Finite Group Databasebeta

← Q8×C3⋊#226 #11922976

Subgroups — Q8×C3⋊#226 #11922976

Order 1152.

498 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 87 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
2#10593625C213yesmaximal
3C4×C3⋊#226C213yesmaximal
4Q8×#226—33nomaximal
5Q8×#216—31nomaximal
6C2×C3⋊#226C2211yescharacteristic
7Q8×C3⋊Dic3—41nomaximal
8C3×C4×SL(2,𝔽3)C2213yes—
9C3⋊#859C2213yes—
10C3×Q82D311yescharacteristic
11Q8×Dic6—31no—
12Q8×SL(2,𝔽3)D313yes—
13#5391—39no—
14C4×#226—39no—
15#5091—33no—
16#5078—33no—
17C4×#216—33no—
18Q8×C3⋊C8—31no—
19#5021—33no—
20C2×C32×Q8—41no—
21C2×C3×SL(2,𝔽3)C2311yescharacteristic
22C3⋊#226Q812yesdirect product factor
23C4⋊(C3⋊Dic3)—43no—
24C4×C3⋊Dic3—43no—
25Q8×Q16—91noSylow
26C2×#226—33no—
27C3×C4⋊Q8—33no—
28C3×C4×Q8—31no—
29C3×C4×Q8C2×D313yes—
30Q8×Dic3—61no—
31Q8×Dic3—123no—
32C2×#216—31no—
33#888—63no—
34#869—33no—
35C4×Dic6—33no—
36C4×SL(2,𝔽3)C2×D319yes—
37#859—39no—
38#835—33no—
39#810—33no—
40#807—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)C2×Q811yescharacteristic
49Q82—91no—
50Q82C3⋊D311yescharacteristic
51#500—93no—
52#477—93no—
53#477—93no—
54C8×Q8—91no—
55C4×Q16—93no—
56C2×C3×Q8—43no—
57C2×C3×Q8—61no—
58C2×C3×Q8S411yescharacteristic
59C2×C3×Q8C22×D311yescharacteristic
60C2×Dic6—31no—
61C2×Dic6—63no—
62C2×SL(2,𝔽3)C22×D313yes—
63#226—36no—
64C3×C4⋊C4—33no—
65C3×C4⋊C4—33no—
66C3×C42—33no—
67#216—32no—
68#216—32no—
69C4⋊Dic3—33no—
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—91no—
82C4⋊Q8—33no—
83C4⋊Q8—93no—
84C4⋊Q8—183no—
85C4×Q8—31no—
86C4×Q8—93no—
87C4×Q8—181no—
88C4×Q8C2×C3⋊D313yes—
89#107—93no—
90#107—96no—
91C4⋊C8—93no—
92#103—93no—
93#103—93no—
94C4×C8—93no—
95C3×Q8—31no—
96C3×Q8—412no—
97C3×Q8—241no—
98C3×Q8#22614yescomplemented
99C3×Q8D3×Q811yescharacteristic
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—32no—
109Dic6—61no—
110Dic6—243no—
111SL(2,𝔽3)D3×Q813yes—
112C3⋊C8—32no—
113C3⋊C8—63no—
114C2×C32—41no—
115C2×C32—41no—
116C2×C32—41no—
117C2×Q8—61no—
118C2×Q8—91no—
119C2×Q8—183no—
120C2×Q8C3⋊S411yescharacteristic
121C2×Q8C22×C3⋊D311yescharacteristic
122Q16—92no—
123Q16—92no—
124C2×C8—91no—
125C2×C8—93no—
126C4⋊C4—33no—
127C4⋊C4—33no—
128C4⋊C4—93no—
129C4⋊C4—183no—
130C4⋊C4—183no—
131C42—33no—
132C42—183no—
133C22×C3—43no—
134C22×C3C22×S411yescharacteristic
135C3×C4—31no—
136C3×C4—31no—
137C3×C4—418no—
138C3×C4—63no—
139C3×C4C2×#22616yes—
140Dic3—62no—
141Dic3—126no—
142Dic3—123no—
143Dic3—249no—
144C32—41noSylow
145Q8—31no—
146Q8—92no—
147Q8—181no—
148Q8—241no—
149Q8—723no—
150Q8C3⋊#22614yesdirect product factor
151Q8Q8×C3⋊D311yescharacteristic
152C2×C4—31no—
153C2×C4—33no—
154C2×C4—181no—
155C2×C4—183no—
156C2×C4C2×C3⋊S413yes—
157C8—92no—
158C8—183no—
159C2×C3—43no—
160C2×C3—43no—
161C2×C3—43no—
162C2×C3C22×#22611yescharacteristic
163C2×C3Q8×S411yescharacteristic
164C2×C3#592511yescharacteristic
165C22C22×C3⋊S411yescharacteristic
166C4—31no—
167C4—31no—
168C4—63no—
169C4—182no—
170C4—363no—
171C4C2×C3⋊#22616yes—
172C3—43no—
173C3Q8×#22611yescharacteristic, complemented
174C2C22×C3⋊#22611yescharacteristic
175C2Q8×C3⋊S411yescharacteristic
176C2#1059658811yescharacteristic
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)C2×Q8
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
22. C3⋊#226Q8direct (N × H)
98. C3×Q8#226semidirect
150. Q8C3⋊#226direct (N × H)
173. C3Q8×#226semidirect

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
#105936252
C4×C3⋊#2262
Q8×#2263
Q8×#2163
Q8×C3⋊Dic34