Finite Group Databasebeta

← #11711822

Subgroups — #11711822

Order 768.

442 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 33 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#73166C211yesmaximal, characteristic, complemented
2#72987C213yesmaximal
3#72986C213yesmaximal, complemented
4#17549—31nomaximal, Sylow
5Q8×S4—41nomaximal
6#376⋊C3C2213yes—
7#4622C2211yescharacteristic
8#4621C2213yes—
9#3523—31no—
10#2984—31no—
11#2276D311yescharacteristic, complemented
12#1509—33no—
13#1506—33no—
14#1503—33no—
15#1500—33no—
16Q8×A4—41no—
17C4×S4—43no—
18A4⋊Q8—43no—
19#95⋊C3C2311yescharacteristic
20#557—33no—
21#557—31no—
22#556—33no—
23#552—63no—
24D4×Q8—31no—
25D4×Q8—61no—
26#541—63no—
27#533—33no—
28#529—33no—
29#519—61no—
30C2×C4×Q8—31no—
31#446—63no—
32C8×Q8—61no—
33#432—33no—
34#431—33no—
35C4×C4⋊C4—33no—
36#376C2×D313yes—
37#328—33no—
38#328—36no—
39#328—33no—
40#327—33no—
41#327—31no—
42C2×S4—41no—
43D3×Q8—161no—
44C4×A4—43no—
45SL(2,ℤ/4)—43no—
46C22×Q8—61no—
47C22×Q8S411yescharacteristic
48C4⋊Q8—33no—
49C4⋊Q8—33no—
50C4⋊Q8—63no—
51#126—63no—
52#126—66no—
53#125—33no—
54#125—63no—
55#125—66no—
56#125—63no—
57#124—33no—
58#123—63no—
59C22⋊Q8—33no—
60C22⋊Q8—63no—
61C22⋊Q8—63no—
62C22⋊Q8—63no—
63#121—31no—
64C4×Q8—31no—
65C4×Q8—33no—
66C4×Q8—31no—
67C4×Q8—33no—
68C4×Q8—33no—
69C4×Q8—36no—
70C4×Q8—63no—
71C4×Q8—63no—
72C4×Q8—61no—
73C4×D4—33no—
74C4×D4—63no—
75#117—63no—
76#117—63no—
77#117—63no—
78C2×C4⋊C4—33no—
79C2×C42—33no—
80C2×C42—61no—
81C4⋊C8—63no—
82C4⋊C8—63no—
83C22⋊C8—31no—
84C22⋊C8—33no—
85#97—63no—
86C4×C8—63no—
87#95—33no—
88#95C22×D311yescharacteristic
89C2×A4—41no—
90S4—42no—
91C3×Q8—161no—
92C4×D3—163no—
93Dic6—163no—
94C2×Q8—31no—
95C2×Q8—33no—
96C2×Q8—33no—
97C2×Q8—63no—
98C2×Q8—123no—
99C2×Q8—121no—
100C2×D4—31no—
101C2×D4—61no—
102C22×C4—33no—
103C22×C4—31no—
104C22×C4—63no—
105C22×C4C2×S413yes—
106C2×C8—61no—
107C2×C8—63no—
108C4⋊C4—33no—
109C4⋊C4—33no—
110C4⋊C4—36no—
111C4⋊C4—33no—
112C4⋊C4—31no—
113C4⋊C4—63no—
114C4⋊C4—63no—
115C4⋊C4—63no—
116C4⋊C4—63no—
117C4⋊C4—63no—
118C4⋊C4—63no—
119C4⋊C4—66no—
120C4⋊C4—63no—
121C22⋊C4—61no—
122C22⋊C4—63no—
123C22⋊C4—63no—
124C22⋊C4—63no—
125C22⋊C4—63no—
126C42—33no—
127C42—33no—
128C42—33no—
129C42—33no—
130C42—66no—
131C42—63no—
132C42—241no—
133C2×D3—161no—
134A4—41no—
135C3×C4—163no—
136Dic3—163no—
137C23—61no—
138C23C22×S411yescharacteristic
139Q8—33no—
140Q8—61no—
141Q8—123no—
142Q8—123no—
143Q8—123no—
144Q8#85711yescharacteristic
145D4—121no—
146D4—121no—
147D4—121no—
148C2×C4—33no—
149C2×C4—33no—
150C2×C4—31no—
151C2×C4—31no—
152C2×C4—33no—
153C2×C4—33no—
154C2×C4—63no—
155C2×C4—63no—
156C2×C4—63no—
157C2×C4—61no—
158C2×C4—63no—
159C2×C4—123no—
160C2×C4—121no—
161C2×C4—123no—
162C8—121no—
163C8—123no—
164C2×C3—161no—
165D3—162no—
166C22—31no—
167C22—31no—
168C22—121no—
169C22—121no—
170C22—121no—
171C22#592411yescharacteristic
172C4—33no—
173C4—61no—
174C4—63no—
175C4—63no—
176C4—61no—
177C4—121no—
178C4—123no—
179C4—123no—
180C4C2×#85713yes—
181C3—161noSylow
182C2—31no—
183C2—31no—
184C2—241no—
185C2C22×#85711yescharacteristic
subgroupG/H
centerC2C22×#857
commutator, cosocle, hypocenter#95⋊C3C23
Frattini, socleC23C22×S4
Fitting#2276D3
prime radical#4622C22
hypercenterQ8#857
LCSG ▷ #95⋊C3
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. #73166C2semidirect
3. #72986C2semidirect
11. #2276D3semidirect

Maximal subgroups

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

subgroupindex
#731662
#729872
#729862
#175493
Q8×S44