Finite Group Databasebeta

← #11710353

Subgroups — #11710353

Order 768.

873 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 873 conjugacy classes of subgroups — too many to draw. Only its 22 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#87551C211yesmaximal, characteristic, complemented
2#60402C311yesmaximal, characteristic, complemented, Sylow
3#5948—41nomaximal
4#458⋊C3—44nomaximal
52−1+6C2×C311yescharacteristic
6#3487—31no—
7#3027—34no—
8#3025—34no—
9#2884—31no—
10#997—44no—
11#997—84no—
12C22⋊SL(2,𝔽3)—41no—
13C22⋊SL(2,𝔽3)—84no—
14#585—34no—
15#585—64no—
16C2×2−1+4—31no—
17C2×2−1+4—64no—
18C22×#41A411yescharacteristic, complemented
19#550—64no—
20#548—64no—
21#535—36no—
22#525—36no—
23#519—31no—
24#458—34no—
25#458A414yescomplemented
26B(4,𝔽2)—312no—
27B(4,𝔽2)—34no—
28#410—312no—
29#41⋊C3—164no—
30C4×A4—161no—
312−1+4—68no—
322−1+4—612no—
332−1+4—612no—
342−1+4—64no—
35H(5,𝔽2)—24no—
36H(5,𝔽2)—34no—
37H(5,𝔽2)—612no—
38H(5,𝔽2)—64no—
39H(5,𝔽2)C2×A414yes—
40C2×#41—36no—
41C2×#41—34no—
42C2×#41—612no—
43C2×#41—612no—
44C2×#41—64no—
45C2×#41—64no—
46C2×#41—612no—
47C22×Q8—24no—
48C22×Q8—34no—
49C22×Q8—64no—
50C22×Q8C2×A411yescharacteristic
51C22×D4—31no—
52C23×C4—31no—
53#127—66no—
54#124—66no—
55#123—34no—
56C22⋊Q8—66no—
57#121—612no—
58#121—66no—
59#120—312no—
60C4×Q8—64no—
61C4×D4—612no—
62#117—36no—
63#99—312no—
64#99—34no—
65C2×A4—161no—
66C2×A4—324no—
67SL(2,𝔽3)—164no—
68SL(2,𝔽3)—324no—
69C24—31no—
70#41—44no—
71#41—624no—
72#41—612no—
73#41—64no—
74#41—64no—
75#41—612no—
76#41—612no—
77#41—624no—
78#41—624no—
79#41—64no—
80#41—124no—
81C2×Q8—34no—
82C2×Q8—36no—
83C2×Q8—612no—
84C2×Q8—612no—
85C2×Q8—612no—
86C2×Q8—64no—
87C2×D4—36no—
88C2×D4—34no—
89C2×D4—66no—
90C2×D4—612no—
91C2×D4—612no—
92C2×D4—612no—
93C2×D4—64no—
94C2×D4—612no—
95C22×C4—31no—
96C22×C4—312no—
97C22×C4—66no—
98C22×C4—64no—
99C22×C4—612no—
100C22×C4C22⋊A411yescharacteristic
101C4⋊C4—66no—
102C4⋊C4—64no—
103C22⋊C4—34no—
104C22⋊C4—612no—
105C22⋊C4—64no—
106C42—66no—
107A4—161no—
108A4—324no—
109C3×C4—641no—
110C23—24no—
111C23—34no—
112C23—63no—
113C23—64no—
114C23—241no—
115C23C2×C22⋊A411yescharacteristic
116Q8—24no—
117Q8—34no—
118Q8—612no—
119Q8—64no—
120Q8#120⋊C314yes—
121D4—612no—
122D4—68no—
123D4—64no—
124D4—612no—
125D4—124no—
126D4—2412no—
127C2×C4—36no—
128C2×C4—34no—
129C2×C4—612no—
130C2×C4—612no—
131C2×C4—61no—
132C2×C4—612no—
133C2×C4—64no—
134C2×C4—64no—
135C2×C4—124no—
136C2×C3—641no—
137C22—31no—
138C22—41no—
139C22—64no—
140C22—84no—
141C22—124no—
142C22—126no—
143C22—121no—
144C22—244no—
145C4—34no—
146C4—41no—
147C4—64no—
148C4—124no—
149C3—641noSylow
150C2—61no—
151C2—124no—
152C2—241no—
153C2#8564511yescharacteristic
subgroupG/H
center, socle, hypercenterC2#85645
commutator, cosocle, hypocenter2−1+6C2×C3
FrattiniC23C2×C22⋊A4
Fitting#60402C3
LCSG ▷ 2−1+6
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #87551C2semidirect
2. #60402C3semidirect
18. C22×#41A4semidirect
25. #458A4semidirect

Maximal subgroups

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

subgroupindex
#875512
#604023
#59484
#458⋊C34