Finite Group Databasebeta

← #10691775

Subgroups — #10691775

Order 768.

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

normal subgroups only

This group has 422 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#79338C211yesmaximal, characteristic
2#79270C211yesmaximal, characteristic
3#70836C211yesmaximal, characteristic, complemented
4#70832C211yesmaximal, characteristic
5#70803C211yesmaximal, characteristic, complemented
6C3×#1287C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#9279—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#4932C2211yescharacteristic
15#4925C2211yescharacteristic
16#4847—22no—
17#4838—21no—
18#4834—21no—
19#4737—22no—
20#4727—22no—
21#4706—22no—
22#4694C411yescharacteristic, complemented
23#4694C411yescharacteristic, complemented
24#4682C411yescharacteristic, complemented
25#4680C411yescharacteristic, complemented
26#4484—21no—
27#4484—21no—
28#4470—22no—
29#4467—22no—
30#3228—31no—
31#2981—31no—
32#1506—31no—
33#1505—31no—
34#1497—31no—
35#1287D311yescharacteristic, complemented
36#1274—31no—
37C2×#235—21no—
38C2×C3×C42C2311yescharacteristic
39#930—21no—
40#930D412yes—
41#926—21no—
42#924—21no—
43#924—21no—
44C2×C4×Dic3—21no—
45C2×ΓL(1,𝔽25)—22no—
46C2×D3×C8—22no—
47#875—21no—
48#874C2×C411yescharacteristic
49C4×D12—21no—
50C4×D12D412yes—
51#870—21no—
52#869C2×C411yescharacteristic
53#869C2×C411yescharacteristic
54#869C2×C411yescharacteristic
55C4×Dic6—21no—
56C4×Dic6—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×C22⋊C8D412yes—
63C3×#97—21no—
64C3×#97—21no—
65C3×C4×C8C2×C411yescharacteristic
66C3×C4×C8C2×C411yescharacteristic
67C3⋊#95—21no—
68C3⋊#95—21no—
69#830—21no—
70#821—22no—
71#821—22no—
72(C2×D3)⋊C8—22no—
73(C2×D3)⋊C8—22no—
74#818—22no—
75#817—22no—
76C4⋊(C3⋊C8)—21no—
77C4×C3⋊C8D411yescharacteristic
78C4×C3⋊C8D411yescharacteristic
79#533—31no—
80#532—31no—
81#500—31no—
82#498—31no—
83#486—31no—
84#486—31no—
85#482—62no—
86#478—62no—
87#436—61no—
88#435—62no—
89C8×D4—61no—
90#432C2×D311yescharacteristic
91#432C2×D311yescharacteristic
92#431—31no—
93#431C2×D311yescharacteristic
94#408—62no—
95#376—31no—
96#328—62no—
97#327—62no—
98Q8⋊C8—61no—
99Q8⋊C8—61no—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C2×#67—22no—
104C22×Dic3—41no—
105#235—82no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12C22⋊C412yes—
109C2×C4×D3—22no—
110C2×Dic6—21no—
111C2×Dic6—21no—
112C2×Dic6—21no—
113C2×C3×C8—21no—
114C2×C3×C8—21no—
115C2×C3×C8—22no—
116C2×C3×C8—41no—
117C3×C42C22×C411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42#4111yescharacteristic
121#217—41no—
122(C2×D3)⋊C4—22no—
123(C2×D3)⋊C4—22no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3C22⋊C412yes—
130Dic3⋊C4—41no—
131Dic3⋊C4—41no—
132C4×Dic3—22no—
133C4×Dic3—41no—
134C4×Dic3—41no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—21no—
137ΓL(1,𝔽25)—82no—
138D3×C8—82no—
139C2×#41—61no—
140C2×M16—62no—
141C22×C8—62no—
142C4⋊Q8—31no—
143C4⋊Q8—31no—
144C4⋊Q8—31no—
145#127—31no—
146#125—61no—
147#124—61no—
148C22⋊Q8—61no—
149C22⋊Q8—61no—
150#121—32no—
151#121—61no—
152C4×Q8—61no—
153C4×Q8—61no—
154C4×D4—32no—
155C4×D4—61no—
156#117—61no—
157C2×C42—61no—
158C2×C42C22×D311yescharacteristic
159#107—62no—
160#106—62no—
161C4⋊C8—21no—
162C4⋊C8—21no—
163C4⋊C8—21no—
164C4⋊C8—61no—
165#102—62no—
166#102—62no—
167C22⋊C8—21no—
168C22⋊C8—21no—
169C22⋊C8—62no—
170C22⋊C8—62no—
171C22⋊C8—61no—
172C22⋊C8D1212yes—
173#97—21no—
174#97—21no—
175C4×C8—31no—
176C4×C8—31no—
177C4×C8C4×D311yescharacteristic
178C4×C8C4×D311yescharacteristic
179#95—61no—
180#95—61no—
181C23×C3#12111yescharacteristic
182C22×D3—22no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4C2×C22⋊C411yescharacteristic
188C2×C3×C4C4×D411yescharacteristic
189C2×C3×C4C4×D411yescharacteristic
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12311yescharacteristic
193#67—82no—
194C2×Dic3—22no—
195C2×Dic3—42no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—81no—
199C2×Dic3—81no—
200D12—42no—
201D12—41no—
202D12—41no—
203D12—81no—
204C4×D3—42no—
205C4×D3—42no—
206Dic6—41no—
207Dic6—41no—
208Dic6—81no—
209Dic6—81no—
210C3×C8—42no—
211C3×C8—41no—
212C3×C8—41no—
213C3×C8—81no—
214C3⋊C8—41no—
215C3⋊C8—41no—
216#41—242no—
217C2×Q8—61no—
218C2×Q8—61no—
219C2×Q8—61no—
220C2×D4—32no—
221C2×D4—62no—
222C2×D4—61no—
223C2×D4—61no—
224C22×C4—62no—
225C22×C4—121no—
226C22×C4C2×D1211yescharacteristic
227C22×C4D3×D411yescharacteristic
228C22×C4#23911yescharacteristic
229M16—242no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—22no—
233C2×C8—41no—
234C2×C8—61no—
235C2×C8—61no—
236C2×C8—242no—
237C4⋊C4—32no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C22⋊C4—62no—
246C22⋊C4—62no—
247C22⋊C4—121no—
248C42—62no—
249C42—121no—
250C42—121no—
251C42C2×C4×D311yescharacteristic
252C42#23511yescharacteristic
253C42D3×D411yescharacteristic
254C42D3×D411yescharacteristic
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3#38611yescharacteristic
259C2×D3—42no—
260C2×D3—42no—
261C2×D3—42no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4#44311yescharacteristic
268C3×C4#44411yescharacteristic
269Dic3—42no—
270Dic3—82no—
271Dic3—81no—
272Dic3—81no—
273C23—62no—
274C23#89511yescharacteristic
275Q8—121no—
276Q8—121no—
277Q8—241no—
278Q8—241no—
279D4—122no—
280D4—121no—
281D4—121no—
282D4—241no—
283D4—242no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—62no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—241no—
295C2×C4—241no—
296C2×C4C4×D1211yescharacteristic
297C2×C4D3×C22⋊C411yescharacteristic
298C2×C4#88211yescharacteristic
299C2×C4#88411yescharacteristic
300C2×C4#89311yescharacteristic
301C2×C4#89411yescharacteristic
302C8—42no—
303C8—41no—
304C8—41no—
305C8—81no—
306C8—121no—
307C8—121no—
308C2×C3—41no—
309C2×C3#187611yescharacteristic
310C2×C3#190411yescharacteristic
311C2×C3#190911yescharacteristic
312D3—82no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—122no—
317C22—122no—
318C22—122no—
319C22#466511yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—122no—
326C4—242no—
327C4—241no—
328C4—241no—
329C4#489411yescharacteristic
330C4#489511yescharacteristic
331C3#927911yescharacteristic, complemented, Sylow
332C2—41no—
333C2—242no—
334C2#6871211yescharacteristic
335C2#6894611yescharacteristic
336C2#6902711yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1287C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1287D3
hypocenterC3#9279
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1287

Complements

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

NH ≅ G/Ntype
3. #70836C2semidirect
5. #70803C2semidirect
6. C3×#1287C2semidirect
7. #67737C2semidirect
22. #4694C4semidirect
23. #4694C4semidirect
24. #4682C4semidirect
25. #4680C4semidirect
35. #1287D3semidirect
331. C3#9279semidirect

Maximal subgroups

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

subgroupindex
#793382
#792702
#708362
#708322
#708032
C3×#12872
#677372
#92793