Finite Group Databasebeta

← #10777916

Subgroups — #10777916

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#83299C211yesmaximal, characteristic
2#83290C211yesmaximal, characteristic, complemented
3#79302C211yesmaximal, characteristic, complemented
4#79277C211yesmaximal, characteristic, complemented
5C3×#1543C211yesmaximal, characteristic, complemented
6#70777C211yesmaximal, characteristic, complemented
7#70745C211yesmaximal, characteristic, complemented
8#68515C211yesmaximal, characteristic, complemented
9#68512C211yesmaximal, characteristic, complemented
10#68503C211yesmaximal, characteristic, complemented
11#68500C211yesmaximal, characteristic, complemented
12#68415C211yesmaximal, characteristic, complemented
13#68412C211yesmaximal, characteristic, complemented
14#68401C211yesmaximal, characteristic, complemented
15#68398C211yesmaximal, characteristic, complemented
16#19159—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18C4×#235C2211yescharacteristic
19#5470C2211yescharacteristic
20C3×#431C2211yescharacteristic
21C3×#431C2211yescharacteristic
22#5093C2211yescharacteristic
23#5091C2211yescharacteristic
24#5084C2211yescharacteristic
25#5080C2211yescharacteristic
26#5076C2211yescharacteristic
27#5075C2211yescharacteristic
28#5072C2211yescharacteristic
29#5070C2211yescharacteristic
30#5041—21no—
31#5041—21no—
32#5037—21no—
33#5035—21no—
34C4×#214—21no—
35C4×#214—21no—
36C4×#214—21no—
37C4×#213—21no—
38#4925C2211yescharacteristic
39C2×C4×C3⋊C8C2211yescharacteristic
40#4851C2211yescharacteristic
41#4849C2211yescharacteristic
42#4846C2211yescharacteristic
43#4841C2211yescharacteristic
44#4838C2211yescharacteristic
45#4838C2211yescharacteristic
46#4831C2211yescharacteristic
47#4831C2211yescharacteristic
48#4733—21no—
49#4730—21no—
50#4730—21no—
51#4723—21no—
52C4×Dic12—21no—
53C4×#204—21no—
54C4×ΓL(1,𝔽25)—21no—
55#4685—21no—
56C3×#333C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#331C2211yescharacteristic
59C3×#331C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4482C2211yescharacteristic
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#1544—31no—
82#1543D311yescharacteristic, complemented
83C2×C3×#41—21no—
84C2×#235D411yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×#127C2311yescharacteristic
87C3×#124—21no—
88C3×#121—21no—
89C3×#121—21no—
90C3×C4×D4—21no—
91C3×C4×D4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#214—21no—
94C2×#214—21no—
95C2×#214—21no—
96C2×#213—21no—
97C4×#67—21no—
98C4×#67—21no—
99#926—21no—
100#926D411yescharacteristic
101#924—21no—
102#924—21no—
103C22×C3⋊C8D411yescharacteristic
104C22×C3⋊C8D411yescharacteristic
105C2×Dic12—21no—
106C2×#204—21no—
107C2×ΓL(1,𝔽25)—21no—
108C2×ΓL(1,𝔽25)—21no—
109C4×D12D411yescharacteristic
110C4×D12C2311yescharacteristic
111#872—21no—
112C42×D3—21no—
113#870—21no—
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116C4×Dic6—21no—
117C4×Dic6—21no—
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#832—21no—
129#832—21no—
130#832—21no—
131#832—21no—
132#830—21no—
133#821—21no—
134(C2×D3)⋊C8—21no—
135(C2×D3)⋊C8—21no—
136#818—21no—
137#817—21no—
138#816—21no—
139#816—21no—
140#816—21no—
141#815—21no—
142#815—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145#810—21no—
146#810—21no—
147#810—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#533—31no—
161#532C2×D311yescharacteristic
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#333C2×D311yescharacteristic
204#333C2×D311yescharacteristic
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#331C2×D311yescharacteristic
210#331C2×D311yescharacteristic
211C3×#41—82no—
212C2×C3×Q8—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—41no—
217C22×C3×C4—41no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4#4111yescharacteristic
221C2×#67—21no—
222#235—41no—
223#235—41no—
224C2×D12C2×D411yescharacteristic
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×Dic6—21no—
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×C22⋊C4—41no—
238C3×C22⋊C4—41no—
239C3×C42C2×D411yescharacteristic
240C3×C42C2×D411yescharacteristic
241C3×C42#4111yescharacteristic
242C3×C42C2411yescharacteristic
243#217—21no—
244#217—41no—
245#214—42no—
246#214—42no—
247#214—42no—
248#213—42no—
249(C2×D3)⋊C4—21no—
250(C2×D3)⋊C4—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3C2×D411yescharacteristic
256C4⋊Dic3C2×D411yescharacteristic
257Dic3⋊C4—21no—
258Dic3⋊C4—21no—
259Dic3⋊C4—41no—
260Dic3⋊C4—41no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—41no—
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
273Dic12—42no—
274#204—42no—
275ΓL(1,𝔽25)—42no—
276ΓL(1,𝔽25)—42no—
277C2×#41—21no—
278C2×#41—31no—
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⋊Q8—31no—
291C4⋊Q8C22×D311yescharacteristic
292#127C22×D311yescharacteristic
293#125—61no—
294#124—21no—
295C22⋊Q8—61no—
296C22⋊Q8—61no—
297#121—21no—
298#121—21no—
299C4×Q8—31no—
300C4×Q8—31no—
301C4×Q8—61no—
302C4×Q8—61no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—31no—
306C4×D4—31no—
307C4×D4—61no—
308C4×D4—61no—
309#117—31no—
310#117—61no—
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—41no—
355C23×C3#12111yescharacteristic
356C22×D3—21no—
357C3×Q8—81no—
358C3×D4—41no—
359C3×D4—41no—
360C3×D4—41no—
361C3×D4—41no—
362C3×D4—81no—
363C3×D4—82no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—41no—
370C2×C3×C4—82no—
371C2×C3×C4#12111yescharacteristic
372C2×C3×C4#12111yescharacteristic
373C2×C3×C4#12111yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377#67—42no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383D12—21no—
384D12—21no—
385C4×D3—42no—
386C4×D3—41no—
387C4×D3—41no—
388C4×D3—42no—
389Dic6—21no—
390Dic6—21no—
391Dic6—41no—
392Dic6—41no—
393Dic6—41no—
394Dic6—41no—
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—82no—
406#41—121no—
407#41—121no—
408C2×Q8—21no—
409C2×Q8—31no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—31no—
416C2×D4—41no—
417C2×D4—61no—
418C22×C4—41no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
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—31no—
448C4⋊C4—31no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—121no—
456C4⋊C4—121no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—61no—
460C22⋊C4—61no—
461C22⋊C4—61no—
462C22⋊C4—121no—
463C42—61no—
464C42—61no—
465C42—61no—
466C42—121no—
467C42#23511yescharacteristic
468C42D3×D411yescharacteristic
469C42D3×D411yescharacteristic
470C42C23×D311yescharacteristic
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—81no—
475C22×C3—81no—
476C22×C3—81no—
477C22×C3C2×#12111yescharacteristic
478C2×D3—41no—
479C2×D3—41no—
480C2×D3—41no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—81no—
485C3×C4#47112yes—
486C3×C4C2×#13511yescharacteristic
487C3×C4#57511yescharacteristic
488C3×C4#57511yescharacteristic
489C3×C4#57511yescharacteristic
490Dic3—41no—
491Dic3—41no—
492Dic3—41no—
493Dic3—81no—
494Dic3—81no—
495C23—41no—
496C23—61no—
497C23#88411yescharacteristic
498Q8—61no—
499Q8—61no—
500Q8—81no—
501Q8—121no—
502Q8—121no—
503Q8—121no—
504Q8—121no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—61no—
510D4—61no—
511D4—81no—
512D4—82no—
513D4—122no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—82no—
524C2×C4—122no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—122no—
528C2×C4—121no—
529C2×C4—121no—
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—161no—
548C2×C3#303511yescharacteristic
549C2×C3#305411yescharacteristic
550C2×C3C2×#47111yescharacteristic
551D3—81no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—121no—
559C22—121no—
560C22—121no—
561C22C2×#88411yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—81no—
566C4—121no—
567C4—121no—
568C4—121no—
569C4—241no—
570C4—241no—
571C4#474812yes—
572C4D3×#13511yescharacteristic
573C4#576811yescharacteristic
574C4#576811yescharacteristic
575C4#576811yescharacteristic
576C3#1915911yescharacteristic, complemented, Sylow
577C2—41no—
578C2—161no—
579C2—241no—
580C2C2×#474811yescharacteristic
581C2#7883911yescharacteristic
582C2#7886111yescharacteristic
subgroupG/H
centerC2×C4#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1543C2
socleC22×C3C2×#121
prime radical#83299C2
hypercenter#1543D3
hypocenterC3#19159
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1543

Complements

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

NH ≅ G/Ntype
2. #83290C2semidirect
3. #79302C2semidirect
4. #79277C2semidirect
5. C3×#1543C2semidirect
6. #70777C2semidirect
7. #70745C2semidirect
8. #68515C2semidirect
9. #68512C2semidirect
10. #68503C2semidirect
11. #68500C2semidirect
12. #68415C2semidirect
13. #68412C2semidirect
14. #68401C2semidirect
15. #68398C2semidirect
82. #1543D3semidirect
576. C3#19159semidirect

Maximal subgroups

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

subgroupindex
#832992
#832902
#793022
#792772
C3×#15432
#707772
#707452
#685152
#685122
#685032
#685002
#684152
#684122
#684012
#683982
#191593