Finite Group Databasebeta

← #10693265

Subgroups — #10693265

Order 768.

436 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 89 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#82444C211yesmaximal, characteristic
2#79338C211yesmaximal, characteristic
3C3×#1467C211yesmaximal, characteristic, complemented
4#71622C211yesmaximal, characteristic, complemented
5#70819C211yesmaximal, characteristic
6#67758C211yesmaximal, characteristic, complemented
7#67756C211yesmaximal, characteristic, complemented
8#9527—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13#5227C412yes—
14#5027—21no—
15#5026—22no—
16#5021—21no—
17#4967C412yes—
18#4926C2211yescharacteristic
19#4925C2211yescharacteristic
20#4924C2211yescharacteristic
21#4838—21no—
22#4834—21no—
23#4727—22no—
24#4706—22no—
25C3×#324—22no—
26#4535—21no—
27#4535—21no—
28#4485—21no—
29#4482—21no—
30#3112—31no—
31#2981—31no—
32#1557—31no—
33#1487—31no—
34#1467D311yescharacteristic, complemented
35#1307—31no—
36#1287—31no—
37C22×C3×Q8C2×C411yescharacteristic
38C2×#235—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C42—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#215—22no—
48#930—21no—
49#930C2×C412yes—
50C2×#208—21no—
51C2×ΓL(1,𝔽25)—22no—
52C2×D3×C8—22no—
53#875—21no—
54#874D411yescharacteristic
55C4×D12—21no—
56C4×D12D412yes—
57#869D411yescharacteristic
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×C22⋊C8D412yes—
62C3×#97—21no—
63C3×#97—21no—
64#835—22no—
65#830—21no—
66#830C2×C412yes—
67(C2×D3)⋊C8—22no—
68(C2×D3)⋊C8—22no—
69#809—22no—
70#807—22no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75#803—21no—
76C4×C3⋊C8D411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78#532—31no—
79C2×C4×Q8C2×D311yescharacteristic
80#480—32no—
81#477—61no—
82#460—61no—
83#448—32no—
84#440—62no—
85#436—61no—
86#435—62no—
87C8×D4—61no—
88#432—31no—
89#432C2×D311yescharacteristic
90#432C2×D311yescharacteristic
91#431—31no—
92#423—31no—
93#408—62no—
94#332—61no—
95#331—61no—
96Q8⋊C8—61no—
97Q8⋊C8—61no—
98#324—22no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—42no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C2×#67—22no—
108#235—82no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12C22⋊C412yes—
112C2×C4×D3—22no—
113C2×Dic6—21no—
114C2×C3×C8—22no—
115C2×C3×C8—41no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C42—41no—
121C3×C42—41no—
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126#215—44no—
127(C2×D3)⋊C4—22no—
128(C2×D3)⋊C4—22no—
129C4⋊Dic3—21no—
130C4⋊Dic3C22⋊C412yes—
131#208—81no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—41no—
136ΓL(1,𝔽25)—82no—
137D3×C8—82no—
138C2×#41—61no—
139C22×Q8C4×D311yescharacteristic
140C2×SD16—62no—
141C2×M16—62no—
142C2×M16—61no—
143C22×C8—62no—
144C4⋊Q8—31no—
145#127—31no—
146#124—61no—
147#121—32no—
148#121—61no—
149C4×Q8—21no—
150C4×Q8—21no—
151C4×Q8—21no—
152C4×Q8—21no—
153C4×D4—32no—
154C4×D4—61no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4C4×D311yescharacteristic
157C2×C42—21no—
158C2×C42C22×D311yescharacteristic
159#107—62no—
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166#103—62no—
167#102—62no—
168C22⋊C8—21no—
169C22⋊C8—32no—
170C22⋊C8—62no—
171C22⋊C8—62no—
172C22⋊C8—61no—
173C22⋊C8#6712yes—
174#97—21no—
175#97—21no—
176#97—61no—
177C4×C8—31no—
178C4×C8—31no—
179C23×C3C4×D411yescharacteristic
180C22×D3—22no—
181C3×Q8—22no—
182C3×Q8—22no—
183C3×Q8—41no—
184C3×Q8—41no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—22no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—81no—
194C2×C3×C4C2×C22⋊C411yescharacteristic
195C2×C3×C4C4×D411yescharacteristic
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12311yescharacteristic
200#67—82no—
201C2×Dic3—22no—
202D12—42no—
203D12—41no—
204D12—41no—
205D12—81no—
206C4×D3—42no—
207C4×D3—42no—
208Dic6—81no—
209C3×C8—42no—
210C3×C8—81no—
211C3⋊C8—41no—
212C3⋊C8—41no—
213C3⋊C8—42no—
214C3⋊C8—81no—
215#41—242no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—42no—
219C2×Q8—61no—
220C2×D4—32no—
221C2×D4—62no—
222C2×D4—61no—
223C2×D4—61no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—62no—
227C22×C4C2×C4×D311yescharacteristic
228C22×C4#23511yescharacteristic
229C22×C4C2×#6711yescharacteristic
230SD16—124no—
231M16—242no—
232M16—241no—
233C2×C8—22no—
234C2×C8—41no—
235C2×C8—62no—
236C2×C8—61no—
237C2×C8—61no—
238C2×C8—121no—
239C2×C8—242no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—32no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—61no—
246C22⋊C4—62no—
247C22⋊C4—62no—
248C42—41no—
249C42—41no—
250C42D3×D411yescharacteristic
251C42D3×D411yescharacteristic
252C42D3×D411yescharacteristic
253C42#23711yescharacteristic
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3#38611yescharacteristic
258C2×D3—42no—
259C2×D3—42no—
260C2×D3—42no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—42no—
267C3×C4—81no—
268C3×C4#47111yescharacteristic
269C3×C4#47311yescharacteristic
270Dic3—42no—
271C23—62no—
272C23C4×#6711yescharacteristic
273Q8—22no—
274Q8—22no—
275Q8—41no—
276Q8—41no—
277Q8—241no—
278D4—122no—
279D4—121no—
280D4—121no—
281D4—241no—
282D4—242no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—62no—
292C2×C4—81no—
293C2×C4—122no—
294C2×C4—122no—
295C2×C4D3×C22⋊C411yescharacteristic
296C2×C4#67⋊C411yescharacteristic
297C2×C4#88311yescharacteristic
298C2×C4#88411yescharacteristic
299C2×C4#93711yescharacteristic
300C2×C4#93811yescharacteristic
301C8—42no—
302C8—81no—
303C8—121no—
304C8—121no—
305C8—122no—
306C8—241no—
307C2×C3—41no—
308C2×C3#189111yescharacteristic
309C2×C3#190411yescharacteristic
310C2×C3#190811yescharacteristic
311D3—82no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—122no—
316C22—122no—
317C22—122no—
318C22#495611yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—42no—
325C4—81no—
326C4—122no—
327C4#475411yescharacteristic
328C4#475511yescharacteristic
329C3#952711yescharacteristic, complemented, Sylow
330C2—41no—
331C2—242no—
332C2#6999211yescharacteristic
333C2#7004911yescharacteristic
334C2#7012311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1467C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1467D3
hypocenterC3#9527
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
3. C3×#1467C2semidirect
4. #71622C2semidirect
6. #67758C2semidirect
7. #67756C2semidirect
34. #1467D3semidirect
329. C3#9527semidirect

Maximal subgroups

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

subgroupindex
#824442
#793382
C3×#14672
#716222
#708192
#677582
#677562
#95273