Finite Group Databasebeta

← C3×#3503⋊C3 #11919754

Subgroups — C3×#3503⋊C3 #11919754

Order 1152.

636 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 166 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)C212yesmaximal, complemented
2C3×#557⋊C3C212yesmaximal, complemented
3C3×#520⋊C3C212yesmaximal
4C3×C4×#41⋊C3C211yesmaximal, characteristic
5C3×#3503C311yesmaximal, characteristic, complemented
6#3503⋊C3C313yesmaximal, direct product factor
7C2×C3×#41⋊C3C2211yescharacteristic
8C32×C4⋊Q8—41nomaximal
9C3×C4×SL(2,𝔽3)C2211yescharacteristic
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11C3×C4×SL(2,𝔽3)C2214yes—
12C3×Q82C2×C312yes—
13C3×#557C2×C312yes—
14C3×#554—32no—
15C3×D4×Q8—32no—
16C3×#544—32no—
17C3×#533—31no—
18C2×C3×C4⋊Q8—31no—
19C3×#520C2×C312yes—
20C3×C4×#41C2×C311yescharacteristic
21Q8×SL(2,𝔽3)C2×C316yescomplemented
22#557⋊C3C2×C316yescomplemented
23#520⋊C3C2×C316yes—
24C4×#41⋊C3C2×C313yes—
25C2×C32×Q8—42no—
26C3×#41⋊C3Q812yescomplemented
27C2×C3×SL(2,𝔽3)C2311yescharacteristic
28C32×C4⋊C4—42no—
29C32×C4⋊C4—42no—
30C32×C42—41no—
31#3503C3211yescharacteristic, complemented, Sylow
32C2×C3×#41C22×C311yescharacteristic
33C22×C3×Q8—32no—
34C2×#41⋊C3C22×C313yes—
35C3×C4⋊Q8—32no—
36C3×C4⋊Q8—32no—
37C3×C4⋊Q8—32no—
38C3×C4⋊Q8—34no—
39C3×C4⋊Q8—31no—
40C3×C4⋊Q8—43no—
41C3×C4⋊Q8A411yescharacteristic
42C3×#125—34no—
43C3×#125—32no—
44C3×C22⋊Q8—38no—
45C3×C22⋊Q8—32no—
46C3×C22⋊Q8—32no—
47C3×C4×Q8—32no—
48C3×C4×Q8—32no—
49C3×C4×Q8C22×C311yescharacteristic
50C3×C4×Q8C22×C314yes—
51C3×C4×Q8C22×C311yescharacteristic
52C3×C4×D4—31no—
53C3×C4×D4—31no—
54C3×C4×D4—34no—
55C3×#117—32no—
56C3×#117—31no—
57C2×C3×C4⋊C4—32no—
58C2×C3×C4⋊C4—32no—
59C2×C3×C42—31no—
60C4×SL(2,𝔽3)C22×C313yes—
61C4×SL(2,𝔽3)C22×C313yes—
62C4×SL(2,𝔽3)C22×C3112yes—
63C32×Q8—84no—
64C2×C32×C4—41no—
65C2×C32×C4—41no—
66C2×C32×C4—41no—
67C2×C32×C4—44no—
68C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
69Q82C2×C3212yes—
70#557C2×C3212yes—
71#554—32no—
72D4×Q8—32no—
73#544—32no—
74#533—31no—
75C2×C4⋊Q8—31no—
76#520C2×C3212yes—
77C4×#41C2×C3211yescharacteristic
78C3×#41—32no—
79C3×#41C3×Q812yescomplemented
80C2×C3×Q8—34no—
81C2×C3×Q8—32no—
82C2×C3×Q8—46no—
83C2×C3×Q8—124no—
84C2×C3×Q8C2×A412yes—
85C2×C3×Q8C23×C311yescharacteristic
86C2×C3×D4—31no—
87C22×C3×C4—31no—
88C22×C3×C4—34no—
89C22×C3×C4—31no—
90C22×C3×C4—31no—
91#41⋊C3C3×Q816yescomplemented
92C2×SL(2,𝔽3)C23×C313yes—
93C3×C4⋊C4—34no—
94C3×C4⋊C4—31no—
95C3×C4⋊C4—31no—
96C3×C4⋊C4—34no—
97C3×C4⋊C4—32no—
98C3×C4⋊C4—32no—
99C3×C4⋊C4—34no—
100C3×C4⋊C4—34no—
101C3×C4⋊C4—46no—
102C3×C4⋊C4—46no—
103C3×C4⋊C4C2×A412yes—
104C3×C4⋊C4C2×A412yes—
105C3×C22⋊C4—32no—
106C3×C22⋊C4—34no—
107C3×C42—31no—
108C3×C42—31no—
109C3×C42—31no—
110C3×C42—34no—
111C3×C42—43no—
112C3×C42C2×A411yescharacteristic
113C22×C32—41no—
114C32×C4—42no—
115C32×C4—42no—
116C32×C4—42no—
117C32×C4—84no—
118C2×#41C22×C3211yescharacteristic
119C22×Q8—32no—
120C4⋊Q8—32no—
121C4⋊Q8—32no—
122C4⋊Q8—32no—
123C4⋊Q8—34no—
124C4⋊Q8—31no—
125C4⋊Q8C3×A411yescharacteristic
126#125—34no—
127#125—32no—
128C22⋊Q8—38no—
129C22⋊Q8—32no—
130C22⋊Q8—32no—
131C4×Q8—32no—
132C4×Q8—32no—
133C4×Q8C22×C3211yescharacteristic
134C4×Q8C22×C3211yescharacteristic
135C4×Q8C22×C3214yes—
136C4×D4—31no—
137C4×D4—31no—
138C4×D4—34no—
139#117—32no—
140#117—31no—
141C2×C4⋊C4—32no—
142C2×C4⋊C4—32no—
143C2×C42—31no—
144C23×C3—31no—
145C3×Q8—24no—
146C3×Q8—31no—
147C3×Q8—64no—
148C3×Q8—812no—
149C3×Q8—124no—
150C3×Q8C2×C3×Q811yescharacteristic
151C3×D4—32no—
152C3×D4—32no—
153C2×C3×C4—31no—
154C2×C3×C4—32no—
155C2×C3×C4—32no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—34no—
159C2×C3×C4—43no—
160C2×C3×C4—43no—
161C2×C3×C4—43no—
162C2×C3×C4—412no—
163C2×C3×C4—62no—
164C2×C3×C4—62no—
165C2×C3×C4—124no—
166C2×C3×C4C22×A414yes—
167C2×C3×C4C22×A411yescharacteristic
168C2×C3×C4C22×A411yescharacteristic
169C2×C3×C4C22×A411yescharacteristic
170SL(2,𝔽3)C2×C3×Q813yescomplemented
171C2×C32—41no—
172C2×C32—41no—
173C2×C32—41no—
174#41—32no—
175#41C32×Q812yescomplemented
176C2×Q8—34no—
177C2×Q8—32no—
178C2×Q8—124no—
179C2×Q8C2×C3×A412yes—
180C2×Q8C23×C3211yescharacteristic
181C2×D4—31no—
182C22×C4—31no—
183C22×C4—34no—
184C22×C4—31no—
185C22×C4—31no—
186C4⋊C4—34no—
187C4⋊C4—31no—
188C4⋊C4—31no—
189C4⋊C4—34no—
190C4⋊C4—32no—
191C4⋊C4—32no—
192C4⋊C4—34no—
193C4⋊C4—34no—
194C4⋊C4C2×C3×A412yes—
195C4⋊C4C2×C3×A412yes—
196C22⋊C4—32no—
197C22⋊C4—34no—
198C42—31no—
199C42—31no—
200C42—31no—
201C42—34no—
202C42C2×C3×A411yescharacteristic
203C22×C3—32no—
204C22×C3—43no—
205C22×C3—61no—
206C22×C3—61no—
207C22×C3C23×A411yescharacteristic
208C3×C4—24no—
209C3×C4—31no—
210C3×C4—31no—
211C3×C4—46no—
212C3×C4—46no—
213C3×C4—46no—
214C3×C4—61no—
215C3×C4—64no—
216C3×C4—61no—
217C3×C4—812no—
218C3×C4Q8×A412yes—
219C3×C4#99412yes—
220C3×C42−1+4⋊C312yes—
221C32—41noSylow
222C23—31no—
223Q8—24no—
224Q8—31no—
225Q8—64no—
226Q8—124no—
227Q8C2×C32×Q811yescharacteristic
228D4—32no—
229D4—32no—
230C2×C4—31no—
231C2×C4—32no—
232C2×C4—32no—
233C2×C4—31no—
234C2×C4—31no—
235C2×C4—34no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—124no—
239C2×C4C22×C3×A414yes—
240C2×C4C22×C3×A411yescharacteristic
241C2×C4C22×C3×A411yescharacteristic
242C2×C4C22×C3×A411yescharacteristic
243C2×C3—43no—
244C2×C3—43no—
245C2×C3—43no—
246C2×C3—62no—
247C2×C3C2×Q8×A411yescharacteristic
248C2×C3C2×#99411yescharacteristic
249C2×C3C2×2−1+4⋊C311yescharacteristic
250C22—32no—
251C22—61no—
252C22—61no—
253C22C23×C3×A411yescharacteristic
254C4—24no—
255C4—31no—
256C4—31no—
257C4—61no—
258C4—64no—
259C4—61no—
260C4C3×Q8×A412yes—
261C4C3×#99412yes—
262C4C3×2−1+4⋊C312yes—
263C3—43no—
264C3#3503⋊C311yescharacteristic, direct product factor
265C2—62no—
266C2C2×C3×Q8×A411yescharacteristic
267C2C2×C3×#99411yescharacteristic
268C2C2×C3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, socleC22×C3C23×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC3×#3503C3
prime radicalC2×C3×#41⋊C3C22
hypercenterC3×C4⋊Q8A4
hypocenterQ8C2×C32×Q8
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×SL(2,𝔽3)C2semidirect
2. C3×#557⋊C3C2semidirect
5. C3×#3503C3semidirect
6. #3503⋊C3C3direct (N × H)
21. Q8×SL(2,𝔽3)C2×C3semidirect
22. #557⋊C3C2×C3semidirect
26. C3×#41⋊C3Q8semidirect
31. #3503C32semidirect
68. C3×SL(2,𝔽3)C2×Q8semidirect
79. C3×#41C3×Q8semidirect
91. #41⋊C3C3×Q8semidirect
170. SL(2,𝔽3)C2×C3×Q8semidirect
175. #41C32×Q8semidirect
264. C3#3503⋊C3direct (N × H)

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
C3×#557⋊C32
C3×#520⋊C32
C3×C4×#41⋊C32
C3×#35033
#3503⋊C33
C32×C4⋊Q84