Finite Group Databasebeta

← #10692825

Subgroups — #10692825

Order 768.

712 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78630C211yesmaximal, characteristic
2#73489C211yesmaximal, characteristic
3#69981C211yesmaximal, characteristic, complemented
4#69537C211yesmaximal, characteristic
5#69533C211yesmaximal, characteristic, complemented
6C3×#1284C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9443—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×D3×M16—21no—
12C2×#875C411yescharacteristic, complemented
13C2×#869C411yescharacteristic, complemented
14C3×#407C2211yescharacteristic
15C3×#381C2211yescharacteristic
16C3×C4×C22⋊C4C2211yescharacteristic
17#5211C2211yescharacteristic
18#5207C2211yescharacteristic
19#5126—21no—
20#4961—21no—
21#4961—21no—
22C2×#806—21no—
23C2×#806—21no—
24#4939C411yescharacteristic, complemented
25#4929C411yescharacteristic, complemented
26#4727—22no—
27#4727—22no—
28#4668—21no—
29#4668—21no—
30#4467—22no—
31#4452—22no—
32#2879—31no—
33#2432—31no—
34#1782—31no—
35#1511—31no—
36#1511—31no—
37#1284—31no—
38#1284D311yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×#67D411yescharacteristic
41C22×#67D411yescharacteristic
42C2×#235—21no—
43C2×#235—21no—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—42no—
47C2×#235—42no—
48C22×D12C2×C411yescharacteristic
49C22×C4×D3D411yescharacteristic
50C22×C4×D3D411yescharacteristic
51C22×Dic6C2×C411yescharacteristic
52C22×Dic6C2×C411yescharacteristic
53C2×C3×M16—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C42C2×C411yescharacteristic
57C2×C3×C42C2×C411yescharacteristic
58C2×#217—21no—
59C2×#217—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3C2×C411yescharacteristic
67#924—21no—
68#924—21no—
69#924—21no—
70#924—21no—
71C2×Dic3⋊C4—21no—
72C2×#208—21no—
73D3×M16—42no—
74D3×M16—42no—
75C2×ΓL(1,𝔽25)—22no—
76C2×D3×C8—22no—
77#875—42no—
78#869—42no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8D412yes—
81C3×#95—21no—
82C3×#95—21no—
83C3×#95—21no—
84C3⋊#95—21no—
85C3⋊#95—21no—
86C3⋊#95—21no—
87C3⋊#95—21no—
88#830—21no—
89#830D412yes—
90(C2×D3)⋊C8—24no—
91#806—42no—
92#806—42no—
93C22×#41—31no—
94C22×M16—61no—
95C2×C4⋊Q8—31no—
96C2×#124—31no—
97C2×C22⋊Q8—31no—
98C2×#99—61no—
99C2×#99—61no—
100#408—62no—
101#408—62no—
102#407—31no—
103#407—61no—
104#407C2×D311yescharacteristic
105#394—61no—
106#394—61no—
107#391—31no—
108#390—31no—
109#385—61no—
110#385—61no—
111#381—31no—
112#381C2×D311yescharacteristic
113C4×C22⋊C4C2×D311yescharacteristic
114#328—62no—
115#328—62no—
116C24×C3C2×D411yescharacteristic
117C23×D3C22⋊C411yescharacteristic
118C23×D3C22⋊C411yescharacteristic
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4C22×C411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C2×#67—21no—
131C2×#67—21no—
132C2×#67—21no—
133C2×#67—21no—
134C2×#67—22no—
135C2×#67—22no—
136C2×#67—41no—
137C2×#67—41no—
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—41no—
141C22×Dic3C22⋊C411yescharacteristic
142C22×Dic3C22⋊C411yescharacteristic
143#235—42no—
144#235—42no—
145#235—42no—
146#235—42no—
147#235—44no—
148#235—44no—
149C2×D12—22no—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12—41no—
153C2×C4×D3—22no—
154C2×C4×D3—21no—
155C2×C4×D3—21no—
156C2×C4×D3—22no—
157C2×C4×D3—21no—
158C2×C4×D3—21no—
159C2×C4×D3—41no—
160C2×C4×D3—41no—
161C2×Dic6—21no—
162C2×Dic6—21no—
163C2×Dic6—22no—
164C2×Dic6—21no—
165C2×Dic6—21no—
166C2×Dic6—41no—
167C2×Dic6—41no—
168C2×Dic6—41no—
169C3×M16—41no—
170C3×M16—41no—
171C2×C3×C8—22no—
172C3×C22⋊C4—42no—
173C3×C22⋊C4—41no—
174C3×C22⋊C4—41no—
175C3×C42—41no—
176C3×C42—41no—
177#217—41no—
178#217—41no—
179#217—41no—
180#217—41no—
181(C2×D3)⋊C4—42no—
182(C2×D3)⋊C4—42no—
183(C2×D3)⋊C4—42no—
184(C2×D3)⋊C4—42no—
185C4⋊Dic3—21no—
186C4⋊Dic3—21no—
187C4⋊Dic3—42no—
188C4⋊Dic3—42no—
189Dic3⋊C4—41no—
190Dic3⋊C4—41no—
191#208—41no—
192#208—41no—
193C2×C3⋊C8—22no—
194ΓL(1,𝔽25)—44no—
195D3×C8—44no—
196C2×#41—61no—
197C2×#41—61no—
198C2×#41—61no—
199C2×#41—61no—
200C2×#41—122no—
201C2×#41—122no—
202C22×Q8—31no—
203C22×Q8—31no—
204C22×D4—31no—
205C22×D4—31no—
206C22×D4—31no—
207C23×C4—31no—
208C23×C4—31no—
209C23×C4C22×D311yescharacteristic
210C2×M16—21no—
211C2×M16—62no—
212C2×M16—61no—
213C2×M16—122no—
214C2×M16—122no—
215C22×C8—62no—
216C4⋊Q8—122no—
217#124—122no—
218C22⋊Q8—61no—
219C22⋊Q8—61no—
220C22⋊Q8—61no—
221C22⋊Q8—61no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—61no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C22⋊C4—61no—
231C2×C22⋊C4—61no—
232C2×C22⋊C4—61no—
233C2×C22⋊C4—61no—
234C2×C42C4×D311yescharacteristic
235C2×C42C4×D311yescharacteristic
236#99—122no—
237#99—122no—
238C22⋊C8—21no—
239C22⋊C8—32no—
240C22⋊C8—64no—
241C22⋊C8—61no—
242C22⋊C8D1212yes—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—61no—
247#95—61no—
248#95—61no—
249#95—61no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3C2×C22⋊C411yescharacteristic
255C23×C3#12111yescharacteristic
256C23×C3#12111yescharacteristic
257C22×D3—21no—
258C22×D3—21no—
259C22×D3—21no—
260C22×D3—21no—
261C22×D3—21no—
262C22×D3—41no—
263C22×D3—41no—
264C22×D3—41no—
265C22×D3—41no—
266C22×D3#9912yes—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—22no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—81no—
282C2×C3×C4—81no—
283C2×C3×C4—81no—
284C2×C3×C4C4×D411yescharacteristic
285C2×C3×C4C4×D411yescharacteristic
286C2×C3×C4#12011yescharacteristic
287C2×C3×C4#12311yescharacteristic
288#67—42no—
289#67—42no—
290#67—42no—
291#67—42no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—81no—
304C2×Dic3—81no—
305C2×Dic3—81no—
306C2×Dic3#9912yes—
307D12—42no—
308D12—42no—
309C4×D3—22no—
310C4×D3—22no—
311C4×D3—24no—
312C4×D3—41no—
313C4×D3—41no—
314C4×D3—42no—
315Dic6—42no—
316Dic6—42no—
317Dic6—81no—
318Dic6—81no—
319C3×C8—42no—
320C3⋊C8—42no—
321C24—31no—
322C24—31no—
323C24D3×D411yescharacteristic
324#41—122no—
325#41—122no—
326#41—124no—
327#41—122no—
328#41—122no—
329#41—124no—
330C2×Q8—62no—
331C2×Q8—61no—
332C2×Q8—61no—
333C2×Q8—61no—
334C2×Q8—61no—
335C2×Q8—121no—
336C2×Q8—121no—
337C2×Q8—121no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—61no—
342C2×D4—61no—
343C2×D4—61no—
344C2×D4—61no—
345C2×D4—61no—
346C2×D4—61no—
347C2×D4—121no—
348C2×D4—121no—
349C2×D4—121no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—31no—
353C22×C4—31no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4—61no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—121no—
366C22×C4—121no—
367C22×C4—121no—
368C22×C4C2×C4×D311yescharacteristic
369C22×C4C2×D1211yescharacteristic
370C22×C4#23511yescharacteristic
371C22×C4D3×D411yescharacteristic
372C22×C4D3×D411yescharacteristic
373C22×C4#23911yescharacteristic
374M16—41no—
375M16—41no—
376M16—124no—
377M16—121no—
378M16—121no—
379C2×C8—22no—
380C2×C8—62no—
381C2×C8—124no—
382C4⋊C4—61no—
383C4⋊C4—61no—
384C4⋊C4—122no—
385C4⋊C4—122no—
386C4⋊C4—121no—
387C4⋊C4—121no—
388C22⋊C4—42no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—122no—
392C22⋊C4—122no—
393C22⋊C4—122no—
394C22⋊C4—122no—
395C22⋊C4—121no—
396C22⋊C4—121no—
397C22⋊C4—121no—
398C22⋊C4—121no—
399C42—41no—
400C42—41no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3#38611yescharacteristic
412C22×C3C2×#9911yescharacteristic
413C22×C3C2×#9911yescharacteristic
414C2×D3—22no—
415C2×D3—41no—
416C2×D3—41no—
417C2×D3—42no—
418C2×D3—42no—
419C2×D3—41no—
420C2×D3—41no—
421C2×D3—41no—
422C2×D3—81no—
423C2×D3—81no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—42no—
429C3×C4—81no—
430C3×C4—161no—
431Dic3—22no—
432Dic3—41no—
433Dic3—81no—
434Dic3—81no—
435Dic3—161no—
436C23—21no—
437C23—21no—
438C23—32no—
439C23—41no—
440C23—41no—
441C23—61no—
442C23—61no—
443C23—61no—
444C23—61no—
445C23—61no—
446C23—121no—
447C23—121no—
448C23—121no—
449C23—121no—
450C23D3×C22⋊C411yescharacteristic
451C23#88411yescharacteristic
452C23#89511yescharacteristic
453Q8—122no—
454Q8—122no—
455Q8—241no—
456Q8—241no—
457D4—122no—
458D4—122no—
459D4—122no—
460D4—122no—
461D4—122no—
462D4—122no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—32no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—64no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—122no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—241no—
499C2×C4—241no—
500C2×C4—241no—
501C2×C4C4×D1211yescharacteristic
502C2×C4#88211yescharacteristic
503C2×C4#89311yescharacteristic
504C2×C4#89411yescharacteristic
505C8—42no—
506C8—122no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—41no—
510C2×C3—41no—
511C2×C3#188411yescharacteristic
512C2×C3#189811yescharacteristic
513C2×C3#190911yescharacteristic
514D3—42no—
515D3—81no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—62no—
525C22—81no—
526C22—81no—
527C22—122no—
528C22—122no—
529C22—121no—
530C22—121no—
531C22—121no—
532C22—121no—
533C22—121no—
534C22—241no—
535C22—241no—
536C22#466511yescharacteristic
537C22D3×#9911yescharacteristic
538C22D3×#9911yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—42no—
544C4—62no—
545C4—81no—
546C4—121no—
547C4—161no—
548C4—241no—
549C4—241no—
550C4—481no—
551C3#944311yescharacteristic, complemented, Sylow
552C2—21no—
553C2—21no—
554C2—41no—
555C2—41no—
556C2—122no—
557C2—241no—
558C2#6871211yescharacteristic
559C2#6877711yescharacteristic
560C2#6897011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1284C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1284D3
hypocenterC3#9443
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
3. #69981C2semidirect
5. #69533C2semidirect
6. C3×#1284C2semidirect
7. #67616C2semidirect
12. C2×#875C4semidirect
13. C2×#869C4semidirect
24. #4939C4semidirect
25. #4929C4semidirect
38. #1284D3semidirect
551. C3#9443semidirect

Maximal subgroups

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

subgroupindex
#786302
#734892
#699812
#695372
#695332
C3×#12842
#676162
#94433