Finite Group Databasebeta

← #11920552

Subgroups — #11920552

Order 1152.

289 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 41 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
1C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
4C8×GL(2,𝔽3)—31nomaximal
5#68212—31nomaximal
6C2×C3⋊GL(2,𝔽3)C411yescharacteristic
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C2×#3710—41nomaximal
9C3⋊#859C411yescharacteristic
10C4×GL(2,𝔽3)—31no—
11C4×GL(2,𝔽3)—61no—
12C4×#215—31no—
13Q8×C3⋊C8D311yescharacteristic, complemented
14#4834—31no—
15C8×SL(2,𝔽3)—31no—
16SL(2,𝔽3)⋊C8D311yescharacteristic, complemented
17C3×Q8⋊C8—31no—
18#4480—31no—
19#4459—31no—
20C8×C3⋊C8—31no—
21C2×C4×C3⋊D3—41no—
22C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
23C3⋊GL(2,𝔽3)C812yescomplemented
24C2×C3×C3⋊C8—41no—
25C2×C3×C3⋊C8—41no—
26#3710—44no—
27C8×SD16—91noSylow
28C2×GL(2,𝔽3)—31no—
29C2×GL(2,𝔽3)—61no—
30C3×C4×Q8C2×D311yescharacteristic
31C2×#215—31no—
32C2×D3×C8—61no—
33C2×D3×C8—121no—
34C4×D12—31no—
35C4×SL(2,𝔽3)—21no—
36C4×SL(2,𝔽3)C2×D311yescharacteristic
37#859—31no—
38#859—61no—
39C3×C4⋊C8—31no—
40C3×C4×C8—31no—
41#835—31no—
42(C2×D3)⋊C8—61no—
43#809—31no—
44#808—31no—
45C4⋊(C3⋊C8)—31no—
46C4×C3⋊C8—31no—
47C4×C3⋊C8—31no—
48C22×C3⋊D3—41no—
49C2×C32×C4—41no—
50C2×C3⋊Dic3—41no—
51C4×C3⋊D3—42no—
52C4×C3⋊D3—42no—
53C3×SL(2,𝔽3)C2×C811yescharacteristic, complemented
54C3×C3⋊C8—42no—
55C3×C3⋊C8—42no—
56C8×Q8—31no—
57C4×SD16—91no—
58C8×D4—91no—
59#334—91no—
60Q8⋊C8—31no—
61D4⋊C8—91no—
62C82—91no—
63C2×C3×Q8C4×D311yescharacteristic
64C2×D12—31no—
65C2×C4×D3—61no—
66C2×C4×D3—121no—
67C2×C4×D3—241no—
68C2×SL(2,𝔽3)—21no—
69C2×SL(2,𝔽3)C4×D311yescharacteristic
70GL(2,𝔽3)—32no—
71GL(2,𝔽3)—62no—
72C2×C3×C8—61no—
73C2×C3×C8—61no—
74C2×C3×C8—121no—
75C3×C4⋊C4—31no—
76C3×C42—31no—
77#215—32no—
78#215—32no—
79(C2×D3)⋊C4—61no—
80C4⋊Dic3—31no—
81C2×C3⋊C8—31no—
82C2×C3⋊C8—31no—
83C2×C3⋊C8—31no—
84C2×C3⋊C8—41no—
85C2×C3⋊C8S411yescharacteristic
86D3×C8—124no—
87D3×C8—122no—
88C22×C32—41no—
89C2×C3⋊D3—42no—
90C2×C3⋊D3—42no—
91C2×C3⋊D3—42no—
92C32×C4—41no—
93C32×C4—41no—
94C3⋊Dic3—42no—
95C2×SD16—91no—
96C22×C8—181no—
97C4×Q8D3211yescharacteristic
98C4×D4—91no—
99#106—91no—
100C4⋊C8—31no—
101C4⋊C8—91no—
102#103—91no—
103#102—91no—
104C22⋊C8—181no—
105C4×C8—31no—
106C4×C8—91no—
107C4×C8—91no—
108C22×D3—61no—
109C22×D3—121no—
110C22×D3—241no—
111C3×Q8—31no—
112C3×Q8D3×C811yescharacteristic, complemented
113C2×C3×C4—31no—
114C2×C3×C4—31no—
115C2×C3×C4—41no—
116C2×C3×C4—81no—
117C2×C3×C4C2×S411yescharacteristic
118C2×Dic3—61no—
119C2×Dic3—121no—
120C2×Dic3—241no—
121D12—32no—
122D12—61no—
123C4×D3—122no—
124C4×D3—122no—
125C4×D3—121no—
126C4×D3—121no—
127C4×D3—242no—
128C4×D3—242no—
129SL(2,𝔽3)—21no—
130SL(2,𝔽3)D3×C811yescharacteristic, complemented
131C3×C8—62no—
132C3×C8—122no—
133C3×C8—121no—
134C3⋊C8—32no—
135C3⋊C8—42no—
136C3⋊C8—61no—
137C3⋊C8—61no—
138C3⋊C8GL(2,𝔽3)12yescomplemented
139C2×C32—41no—
140C2×C32—41no—
141C2×C32—41no—
142C3⋊D3—44no—
143C2×Q8#62411yescharacteristic
144C2×D4—91no—
145C22×C4—181no—
146SD16—92no—
147SD16—92no—
148C2×C8—31no—
149C2×C8—61no—
150C2×C8—61no—
151C2×C8—91no—
152C2×C8—91no—
153C2×C8—91no—
154C2×C8—362no—
155C4⋊C4—31no—
156C4⋊C4—91no—
157C22⋊C4—181no—
158C42—31no—
159C22×C3—41no—
160C22×C3—81no—
161C22×C3C4×S411yescharacteristic
162C2×D3—62no—
163C2×D3—122no—
164C2×D3—122no—
165C2×D3—122no—
166C2×D3—121no—
167C2×D3—121no—
168C2×D3—242no—
169C2×D3—242no—
170C2×D3—242no—
171C3×C4—31no—
172C3×C4—31no—
173C3×C4—41no—
174C3×C4—41no—
175C3×C4—61no—
176C3×C4—81no—
177C3×C4—81no—
178C3×C4C2×GL(2,𝔽3)11yescharacteristic
179C3×C4#98511yescharacteristic
180Dic3—122no—
181Dic3—121no—
182Dic3—242no—
183C32—41noSylow
184C23—181no—
185Q8—31no—
186Q8#371011yescharacteristic, complemented
187D4—92no—
188D4—181no—
189C2×C4—31no—
190C2×C4—31no—
191C2×C4—181no—
192C2×C4—361no—
193C2×C4—361no—
194C2×C4D3×S411yescharacteristic
195C8—32no—
196C8—62no—
197C8—92no—
198C8—121no—
199C8—181no—
200C8—181no—
201C2×C3—41no—
202C2×C3—41no—
203C2×C3—41no—
204C2×C3—81no—
205C2×C3—81no—
206C2×C3—81no—
207C2×C3C4×GL(2,𝔽3)11yescharacteristic
208C2×C3C8×S411yescharacteristic
209C2×C3#540411yescharacteristic
210D3—124no—
211D3—122no—
212D3—244no—
213C22—182no—
214C22—361no—
215C22—361no—
216C22(C3⋊S4)⋊C411yescharacteristic
217C4—31no—
218C4—31no—
219C4—61no—
220C4—361no—
221C4#6419811yescharacteristic
222C4D3×GL(2,𝔽3)11yescharacteristic
223C3—41no—
224C3—81no—
225C3C8×GL(2,𝔽3)11yescharacteristic, complemented
226C2—362no—
227C2(C3⋊S4)⋊C811yescharacteristic
228C2#1059321411yescharacteristic
229C2#1059322011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C8
FittingC3×C4×Q8C2×D3
socleC22×C3C4×S4
prime radicalC2×C3⋊GL(2,𝔽3)C4
cosocleC3×C4×SL(2,𝔽3)C22
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3⋊C8×SL(2,𝔽3)C2semidirect
3. C3×SL(2,𝔽3)⋊C8C2semidirect
13. Q8×C3⋊C8D3semidirect
16. SL(2,𝔽3)⋊C8D3semidirect
23. C3⋊GL(2,𝔽3)C8semidirect
53. C3×SL(2,𝔽3)C2×C8semidirect
112. C3×Q8D3×C8semidirect
130. SL(2,𝔽3)D3×C8semidirect
138. C3⋊C8GL(2,𝔽3)semidirect
186. Q8#3710semidirect
225. C3C8×GL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C4×C3⋊GL(2,𝔽3)2
C3⋊C8×SL(2,𝔽3)2
C3×SL(2,𝔽3)⋊C82
C8×GL(2,𝔽3)3
#682123
C2×#37104