Finite Group Databasebeta

← #11922977

Subgroups — #11922977

Order 1152.

502 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C3×Q8×SL(2,𝔽3)C211yesmaximal, characteristic
2C4⋊(C3⋊#226)C213yesmaximal
3C4×C3⋊#226C213yesmaximal
4#85428—33nomaximal
5#83707—31nomaximal
6C2×C3⋊#226C2213yes—
7Q8×C3⋊Dic3—41nomaximal
8C3×C4×SL(2,𝔽3)C2213yes—
9C3⋊#859C2211yescharacteristic
10C3×Q82D311yescharacteristic
11#5570—31no—
12Q8×SL(2,𝔽3)D313yes—
13C4⋊#226—39no—
14C4×#226—39no—
15#5093—33no—
16#5091—33no—
17#5031—33no—
18C4×#216—33no—
19Q8×C3⋊C8—31no—
20C2×C32×Q8—41no—
21C2×C3×SL(2,𝔽3)C2311yescharacteristic
22C3⋊#226—23no—
23C4⋊(C3⋊Dic3)—43no—
24C4×C3⋊Dic3—43no—
25#3363—91noSylow
26C2×#226—39no—
27C3×C4⋊Q8—33no—
28C3×C4×Q8—31no—
29C3×C4×Q8C2×D313yes—
30Q8×Dic3—61no—
31Q8×Dic3—123no—
32C2×#216—33no—
33#890—63no—
34#869—33no—
35C4×Dic6—33no—
36C4×SL(2,𝔽3)C2×D319yes—
37#859—33no—
38#835—31no—
39#810—33no—
40#810—36no—
41#807—33no—
42C4⋊(C3⋊C8)—33no—
43C4×C3⋊C8—33no—
44C32×Q8—44no—
45C2×C32×C4—43no—
46C2×C3⋊Dic3—41no—
47C2×C3⋊Dic3—43no—
48C3×SL(2,𝔽3)#4111yescharacteristic
49Q82C3⋊D311yescharacteristic
50#557—91no—
51#487—93no—
52#477—93no—
53#462—93no—
54C8×Q8—91no—
55C4×Q16—93no—
56C2×C3×Q8—43no—
57C2×C3×Q8—61no—
58C2×C3×Q8S411yescharacteristic
59C2×C3×Q8C22×D311yescharacteristic
60C2×Dic6—33no—
61C2×SL(2,𝔽3)C22×D313yes—
62#226—69no—
63C3×C4⋊C4—33no—
64C3×C4⋊C4—33no—
65C3×C42—33no—
66#216—63no—
67#216—63no—
68C4⋊Dic3—31no—
69C4⋊Dic3—63no—
70C4⋊Dic3—63no—
71C4⋊Dic3—129no—
72Dic3⋊C4—63no—
73C4×Dic3—63no—
74C4×Dic3—129no—
75C2×C3⋊C8—31no—
76C2×C3⋊C8—33no—
77C22×C32—41no—
78C32×C4—46no—
79C3⋊Dic3—42no—
80C3⋊Dic3—83no—
81C2×Q16—93no—
82C4⋊Q8—33no—
83C4⋊Q8—93no—
84#125—183no—
85C4×Q8—31no—
86C4×Q8—93no—
87C4×Q8—181no—
88C4×Q8C2×C3⋊D313yes—
89#107—93no—
90C4⋊C8—93no—
91#103—93no—
92#103—96no—
93#103—91no—
94C4×C8—93no—
95C3×Q8—31no—
96C3×Q8—412no—
97C3×Q8—241no—
98C3×Q8#22614yescomplemented
99C3×Q8#23911yescharacteristic
100C2×C3×C4—31no—
101C2×C3×C4—33no—
102C2×C3×C4—49no—
103C2×C3×C4C2×S413yes—
104C2×Dic3—61no—
105C2×Dic3—63no—
106C2×Dic3—123no—
107C2×Dic3—129no—
108Dic6—63no—
109Dic6—63no—
110SL(2,𝔽3)#23913yes—
111C3⋊C8—61no—
112C3⋊C8—63no—
113C2×C32—41no—
114C2×C32—41no—
115C2×C32—41no—
116C2×Q8—61no—
117C2×Q8—93no—
118C2×Q8C3⋊S411yescharacteristic
119C2×Q8C22×C3⋊D311yescharacteristic
120Q16—183no—
121Q16—183no—
122C2×C8—91no—
123C2×C8—93no—
124C4⋊C4—33no—
125C4⋊C4—33no—
126C4⋊C4—91no—
127C4⋊C4—183no—
128C4⋊C4—183no—
129C4⋊C4—183no—
130C42—33no—
131C42—183no—
132C22×C3—43no—
133C22×C3C22×S411yescharacteristic
134C3×C4—31no—
135C3×C4—31no—
136C3×C4—418no—
137C3×C4—63no—
138C3×C4C2×#22616yes—
139Dic3—126no—
140Dic3—123no—
141Dic3—121no—
142Dic3—249no—
143C32—41noSylow
144Q8—31no—
145Q8—183no—
146Q8—183no—
147Q8—241no—
148Q8C3⋊#22614yescomplemented
149Q8#383211yescharacteristic
150C2×C4—31no—
151C2×C4—33no—
152C2×C4—181no—
153C2×C4—183no—
154C2×C4C2×C3⋊S413yes—
155C8—183no—
156C8—181no—
157C2×C3—43no—
158C2×C3—43no—
159C2×C3—43no—
160C2×C3C22×#22611yescharacteristic
161C2×C3#591911yescharacteristic
162C2×C3#592411yescharacteristic
163C22C22×C3⋊S411yescharacteristic
164C4—31no—
165C4—31no—
166C4—63no—
167C4—363no—
168C4—361no—
169C4C2×C3⋊#22616yes—
170C3—43no—
171C3#8542811yescharacteristic, complemented
172C2C22×C3⋊#22611yescharacteristic
173C2#1059658211yescharacteristic
174C2#1059658711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, prime radical, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×Q82D3
socleC22×C3C22×S4
hypercenterC2×Q8C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
98. C3×Q8#226semidirect
148. Q8C3⋊#226semidirect
171. C3#85428semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×SL(2,𝔽3)2
C4⋊(C3⋊#226)2
C4×C3⋊#2262
#854283
#837073
Q8×C3⋊Dic34