Finite Group Databasebeta

← #3444⋊C3 #85648

Subgroups — #3444⋊C3 #85648

Order 384.

232 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 17 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⋊(C42⋊C3)C211yesmaximal, characteristic, complemented
2#3444C311yesmaximal, characteristic, complemented, Sylow
3#127⋊C3—41nomaximal
4#126⋊C3—42nomaximal
5#120⋊C3—41nomaximal
6C2×C42⋊C3—41nomaximal
7#533—31no—
8#532—31no—
9#529—32no—
10#529—32no—
11#525—31no—
12#525—31no—
13C4×#41—31no—
14C4×#41—31no—
15C22×C42C2×C311yescharacteristic, complemented
16C22⋊A4—41no—
17C42⋊C3—41no—
18C42⋊C3—42no—
19C42⋊C3—41no—
20C2×#41—31no—
21C2×#41—31no—
22C23×C4—31no—
23C4⋊Q8—31no—
24#127A411yescharacteristic, complemented
25#126—32no—
26#126A412yescomplemented
27#125—32no—
28#124—31no—
29#124—31no—
30#123—32no—
31#123—31no—
32#123—31no—
33C22⋊Q8—32no—
34C22⋊Q8—32no—
35#121—32no—
36#121—32no—
37#120—31no—
38#120A411yescharacteristic
39C4×Q8—32no—
40C4×Q8—32no—
41C4×D4—32no—
42C4×D4—32no—
43C4×D4—32no—
44C4×D4—32no—
45C4×D4—32no—
46C4×D4—32no—
47#117—31no—
48#117—31no—
49#117—31no—
50#117—32no—
51#117—31no—
52C2×C42—31no—
53C2×C42—31no—
54C2×C42—32no—
55C2×C42—31no—
56C2×C42A411yescharacteristic, complemented
57C2×A4—161no—
58C24C2×A411yescharacteristic
59#41—64no—
60#41—64no—
61C2×Q8—31no—
62C2×Q8—31no—
63C2×D4—31no—
64C2×D4—32no—
65C2×D4—31no—
66C2×D4—31no—
67C2×D4—31no—
68C22×C4—31no—
69C22×C4—32no—
70C22×C4—32no—
71C22×C4—31no—
72C22×C4—32no—
73C22×C4—31no—
74C22×C4—31no—
75C22×C4—64no—
76C4⋊C4—32no—
77C4⋊C4—32no—
78C4⋊C4—32no—
79C4⋊C4—32no—
80C4⋊C4—32no—
81C4⋊C4—32no—
82C22⋊C4—31no—
83C22⋊C4—32no—
84C22⋊C4—32no—
85C22⋊C4—32no—
86C22⋊C4—32no—
87C22⋊C4—32no—
88C22⋊C4—31no—
89C42—31no—
90C42—31no—
91C42—32no—
92C42—31no—
93C42—31no—
94C42—32no—
95C42C2×A411yescharacteristic
96C42C2×A412yes—
97C42C2×A411yescharacteristic
98A4—161no—
99A4—322no—
100C23—31no—
101C23—31no—
102C23—62no—
103C23C22⋊A411yescharacteristic
104Q8—62no—
105Q8—62no—
106D4—62no—
107D4—62no—
108D4—62no—
109D4—64no—
110D4—62no—
111C2×C4—34no—
112C2×C4—31no—
113C2×C4—31no—
114C2×C4—32no—
115C2×C4—32no—
116C2×C4—31no—
117C2×C4—31no—
118C2×C4—64no—
119C2×C4—64no—
120C2×C4—62no—
121C2×C4—64no—
122C2×C4—62no—
123C2×C4—62no—
124C2×C3—641no—
125C22—22no—
126C22—32no—
127C22—61no—
128C22—62no—
129C22—61no—
130C22—62no—
131C22—62no—
132C22C2×C22⋊A411yescharacteristic
133C4—32no—
134C4—32no—
135C4—62no—
136C4—62no—
137C4—61no—
138C4—61no—
139C3—641noSylow
140C2—31no—
141C2—41no—
142C2—62no—
143C2—121no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22×C42C2×C3
Frattini, socleC22C2×C22⋊A4
Fitting#3444C3
LCSG ▷ C22×C42
UCSC1

Complements

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

NH ≅ G/Ntype
1. C22⋊(C42⋊C3)C2semidirect
2. #3444C3semidirect
15. C22×C42C2×C3semidirect
24. #127A4semidirect
26. #126A4semidirect
56. C2×C42A4semidirect

Maximal subgroups

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

subgroupindex
C22⋊(C42⋊C3)2
#34443
#127⋊C34
#126⋊C34
#120⋊C34
C2×C42⋊C34