Finite Group Databasebeta

← #10708596

Subgroups — #10708596

Order 768.

356 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 59 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#81808C211yesmaximal, characteristic
2#71298C211yesmaximal, characteristic, complemented
3#69306C212yesmaximal, complemented
4C3×#1358C211yesmaximal, characteristic, complemented
5#67568C212yesmaximal, complemented
6#12313—31nomaximal, Sylow
7#5766—22no—
8C3×#494C2211yescharacteristic
9#5192—22no—
10#5095—22no—
11#5078—22no—
12#4909—22no—
13#4700C2212yes—
14#4699C2211yescharacteristic
15C3×#364C2212yes—
16#4517—22no—
17C8×C3⋊C8C2211yescharacteristic
18#3293—31no—
19#2214—32no—
20#1713—31no—
21#1358D311yescharacteristic, complemented
22#1335—32no—
23#1007—22no—
24C2×C3×Q16—22no—
25C3×C4⋊Q8D412yes—
26#942—24no—
27#919—42no—
28#918—44no—
29D3×Q16—42no—
30C2×Dic12—22no—
31#904D412yes—
32#901—22no—
33#869—21no—
34C3×M32—22no—
35C3×C4×C8C2311yescharacteristic
36#836—22no—
37#819D412yes—
38#807—44no—
39#805—22no—
40C4×C3⋊C8—21no—
41Dic24—42no—
42#800—42no—
43#578—62no—
44#510—62no—
45#500—62no—
46#494—31no—
47#494C2×D311yescharacteristic
48#473—62no—
49#456—62no—
50#443—32no—
51#364C2×D312yes—
52#362—62no—
53C82—31no—
54C2×C3×Q8—22no—
55#239—44no—
56D3×Q8—44no—
57#235—22no—
58#235—82no—
59C2×Dic6—41no—
60C3×Q16—42no—
61C3×Q16—42no—
62C2×C3×C8C2×D412yes—
63C3×C4⋊C4—42no—
64C3×C42C2×D411yescharacteristic
65#216—44no—
66#215—44no—
67C4⋊Dic3—42no—
68#208—22no—
69C2×C3⋊C8—22no—
70Dic12—42no—
71Dic12—42no—
72Dic12D812yes—
73D24D812yes—
74#204—22no—
75ΓL(1,𝔽25)—42no—
76D3×C8—42no—
77C3×C16—42no—
782−1+4—62no—
79#137—64no—
80#137—124no—
81#135—32no—
82#135—122no—
83C2×Q16—22no—
84C2×Q16—62no—
85C2×Q16—122no—
86#131—62no—
87C4⋊Q8—61no—
88C4⋊Q8D1212yes—
89Q32—122no—
90SD32—122no—
91M32—22no—
92#108—32no—
93#107—124no—
94#104—62no—
95#101—62no—
96C4×C8—61no—
97C4×C8C22×D311yescharacteristic
98C3×Q8—44no—
99C2×C3×C4—21no—
100C2×C3×C4—42no—
101C2×C3×C4#12011yescharacteristic
102#67—42no—
103C2×Dic3—81no—
104D12—22no—
105D12—82no—
106C4×D3—42no—
107C4×D3—84no—
108Dic6—22no—
109Dic6—82no—
110Dic6—82no—
111C3×C8C2×D812yes—
112C3×C8AGL(1,ℤ/8)12yes—
113C3⋊C8—24no—
114C3⋊C8—42no—
115#41—62no—
116#41—124no—
117#41—242no—
118C2×Q8—22no—
119C2×Q8—124no—
120C2×Q8—121no—
121Q16—32no—
122Q16—42no—
123Q16—42no—
124Q16—124no—
125Q16—122no—
126Q16—122no—
127SD16—62no—
128SD16—124no—
129D8—32no—
130M16—62no—
131M16—122no—
132C2×C8—62no—
133C2×C8—122no—
134C2×C8D3×D412yes—
135C4⋊C4—42no—
136C4⋊C4—122no—
137C42C2×D1211yescharacteristic
138C16—42no—
139C22×C3#45311yescharacteristic
140C2×D3—42no—
141C3×C4—22no—
142C3×C4—82no—
143C3×C4#44712yes—
144Dic3—42no—
145Dic3—161no—
146Q8—44no—
147Q8—62no—
148Q8—242no—
149Q8—242no—
150D4—62no—
151D4—122no—
152D4—242no—
153C2×C4—21no—
154C2×C4—42no—
155C2×C4—122no—
156C2×C4—244no—
157C2×C4—241no—
158C2×C4#88211yescharacteristic
159C8—64no—
160C8—122no—
161C8D3×D812yes—
162C8#91412yes—
163C2×C3—21no—
164C2×C3#161911yescharacteristic
165D3—82no—
166C22—122no—
167C22#474711yescharacteristic
168C4—22no—
169C4—82no—
170C4—122no—
171C4—481no—
172C4#477312yes—
173C3#1231311yescharacteristic, complemented, Sylow
174C2—21no—
175C2—242no—
176C2#6839011yescharacteristic
subgroupG/H
centerC2#68390
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×#1358C2
socleC2×C3#1619
prime radical#81808C2
hypercenter#1358D3
hypocenterC3#12313
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1358

Complements

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

NH ≅ G/Ntype
2. #71298C2semidirect
3. #69306C2semidirect
4. C3×#1358C2semidirect
5. #67568C2semidirect
21. #1358D3semidirect
173. C3#12313semidirect

Maximal subgroups

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

subgroupindex
#818082
#712982
#693062
C3×#13582
#675682
#123133