Finite Group Databasebeta

← #11921056

Subgroups — #11921056

Order 1152.

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

normal subgroups only

This group has 350 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, complemented
2C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic, complemented
4#73100—33nomaximal
5#71939—31nomaximal
6C2×C3⋊GL(2,𝔽3)—21no—
7C2×#3755—41nomaximal
8C3×C4×SL(2,𝔽3)—21no—
9C3×C4×SL(2,𝔽3)C2211yescharacteristic
10C3×Q82D311yescharacteristic, complemented
11Q8×SL(2,𝔽3)D313yescomplemented
12C4⋊GL(2,𝔽3)—33no—
13#5085—31no—
14#5027—31no—
15#5027—31no—
16SL(2,𝔽3)⋊C8—33no—
17#4546—31no—
18#4535—31no—
19#4535—31no—
20C2×C32×Q8—41no—
21C2×C4⋊(C3⋊D3)—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23C3⋊GL(2,𝔽3)—41no—
24#3755—44no—
25C2×C3⋊(C3⋊C8)—41no—
26#1623—91noSylow
27C2×GL(2,𝔽3)—63no—
28C3×C4⋊Q8—31no—
29C3×C4⋊Q8—61no—
30C3×C4×Q8—21no—
31C3×C4×Q8—31no—
32C3×C4×Q8C2×D311yescharacteristic
33C2×#215—62no—
34C2×#215—61no—
35C2×#215—61no—
36C2×#215—123no—
37#874—31no—
38C4×SL(2,𝔽3)—23no—
39C4×SL(2,𝔽3)C2×D313yes—
40#809—61no—
41#809—61no—
42C4⋊(C3⋊C8)—31no—
43C4⋊(C3⋊C8)—31no—
44C4×C3⋊C8—31no—
45C22×C3⋊D3—81no—
46C32×Q8—42no—
47C32×Q8—81no—
48C2×C32×C4—41no—
49C2×C32×C4—81no—
50C4⋊(C3⋊D3)—42no—
51C4⋊(C3⋊D3)—81no—
52C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
53C3⋊(C3⋊C8)—42no—
54Q82C3⋊D311yescharacteristic, complemented
55#492—91no—
56#460—91no—
57#460—91no—
58#333—91no—
59Q8⋊C8—91no—
60Q8⋊C8—91no—
61C2×C3×Q8—32no—
62C2×C3×Q8—43no—
63C2×C3×Q8#6711yescharacteristic
64C2×C3×Q8S411yescharacteristic
65C2×D12—61no—
66C2×D12—61no—
67C2×D12—61no—
68C2×D12—123no—
69C2×SL(2,𝔽3)#6713yes—
70GL(2,𝔽3)—123no—
71C3×C4⋊C4—31no—
72C3×C4⋊C4—31no—
73C3×C4⋊C4—61no—
74C3×C4⋊C4—61no—
75C3×C42—31no—
76C3×C42—61no—
77#215—124no—
78#215—121no—
79#215—1212no—
80#215—121no—
81#215—122no—
82C2×C3⋊C8—61no—
83C2×C3⋊C8—61no—
84C2×C3⋊C8—61no—
85C2×C3⋊C8—123no—
86C22×C32—41no—
87C2×C3⋊D3—82no—
88C2×C3⋊D3—161no—
89C2×C3⋊D3—161no—
90C32×C4—41no—
91C32×C4—41no—
92C32×C4—82no—
93C2×SD16—182no—
94C2×SD16—181no—
95C2×SD16—181no—
96C4⋊Q8—31no—
97C4⋊Q8—61no—
98#127—91no—
99C4×Q8—21no—
100C4×Q8—31no—
101C4×Q8C2×C3⋊D311yescharacteristic
102C4⋊C8—91no—
103C4⋊C8—91no—
104#102—181no—
105#102—181no—
106C4×C8—91no—
107C22×D3—121no—
108C22×D3—243no—
109C3×Q8—21no—
110C3×Q8—31no—
111C3×Q8—46no—
112C3×Q8—83no—
113C3×Q8—122no—
114C3×Q8#21511yescharacteristic, complemented
115C3×Q8GL(2,𝔽3)12yescomplemented
116C2×C3×C4—21no—
117C2×C3×C4—31no—
118C2×C3×C4—31no—
119C2×C3×C4—43no—
120C2×C3×C4—61no—
121C2×C3×C4—83no—
122C2×C3×C4C2×S411yescharacteristic
123D12—121no—
124D12—121no—
125D12—126no—
126D12—122no—
127D12—121no—
128D12—121no—
129D12—243no—
130SL(2,𝔽3)#21513yescomplemented
131C3⋊C8—62no—
132C3⋊C8—121no—
133C3⋊C8—126no—
134C3⋊C8—121no—
135C2×C32—41no—
136C2×C32—41no—
137C2×C32—41no—
138C3⋊D3—162no—
139C2×Q8—32no—
140C2×Q8#63811yescharacteristic
141C2×Q8C3⋊S411yescharacteristic
142C2×D4—181no—
143C2×D4—181no—
144C2×D4—181no—
145SD16—364no—
146SD16—361no—
147SD16—361no—
148SD16—362no—
149C2×C8—181no—
150C2×C8—181no—
151C2×C8—181no—
152C4⋊C4—31no—
153C4⋊C4—31no—
154C4⋊C4—61no—
155C4⋊C4—61no—
156C42—31no—
157C42—61no—
158C22×C3—43no—
159C22×C3GL(2,ℤ/4)11yescharacteristic
160C2×D3—246no—
161C2×D3—241no—
162C2×D3—241no—
163C2×D3—241no—
164C2×D3—483no—
165C2×D3—483no—
166C3×C4—22no—
167C3×C4—31no—
168C3×C4—31no—
169C3×C4—43no—
170C3×C4—43no—
171C3×C4—61no—
172C3×C4—86no—
173C3×C4—121no—
174C3×C4C2×GL(2,𝔽3)11yescharacteristic
175C3×C4#98311yescharacteristic
176C32—41noSylow
177C23—361no—
178Q8—21no—
179Q8—31no—
180Q8—122no—
181Q8#375511yescharacteristic, complemented
182Q8C3⋊GL(2,𝔽3)12yescomplemented
183D4—361no—
184D4—361no—
185D4—362no—
186D4—361no—
187D4—361no—
188C2×C4—21no—
189C2×C4—31no—
190C2×C4—31no—
191C2×C4—61no—
192C2×C4C2×C3⋊S411yescharacteristic
193C8—182no—
194C8—361no—
195C8—361no—
196C2×C3—43no—
197C2×C3—43no—
198C2×C3—43no—
199C2×C3#541711yescharacteristic
200C2×C3#542111yescharacteristic
201C2×C3#542911yescharacteristic
202D3—486no—
203D3—481no—
204C22—721no—
205C22—721no—
206C22—721no—
207C22#6426911yescharacteristic
208C4—22no—
209C4—31no—
210C4—31no—
211C4—61no—
212C4—121no—
213C4C2×C3⋊GL(2,𝔽3)11yescharacteristic
214C4#6426411yescharacteristic
215C3—43no—
216C3#7310011yescharacteristic, complemented
217C2—1441no—
218C2#1059365111yescharacteristic
219C2#1059365511yescharacteristic
220C2#1059366311yescharacteristic
subgroupG/H
centerC22#64269
commutator, cosocleC3×C4×SL(2,𝔽3)C22
FrattiniC2×C4C2×C3⋊S4
FittingC3×Q82D3
socleC22×C3GL(2,ℤ/4)
prime radicalC4⋊(C3⋊GL(2,𝔽3))C2
hypercenterC2×Q8C3⋊S4
hypocenterC3×SL(2,𝔽3)SD16
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
1. C3×Q8×SL(2,𝔽3)C2semidirect
3. C3⋊(SL(2,𝔽3)⋊C8)C2semidirect
10. C3×Q82D3semidirect
11. Q8×SL(2,𝔽3)D3semidirect
52. C3×SL(2,𝔽3)SD16semidirect
54. Q82C3⋊D3semidirect
114. C3×Q8#215semidirect
115. C3×Q8GL(2,𝔽3)semidirect
130. SL(2,𝔽3)#215semidirect
181. Q8#3755semidirect
182. Q8C3⋊GL(2,𝔽3)semidirect
216. C3#73100semidirect

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
C3⋊(SL(2,𝔽3)⋊C8)2
#731003
#719393
C2×#37554