×D411yescharacteristic
245C3×C42#4111yescharacteristic
246C3×C42C2411yescharacteristic
247#217—21no—
248#216—42no—
249#215—42no—
250#215—42no—
251#215—42no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—21no—
254(C2×D3)⋊C4—41no—
255(C2×D3)⋊C4—41no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3C2×D411yescharacteristic
259C4⋊Dic3C2×D411yescharacteristic
260Dic3⋊C4—21no—
261Dic3⋊C4—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C2×C3⋊C8—22no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—41no—
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273D24—42no—
274#204—42no—
275ΓL(1,𝔽25)—42no—
276ΓL(1,𝔽25)—42no—
277C2×#41—31no—
278C2×#41—61no—
279C2×Q16—61no—
280C2×SD16—61no—
281C2×SD16—61no—
282C2×SD16—61no—
283C2×SD16—61no—
284C2×D8—61no—
285C2×M16—61no—
286C2×M16—61no—
287C22×C8—31no—
288C22×C8—31no—
289C4⋊Q8—31no—
290C4⋊Q8C22×D311yescharacteristic
291C4⋊Q8C22×D311yescharacteristic
292#127—31no—
293#125—21no—
294#124—61no—
295C22⋊Q8—21no—
296C22⋊Q8—21no—
297#121—61no—
298#121—61no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—31no—
302C4×Q8—31no—
303C4×D4—31no—
304C4×D4—31no—
305C4×D4—61no—
306C4×D4—61no—
307C4×D4—61no—
308C4×D4—61no—
309#117—21no—
310#117—31no—
311#117—61no—
312C2×C42—61no—
313C2×C42C22×D311yescharacteristic
314#107—61no—
315#107—61no—
316#106—61no—
317#106—61no—
318#106—61no—
319#106—61no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8C22×D311yescharacteristic
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328#103—61no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340C22⋊C8—21no—
341C22⋊C8—21no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345#97—61no—
346#97—61no—
347C4×C8—21no—
348C4×C8—21no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—31no—
352C4×C8—31no—
353C4×C8—61no—
354C23×C3#12111yescharacteristic
355C22×D3—21no—
356C22×D3—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4#12111yescharacteristic
369C2×C3×C4#12111yescharacteristic
370C2×C3×C4#12111yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374#67—42no—
375#67—82no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—41no—
380D12—21no—
381D12—21no—
382D12—41no—
383D12—41no—
384D12—41no—
385D12—41no—
386D12—81no—
387C4×D3—42no—
388C4×D3—41no—
389C4×D3—41no—
390C4×D3—42no—
391C4×D3—82no—
392Dic6—21no—
393Dic6—21no—
394Dic6—81no—
395C3×C8—41no—
396C3×C8—41no—
397C3×C8—41no—
398C3×C8—41no—
399C3⋊C8—21no—
400C3⋊C8—21no—
401C3⋊C8—21no—
402C3⋊C8—41no—
403C3⋊C8—41no—
404C3⋊C8#13512yes—
405#41—121no—
406#41—121no—
407#41—242no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—31no—
411C2×Q8—61no—
412C2×D4—31no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—121no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—121no—
422C22×C4#23511yescharacteristic
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D411yescharacteristic
425Q16—122no—
426SD16—122no—
427SD16—122no—
428SD16—122no—
429SD16—122no—
430D8—122no—
431M16—122no—
432M16—122no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—31no—
438C2×C8—31no—
439C2×C8—31no—
440C2×C8—31no—
441C2×C8—62no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—121no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—31no—
450C4⋊C4—31no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C22⋊C4—41no—
458C22⋊C4—61no—
459C22⋊C4—61no—
460C22⋊C4—61no—
461C22⋊C4—121no—
462C22⋊C4—121no—
463C42—41no—
464C42—61no—
465C42—61no—
466C42—61no—
467C42#23511yescharacteristic
468C42D3×D411yescharacteristic
469C42D3×D411yescharacteristic
470C42C23×D311yescharacteristic
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3C2×#12111yescharacteristic
475C2×D3—41no—
476C2×D3—41no—
477C2×D3—41no—
478C2×D3—81no—
479C2×D3—81no—
480C2×D3—81no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—81no—
485C3×C4—81no—
486C3×C4#47112yes—
487C3×C4C2×#13511yescharacteristic
488C3×C4#57511yescharacteristic
489C3×C4#57511yescharacteristic
490C3×C4#57511yescharacteristic
491Dic3—41no—
492Dic3—41no—
493Dic3—41no—
494Dic3—81no—
495C23—61no—
496C23—121no—
497C23#88411yescharacteristic
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—41no—
502Q8—61no—
503Q8—61no—
504Q8—241no—
505D4—61no—
506D4—61no—
507D4—122no—
508D4—121no—
509D4—121no—
510D4—121no—
511D4—121no—
512D4—241no—
513D4—242no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—122no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—122no—
528C2×C4—121no—
529C2×C4—242no—
530C2×C4#88411yescharacteristic
531C2×C4#88411yescharacteristic
532C2×C4#88411yescharacteristic
533C2×C4C2×#23511yescharacteristic
534C2×C4C2×D3×D411yescharacteristic
535C2×C4C2×D3×D411yescharacteristic
536C8—32no—
537C8—41no—
538C8—41no—
539C8—41no—
540C8—41no—
541C8—61no—
542C8—61no—
543C8—61no—
544C8—121no—
545C8—121no—
546C2×C3—41no—
547C2×C3#303511yescharacteristic
548C2×C3#305411yescharacteristic
549C2×C3C2×#47111yescharacteristic
550D3—81no—
551D3—161no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—121no—
556C22—121no—
557C22—121no—
558C22—241no—
559C22—241no—
560C22—241no—
561C22C2×#88411yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—81no—
566C4—81no—
567C4—121no—
568C4—121no—
569C4—121no—
570C4—241no—
571C4#475412yes—
572C4D3×#13511yescharacteristic
573C4#576811yescharacteristic
574C4#576811yescharacteristic
575C4#576811yescharacteristic
576C3#1915911yescharacteristic, complemented, Sylow
577C2—41no—
578C2—241no—
579C2—481no—
580C2C2×#475411yescharacteristic
581C2#7883911yescharacteristic
582C2#7886111yescharacteristic
subgroupG/H
centerC2×C4#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1544C2
socleC22×C3C2×#121
prime radical#79301C2
hypercenter#1544D3
hypocenterC3#19159
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1544

Complements

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

NH ≅ G/Ntype
1. #83352C2semidirect
2. #83343C2semidirect
4. #79277C2semidirect
5. C3×#1544C2semidirect
6. #70773C2semidirect
7. #70745C2semidirect
8. #68479C2semidirect
9. #68478C2semidirect
10. #68474C2semidirect
11. #68472C2semidirect
12. #68455C2semidirect
13. #68454C2semidirect
14. #68450C2semidirect
15. #68448C2semidirect
81. #1544D3semidirect
576. C3#19159semidirect

Maximal subgroups

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

subgroupindex
#833522
#833432
#793012
#792772
C3×#15442
#707732
#707452
#684792
#684782
#684742
#684722
#684552
#684542
#684502
#684482
#191593