Finite Group Databasebeta

← #11711781

Subgroups — #11711781

Order 768.

1355 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 1355 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
1C22×Q8×A4C211yesmaximal, characteristic, complemented
2C2×(C2×S4)⋊C4C213yesmaximal
3A4⋊#95C213yesmaximal, complemented
4#14873—31nomaximal, Sylow
5C23×S4C2211yescharacteristic
6C2×Q8×A4—26no—
7C22×C4×A4C2213yes—
8C22×SL(2,ℤ/4)C2213yes—
9(C2×S4)⋊C4—26no—
10#5231—41nomaximal
11C24×Q8D311yescharacteristic, complemented
12C2×#393—33no—
13C2×#393—31no—
14C2×#386—33no—
15#1446—33no—
16#1446—31no—
17C22⋊#95—33no—
18C23×A4C2311yescharacteristic
19C22×S4—23no—
20C22×S4—23no—
21C22×S4Q812yescomplemented
22C22×C3×Q8—41no—
23Q8×A4—28no—
24C2×C4×A4—26no—
25C2×C4×A4D416yes—
26C2×SL(2,ℤ/4)—23no—
27C2×SL(2,ℤ/4)—26no—
28C2×(C2×D3)⋊C4—43no—
29C3⋊#95—43no—
30C23×Q8—26no—
31C23×Q8—33no—
32C23×Q8—31no—
33C23×Q8—66no—
34C23×D4—31no—
35C24×C4C2×D313yes—
36C22×C22⋊C4—33no—
37C22×C22⋊C4—33no—
38C22×C22⋊C4—33no—
39#393—612no—
40#393—63no—
41#393—61no—
42#386—66no—
43#386—66no—
44C2×#95—36no—
45C2×#95—33no—
46C2×#95—33no—
47C2×#95—33no—
48C2×#95—33no—
49C23×D3—41no—
50C22×A4C2×D413yes—
51C22×A4C2×Q811yescharacteristic
52C22×A4#4113yes—
53C2×S4—26no—
54C2×S4—41no—
55C2×S4—43no—
56C2×C3×Q8—86no—
57C22×C3×C4—43no—
58C22×Dic3—43no—
59C4×A4—26no—
60SL(2,ℤ/4)—43no—
61(C2×D3)⋊C4—86no—
62C25—31no—
63C25C22×D311yescharacteristic
64C22×Q8—28no—
65C22×Q8—312no—
66C22×Q8—33no—
67C22×Q8—61no—
68C22×Q8—66no—
69C22×Q8—66no—
70C22×Q8—612no—
71C22×Q8—66no—
72C22×Q8—612no—
73C22×Q8—624no—
74C22×Q8—616no—
75C22×Q8S411yescharacteristic, complemented
76C22×D4—32no—
77C22×D4—32no—
78C22×D4—63no—
79C22×D4—63no—
80C22×D4—63no—
81C22×D4—63no—
82C23×C4—26no—
83C23×C4—33no—
84C23×C4—33no—
85C23×C4—36no—
86C23×C4—31no—
87C23×C4—33no—
88C23×C4—33no—
89C23×C4—66no—
90C23×C4#6716yes—
91C2×C22⋊C4—66no—
92C2×C22⋊C4—66no—
93C2×C22⋊C4—612no—
94C2×C22⋊C4—63no—
95C2×C22⋊C4—66no—
96C2×C22⋊C4—66no—
97C2×C22⋊C4—63no—
98C2×C22⋊C4—66no—
99C2×C22⋊C4—63no—
100C2×C22⋊C4—63no—
101#95—66no—
102#95—66no—
103#95—612no—
104#95—63no—
105#95—66no—
106#95—63no—
107C23×C3—41no—
108C22×D3—42no—
109C22×D3—83no—
110C22×D3—83no—
111C2×A4#12011yescharacteristic
112C2×A4C22⋊Q813yes—
113C2×A4#12413yes—
114S4—42no—
115C3×Q8—88no—
116C2×C3×C4—46no—
117C2×C3×C4—86no—
118C2×Dic3—83no—
119C2×Dic3—86no—
120C24—31no—
121C24—33no—
122C24—33no—
123C24—31no—
124C24—32no—
125C24—63no—
126C24—63no—
127C24—63no—
128C24—61no—
129C24—63no—
130C24—61no—
131C24D3×Q811yescharacteristic
132C24#23913yes—
133C24C2×#6713yes—
134C2×Q8—26no—
135C2×Q8—612no—
136C2×Q8—648no—
137C2×Q8—612no—
138C2×Q8—68no—
139C2×Q8—624no—
140C2×Q8—624no—
141C2×Q8—624no—
142C2×Q8—66no—
143C2×D4—66no—
144C2×D4—66no—
145C2×D4—66no—
146C2×D4—66no—
147C2×D4—126no—
148C2×D4—123no—
149C2×D4—123no—
150C2×D4—123no—
151C2×D4—121no—
152C22×C4—26no—
153C22×C4—36no—
154C22×C4—312no—
155C22×C4—33no—
156C22×C4—36no—
157C22×C4—36no—
158C22×C4—32no—
159C22×C4—66no—
160C22×C4—66no—
161C22×C4—63no—
162C22×C4—66no—
163C22×C4—612no—
164C22×C4—612no—
165C22×C4—66no—
166C22×C4—63no—
167C22×C4—63no—
168C22×C4—66no—
169C22×C4—63no—
170C22×C4—63no—
171C22×C4—63no—
172C22×C4—66no—
173C22×C4—66no—
174C22×C4—63no—
175C22×C4—63no—
176C22×C4—66no—
177C22×C4—63no—
178C22×C4C2×S413yes—
179C22⋊C4—1212no—
180C22⋊C4—126no—
181C22⋊C4—126no—
182C22⋊C4—123no—
183C22⋊C4—123no—
184C22⋊C4—126no—
185C22×C3—41no—
186C22×C3—43no—
187C22×C3—43no—
188C2×D3—86no—
189C2×D3—163no—
190C2×D3—161no—
191A4#39311yescharacteristic, complemented
192C3×C4—86no—
193Dic3—163no—
194C23—31no—
195C23—31no—
196C23—33no—
197C23—33no—
198C23—33no—
199C23—33no—
200C23—36no—
201C23—33no—
202C23—33no—
203C23—33no—
204C23—33no—
205C23—33no—
206C23—61no—
207C23—63no—
208C23—63no—
209C23—66no—
210C23—66no—
211C23—62no—
212C23—123no—
213C23—123no—
214C23—121no—
215C23—123no—
216C23—123no—
217C23—126no—
218C23—123no—
219C23—123no—
220C23—123no—
221C23#94613yes—
222C23#94713yes—
223C23#95311yescharacteristic
224C23C22×S411yescharacteristic
225Q8—28no—
226Q8—624no—
227Q8—616no—
228D4—126no—
229D4—126no—
230D4—122no—
231D4—122no—
232C2×C4—26no—
233C2×C4—312no—
234C2×C4—36no—
235C2×C4—66no—
236C2×C4—66no—
237C2×C4—612no—
238C2×C4—612no—
239C2×C4—66no—
240C2×C4—66no—
241C2×C4—66no—
242C2×C4—123no—
243C2×C4—121no—
244C2×C4—123no—
245C2×C4—123no—
246C2×C4—123no—
247C2×C4—126no—
248C2×C4—123no—
249C2×C4—126no—
250C2×C4—126no—
251C2×C4—123no—
252C2×C4—126no—
253C2×C4—123no—
254C2×C4GL(2,ℤ/4)16yes—
255C2×C3—43no—
256C2×C3—43no—
257C2×C3—41no—
258D3—162no—
259C22—33no—
260C22—33no—
261C22—31no—
262C22—33no—
263C22—33no—
264C22—33no—
265C22—33no—
266C22—33no—
267C22—36no—
268C22—33no—
269C22—33no—
270C22—31no—
271C22—63no—
272C22—63no—
273C22—61no—
274C22—122no—
275C22—126no—
276C22—126no—
277C22—243no—
278C22—241no—
279C22—243no—
280C22—241no—
281C22#523111yescharacteristic, complemented
282C22Q8×S411yescharacteristic
283C22#591913yes—
284C22C2×GL(2,ℤ/4)13yes—
285C4—26no—
286C4—66no—
287C4—122no—
288C4—243no—
289C4—243no—
290C3—41noSylow
291C2—31no—
292C2—33no—
293C2—33no—
294C2—31no—
295C2—242no—
296C2#8550613yes—
297C2#8550813yes—
298C2#8554611yescharacteristic
subgroupG/H
center, FrattiniC23C22×S4
commutator, cosocleC23×A4C23
FittingC24×Q8D3
socleC25C22×D3
prime radicalC23×S4C22
hypercenterC22×Q8S4
hypocenterA4#393
LCSG ▷ C23×A4 ▷ A4
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×Q8×A4C2semidirect
3. A4⋊#95C2semidirect
11. C24×Q8D3semidirect
21. C22×S4Q8semidirect
75. C22×Q8S4semidirect
191. A4#393semidirect
281. C22#5231semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×A42
C2×(C2×S4)⋊C42
A4⋊#952
#148733
#52314