Finite Group Databasebeta

← #11711779

Subgroups — #11711779

Order 768.

1025 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 1025 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C2×(C2×S4)⋊C4C213yesmaximal
2C2×C4⋊SL(2,ℤ/4)C213yesmaximal, complemented
3A4×#95C211yesmaximal, characteristic, complemented
4#14836—31nomaximal, Sylow
5C23×S4C2211yescharacteristic
6C22×C4×A4C2213yes—
7C22×SL(2,ℤ/4)C2213yes—
8(C2×S4)⋊C4—26no—
9C4⋊SL(2,ℤ/4)—26no—
10#4677—41nomaximal
11C2×#399—31no—
12C2×#386—33no—
13C2×#380—33no—
14C22×#95D311yescharacteristic, complemented
15#1445—33no—
16#1444—31no—
17#1439—33no—
18C23×A4C2311yescharacteristic
19C22×S4—23no—
20C22×S4—23no—
21C22×S4Q812yes—
22C2×C4×A4—26no—
23C2×C4×A4D416yes—
24C2×SL(2,ℤ/4)—26no—
25C2×SL(2,ℤ/4)—23no—
26C2×(C2×D3)⋊C4—43no—
27C2×C4⋊Dic3—43no—
28C3×#95—41no—
29C23×D4—31no—
30C24×C4C2×D313yes—
31C22×C4⋊C4—33no—
32C22×C4⋊C4—33no—
33C22×C22⋊C4—33no—
34C22×C22⋊C4—33no—
35C22×C22⋊C4—33no—
36#399—63no—
37#399—61no—
38#386—66no—
39#386—66no—
40#380—66no—
41#380—66no—
42C2×#95—31no—
43C2×#95—33no—
44C2×#95—36no—
45C2×#95—33no—
46C2×#95—33no—
47C23×D3—41no—
48C22×A4C2×D413yes—
49C22×A4C2×Q811yescharacteristic
50C22×A4#4113yes—
51C2×S4—26no—
52C2×S4—43no—
53C2×S4—41no—
54C22×C3×C4—43no—
55C22×Dic3—43no—
56C4×A4—26no—
57SL(2,ℤ/4)—43no—
58(C2×D3)⋊C4—86no—
59C4⋊Dic3—86no—
60C25—31no—
61C25C22×D311yescharacteristic
62C22×D4—32no—
63C22×D4—32no—
64C22×D4—63no—
65C22×D4—63no—
66C22×D4—63no—
67C22×D4—63no—
68C23×C4—26no—
69C23×C4—33no—
70C23×C4—33no—
71C23×C4—36no—
72C23×C4—31no—
73C23×C4—33no—
74C23×C4—33no—
75C23×C4—66no—
76C23×C4D1216yes—
77C2×C4⋊C4—66no—
78C2×C4⋊C4—66no—
79C2×C4⋊C4—66no—
80C2×C4⋊C4—63no—
81C2×C4⋊C4—63no—
82C2×C4⋊C4—126no—
83C2×C22⋊C4—66no—
84C2×C22⋊C4—66no—
85C2×C22⋊C4—66no—
86C2×C22⋊C4—63no—
87C2×C22⋊C4—66no—
88C2×C22⋊C4—63no—
89C2×C22⋊C4—66no—
90C2×C22⋊C4—63no—
91C2×C22⋊C4—63no—
92C2×C22⋊C4—126no—
93#95—33no—
94#95—61no—
95#95—612no—
96#95—66no—
97#95—66no—
98#95S411yescharacteristic, complemented
99C23×C3—41no—
100C22×D3—42no—
101C22×D3—83no—
102C22×D3—83no—
103C2×A4C22⋊Q813yes—
104C2×A4#12313yes—
105C2×A4#12711yescharacteristic
106S4—42no—
107C2×C3×C4—46no—
108C2×C3×C4—86no—
109C2×Dic3—86no—
110C2×Dic3—83no—
111C24—31no—
112C24—33no—
113C24—33no—
114C24—31no—
115C24—32no—
116C24—63no—
117C24—63no—
118C24—63no—
119C24—63no—
120C24—61no—
121C24—61no—
122C24C2×D1213yes—
123C24#23713yes—
124C24D3×Q811yescharacteristic
125C2×D4—66no—
126C2×D4—66no—
127C2×D4—66no—
128C2×D4—66no—
129C2×D4—123no—
130C2×D4—123no—
131C2×D4—126no—
132C2×D4—121no—
133C2×D4—123no—
134C22×C4—26no—
135C22×C4—33no—
136C22×C4—36no—
137C22×C4—36no—
138C22×C4—36no—
139C22×C4—32no—
140C22×C4—66no—
141C22×C4—66no—
142C22×C4—63no—
143C22×C4—66no—
144C22×C4—66no—
145C22×C4—612no—
146C22×C4—66no—
147C22×C4—63no—
148C22×C4—63no—
149C22×C4—63no—
150C22×C4—63no—
151C22×C4—66no—
152C22×C4—66no—
153C22×C4—63no—
154C22×C4—66no—
155C22×C4—66no—
156C22×C4—63no—
157C22×C4—63no—
158C22×C4—63no—
159C22×C4—126no—
160C22×C4C2×S413yes—
161C4⋊C4—126no—
162C4⋊C4—1212no—
163C4⋊C4—126no—
164C22⋊C4—126no—
165C22⋊C4—1212no—
166C22⋊C4—123no—
167C22⋊C4—126no—
168C22⋊C4—123no—
169C22⋊C4—126no—
170C22×C3—43no—
171C22×C3—43no—
172C22×C3—41no—
173C2×D3—86no—
174C2×D3—161no—
175C2×D3—163no—
176A4#39911yescharacteristic, complemented
177C3×C4—86no—
178Dic3—163no—
179C23—33no—
180C23—33no—
181C23—33no—
182C23—33no—
183C23—31no—
184C23—31no—
185C23—33no—
186C23—33no—
187C23—36no—
188C23—33no—
189C23—33no—
190C23—33no—
191C23—63no—
192C23—61no—
193C23—63no—
194C23—66no—
195C23—66no—
196C23—62no—
197C23—123no—
198C23—121no—
199C23—123no—
200C23—123no—
201C23—126no—
202C23—123no—
203C23—123no—
204C23—123no—
205C23—123no—
206C23#87411yescharacteristic
207C23#88613yes—
208C23#89713yes—
209C23C22×S411yescharacteristic
210D4—122no—
211D4—126no—
212D4—126no—
213D4—122no—
214C2×C4—26no—
215C2×C4—36no—
216C2×C4—66no—
217C2×C4—66no—
218C2×C4—612no—
219C2×C4—66no—
220C2×C4—66no—
221C2×C4—66no—
222C2×C4—66no—
223C2×C4—126no—
224C2×C4—121no—
225C2×C4—123no—
226C2×C4—123no—
227C2×C4—123no—
228C2×C4—123no—
229C2×C4—126no—
230C2×C4—123no—
231C2×C4—126no—
232C2×C4—123no—
233C2×C4—126no—
234C2×C4—123no—
235C2×C4—126no—
236C2×C4C4⋊S416yes—
237C2×C3—41no—
238C2×C3—43no—
239C2×C3—43no—
240D3—162no—
241C22—31no—
242C22—33no—
243C22—33no—
244C22—31no—
245C22—33no—
246C22—33no—
247C22—33no—
248C22—36no—
249C22—33no—
250C22—33no—
251C22—33no—
252C22—33no—
253C22—63no—
254C22—61no—
255C22—63no—
256C22—122no—
257C22—126no—
258C22—126no—
259C22—241no—
260C22—243no—
261C22—241no—
262C22—243no—
263C22#467711yescharacteristic, complemented
264C22C2×C4⋊S413yes—
265C22#591413yes—
266C22Q8×S411yescharacteristic
267C4—26no—
268C4—66no—
269C4—122no—
270C4—243no—
271C4—243no—
272C3—41noSylow
273C2—31no—
274C2—31no—
275C2—33no—
276C2—33no—
277C2—242no—
278C2#8528911yescharacteristic
279C2#8530113yes—
280C2#8531213yes—
subgroupG/H
center, FrattiniC23C22×S4
commutator, cosocleC23×A4C23
FittingC22×#95D3
socleC25C22×D3
prime radicalC23×S4C22
hypercenter#95S4
hypocenterA4#399
LCSG ▷ C23×A4 ▷ A4
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊SL(2,ℤ/4)C2semidirect
3. A4×#95C2semidirect
14. C22×#95D3semidirect
98. #95S4semidirect
176. A4#399semidirect
263. C22#4677semidirect

Maximal subgroups

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

subgroupindex
C2×(C2×S4)⋊C42
C2×C4⋊SL(2,ℤ/4)2
A4×#952
#148363
#46774