Finite Group Databasebeta

← #10777942

Subgroups — #10777942

Order 768.

598 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83352C211yesmaximal, characteristic
2#83344C211yesmaximal, characteristic, complemented
3#79302C211yesmaximal, characteristic
4#79277C211yesmaximal, characteristic
5C3×#1544C211yesmaximal, characteristic, complemented
6#70777C211yesmaximal, characteristic, complemented
7#70745C211yesmaximal, characteristic, complemented
8#68477C211yesmaximal, characteristic, complemented
9#68476C211yesmaximal, characteristic, complemented
10#68475C211yesmaximal, characteristic, complemented
11#68473C211yesmaximal, characteristic, complemented
12#68453C211yesmaximal, characteristic, complemented
13#68452C211yesmaximal, characteristic, complemented
14#68451C211yesmaximal, characteristic, complemented
15#68449C211yesmaximal, characteristic, complemented
16#19161—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C4×#235C2211yescharacteristic
19#5470C2211yescharacteristic
20C3×#431C2211yescharacteristic
21C3×#431C2211yescharacteristic
22#5094C2211yescharacteristic
23#5093C2211yescharacteristic
24#5092C2211yescharacteristic
25#5090C2211yescharacteristic
26#5087C2211yescharacteristic
27#5083C2211yescharacteristic
28#5079C2211yescharacteristic
29#5078C2211yescharacteristic
30#5051—21no—
31#5051—21no—
32#5047—21no—
33#5045—21no—
34C4×#216—21no—
35C4×#216—21no—
36C4×#216—21no—
37C4×#215—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×#332C2211yescharacteristic
57C3×#332C2211yescharacteristic
58C3×#332C2211yescharacteristic
59C3×#332C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4482C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3170—31no—
69#3107—31no—
70#3107—31no—
71#2965—31no—
72#1700—31no—
73#1700—31no—
74#1699—31no—
75#1699—31no—
76#1694—31no—
77#1693—31no—
78#1690—31no—
79#1689—31no—
80#1564—31no—
81#1544—31no—
82#1544D311yescharacteristic, complemented
83C2×#235D411yescharacteristic
84C3×C4⋊Q8C2311yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×#125—21no—
87C3×C22⋊Q8—21no—
88C3×C22⋊Q8—21no—
89C3×C4×Q8—21no—
90C3×C4×Q8—21no—
91C3×#117—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#216—21no—
94C2×#216—21no—
95C2×#216—21no—
96C2×#215—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#835—21no—
129#835—21no—
130#835—21no—
131#835—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#807—21no—
151#807—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#533C2×D311yescharacteristic
162C4×#41—31no—
163#500—31no—
164#494—31no—
165#487—31no—
166#487—31no—
167#485—61no—
168#485—61no—
169#475—31no—
170#475—31no—
171#474—31no—
172#474—31no—
173#467—61no—
174#467—61no—
175#462—31no—
176#462—31no—
177#461—31no—
178#461—31no—
179#452—61no—
180#452—61no—
181#446—31no—
182#446—31no—
183C4×Q16—61no—
184C4×Q16—61no—
185C4×Q16—61no—
186C4×Q16—61no—
187C4×SD16—61no—
188C4×SD16—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#332—31no—
204#332—31no—
205#332—31no—
206#332—31no—
207#332C2×D311yescharacteristic
208#332C2×D311yescharacteristic
209#332C2×D311yescharacteristic
210#332C2×D311yescharacteristic
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4#4111yescharacteristic
216C2×#67—21no—
217#235—41no—
218#235—41no—
219C2×D12C2×D411yescharacteristic
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6C2×D411yescharacteristic
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—41no—
235C3×C4⋊C4—41no—
236C3×C22⋊C4—41no—
237C3×C42—41no—
238C3×C42C2×D411yescharacteristic
239C3×C42C2×D411yescharacteristic
240C3×C42#4111yescharacteristic
241C3×C42C2411yescharacteristic
242#217—21no—
243#217—41no—
244#216—42no—
245#216—42no—
246#216—42no—
247#215—42no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3C2×D411yescharacteristic
255C4⋊Dic3C2×D411yescharacteristic
256Dic3⋊C4—21no—
257Dic3⋊C4—21no—
258Dic3⋊C4—41no—
259Dic3⋊C4—41no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—41no—
264C2×C3⋊C8—22no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—41no—
268C2×C3⋊C8C2×D411yescharacteristic
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272Dic12—42no—
273#204—42no—
274ΓL(1,𝔽25)—42no—
275ΓL(1,𝔽25)—42no—
276C2×#41—31no—
277C2×Q16—61no—
278C2×Q16—61no—
279C2×Q16—61no—
280C2×Q16—61no—
281C2×SD16—61no—
282C2×SD16—61no—
283C2×M16—61no—
284C2×M16—61no—
285C22×C8—31no—
286C22×C8—31no—
287C4⋊Q8—31no—
288C4⋊Q8—31no—
289C4⋊Q8C22×D311yescharacteristic
290C4⋊Q8C22×D311yescharacteristic
291#125—21no—
292#125—61no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—61no—
296C22⋊Q8—61no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—31no—
300C4×Q8—31no—
301C4×Q8—61no—
302C4×Q8—61no—
303C4×D4—31no—
304C4×D4—31no—
305C4×D4—61no—
306C4×D4—61no—
307#117—21no—
308#117—31no—
309#117—61no—
310#117—61no—
311C2×C42—61no—
312C2×C42C22×D311yescharacteristic
313#107—61no—
314#107—61no—
315#107—61no—
316#107—61no—
317#106—61no—
318#106—61no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—61no—
322C4⋊C8—61no—
323C4⋊C8C22×D311yescharacteristic
324C4⋊C8C22×D311yescharacteristic
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—61no—
328#103—61no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#102—61no—
338#102—61no—
339C22⋊C8—21no—
340C22⋊C8—21no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344#97—61no—
345#97—61no—
346C4×C8—21no—
347C4×C8—21no—
348C4×C8—31no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—31no—
352C4×C8—61no—
353C23×C3#12111yescharacteristic
354C22×D3—21no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4#12111yescharacteristic
367C2×C3×C4#12111yescharacteristic
368C2×C3×C4#12111yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372#67—42no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378D12—21no—
379D12—21no—
380C4×D3—42no—
381C4×D3—41no—
382C4×D3—41no—
383C4×D3—42no—
384Dic6—21no—
385Dic6—21no—
386Dic6—41no—
387Dic6—41no—
388Dic6—41no—
389Dic6—41no—
390C3×C8—41no—
391C3×C8—41no—
392C3×C8—41no—
393C3×C8—41no—
394C3⋊C8—21no—
395C3⋊C8—21no—
396C3⋊C8—21no—
397C3⋊C8—41no—
398C3⋊C8—41no—
399C3⋊C8#13512yes—
400#41—121no—
401#41—121no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—31no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×D4—31no—
408C2×D4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4#23511yescharacteristic
413C22×C4D3×D411yescharacteristic
414C22×C4D3×D411yescharacteristic
415Q16—122no—
416Q16—122no—
417Q16—122no—
418Q16—122no—
419SD16—122no—
420SD16—122no—
421M16—122no—
422M16—122no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—31no—
428C2×C8—31no—
429C2×C8—31no—
430C2×C8—31no—
431C2×C8—62no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—121no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—31no—
440C4⋊C4—31no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—121no—
450C4⋊C4—121no—
451C22⋊C4—41no—
452C22⋊C4—61no—
453C22⋊C4—61no—
454C22⋊C4—61no—
455C22⋊C4—121no—
456C42—41no—
457C42—61no—
458C42—61no—
459C42—61no—
460C42—121no—
461C42#23511yescharacteristic
462C42D3×D411yescharacteristic
463C42D3×D411yescharacteristic
464C42C23×D311yescharacteristic
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3C2×#12111yescharacteristic
469C2×D3—41no—
470C2×D3—41no—
471C2×D3—41no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4#47312yes—
478C3×C4C2×#13511yescharacteristic
479C3×C4#57511yescharacteristic
480C3×C4#57511yescharacteristic
481C3×C4#57511yescharacteristic
482Dic3—41no—
483Dic3—41no—
484Dic3—41no—
485Dic3—81no—
486Dic3—81no—
487C23—61no—
488C23#88411yescharacteristic
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493Q8—61no—
494Q8—61no—
495Q8—121no—
496Q8—121no—
497Q8—121no—
498Q8—121no—
499D4—61no—
500D4—61no—
501D4—122no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—122no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—122no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4#88411yescharacteristic
519C2×C4#88411yescharacteristic
520C2×C4#88411yescharacteristic
521C2×C4C2×#23511yescharacteristic
522C2×C4C2×D3×D411yescharacteristic
523C2×C4C2×D3×D411yescharacteristic
524C8—32no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—61no—
530C8—61no—
531C8—61no—
532C8—121no—
533C8—121no—
534C2×C3—41no—
535C2×C3#303511yescharacteristic
536C2×C3#305411yescharacteristic
537C2×C3C2×#47311yescharacteristic
538D3—81no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—121no—
543C22—121no—
544C22—121no—
545C22C2×#88411yescharacteristic
546C4—21no—
547C4—21no—
548C4—21no—
549C4—81no—
550C4—81no—
551C4—121no—
552C4—121no—
553C4—121no—
554C4—241no—
555C4—241no—
556C4#475512yes—
557C4D3×#13511yescharacteristic
558C4#576811yescharacteristic
559C4#576811yescharacteristic
560C4#576811yescharacteristic
561C3#1916111yescharacteristic, complemented, Sylow
562C2—41no—
563C2—241no—
564C2C2×#475511yescharacteristic
565C2#7883911yescharacteristic
566C2#7886111yescharacteristic
subgroupG/H
centerC2×C4#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1544C2
socleC22×C3C2×#121
prime radicalC2×#235D4
hypercenter#1544D3
hypocenterC3#19161
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
2. #83344C2semidirect
5. C3×#1544C2semidirect
6. #70777C2semidirect
7. #70745C2semidirect
8. #68477C2semidirect
9. #68476C2semidirect
10. #68475C2semidirect
11. #68473C2semidirect
12. #68453C2semidirect
13. #68452C2semidirect
14. #68451C2semidirect
15. #68449C2semidirect
82. #1544D3semidirect
561. C3#19161semidirect

Maximal subgroups

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

subgroupindex
#833522
#833442
#793022
#792772
C3×#15442
#707772
#707452
#684772
#684762
#684752
#684732
#684532
#684522
#684512
#684492
#191613