Finite Group Databasebeta

← #11921055

Subgroups — #11921055

Order 1152.

302 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 49 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#10593620C211yesmaximal, characteristic
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic
4#73099—33nomaximal
5#71941—31nomaximal
6C2×#3756—41nomaximal
7C3×C4×SL(2,𝔽3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C3⋊#859—21no—
10C3×Q82D311yescharacteristic
11Q8×SL(2,𝔽3)D313yes—
12#5386—33no—
13#5079—31no—
14#5031—31no—
15#5020—31no—
16SL(2,𝔽3)⋊C8—33no—
17#4546—31no—
18#4535—31no—
19#4535—31no—
20C2×C32×Q8—41no—
21C2×#634—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23#3756—44no—
24C2×C3⋊(C3⋊C8)—41no—
25#1645—91noSylow
26C3×C4⋊Q8—31no—
27C3×C4⋊Q8—61no—
28C3×C4×Q8—21no—
29C3×C4×Q8—31no—
30C3×C4×Q8C2×D311yescharacteristic
31C2×#216—61no—
32C2×#216—123no—
33#869—31no—
34C4×SL(2,𝔽3)—23no—
35C4×SL(2,𝔽3)C2×D313yes—
36#859—63no—
37#835—62no—
38#835—61no—
39#810—61no—
40#808—61no—
41C4⋊(C3⋊C8)—31no—
42C4⋊(C3⋊C8)—31no—
43C4×C3⋊C8—31no—
44C32×Q8—42no—
45C32×Q8—81no—
46C2×C32×C4—41no—
47C2×C32×C4—81no—
48C2×C3⋊Dic3—81no—
49#634—42no—
50#634—81no—
51C3×SL(2,𝔽3)Q1611yescharacteristic, complemented
52C3⋊(C3⋊C8)—42no—
53Q82C3⋊D311yescharacteristic
54#487—91no—
55#475—91no—
56#462—91no—
57#333—91no—
58Q8⋊C8—91no—
59Q8⋊C8—91no—
60C2×C3×Q8—32no—
61C2×C3×Q8—43no—
62C2×C3×Q8#6711yescharacteristic
63C2×C3×Q8S411yescharacteristic
64C2×Dic6—61no—
65C2×Dic6—123no—
66C2×SL(2,𝔽3)#6713yes—
67C3×C4⋊C4—31no—
68C3×C4⋊C4—31no—
69C3×C4⋊C4—61no—
70C3×C4⋊C4—61no—
71C3×C42—31no—
72C3×C42—61no—
73#216—1212no—
74#216—122no—
75C4⋊Dic3—61no—
76C4⋊Dic3—61no—
77C2×C3⋊C8—61no—
78C2×C3⋊C8—61no—
79C2×C3⋊C8—61no—
80C2×C3⋊C8—123no—
81C22×C32—41no—
82C32×C4—41no—
83C32×C4—41no—
84C32×C4—82no—
85C3⋊Dic3—82no—
86C2×Q16—181no—
87C4⋊Q8—31no—
88C4⋊Q8—61no—
89C4⋊Q8—91no—
90C4×Q8—21no—
91C4×Q8—31no—
92C4×Q8C2×C3⋊D311yescharacteristic
93#106—181no—
94C4⋊C8—91no—
95C4⋊C8—91no—
96#103—181no—
97#103—182no—
98#103—181no—
99C4×C8—91no—
100C3×Q8—21no—
101C3×Q8—31no—
102C3×Q8—46no—
103C3×Q8—83no—
104C3×Q8—122no—
105C3×Q8#21611yescharacteristic, complemented
106C3×Q8GL(2,𝔽3)12yes—
107C2×C3×C4—21no—
108C2×C3×C4—31no—
109C2×C3×C4—31no—
110C2×C3×C4—43no—
111C2×C3×C4—61no—
112C2×C3×C4—83no—
113C2×C3×C4C2×S411yescharacteristic
114C2×Dic3—121no—
115C2×Dic3—243no—
116Dic6—126no—
117Dic6—121no—
118Dic6—121no—
119Dic6—243no—
120SL(2,𝔽3)#21613yescomplemented
121C3⋊C8—121no—
122C3⋊C8—126no—
123C3⋊C8—121no—
124C3⋊C8—121no—
125C2×C32—41no—
126C2×C32—41no—
127C2×C32—41no—
128C2×Q8—32no—
129C2×Q8—181no—
130C2×Q8#63811yescharacteristic
131C2×Q8C3⋊S411yescharacteristic
132Q16—362no—
133C2×C8—181no—
134C2×C8—181no—
135C2×C8—181no—
136C4⋊C4—31no—
137C4⋊C4—31no—
138C4⋊C4—61no—
139C4⋊C4—61no—
140C4⋊C4—181no—
141C4⋊C4—181no—
142C42—31no—
143C42—61no—
144C22×C3—43no—
145C22×C3GL(2,ℤ/4)11yescharacteristic
146C3×C4—22no—
147C3×C4—31no—
148C3×C4—31no—
149C3×C4—43no—
150C3×C4—43no—
151C3×C4—61no—
152C3×C4—86no—
153C3×C4—121no—
154C3×C4C2×GL(2,𝔽3)11yescharacteristic
155C3×C4#98311yescharacteristic
156Dic3—246no—
157Dic3—241no—
158C32—41noSylow
159Q8—21no—
160Q8—31no—
161Q8—122no—
162Q8—361no—
163Q8—361no—
164Q8#375611yescharacteristic, complemented
165Q8C3⋊GL(2,𝔽3)12yes—
166C2×C4—21no—
167C2×C4—31no—
168C2×C4—31no—
169C2×C4—61no—
170C2×C4—361no—
171C2×C4C2×C3⋊S411yescharacteristic
172C8—361no—
173C8—361no—
174C8—361no—
175C2×C3—43no—
176C2×C3—43no—
177C2×C3—43no—
178C2×C3#541611yescharacteristic
179C2×C3#542111yescharacteristic
180C2×C3#542811yescharacteristic
181C22#6426911yescharacteristic
182C4—22no—
183C4—31no—
184C4—31no—
185C4—61no—
186C4—121no—
187C4—721no—
188C4C2×C3⋊GL(2,𝔽3)11yescharacteristic
189C4#6426411yescharacteristic
190C3—43no—
191C3#7309911yescharacteristic, complemented
192C2#1059365011yescharacteristic
193C2#1059365511yescharacteristic
194C2#1059366211yescharacteristic
subgroupG/H
centerC22#64269
commutator, cosocleC3×C4×SL(2,𝔽3)C22
FrattiniC2×C4C2×C3⋊S4
FittingC3×Q82D3
socleC22×C3GL(2,ℤ/4)
prime radicalC2×C3×SL(2,𝔽3)D4
hypercenterC2×Q8C3⋊S4
hypocenterC3×SL(2,𝔽3)Q16
LCSG ▷ C3×C4×SL(2,𝔽3) ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
51. C3×SL(2,𝔽3)Q16semidirect
105. C3×Q8#216semidirect
120. SL(2,𝔽3)#216semidirect
164. Q8#3756semidirect
191. C3#73099semidirect

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
#105936202
C3⋊(SL(2,𝔽3)⋊C8)2
#730993
#719413
C2×#37564