Finite Group Databasebeta

← C22×#64264 #11923722

Subgroups — C22×#64264 #11923722

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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#64264C2112yesmaximal, direct product factor
3C22×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C22×C3⋊#226C211yesmaximal, characteristic, complemented
5C22×#983—33nomaximal
6C22×#942—31nomaximal
7C22×#638—41nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C22×C3×SL(2,𝔽3)C2216yescomplemented
10#64264C22116yesdirect product factor
11C2×C3⋊GL(2,𝔽3)C2216yescomplemented
12C2×C3⋊#226C2216yescomplemented
13C23×C3×Q8D311yescharacteristic, complemented
14C22×#235—31no—
15C23×SL(2,𝔽3)D313yescomplemented
16C2×#983—336no—
17C22×GL(2,𝔽3)—33no—
18C22×#226—33no—
19C22×#216—31no—
20C22×#216—31no—
21C2×#942—312no—
22C2×#942—312no—
23C22×#215—31no—
24C22×#215—31no—
25C22×#208—31no—
26C24×C32—41no—
27C23×C3⋊D3—41no—
28C2×#638—412no—
29C22×C3⋊Dic3—41no—
30C2×C3×SL(2,𝔽3)C2313yes—
31C2×C3×SL(2,𝔽3)C2314yescomplemented
32C3⋊GL(2,𝔽3)C2314yescomplemented
33C3⋊#226C2314yescomplemented
34C22×#137—91noSylow
35C22×C3×Q8—31no—
36C22×C3×Q8—36no—
37C22×C3×Q8C2×D311yescharacteristic
38C22×C3×Q8C2×D316yescomplemented
39C23×C3×C4—31no—
40C22×#67—61no—
41C22×#67—123no—
42C2×#235—312no—
43C2×#235—66no—
44C22×D12—31no—
45C22×C4×D3—61no—
46C22×Dic6—31no—
47C22×SL(2,𝔽3)C2×D313yes—
48C22×SL(2,𝔽3)C2×D3118yescomplemented
49#983—348no—
50C2×GL(2,𝔽3)—318no—
51C2×#226—318no—
52C2×#216—36no—
53C2×#216—312no—
54C2×#216—36no—
55#942—348no—
56#942—316no—
57C2×#215—36no—
58C2×#215—312no—
59C2×#215—36no—
60C2×#208—312no—
61C22×C3⋊C8—31no—
62C22×C3⋊C8—31no—
63C23×C32—41no—
64C23×C32—46no—
65C23×C32—84no—
66C22×C3⋊D3—46no—
67C22×C3⋊D3—84no—
68#638—416no—
69C2×C3⋊Dic3—46no—
70C3×SL(2,𝔽3)C2411yescharacteristic
71C22×#41—91no—
72C23×Q8C3⋊D311yescharacteristic, complemented
73C2×#137—912no—
74C2×#137—912no—
75C22×Q16—91no—
76C22×Q16—91no—
77C22×SD16—91no—
78C22×SD16—91no—
79C22×M16—91no—
80C24×C3—43no—
81C24×C3S411yescharacteristic
82C23×D3—61no—
83C23×D3—123no—
84C2×C3×Q8—33no—
85C2×C3×Q8—36no—
86C2×C3×Q8—312no—
87C2×C3×Q8—63no—
88C2×C3×Q8—64no—
89C2×C3×Q8C22×D313yes—
90C2×C3×Q8C22×D314yescomplemented
91C22×C3×C4—31no—
92C22×C3×C4—36no—
93C22×C3×C4—31no—
94C22×C3×C4—36no—
95C2×#67—612no—
96C2×#67—1236no—
97C22×Dic3—61no—
98C22×Dic3—123no—
99#235—316no—
100#235—624no—
101C2×D12—36no—
102C2×D12—63no—
103C2×C4×D3—612no—
104C2×Dic6—36no—
105C2×Dic6—63no—
106C2×SL(2,𝔽3)C22×D319yes—
107C2×SL(2,𝔽3)C22×D3112yescomplemented
108GL(2,𝔽3)—312no—
109#226—312no—
110#216—312no—
111#216—316no—
112#216—34no—
113#215—312no—
114#215—316no—
115#215—34no—
116#208—316no—
117C2×C3⋊C8—36no—
118C2×C3⋊C8—36no—
119C22×C32—44no—
120C22×C32—43no—
121C22×C32—44no—
122C22×C32—812no—
123C2×C3⋊D3—44no—
124C2×C3⋊D3—812no—
125C3⋊Dic3—44no—
126C2×#41—912no—
127C2×#41—186no—
128C22×Q8—31no—
129C22×Q8—36no—
130C22×Q8—91no—
131C22×Q8C2×C3⋊D311yescharacteristic
132C22×Q8C2×C3⋊D316yescomplemented
133C22×D4—91no—
134C22×D4—181no—
135C23×C4—31no—
136C23×C4—181no—
137#137—948no—
138#137—916no—
139C2×Q16—96no—
140C2×Q16—912no—
141C2×Q16—96no—
142C2×SD16—96no—
143C2×SD16—912no—
144C2×SD16—96no—
145C2×M16—912no—
146C22×C8—91no—
147C22×C8—91no—
148C23×C3—24no—
149C23×C3—418no—
150C23×C3—43no—
151C23×C3—812no—
152C23×C3C2×S416yes—
153C23×C3C2×S411yescharacteristic
154C22×D3—66no—
155C22×D3—1218no—
156C22×D3—124no—
157C22×D3—2412no—
158C3×Q8—33no—
159C3×Q8—34no—
160C3×Q8—61no—
161C3×Q8—66no—
162C3×Q8C23×D311yescharacteristic
163C2×C3×C4—33no—
164C2×C3×C4—34no—
165C2×C3×C4—33no—
166C2×C3×C4—36no—
167C2×C3×C4—312no—
168#67—616no—
169#67—1248no—
170C2×Dic3—66no—
171C2×Dic3—1218no—
172D12—34no—
173D12—66no—
174C4×D3—616no—
175Dic6—34no—
176Dic6—66no—
177SL(2,𝔽3)C23×D313yes—
178C3⋊C8—34no—
179C3⋊C8—34no—
180C2×C32—46no—
181C2×C32—41no—
182C2×C32—84no—
183C3⋊D3—84no—
184C24—181no—
185C24C3⋊S411yescharacteristic
186#41—916no—
187#41—1824no—
188C2×Q8—33no—
189C2×Q8—36no—
190C2×Q8—312no—
191C2×Q8—63no—
192C2×Q8—64no—
193C2×Q8—96no—
194C2×Q8—183no—
195C2×Q8C22×C3⋊D313yes—
196C2×Q8C22×C3⋊D314yescomplemented
197C2×D4—96no—
198C2×D4—183no—
199C2×D4—1812no—
200C22×C4—36no—
201C22×C4—36no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—1812no—
205C22×C4—181no—
206Q16—912no—
207Q16—916no—
208Q16—94no—
209SD16—912no—
210SD16—916no—
211SD16—94no—
212M16—916no—
213C2×C8—96no—
214C2×C8—96no—
215C22×C3—212no—
216C22×C3—412no—
217C22×C3—49no—
218C22×C3—412no—
219C22×C3—836no—
220C22×C3#98314yescomplemented
221C22×C3C22×S414yes—
222C22×C3C22×S413yes—
223C2×D3—64no—
224C2×D3—1212no—
225C2×D3—1212no—
226C2×D3—2436no—
227C3×C4—31no—
228C3×C4—33no—
229C3×C4—34no—
230Dic3—64no—
231Dic3—1212no—
232C32—41noSylow
233C23—24no—
234C23—186no—
235C23—364no—
236C23C2×C3⋊S416yes—
237C23C2×C3⋊S411yescharacteristic
238Q8—33no—
239Q8—34no—
240Q8—66no—
241Q8—61no—
242Q8—94no—
243Q8—186no—
244Q8C23×C3⋊D311yescharacteristic
245D4—94no—
246D4—186no—
247D4—1816no—
248C2×C4—33no—
249C2×C4—34no—
250C2×C4—36no—
251C2×C4—33no—
252C2×C4—312no—
253C2×C4—1816no—
254C2×C4—186no—
255C8—94no—
256C8—94no—
257C2×C3—24no—
258C2×C3—43no—
259C2×C3—418no—
260C2×C3—812no—
261C2×C3C2×#98316yescomplemented
262C2×C3C23×S411yescharacteristic
263D3—124no—
264D3—2412no—
265C22—212no—
266C22—184no—
267C22—3612no—
268C22#6426414yesdirect product factor
269C22C22×C3⋊S414yes—
270C22C22×C3⋊S413yes—
271C4—31no—
272C4—34no—
273C4—33no—
274C4—184no—
275C3—43no—
276C3C22×#98311yescharacteristic, complemented
277C2—24no—
278C2—364no—
279C2C2×#6426416yesdirect product factor
280C2C23×C3⋊S411yescharacteristic
subgroupG/H
centerC23C2×C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C24
FrattiniC2C23×C3⋊S4
FittingC23×C3×Q8D3
socleC23×C3C2×S4
hypercenterC24C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C3×SL(2,𝔽3)C2semidirect
2. C2×#64264C2direct (N × H)
3. C22×C3⋊GL(2,𝔽3)C2semidirect
4. C22×C3⋊#226C2semidirect
9. C22×C3×SL(2,𝔽3)C22semidirect
10. #64264C22direct (N × H)
11. C2×C3⋊GL(2,𝔽3)C22semidirect
12. C2×C3⋊#226C22semidirect
13. C23×C3×Q8D3semidirect
15. C23×SL(2,𝔽3)D3semidirect
31. C2×C3×SL(2,𝔽3)C23semidirect
32. C3⋊GL(2,𝔽3)C23semidirect
33. C3⋊#226C23semidirect
38. C22×C3×Q8C2×D3semidirect
48. C22×SL(2,𝔽3)C2×D3semidirect
72. C23×Q8C3⋊D3semidirect
90. C2×C3×Q8C22×D3semidirect
107. C2×SL(2,𝔽3)C22×D3semidirect
132. C22×Q8C2×C3⋊D3semidirect
196. C2×Q8C22×C3⋊D3semidirect
220. C22×C3#983semidirect
261. C2×C3C2×#983semidirect
268. C22#64264direct (N × H)
276. C3C22×#983semidirect
279. C2C2×#64264direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C2×#642642
C22×C3⋊GL(2,𝔽3)2
C22×C3⋊#2262
C22×#9833
C22×#9423
C22×#6384