Finite Group Databasebeta

← C22×#64266 #11923727

Subgroups — C22×#64266 #11923727

Order 1152.

1976 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1976 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×#64266C2112yesmaximal, direct product factor
3C22×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C22×C3⋊#226C211yesmaximal, characteristic, complemented
5C22×#985—33nomaximal
6C22×#950—31nomaximal
7C22×C4×C3⋊D3—41nomaximal
8C2×C3×#41⋊C3C2216yescomplemented
9C22×C3×SL(2,𝔽3)C2211yescharacteristic
10#64266C22116yesdirect product factor
11C2×C3⋊GL(2,𝔽3)C2216yescomplemented
12C2×C3⋊#226C2216yescomplemented
13C22×C3×#41D311yescharacteristic, complemented
14C22×#235—31no—
15C22×#41⋊C3D313yescomplemented
16C2×#985—336no—
17C22×GL(2,𝔽3)—33no—
18C22×#226—33no—
19C2×#950—312no—
20C2×#950—312no—
21C22×#216—31no—
22C22×#215—31no—
23C22×#214—31no—
24C22×#213—31no—
25C23×C3⋊C8—31no—
26C23×C3⋊D3—41no—
27C22×C32×C4—41no—
28C22×C3⋊Dic3—41no—
29C2×C4×C3⋊D3—412no—
30C3×#41⋊C3C2314yescomplemented
31C2×C3×SL(2,𝔽3)C2313yes—
32C3⋊GL(2,𝔽3)C2314yescomplemented
33C3⋊#226C2314yescomplemented
34C22×#135—91noSylow
35C2×C3×#41—36no—
36C2×C3×#41C2×D316yescomplemented
37C22×C3×Q8C2×D311yescharacteristic
38C22×C3×D4—31no—
39C23×C3×C4—31no—
40C22×#67—61no—
41C2×#235—312no—
42C2×#235—66no—
43C22×D12—31no—
44C22×C4×D3—61no—
45C22×C4×D3—123no—
46C22×Dic6—31no—
47C2×#41⋊C3C2×D3118yescomplemented
48C22×SL(2,𝔽3)C2×D313yes—
49#985—348no—
50C2×GL(2,𝔽3)—318no—
51C2×#226—318no—
52#950—348no—
53#950—316no—
54C2×#216—36no—
55C2×#216—36no—
56C2×#215—36no—
57C2×#215—36no—
58C2×#214—312no—
59C2×#213—312no—
60C22×C3⋊C8—31no—
61C22×C3⋊C8—31no—
62C22×C3⋊C8—312no—
63C23×C32—41no—
64C22×C3⋊D3—46no—
65C22×C3⋊D3—48no—
66C2×C32×C4—46no—
67C2×C3⋊Dic3—46no—
68C4×C3⋊D3—416no—
69C3×SL(2,𝔽3)C2411yescharacteristic
70C22×#41—91no—
71C22×#41C3⋊D311yescharacteristic, complemented
72C2×#135—912no—
73C2×#135—912no—
74C22×Q16—91no—
75C22×SD16—91no—
76C22×SD16—91no—
77C22×D8—91no—
78C23×C8—91no—
79C24×C3—31no—
80C23×D3—61no—
81C23×D3—123no—
82C3×#41—312no—
83C3×#41—64no—
84C3×#41C22×D314yescomplemented
85C2×C3×Q8—33no—
86C2×C3×Q8C22×D313yes—
87C2×C3×D4—36no—
88C2×C3×D4—63no—
89C22×C3×C4—31no—
90C22×C3×C4—36no—
91C22×C3×C4—36no—
92C22×C3×C4—43no—
93C22×C3×C4S411yescharacteristic
94C2×#67—612no—
95C22×Dic3—61no—
96C22×Dic3—123no—
97#235—316no—
98#235—624no—
99C2×D12—36no—
100C2×D12—63no—
101C2×C4×D3—612no—
102C2×C4×D3—1236no—
103C2×Dic6—36no—
104C2×Dic6—63no—
105#41⋊C3C22×D3112yescomplemented
106C2×SL(2,𝔽3)C22×D319yes—
107GL(2,𝔽3)—312no—
108#226—312no—
109#216—312no—
110#216—34no—
111#215—312no—
112#215—34no—
113#214—316no—
114#213—316no—
115C2×C3⋊C8—36no—
116C2×C3⋊C8—36no—
117C2×C3⋊C8—316no—
118C22×C32—44no—
119C22×C32—43no—
120C2×C3⋊D3—44no—
121C2×C3⋊D3—424no—
122C32×C4—44no—
123C3⋊Dic3—44no—
124C2×#41—36no—
125C2×#41—912no—
126C2×#41—186no—
127C2×#41C2×C3⋊D316yescomplemented
128C22×Q8—91no—
129C22×Q8C2×C3⋊D311yescharacteristic
130C22×D4—31no—
131C22×D4—91no—
132C22×D4—181no—
133C23×C4—31no—
134C23×C4—181no—
135#135—948no—
136#135—916no—
137C2×Q16—96no—
138C2×Q16—96no—
139C2×SD16—912no—
140C2×SD16—96no—
141C2×SD16—96no—
142C2×D8—912no—
143C22×C8—91no—
144C22×C8—91no—
145C22×C8—912no—
146C23×C3—36no—
147C23×C3—43no—
148C23×C3—64no—
149C23×C3C2×S411yescharacteristic
150C22×D3—66no—
151C22×D3—1218no—
152C22×D3—1224no—
153C22×D3—124no—
154C3×Q8—33no—
155C3×Q8—61no—
156C3×Q8C23×D311yescharacteristic
157C3×D4—34no—
158C3×D4—66no—
159C2×C3×C4—33no—
160C2×C3×C4—33no—
161C2×C3×C4—34no—
162C2×C3×C4—312no—
163C2×C3×C4—418no—
164C2×C3×C4C2×S416yes—
165#67—616no—
166C2×Dic3—66no—
167C2×Dic3—1218no—
168D12—34no—
169D12—66no—
170C4×D3—616no—
171C4×D3—1248no—
172Dic6—34no—
173Dic6—66no—
174SL(2,𝔽3)C23×D313yes—
175C3⋊C8—34no—
176C3⋊C8—34no—
177C2×C32—46no—
178C2×C32—41no—
179C3⋊D3—48no—
180C24—31no—
181C24—181no—
182#41—312no—
183#41—64no—
184#41—916no—
185#41—1824no—
186#41C22×C3⋊D314yescomplemented
187C2×Q8—33no—
188C2×Q8—96no—
189C2×Q8—183no—
190C2×Q8C22×C3⋊D313yes—
191C2×D4—36no—
192C2×D4—63no—
193C2×D4—96no—
194C2×D4—183no—
195C2×D4—1812no—
196C22×C4—31no—
197C22×C4—36no—
198C22×C4—36no—
199C22×C4—181no—
200C22×C4—1812no—
201C22×C4C3⋊S411yescharacteristic
202Q16—912no—
203Q16—94no—
204SD16—916no—
205SD16—912no—
206SD16—94no—
207D8—916no—
208C2×C8—96no—
209C2×C8—96no—
210C2×C8—916no—
211C22×C3—34no—
212C22×C3—412no—
213C22×C3—49no—
214C22×C3—612no—
215C22×C3#98514yescomplemented
216C22×C3C22×S413yes—
217C2×D3—64no—
218C2×D3—1212no—
219C2×D3—1272no—
220C2×D3—1212no—
221C3×C4—31no—
222C3×C4—33no—
223C3×C4—412no—
224C3×C4C22×S414yes—
225Dic3—64no—
226Dic3—1212no—
227C32—41noSylow
228C23—36no—
229C23—64no—
230C23—186no—
231C23—364no—
232C23C2×C3⋊S411yescharacteristic
233Q8—33no—
234Q8—61no—
235Q8—94no—
236Q8—186no—
237Q8C23×C3⋊D311yescharacteristic
238D4—34no—
239D4—66no—
240D4—94no—
241D4—186no—
242D4—1816no—
243C2×C4—33no—
244C2×C4—33no—
245C2×C4—34no—
246C2×C4—312no—
247C2×C4—186no—
248C2×C4—1816no—
249C2×C4C2×C3⋊S416yes—
250C8—94no—
251C8—94no—
252C2×C3—43no—
253C2×C3—418no—
254C2×C3—64no—
255C2×C3C2×#98516yescomplemented
256C2×C3C23×S411yescharacteristic
257D3—1224no—
258D3—124no—
259C22—34no—
260C22—612no—
261C22—184no—
262C22—3612no—
263C22#6426614yesdirect product factor
264C22C22×C3⋊S413yes—
265C4—31no—
266C4—33no—
267C4—184no—
268C4C22×C3⋊S414yes—
269C3—43no—
270C3C22×#98511yescharacteristic, complemented
271C2—64no—
272C2—364no—
273C2C2×#6426616yesdirect product factor
274C2C23×C3⋊S411yescharacteristic
subgroupG/H
center, hypercenterC22×C4C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C24
FrattiniC2C23×C3⋊S4
FittingC22×C3×#41D3
socleC23×C3C2×S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41⋊C3C2semidirect
2. C2×#64266C2direct (N × H)
3. C22×C3⋊GL(2,𝔽3)C2semidirect
4. C22×C3⋊#226C2semidirect
8. C2×C3×#41⋊C3C22semidirect
10. #64266C22direct (N × H)
11. C2×C3⋊GL(2,𝔽3)C22semidirect
12. C2×C3⋊#226C22semidirect
13. C22×C3×#41D3semidirect
15. C22×#41⋊C3D3semidirect
30. C3×#41⋊C3C23semidirect
32. C3⋊GL(2,𝔽3)C23semidirect
33. C3⋊#226C23semidirect
36. C2×C3×#41C2×D3semidirect
47. C2×#41⋊C3C2×D3semidirect
71. C22×#41C3⋊D3semidirect
84. C3×#41C22×D3semidirect
105. #41⋊C3C22×D3semidirect
127. C2×#41C2×C3⋊D3semidirect
186. #41C22×C3⋊D3semidirect
215. C22×C3#985semidirect
255. C2×C3C2×#985semidirect
263. C22#64266direct (N × H)
270. C3C22×#985semidirect
273. C2C2×#64266direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×#642662
C22×C3⋊GL(2,𝔽3)2
C22×C3⋊#2262
C22×#9853
C22×#9503
C22×C4×C3⋊D34