Finite Group Databasebeta

← #72222714

Subgroups — #72222714

Order 2304.

302 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 29 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
1SL(2,𝔽3)⋊(C42⋊C3)C211yesmaximal, characteristic, complemented
2#11709914C311yesmaximal, characteristic, complemented
3#11709913C312yesmaximal, complemented
4#11709912C311yesmaximal, characteristic, complemented
5C2×A4×SL(2,𝔽3)—41nomaximal
6C3×#396⋊C3—41nomaximal
7#73286C2×C312yescomplemented
8#2645⋊C3—31no—
9#2401⋊C3—31no—
10C4⋊C4×SL(2,𝔽3)—31no—
11C22⋊C4×SL(2,𝔽3)—31no—
12#73167C2×C311yescharacteristic, complemented
13#73166C2×C311yescharacteristic, complemented
14A4×SL(2,𝔽3)—41no—
15C3×#95⋊C3—41no—
16#35920C3211yescharacteristic, complemented, Sylow
17C2×C22⋊SL(2,𝔽3)—42no—
18C2×Q8×A4—41no—
19C23×SL(2,𝔽3)A411yescharacteristic
20C2×#126⋊C3—122no—
21C2×C4×SL(2,𝔽3)—31no—
22C2×C4×SL(2,𝔽3)—31no—
23C3×#396—41no—
24#401⋊C3—31no—
25#396⋊C3—42no—
26#396⋊C3A411yescharacteristic
27#376⋊C3—31no—
28C22×C3×A4—161no—
29C2×C3×SL(2,𝔽3)—161no—
30#2645—31no—
31#2511—91no—
32#2496—91no—
33#2401—31no—
34Q8×C4⋊C4—31no—
35Q8×C22⋊C4—31no—
36#2276C2×C3211yescharacteristic, complemented
37C22⋊SL(2,𝔽3)—42no—
38Q8×A4—41no—
39Q8×A4—121no—
40C22×SL(2,𝔽3)—61no—
41C22×SL(2,𝔽3)C2×A411yescharacteristic
42C2×C4×A4—121no—
43C2×C3×C4⋊C4—121no—
44C2×C3×C22⋊C4—121no—
45#126⋊C3—124no—
46C4×SL(2,𝔽3)—61no—
47C4×SL(2,𝔽3)—61no—
48C2×C42⋊C3—122no—
49C3×#95—41no—
50#95⋊C3—42no—
51#95⋊C3C2×A411yescharacteristic
52C2×C3×A4—162no—
53C2×C3×A4—161no—
54C3×SL(2,𝔽3)—161no—
55C23×Q8C3×A411yescharacteristic
56C2×#126—32no—
57C2×C4×Q8—31no—
58C2×C4×Q8—31no—
59#401—31no—
60#400—91no—
61#400—92no—
62#397—91no—
63#396C3×A411yescharacteristic
64#391—91no—
65#391—91no—
66#389—91no—
67#389—91no—
68#387—92no—
69#384—91no—
70#380—91no—
71C4×C4⋊C4—91no—
72C4×C4⋊C4—91no—
73C4×C22⋊C4—91no—
74#376—31no—
75C24×C3—41no—
76C22×A4—41no—
77C22×A4—162no—
78C2×C3×Q8—161no—
79C22×C3×C4—121no—
80C22×C3×C4—121no—
81C2×SL(2,𝔽3)—31no—
82C2×SL(2,𝔽3)—41no—
83C2×SL(2,𝔽3)—162no—
84C2×SL(2,𝔽3)—482no—
85C4×A4—121no—
86C4×A4—121no—
87C3×C4⋊C4—242no—
88C3×C22⋊C4—242no—
89C42⋊C3—122no—
90C22×C32—641no—
91C3×A4—161no—
92C22×Q8—31no—
93C22×Q8—61no—
94C22×Q8—181no—
95C22×Q8C2×C3×A411yescharacteristic
96C23×C4—31no—
97#126—124no—
98C4×Q8—61no—
99C4×Q8—61no—
100C4×Q8—181no—
101C4×Q8—181no—
102C2×C4⋊C4—31no—
103C2×C4⋊C4—91no—
104C2×C4⋊C4—91no—
105C2×C4⋊C4—91no—
106C2×C4⋊C4—92no—
107C2×C22⋊C4—31no—
108C2×C22⋊C4—91no—
109C2×C42—32no—
110C2×C42—91no—
111C2×C42—91no—
112#95—91no—
113#95—91no—
114#95—91no—
115#95—91no—
116#95—91no—
117#95—92no—
118#95C2×C3×A411yescharacteristic
119C23×C3—41no—
120C23×C3—42no—
121C23×C3—241no—
122C23×C3—241no—
123C2×A4—41no—
124C2×A4—42no—
125C2×A4—162no—
126C2×A4—164no—
127C3×Q8—161no—
128C3×Q8—481no—
129C2×C3×C4—122no—
130C2×C3×C4—241no—
131C2×C3×C4—241no—
132C2×C3×C4—241no—
133C2×C3×C4—241no—
134C2×C3×C4—241no—
135C2×C3×C4—481no—
136SL(2,𝔽3)—162no—
137SL(2,𝔽3)—482no—
138SL(2,𝔽3)#126⋊C311yescharacteristic, complemented
139C2×C32—641no—
140C2×C32—642no—
141C24A4211yescharacteristic
142C2×Q8—31no—
143C2×Q8—41no—
144C2×Q8—91no—
145C2×Q8—122no—
146C2×Q8—181no—
147C2×Q8—181no—
148C2×Q8—361no—
149C22×C4—31no—
150C22×C4—31no—
151C22×C4—31no—
152C22×C4—31no—
153C22×C4—91no—
154C22×C4—91no—
155C22×C4—181no—
156C22×C4—181no—
157C4⋊C4—62no—
158C4⋊C4—182no—
159C4⋊C4—181no—
160C4⋊C4—181no—
161C4⋊C4—181no—
162C4⋊C4—181no—
163C4⋊C4—362no—
164C22⋊C4—62no—
165C22⋊C4—182no—
166C42—122no—
167C42—181no—
168C42—181no—
169C42—181no—
170C42—181no—
171C22×C3—41no—
172C22×C3—121no—
173C22×C3—121no—
174C22×C3—161no—
175C22×C3—161no—
176C22×C3—242no—
177C22×C3—481no—
178C22×C3—642no—
179A4—41no—
180A4—162no—
181C3×C4—242no—
182C3×C4—481no—
183C3×C4—481no—
184C3×C4—481no—
185C32—641noSylow
186C23—61no—
187C23—61no—
188C23Ω+(4,𝔽3)12yes—
189C23C2×A4211yescharacteristic
190Q8—32no—
191Q8—91no—
192Q8—121no—
193Q8—361no—
194Q8C3×#126⋊C311yescharacteristic
195C2×C4—32no—
196C2×C4—61no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—91no—
202C2×C4—91no—
203C2×C4—121no—
204C2×C4—181no—
205C2×C4—181no—
206C2×C4—181no—
207C2×C4—181no—
208C2×C4—181no—
209C2×C4—181no—
210C2×C4—181no—
211C2×C4—361no—
212C2×C3—41no—
213C2×C3—121no—
214C2×C3—121no—
215C2×C3—162no—
216C2×C3—162no—
217C2×C3—161no—
218C2×C3—644no—
219C2×C3—642no—
220C22—31no—
221C22—31no—
222C22—41no—
223C22—62no—
224C22—121no—
225C22C2×Ω+(4,𝔽3)11yescharacteristic
226C4—31no—
227C4—62no—
228C4—91no—
229C4—121no—
230C4—121no—
231C4—361no—
232C4—361no—
233C3—41no—
234C3—161no—
235C3—642no—
236C2—31no—
237C2—31no—
238C2—42no—
239C2A4×#126⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2A4×#126⋊C3
commutator, cosocle, hypocenter#2276C2×C32
Frattini, socleC23C2×A42
Fitting#35920C32
LCSG ▷ #2276
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. SL(2,𝔽3)⋊(C42⋊C3)C2semidirect
2. #11709914C3semidirect
3. #11709913C3semidirect
4. #11709912C3semidirect
7. #73286C2×C3semidirect
12. #73167C2×C3semidirect
13. #73166C2×C3semidirect
16. #35920C32semidirect
36. #2276C2×C32semidirect
138. SL(2,𝔽3)#126⋊C3semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)⋊(C42⋊C3)2
#117099143
#117099133
#117099123
C2×A4×SL(2,𝔽3)4
C3×#396⋊C34