Finite Group Databasebeta

← #11711782

Subgroups — #11711782

Order 768.

1249 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 1249 conjugacy classes of subgroups — too many to draw. Only its 95 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
2C2×SL(2,ℤ/4)⋊C4C213yesmaximal
3A4⋊#95C213yesmaximal
4#14874—31nomaximal, Sylow
5C2×Q8×A4—26no—
6C22×C4×A4C2213yes—
7C22×SL(2,ℤ/4)C2211yescharacteristic
8C22×SL(2,ℤ/4)C2213yes—
9SL(2,ℤ/4)⋊C4—26no—
10#5230—41nomaximal
11C24×Q8D311yescharacteristic
12C2×#395—33no—
13C2×#395—31no—
14C2×#385—33no—
15#1446—33no—
16#1446—31no—
17C22⋊#95—33no—
18C23×A4C2311yescharacteristic
19C22×C3×Q8—41no—
20Q8×A4—28no—
21C2×C4×A4—26no—
22C2×C4×A4D416yes—
23C2×SL(2,ℤ/4)—23no—
24C2×SL(2,ℤ/4)—23no—
25C2×SL(2,ℤ/4)—23no—
26C2×SL(2,ℤ/4)Q816yescomplemented
27C2×Dic3⋊C4—43no—
28C3⋊#95—43no—
29C23×Q8—26no—
30C23×Q8—33no—
31C23×Q8—31no—
32C23×Q8—66no—
33C24×C4C2×D313yes—
34C22×C4⋊C4—33no—
35C22×C4⋊C4—33no—
36C22×C22⋊C4—33no—
37C22×C22⋊C4—31no—
38#395—612no—
39#395—63no—
40#395—61no—
41#385—66no—
42#385—66no—
43C2×#95—36no—
44C2×#95—33no—
45C2×#95—33no—
46C2×#95—33no—
47C2×#95—33no—
48C22×A4C2×D413yes—
49C22×A4C2×Q813yes—
50C22×A4#4111yescharacteristic
51C2×C3×Q8—86no—
52C22×C3×C4—43no—
53C22×Dic3—43no—
54C22×Dic3—41no—
55C4×A4—26no—
56SL(2,ℤ/4)—26no—
57SL(2,ℤ/4)—41no—
58Dic3⋊C4—86no—
59C25C22×D311yescharacteristic
60C22×Q8—28no—
61C22×Q8—312no—
62C22×Q8—33no—
63C22×Q8—61no—
64C22×Q8—66no—
65C22×Q8—66no—
66C22×Q8—612no—
67C22×Q8—66no—
68C22×Q8—612no—
69C22×Q8—624no—
70C22×Q8—616no—
71C22×Q8S411yescharacteristic
72C23×C4—26no—
73C23×C4—33no—
74C23×C4—33no—
75C23×C4—36no—
76C23×C4—33no—
77C23×C4—33no—
78C23×C4—31no—
79C23×C4—31no—
80C23×C4—66no—
81C23×C4#6716yes—
82C2×C4⋊C4—66no—
83C2×C4⋊C4—66no—
84C2×C4⋊C4—612no—
85C2×C4⋊C4—63no—
86C2×C4⋊C4—66no—
87C2×C4⋊C4—63no—
88C2×C22⋊C4—36no—
89C2×C22⋊C4—36no—
90C2×C22⋊C4—63no—
91C2×C22⋊C4—63no—
92C2×C22⋊C4—63no—
93C2×C22⋊C4—63no—
94C2×C22⋊C4—63no—
95C2×C22⋊C4—63no—
96#95—66no—
97#95—66no—
98#95—612no—
99#95—63no—
100#95—66no—
101#95—63no—
102C23×C3—41no—
103C2×A4#12011yescharacteristic
104C2×A4C22⋊Q813yes—
105C2×A4C4⋊Q813yes—
106C3×Q8—88no—
107C2×C3×C4—46no—
108C2×C3×C4—86no—
109C2×Dic3—46no—
110C2×Dic3—83no—
111C2×Dic3—83no—
112C2×Dic3—83no—
113C24—31no—
114C24—33no—
115C24—31no—
116C24—33no—
117C24D3×Q813yes—
118C24#23911yescharacteristic
119C24C2×#6713yes—
120C2×Q8—26no—
121C2×Q8—612no—
122C2×Q8—648no—
123C2×Q8—612no—
124C2×Q8—68no—
125C2×Q8—624no—
126C2×Q8—624no—
127C2×Q8—624no—
128C2×Q8—66no—
129C22×C4—26no—
130C22×C4—36no—
131C22×C4—312no—
132C22×C4—33no—
133C22×C4—36no—
134C22×C4—36no—
135C22×C4—36no—
136C22×C4—36no—
137C22×C4—66no—
138C22×C4—66no—
139C22×C4—63no—
140C22×C4—66no—
141C22×C4—612no—
142C22×C4—612no—
143C22×C4—66no—
144C22×C4—66no—
145C22×C4—63no—
146C22×C4—63no—
147C22×C4—63no—
148C22×C4—63no—
149C22×C4—63no—
150C22×C4—66no—
151C22×C4—63no—
152C22×C4—63no—
153C22×C4—63no—
154C22×C4—63no—
155C22×C4—63no—
156C22×C4—63no—
157C22×C4—63no—
158C22×C4—61no—
159C22×C4—61no—
160C22×C4C2×S413yes—
161C4⋊C4—1212no—
162C4⋊C4—126no—
163C4⋊C4—126no—
164C22⋊C4—66no—
165C22⋊C4—66no—
166C22⋊C4—66no—
167C22⋊C4—66no—
168C22⋊C4—123no—
169C22⋊C4—121no—
170C22×C3—43no—
171C22×C3—41no—
172C22×C3—43no—
173A4#39511yescharacteristic, complemented
174C3×C4—86no—
175Dic3—86no—
176Dic3—161no—
177C23—33no—
178C23—33no—
179C23—31no—
180C23—31no—
181C23—33no—
182C23—33no—
183C23—33no—
184C23—33no—
185C23—33no—
186C23—33no—
187C23—36no—
188C23—33no—
189C23—61no—
190C23—63no—
191C23—63no—
192C23#94413yes—
193C23#94613yes—
194C23#95311yescharacteristic
195C23C22×S411yescharacteristic
196Q8—28no—
197Q8—624no—
198Q8—616no—
199C2×C4—26no—
200C2×C4—312no—
201C2×C4—36no—
202C2×C4—66no—
203C2×C4—66no—
204C2×C4—612no—
205C2×C4—612no—
206C2×C4—66no—
207C2×C4—66no—
208C2×C4—66no—
209C2×C4—66no—
210C2×C4—612no—
211C2×C4—66no—
212C2×C4—66no—
213C2×C4—123no—
214C2×C4—126no—
215C2×C4—123no—
216C2×C4—123no—
217C2×C4—123no—
218C2×C4—123no—
219C2×C4—123no—
220C2×C4—121no—
221C2×C4—121no—
222C2×C4—123no—
223C2×C4—123no—
224C2×C4—123no—
225C2×C4—123no—
226C2×C4GL(2,ℤ/4)16yes—
227C2×C3—43no—
228C2×C3—41no—
229C2×C3—43no—
230C22—33no—
231C22—33no—
232C22—31no—
233C22—33no—
234C22—33no—
235C22—33no—
236C22—33no—
237C22—33no—
238C22—31no—
239C22—36no—
240C22—33no—
241C22—33no—
242C22—61no—
243C22—63no—
244C22—63no—
245C22#523011yescharacteristic, complemented
246C22Q8×S413yes—
247C22#591911yescharacteristic
248C22C2×GL(2,ℤ/4)13yes—
249C4—26no—
250C4—66no—
251C4—126no—
252C4—126no—
253C4—241no—
254C4—241no—
255C3—41noSylow
256C2—33no—
257C2—31no—
258C2—31no—
259C2—33no—
260C2#8550213yes—
261C2#8550613yes—
262C2#8554611yescharacteristic
subgroupG/H
center, FrattiniC23C22×S4
commutator, prime radical, cosocleC23×A4C23
FittingC24×Q8D3
socleC25C22×D3
hypercenterC22×Q8S4
hypocenterA4#395
LCSG ▷ C23×A4 ▷ A4
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
26. C2×SL(2,ℤ/4)Q8semidirect
173. A4#395semidirect
245. C22#5230semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×A42
C2×SL(2,ℤ/4)⋊C42
A4⋊#952
#148743
#52304