Finite Group Databasebeta

← #10691665

Subgroups — #10691665

Order 768.

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

normal subgroups only

This group has 478 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#82737C211yesmaximal, characteristic
2#78605C211yesmaximal, characteristic
3C3×#1472C211yesmaximal, characteristic, complemented
4C2×#4538C211yesmaximal, characteristic, complemented
5#69976C211yesmaximal, characteristic
6#67688C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9263—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10#5791C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13#5228C411yescharacteristic, complemented
14#5227C411yescharacteristic, complemented
15C2×C3⋊#95C2211yescharacteristic
16C2×#830C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5111—21no—
19C8×#67—21no—
20#5051—21no—
21#5048—21no—
22#5047—21no—
23#5045—21no—
24#4971C411yescharacteristic, complemented
25#4967C411yescharacteristic, complemented
26#4730—21no—
27#4727—21no—
28#67⋊C8—21no—
29#4720—21no—
30C3×#324—21no—
31C3×#324—21no—
32#4538—21no—
33#4538—21no—
34#4470—21no—
35#4467—21no—
36#3190—31no—
37#2977—31no—
38#1504—31no—
39C2×#328—31no—
40#1472D311yescharacteristic, complemented
41#1298—31no—
42#1280—31no—
43C22×C3×Q8C2×C411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×Dic3—21no—
46C2×#235D411yescharacteristic
47C2×C3×M16—21no—
48C22×C3×C8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4C2×C411yescharacteristic
55C2×C3×C22⋊C4—21no—
56#953—21no—
57C2×#216—21no—
58C2×#215—21no—
59#930C2×C411yescharacteristic
60#930D411yescharacteristic
61#929—21no—
62C4×#67—21no—
63C4×#67—21no—
64#926D411yescharacteristic
65#924C2×C411yescharacteristic
66#924D411yescharacteristic
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C2×ΓL(1,𝔽25)—21no—
70C2×D3×C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8D411yescharacteristic
75C3×C22⋊C8D411yescharacteristic
76#835—21no—
77#835—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80#830—21no—
81#830—21no—
82#830C2×C411yescharacteristic
83#830C2×C411yescharacteristic
84#830D411yescharacteristic
85#830D411yescharacteristic
86(C2×D3)⋊C8—21no—
87(C2×D3)⋊C8—21no—
88#815—21no—
89Dic3⋊C8—21no—
90Dic3⋊C8—21no—
91C8×Dic3—21no—
92#810—21no—
93#809—21no—
94#808—21no—
95#807—21no—
96#525—31no—
97C2×C22⋊Q8C2×D311yescharacteristic
98#485—61no—
99#483—31no—
100#480—31no—
101#467—61no—
102#465—61no—
103#452—61no—
104#451—31no—
105#448—31no—
106#435—61no—
107#435—61no—
108C8×D4—61no—
109C8×D4—61no—
110#433—61no—
111#408—61no—
112#407C2×D311yescharacteristic
113C2×C22⋊C8—31no—
114C2×C22⋊C8—31no—
115C2×C22⋊C8C2×D311yescharacteristic
116C2×#95—31no—
117#328—61no—
118#328—61no—
119#328—61no—
120#327—61no—
121#324—21no—
122#324—21no—
123C24×C3#4111yescharacteristic
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—81no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—81no—
130C22×C3×C4C22×C411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C2×#67—21no—
137C2×#67—21no—
138C2×#67—21no—
139C22×Dic3—22no—
140C22×Dic3—41no—
141C22×Dic3—41no—
142C22×Dic3—81no—
143C22×Dic3—81no—
144#235—81no—
145C2×D12C22⋊C411yescharacteristic
146C2×C4×D3—21no—
147C2×Dic6C22⋊C411yescharacteristic
148C3×M16—81no—
149C2×C3×C8—21no—
150C2×C3×C8—21no—
151C2×C3×C8—41no—
152C2×C3×C8—81no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159#217—21no—
160#217—21no—
161#217—21no—
162#216—81no—
163#215—81no—
164(C2×D3)⋊C4—21no—
165(C2×D3)⋊C4—21no—
166C4⋊Dic3C22⋊C411yescharacteristic
167C4⋊Dic3C22⋊C411yescharacteristic
168Dic3⋊C4—21no—
169Dic3⋊C4—21no—
170C4×Dic3—21no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—81no—
176C2×C3⋊C8—81no—
177ΓL(1,𝔽25)—81no—
178D3×C8—81no—
179C2×#41—31no—
180C22×Q8C4×D311yescharacteristic
181C23×C4—61no—
182C23×C4C22×D311yescharacteristic
183C2×Q16—61no—
184C2×SD16—61no—
185C2×M16—21no—
186C2×M16—61no—
187C22×C8—21no—
188C22×C8—61no—
189C22×C8—61no—
190C22×C8—61no—
191#123—61no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8—31no—
197C22⋊Q8—31no—
198#121—31no—
199#121—31no—
200#120—61no—
201C4×D4—61no—
202C4×D4—61no—
203#117—31no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C4×D311yescharacteristic
206C2×C22⋊C4—21no—
207#107—61no—
208#106—61no—
209C4⋊C8—61no—
210C4⋊C8—61no—
211#103—61no—
212#103—61no—
213#103—61no—
214#102—61no—
215C22⋊C8—21no—
216C22⋊C8—21no—
217C22⋊C8—21no—
218C22⋊C8—31no—
219C22⋊C8—31no—
220C22⋊C8—31no—
221C22⋊C8—31no—
222C22⋊C8—61no—
223C22⋊C8—61no—
224C22⋊C8—61no—
225C22⋊C8—61no—
226C22⋊C8#6711yescharacteristic
227C22⋊C8#6711yescharacteristic
228#97—61no—
229C4×C8—61no—
230#95—61no—
231#95—61no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—41no—
237C23×C3C4×D411yescharacteristic
238C23×C3#12111yescharacteristic
239C23×C3#12311yescharacteristic
240C22×D3—21no—
241C3×Q8—41no—
242C3×Q8—41no—
243C3×Q8—81no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—81no—
255C2×C3×C4C2×C22⋊C411yescharacteristic
256C2×C3×C4C4×D411yescharacteristic
257C2×C3×C4#12011yescharacteristic
258C2×C3×C4#12111yescharacteristic
259#67—41no—
260#67—41no—
261#67—41no—
262#67—41no—
263#67—81no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—42no—
268C2×Dic3—81no—
269C2×Dic3—82no—
270C2×Dic3—81no—
271C2×Dic3—82no—
272D12—41no—
273C4×D3—41no—
274C4×D3—41no—
275Dic6—41no—
276C3×C8—41no—
277C3×C8—41no—
278C3×C8—81no—
279C3⋊C8—41no—
280C3⋊C8—41no—
281C3⋊C8—41no—
282C3⋊C8—41no—
283C24#23511yescharacteristic
284#41—241no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—31no—
288C2×Q8—81no—
289C2×D4—31no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—61no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—61no—
296C22×C4—62no—
297C22×C4—81no—
298C22×C4—121no—
299C22×C4—121no—
300C22×C4—241no—
301C22×C4—241no—
302C22×C4C2×C4×D311yescharacteristic
303C22×C4D3×D411yescharacteristic
304C22×C4D3×D411yescharacteristic
305C22×C4D3×D411yescharacteristic
306C22×C4#23711yescharacteristic
307C22×C4C2×#6711yescharacteristic
308Q16—241no—
309SD16—241no—
310M16—81no—
311M16—241no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—41no—
315C2×C8—61no—
316C2×C8—61no—
317C2×C8—61no—
318C2×C8—61no—
319C2×C8—81no—
320C2×C8—241no—
321C2×C8—241no—
322C2×C8—241no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—31no—
326C4⋊C4—31no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—61no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—61no—
337C22⋊C4—61no—
338C42—61no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3#38611yescharacteristic
351C22×C3#44311yescharacteristic
352C22×C3#44411yescharacteristic
353C22×C3#47111yescharacteristic
354C22×C3#47311yescharacteristic
355C2×D3—41no—
356C2×D3—41no—
357C2×D3—41no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—81no—
363C3×C4—81no—
364Dic3—41no—
365Dic3—41no—
366Dic3—41no—
367Dic3—82no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—41no—
373C23—61no—
374C23#88311yescharacteristic
375C23#88411yescharacteristic
376C23C4×#6711yescharacteristic
377Q8—41no—
378Q8—41no—
379Q8—81no—
380Q8—121no—
381D4—121no—
382D4—121no—
383D4—121no—
384D4—121no—
385D4—121no—
386D4—241no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—81no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—122no—
404C2×C4—241no—
405C2×C4—242no—
406C2×C4—241no—
407C2×C4—242no—
408C2×C4D3×C22⋊C411yescharacteristic
409C2×C4#67⋊C411yescharacteristic
410C2×C4#93711yescharacteristic
411C2×C4#93811yescharacteristic
412C8—41no—
413C8—41no—
414C8—81no—
415C8—121no—
416C8—121no—
417C8—121no—
418C8—121no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—41no—
424C2×C3#187611yescharacteristic
425C2×C3#190811yescharacteristic
426C2×C3#191311yescharacteristic
427D3—81no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—121no—
440C22—121no—
441C22—121no—
442C22#475411yescharacteristic
443C22#475511yescharacteristic
444C22#495611yescharacteristic
445C22#514411yescharacteristic
446C22#514511yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—81no—
452C4—81no—
453C4—121no—
454C4—121no—
455C4—121no—
456C4—242no—
457C3#926311yescharacteristic, complemented, Sylow
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—41no—
463C2—241no—
464C2#7001111yescharacteristic
465C2#7003411yescharacteristic
466C2#7012311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1472C2
socleC22×C3#386
prime radical#5791C22
cosocleC23×C3×C4C23
hypercenter#1472D3
hypocenterC3#9263
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1472

Complements

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

NH ≅ G/Ntype
3. C3×#1472C2semidirect
4. C2×#4538C2semidirect
6. #67688C2semidirect
7. #67674C2semidirect
13. #5228C4semidirect
14. #5227C4semidirect
24. #4971C4semidirect
25. #4967C4semidirect
40. #1472D3semidirect
457. C3#9263semidirect

Maximal subgroups

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

subgroupindex
#827372
#786052
C3×#14722
C2×#45382
#699762
#676882
#676742
#92633