Finite Group Databasebeta

← #10690293

Subgroups — #10690293

Order 768.

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

normal subgroups only

This group has 676 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#84080C211yesmaximal, characteristic
2#78635C211yesmaximal, characteristic, complemented
3C3×#1789C211yesmaximal, characteristic, complemented
4#72136C211yesmaximal, characteristic, complemented
5#69443C211yesmaximal, characteristic, complemented
6#67688C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9037—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×#950—21no—
11#5791C2211yescharacteristic
12C2×C3×#104—21no—
13C2×C3×#104—21no—
14C3×#407C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16#5239—21no—
17#5238—21no—
18#5233—21no—
19#5228C411yescharacteristic, complemented
20#5227C411yescharacteristic, complemented
21#5218C411yescharacteristic, complemented
22#5217C411yescharacteristic, complemented
23C2×#830—21no—
24C2×#830—21no—
25C2×#830C2211yescharacteristic
26C2×#830C2211yescharacteristic
27C2×#830C2211yescharacteristic
28#5128—21no—
29#5127—21no—
30#4730—21no—
31#4727—21no—
32#67⋊C8—21no—
33#4720—21no—
34#4452—21no—
35#4451—21no—
36#3002—31no—
37#2976—31no—
38#1789D311yescharacteristic, complemented
39#1780—31no—
40#1501—31no—
41#1298—31no—
42#1282—31no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—21no—
46C2×C3×#41—21no—
47C2×C3×#41—42no—
48C2×C3×#41—42no—
49C22×C3×Q8C2×C411yescharacteristic
50C22×C3×D4—21no—
51C22×C3×D4C2×C411yescharacteristic
52C23×C3×C4—21no—
53C23×C3×C4C2311yescharacteristic
54C2×#235D411yescharacteristic
55C2×C3×M16D411yescharacteristic
56C2×C3×M16D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C42—21no—
59#953—21no—
60#950—42no—
61#950—42no—
62C2×#216—21no—
63C2×#215—21no—
64C2×#214—21no—
65C2×#213—21no—
66#930C2×C411yescharacteristic
67#930D411yescharacteristic
68#929—21no—
69C4×#67—21no—
70C4×#67—21no—
71#926D411yescharacteristic
72#924C2×C411yescharacteristic
73#924D411yescharacteristic
74C22×C3⋊C8—21no—
75C22×C3⋊C8—21no—
76C22×C3⋊C8D411yescharacteristic
77C22×C3⋊C8D411yescharacteristic
78C2×ΓL(1,𝔽25)—21no—
79C2×D3×C8—21no—
80C3×#104—42no—
81C3×#104—42no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8—21no—
84C3×#95—21no—
85#835—21no—
86#835—21no—
87#832—21no—
88#832—21no—
89#830—21no—
90#830—21no—
91#830—21no—
92#830—21no—
93#830—21no—
94#830—42no—
95#830—42no—
96#830C2×C411yescharacteristic
97#830C2×C411yescharacteristic
98(C2×D3)⋊C8—21no—
99(C2×D3)⋊C8—21no—
100#815—21no—
101Dic3⋊C8—21no—
102Dic3⋊C8—21no—
103C8×Dic3—21no—
104C22×#41C2×D311yescharacteristic
105C2×#135—61no—
106#525—31no—
107#452—61no—
108#451—31no—
109#450—31no—
110#449—61no—
111#448—31no—
112#447—31no—
113#436—61no—
114#435—61no—
115#435—61no—
116C8×D4—61no—
117#433—61no—
118C2×#104—21no—
119C2×#104—21no—
120#416—61no—
121#408—61no—
122#407C2×D311yescharacteristic
123C2×C22⋊C8—31no—
124C2×C22⋊C8—31no—
125C2×C22⋊C8—31no—
126C2×C22⋊C8—61no—
127C2×C22⋊C8—61no—
128C4×C22⋊C4C2×D311yescharacteristic
129#328—61no—
130#327—61no—
131C24×C3—21no—
132C24×C3#4111yescharacteristic
133C3×#41—22no—
134C3×#41—22no—
135C3×#41—22no—
136C3×#41—22no—
137C3×#41—41no—
138C3×#41—41no—
139C3×#41—41no—
140C3×#41—41no—
141C3×#41—42no—
142C3×#41—42no—
143C3×#41—42no—
144C3×#41—42no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—42no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4—41no—
154C2×C3×D4—41no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—41no—
159C22×C3×C4—41no—
160C22×C3×C4—41no—
161C22×C3×C4—41no—
162C22×C3×C4—41no—
163C22×C3×C4—41no—
164C22×C3×C4—41no—
165C22×C3×C4C22×C411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4#4111yescharacteristic
171C2×#67—21no—
172C2×#67—21no—
173C2×#67—21no—
174#235—41no—
175#235—41no—
176C2×D12C22⋊C411yescharacteristic
177C2×C4×D3—21no—
178C2×Dic6C22⋊C411yescharacteristic
179C3×M16—41no—
180C3×M16—41no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—41no—
185C3×C42—41no—
186C3×C42—41no—
187#217—21no—
188#217—21no—
189#217—21no—
190#216—42no—
191#215—42no—
192#214—42no—
193#213—42no—
194(C2×D3)⋊C4—21no—
195(C2×D3)⋊C4—21no—
196C4⋊Dic3C22⋊C411yescharacteristic
197C4⋊Dic3C22⋊C411yescharacteristic
198Dic3⋊C4—21no—
199Dic3⋊C4—21no—
200C4×Dic3—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—41no—
206C2×C3⋊C8—41no—
207C2×C3⋊C8—41no—
208C2×C3⋊C8—41no—
209C2×C3⋊C8—41no—
210C2×C3⋊C8—41no—
211C2×C3⋊C8—41no—
212ΓL(1,𝔽25)—81no—
213D3×C8—81no—
214C2×#41—21no—
215C2×#41—21no—
216C2×#41—21no—
217C2×#41—21no—
218C2×#41—31no—
219C2×#41—42no—
220C2×#41—42no—
221C22×Q8C4×D311yescharacteristic
222C22×D4—21no—
223C22×D4C4×D311yescharacteristic
224C23×C4—21no—
225C23×C4C22×D311yescharacteristic
226#135—122no—
227#135—122no—
228C2×Q16—61no—
229C2×SD16—61no—
230C2×SD16—61no—
231C2×D8—61no—
232C2×M16—61no—
233C2×M16#6711yescharacteristic
234C2×M16#6711yescharacteristic
235C22×C8—31no—
236C22×C8—31no—
237C22×C8—61no—
238C22×C8—61no—
239C22×C8—61no—
240#123—61no—
241C22⋊Q8—31no—
242C22⋊Q8—31no—
243#121—31no—
244#121—31no—
245#120—61no—
246C4×D4—61no—
247C4×D4—61no—
248#117—31no—
249C2×C22⋊C4—21no—
250C2×C42—21no—
251C4⋊C8—61no—
252C4⋊C8—61no—
253#104—42no—
254#104—42no—
255#103—61no—
256#103—61no—
257#102—61no—
258#102—61no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C22⋊C8—31no—
262C22⋊C8—31no—
263C22⋊C8—61no—
264C22⋊C8—61no—
265C22⋊C8—61no—
266C22⋊C8—61no—
267C22⋊C8—61no—
268C22⋊C8—61no—
269C22⋊C8—61no—
270C22⋊C8—122no—
271C22⋊C8—122no—
272#97—61no—
273C4×C8—61no—
274#95—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—22no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3C4×D411yescharacteristic
287C23×C3#12111yescharacteristic
288C23×C3#12311yescharacteristic
289C22×D3—21no—
290C3×Q8—22no—
291C3×Q8—22no—
292C3×Q8—41no—
293C3×Q8—41no—
294C3×D4—22no—
295C3×D4—22no—
296C3×D4—41no—
297C3×D4—41no—
298C3×D4—42no—
299C3×D4—42no—
300C3×D4—42no—
301C3×D4—42no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—81no—
320C2×C3×C4—81no—
321C2×C3×C4C2×C22⋊C411yescharacteristic
322C2×C3×C4C4×D411yescharacteristic
323C2×C3×C4#12011yescharacteristic
324C2×C3×C4#12111yescharacteristic
325#67—41no—
326#67—41no—
327#67—81no—
328#67—81no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332D12—41no—
333C4×D3—41no—
334C4×D3—41no—
335Dic6—41no—
336C3×C8—41no—
337C3×C8—41no—
338C3⋊C8—41no—
339C3⋊C8—41no—
340C3⋊C8—41no—
341C3⋊C8—41no—
342C3⋊C8—81no—
343C24—21no—
344C24#23711yescharacteristic
345#41—22no—
346#41—22no—
347#41—22no—
348#41—22no—
349#41—41no—
350#41—41no—
351#41—41no—
352#41—41no—
353#41—42no—
354#41—42no—
355#41—42no—
356#41—42no—
357#41—121no—
358#41—121no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—31no—
362C2×Q8—42no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—31no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—61no—
372C2×D4—61no—
373C2×D4—61no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—61no—
385C22×C4C2×C4×D311yescharacteristic
386C22×C4#23511yescharacteristic
387C22×C4D3×D411yescharacteristic
388C22×C4D3×D411yescharacteristic
389C22×C4D3×D411yescharacteristic
390C22×C4C2×#6711yescharacteristic
391Q16—122no—
392SD16—122no—
393SD16—122no—
394D8—122no—
395M16—41no—
396M16—41no—
397M16—241no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—61no—
401C2×C8—61no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—121no—
405C2×C8—121no—
406C2×C8—121no—
407C2×C8—121no—
408C2×C8—121no—
409C2×C8—121no—
410C2×C8—121no—
411C2×C8—241no—
412C4⋊C4—31no—
413C4⋊C4—31no—
414C4⋊C4—61no—
415C4⋊C4—61no—
416C22⋊C4—22no—
417C22⋊C4—41no—
418C22⋊C4—61no—
419C22⋊C4—61no—
420C22⋊C4—61no—
421C22⋊C4—61no—
422C22⋊C4—61no—
423C42—41no—
424C42—41no—
425C42—61no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—81no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3—81no—
443C22×C3#38611yescharacteristic
444C22×C3#44311yescharacteristic
445C22×C3#44411yescharacteristic
446C2×D3—41no—
447C2×D3—41no—
448C2×D3—41no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—42no—
453C3×C4—82no—
454C3×C4B(4,𝔽2)11yescharacteristic
455C3×C4B(4,𝔽2)11yescharacteristic
456Dic3—41no—
457Dic3—41no—
458Dic3—41no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—61no—
471C23#67⋊C411yescharacteristic
472C23#88311yescharacteristic
473C23#93811yescharacteristic
474Q8—22no—
475Q8—22no—
476Q8—41no—
477Q8—41no—
478Q8—121no—
479D4—22no—
480D4—22no—
481D4—41no—
482D4—41no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—121no—
488D4—121no—
489D4—121no—
490D4—241no—
491D4—241no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4D3×C22⋊C411yescharacteristic
517C2×C4#88411yescharacteristic
518C2×C4C4×#6711yescharacteristic
519C2×C4#93711yescharacteristic
520C8—41no—
521C8—41no—
522C8—121no—
523C8—121no—
524C8—121no—
525C8—121no—
526C8—241no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3—82no—
532C2×C3#187611yescharacteristic
533C2×C3#188111yescharacteristic
534C2×C3#188911yescharacteristic
535D3—81no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—121no—
554C22—121no—
555C22—121no—
556C22#482311yescharacteristic
557C22#482411yescharacteristic
558C22#495611yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—42no—
563C4—82no—
564C4—121no—
565C4—121no—
566C4—121no—
567C4#503211yescharacteristic
568C4#503211yescharacteristic
569C3#903711yescharacteristic, complemented, Sylow
570C2—21no—
571C2—21no—
572C2—41no—
573C2—41no—
574C2—82no—
575C2—241no—
576C2#6949911yescharacteristic
577C2#7039011yescharacteristic
578C2#7041611yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1789C2
socleC22×C3#386
prime radical#84080C2
cosocleC23×C3×C4C23
hypercenter#1789D3
hypocenterC3#9037
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1789

Complements

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

NH ≅ G/Ntype
2. #78635C2semidirect
3. C3×#1789C2semidirect
4. #72136C2semidirect
5. #69443C2semidirect
6. #67688C2semidirect
7. #67617C2semidirect
19. #5228C4semidirect
20. #5227C4semidirect
21. #5218C4semidirect
22. #5217C4semidirect
38. #1789D3semidirect
569. C3#9037semidirect

Maximal subgroups

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

subgroupindex
#840802
#786352
C3×#17892
#721362
#694432
#676882
#676172
#90373