Finite Group Databasebeta

← Q8×C3⋊GL(2,𝔽3) #11922978

Subgroups — Q8×C3⋊GL(2,𝔽3) #11922978

Order 1152.

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

normal subgroups only

This group has 704 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, complemented
2C4×C3⋊GL(2,𝔽3)C213yesmaximal
3#10593620C213yesmaximal, complemented
4Q8×GL(2,𝔽3)—33nomaximal
5Q8×#215—31nomaximal
6C2×Q8×C3⋊D3—41nomaximal
7C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
8C3×C4×SL(2,𝔽3)C2213yes—
9C3⋊#859C2213yes—
10C3×Q82D311yescharacteristic, complemented
11Q8×D12—31no—
12Q8×SL(2,𝔽3)D313yescomplemented
13C4×GL(2,𝔽3)—39no—
14#5386—39no—
15#5084—33no—
16#5080—33no—
17C4×#215—33no—
18Q8×C3⋊C8—31no—
19#5020—33no—
20C2×C32×Q8—41no—
21Q8×C3⋊D3—48no—
22C2×C4×C3⋊D3—43no—
23C2×#634—43no—
24C2×C3×SL(2,𝔽3)C2311yescharacteristic
25C3⋊GL(2,𝔽3)Q812yesdirect product factor
26Q8×SD16—91noSylow
27C2×D3×Q8—61no—
28C2×D3×Q8—123no—
29C2×GL(2,𝔽3)—33no—
30C3×C4⋊Q8—33no—
31C3×C4×Q8—31no—
32C3×C4×Q8C2×D313yes—
33C2×#215—31no—
34#897—63no—
35C4×D12—33no—
36#869—33no—
37C4×SL(2,𝔽3)C2×D319yes—
38#859—39no—
39#835—33no—
40#809—33no—
41#808—36no—
42#808—33no—
43C4⋊(C3⋊C8)—33no—
44C4×C3⋊C8—33no—
45C22×C3⋊D3—41no—
46C32×Q8—44no—
47C2×C32×C4—43no—
48C2×C3⋊Dic3—43no—
49C4×C3⋊D3—412no—
50#634—412no—
51C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
52Q82C3⋊D311yescharacteristic, complemented
53D4×Q8—91no—
54#498—93no—
55#476—93no—
56#475—93no—
57C8×Q8—91no—
58C4×SD16—93no—
59C2×C3×Q8—43no—
60C2×C3×Q8—61no—
61C2×C3×Q8S411yescharacteristic
62C2×C3×Q8C22×D311yescharacteristic
63D3×Q8—1224no—
64D3×Q8—124no—
65C2×D12—31no—
66C2×C4×D3—63no—
67C2×C4×D3—129no—
68C2×Dic6—63no—
69C2×Dic6—129no—
70C2×SL(2,𝔽3)C22×D313yes—
71GL(2,𝔽3)—36no—
72C3×C4⋊C4—33no—
73C3×C4⋊C4—33no—
74C3×C42—33no—
75#215—32no—
76#215—32no—
77(C2×D3)⋊C4—63no—
78C4⋊Dic3—33no—
79C4⋊Dic3—63no—
80C2×C3⋊C8—31no—
81C2×C3⋊C8—33no—
82C22×C32—41no—
83C2×C3⋊D3—42no—
84C2×C3⋊D3—42no—
85C2×C3⋊D3—42no—
86C32×C4—46no—
87C3⋊Dic3—46no—
88C22×Q8—181no—
89C2×SD16—91no—
90C4⋊Q8—33no—
91C4⋊Q8—93no—
92C22⋊Q8—183no—
93C4×Q8—31no—
94C4×Q8C2×C3⋊D313yes—
95C4×D4—93no—
96#106—93no—
97#106—96no—
98C4⋊C8—93no—
99#103—93no—
100#102—93no—
101C4×C8—93no—
102C22×D3—61no—
103C22×D3—123no—
104C3×Q8—31no—
105C3×Q8—412no—
106C3×Q8—241no—
107C3×Q8GL(2,𝔽3)14yescomplemented
108C3×Q8D3×Q811yescharacteristic, complemented
109C2×C3×C4—31no—
110C2×C3×C4—33no—
111C2×C3×C4—49no—
112C2×C3×C4C2×S413yes—
113C2×Dic3—63no—
114C2×Dic3—129no—
115D12—32no—
116D12—61no—
117C4×D3—1236no—
118C4×D3—126no—
119Dic6—1236no—
120Dic6—126no—
121SL(2,𝔽3)D3×Q813yescomplemented
122C3⋊C8—32no—
123C3⋊C8—63no—
124C2×C32—41no—
125C2×C32—41no—
126C2×C32—41no—
127C3⋊D3—44no—
128C2×Q8—61no—
129C2×Q8—183no—
130C2×Q8—364no—
131C2×Q8C3⋊S411yescharacteristic
132C2×Q8C22×C3⋊D311yescharacteristic
133C2×D4—91no—
134C22×C4—183no—
135SD16—92no—
136SD16—92no—
137C2×C8—91no—
138C2×C8—93no—
139C4⋊C4—33no—
140C4⋊C4—33no—
141C4⋊C4—93no—
142C4⋊C4—183no—
143C22⋊C4—183no—
144C42—33no—
145C22×C3—43no—
146C22×C3C22×S411yescharacteristic
147C2×D3—62no—
148C2×D3—126no—
149C2×D3—126no—
150C2×D3—126no—
151C2×D3—121no—
152C2×D3—121no—
153C3×C4—31no—
154C3×C4—31no—
155C3×C4—418no—
156C3×C4—63no—
157C3×C4C2×GL(2,𝔽3)16yes—
158Dic3—1218no—
159Dic3—123no—
160C32—41noSylow
161C23—181no—
162Q8—31no—
163Q8—241no—
164Q8—366no—
165Q8C3⋊GL(2,𝔽3)14yesdirect product factor
166Q8Q8×C3⋊D311yescharacteristic, complemented
167D4—92no—
168D4—181no—
169C2×C4—31no—
170C2×C4—33no—
171C2×C4—183no—
172C2×C4—366no—
173C2×C4C2×C3⋊S413yes—
174C8—92no—
175C8—183no—
176C2×C3—43no—
177C2×C3—43no—
178C2×C3—43no—
179C2×C3C22×GL(2,𝔽3)11yescharacteristic
180C2×C3Q8×S411yescharacteristic
181C2×C3#592411yescharacteristic
182D3—1212no—
183D3—122no—
184C22—182no—
185C22—361no—
186C22—361no—
187C22C22×C3⋊S411yescharacteristic
188C4—31no—
189C4—31no—
190C4—63no—
191C4—363no—
192C4C2×C3⋊GL(2,𝔽3)16yes—
193C3—43no—
194C3Q8×GL(2,𝔽3)11yescharacteristic, complemented
195C2—362no—
196C2C22×C3⋊GL(2,𝔽3)11yescharacteristic
197C2Q8×C3⋊S411yescharacteristic
198C2#1059658711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×Q82D3
socleC22×C3C22×S4
prime radicalC2×C3⋊GL(2,𝔽3)C22
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
1. C3×Q8×SL(2,𝔽3)C2semidirect
3. #10593620C2semidirect
10. C3×Q82D3semidirect
12. Q8×SL(2,𝔽3)D3semidirect
25. C3⋊GL(2,𝔽3)Q8direct (N × H)
51. C3×SL(2,𝔽3)C2×Q8semidirect
52. Q82C3⋊D3semidirect
107. C3×Q8GL(2,𝔽3)semidirect
108. C3×Q8D3×Q8semidirect
121. SL(2,𝔽3)D3×Q8semidirect
165. Q8C3⋊GL(2,𝔽3)direct (N × H)
166. Q8Q8×C3⋊D3semidirect
194. C3Q8×GL(2,𝔽3)semidirect

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⋊GL(2,𝔽3)2
#105936202
Q8×GL(2,𝔽3)3
Q8×#2153
C2×Q8×C3⋊D34