Finite Group Databasebeta

← C2×C4×C3⋊GL(2,𝔽3) #11922872

Subgroups — C2×C4×C3⋊GL(2,𝔽3) #11922872

Order 1152.

1056 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 121 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⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×C3⋊#859C211yesmaximal, characteristic, complemented
4C4×C3⋊GL(2,𝔽3)C214yesmaximal, direct product factor
5C2×C4×GL(2,𝔽3)—33nomaximal
6C2×C4×#215—31nomaximal
7C22×C4×C3⋊D3—41nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3⋊GL(2,𝔽3)C414yesdirect product factor
10C2×C3⋊GL(2,𝔽3)C2212yes—
11C3×C4×SL(2,𝔽3)C2212yescomplemented
12C3⋊#859C2212yescomplemented
13C22×GL(2,𝔽3)—33no—
14C2×C3×C4×Q8D311yescharacteristic, complemented
15C22×#215—31no—
16C2×C4×D12—31no—
17C2×C4×SL(2,𝔽3)D313yescomplemented
18C2×#859—33no—
19C4×GL(2,𝔽3)—312no—
20C2×#835—31no—
21C4×#215—34no—
22C4×#215—34no—
23C2×#809—31no—
24C2×#808—31no—
25C2×C4×C3⋊C8—31no—
26C23×C3⋊D3—41no—
27C22×C32×C4—41no—
28C22×C3⋊Dic3—41no—
29C2×C4×C3⋊D3—44no—
30C2×C4×C3⋊D3—48no—
31C2×C3×SL(2,𝔽3)C2×C412yescomplemented
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3⋊GL(2,𝔽3)C2×C414yesdirect product factor
34C2×C4×SD16—91noSylow
35C22×C3×Q8C2×D311yescharacteristic
36C22×D12—31no—
37C22×C4×D3—61no—
38C22×C4×D3—123no—
39C22×SL(2,𝔽3)C2×D313yes—
40C2×GL(2,𝔽3)—312no—
41C2×GL(2,𝔽3)—36no—
42C3×C4×Q8—32no—
43C3×C4×Q8C2×D312yescomplemented
44C2×C3×C4⋊C4—31no—
45C2×C3×C42—31no—
46C2×#215—32no—
47C2×#215—34no—
48C2×#215—34no—
49C2×#215—32no—
50C2×(C2×D3)⋊C4—61no—
51C2×C4⋊Dic3—31no—
52C22×C3⋊C8—31no—
53C22×C3⋊C8—31no—
54C4×D12—34no—
55C4×D12—62no—
56C4×SL(2,𝔽3)C2×D316yescomplemented
57#859—36no—
58#835—32no—
59#835—32no—
60#809—34no—
61#808—34no—
62C4×C3⋊C8—34no—
63C23×C32—41no—
64C22×C3⋊D3—44no—
65C22×C3⋊D3—42no—
66C22×C3⋊D3—44no—
67C22×C3⋊D3—44no—
68C2×C32×C4—44no—
69C2×C32×C4—42no—
70C2×C3⋊Dic3—42no—
71C2×C3⋊Dic3—44no—
72C4×C3⋊D3—416no—
73C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
74C22×SD16—91no—
75C2×C4×Q8C3⋊D311yescharacteristic, complemented
76C2×C4×D4—91no—
77C4×SD16—94no—
78C4×SD16—94no—
79C2×#106—91no—
80C2×#103—91no—
81C2×#102—91no—
82C2×C4×C8—91no—
83C23×D3—61no—
84C23×D3—123no—
85C2×C3×Q8—31no—
86C2×C3×Q8—32no—
87C2×C3×Q8C4×D312yescomplemented
88C2×C3×Q8C22×D311yescharacteristic
89C22×C3×C4—31no—
90C22×C3×C4—31no—
91C22×C3×C4—43no—
92C22×C3×C4S411yescharacteristic
93C22×Dic3—61no—
94C22×Dic3—123no—
95C2×D12—32no—
96C2×D12—34no—
97C2×D12—61no—
98C2×D12—62no—
99C2×C4×D3—64no—
100C2×C4×D3—1212no—
101C2×C4×D3—1224no—
102C2×C4×D3—124no—
103C2×SL(2,𝔽3)C4×D316yescomplemented
104C2×SL(2,𝔽3)C22×D313yes—
105GL(2,𝔽3)—312no—
106C3×C4⋊C4—32no—
107C3×C4⋊C4—61no—
108C3×C42—32no—
109C3×C42—32no—
110#215—38no—
111#215—34no—
112#215—34no—
113(C2×D3)⋊C4—64no—
114C4⋊Dic3—32no—
115C4⋊Dic3—61no—
116C2×C3⋊C8—32no—
117C2×C3⋊C8—34no—
118C2×C3⋊C8—32no—
119C2×C3⋊C8—62no—
120C22×C32—42no—
121C22×C32—42no—
122C22×C32—42no—
123C22×C32—41no—
124C2×C3⋊D3—44no—
125C2×C3⋊D3—44no—
126C2×C3⋊D3—416no—
127C2×C3⋊D3—44no—
128C32×C4—44no—
129C3⋊Dic3—44no—
130C22×Q8C2×C3⋊D311yescharacteristic
131C22×D4—91no—
132C23×C4—181no—
133C2×SD16—94no—
134C2×SD16—92no—
135C2×SD16—94no—
136C2×SD16—92no—
137C22×C8—91no—
138C22×C8—91no—
139C4×Q8—32no—
140C4×Q8C2×C3⋊D312yescomplemented
141C4×D4—94no—
142C4×D4—182no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—91no—
145C2×C22⋊C4—181no—
146C2×C42—31no—
147#106—94no—
148#103—92no—
149#103—92no—
150#102—94no—
151C4×C8—94no—
152C23×C3—43no—
153C23×C3C2×S411yescharacteristic
154C22×D3—64no—
155C22×D3—62no—
156C22×D3—126no—
157C22×D3—1212no—
158C22×D3—1212no—
159C22×D3—1212no—
160C22×D3—122no—
161C22×D3—122no—
162C3×Q8—32no—
163C3×Q8—31no—
164C3×Q8—61no—
165C3×Q8C2×C4×D311yescharacteristic, complemented
166C2×C3×C4—32no—
167C2×C3×C4—32no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—32no—
171C2×C3×C4—46no—
172C2×C3×C4—412no—
173C2×C3×C4—62no—
174C2×C3×C4GL(2,𝔽3)14yescomplemented
175C2×C3×C4C2×S412yes—
176C2×Dic3—62no—
177C2×Dic3—126no—
178C2×Dic3—1212no—
179C2×Dic3—122no—
180D12—34no—
181D12—62no—
182D12—64no—
183C4×D3—1248no—
184C4×D3—128no—
185SL(2,𝔽3)C2×C4×D313yescomplemented
186C3⋊C8—34no—
187C3⋊C8—62no—
188C2×C32—44no—
189C2×C32—41no—
190C2×C32—41no—
191C2×C32—41no—
192C3⋊D3—48no—
193C24—181no—
194C2×Q8—31no—
195C2×Q8—32no—
196C2×Q8C4×C3⋊D312yescomplemented
197C2×Q8C22×C3⋊D311yescharacteristic
198C2×D4—92no—
199C2×D4—94no—
200C2×D4—181no—
201C2×D4—182no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—181no—
205C22×C4—184no—
206C22×C4—364no—
207C22×C4C3⋊S411yescharacteristic
208SD16—98no—
209SD16—94no—
210SD16—94no—
211C2×C8—92no—
212C2×C8—94no—
213C2×C8—92no—
214C2×C8—182no—
215C4⋊C4—32no—
216C4⋊C4—61no—
217C4⋊C4—92no—
218C4⋊C4—181no—
219C22⋊C4—184no—
220C42—32no—
221C42—32no—
222C22×C3—46no—
223C22×C3—46no—
224C22×C3—46no—
225C22×C3—43no—
226C22×C3C4×S412yes—
227C22×C3C2×GL(2,𝔽3)12yes—
228C22×C3#98512yes—
229C22×C3C22×S411yescharacteristic
230C2×D3—64no—
231C2×D3—1212no—
232C2×D3—1212no—
233C2×D3—1248no—
234C2×D3—1212no—
235C2×D3—128no—
236C2×D3—122no—
237C2×D3—122no—
238C3×C4—31no—
239C3×C4—31no—
240C3×C4—32no—
241C3×C4—412no—
242C3×C4—62no—
243C3×C4C2×GL(2,𝔽3)14yescomplemented
244Dic3—1212no—
245Dic3—122no—
246C32—41noSylow
247C23—184no—
248C23—182no—
249C23—362no—
250C23—362no—
251C23C2×C3⋊S411yescharacteristic
252Q8—32no—
253Q8—31no—
254Q8—61no—
255Q8C2×C4×C3⋊D311yescharacteristic, complemented
256D4—94no—
257D4—182no—
258D4—184no—
259C2×C4—32no—
260C2×C4—32no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—32no—
264C2×C4—62no—
265C2×C4—182no—
266C2×C4—362no—
267C2×C4—368no—
268C2×C4C3⋊GL(2,𝔽3)14yesdirect product factor
269C2×C4C2×C3⋊S412yes—
270C8—94no—
271C8—182no—
272C2×C3—412no—
273C2×C3—43no—
274C2×C3—43no—
275C2×C3—43no—
276C2×C3C4×GL(2,𝔽3)14yescomplemented
277C2×C3C2×C4×S411yescharacteristic
278C2×C3C22×GL(2,𝔽3)11yescharacteristic
279C2×C3C2×#98511yescharacteristic
280D3—1224no—
281D3—124no—
282C22—184no—
283C22—368no—
284C22—362no—
285C22—362no—
286C22C4×C3⋊S412yes—
287C22C2×C3⋊GL(2,𝔽3)12yes—
288C22#6426612yes—
289C22C22×C3⋊S411yescharacteristic
290C4—31no—
291C4—31no—
292C4—32no—
293C4—62no—
294C4—362no—
295C4C2×C3⋊GL(2,𝔽3)14yesdirect product factor
296C3—43no—
297C3C2×C4×GL(2,𝔽3)11yescharacteristic, complemented
298C2—364no—
299C2C4×C3⋊GL(2,𝔽3)14yesdirect product factor
300C2C2×C4×C3⋊S411yescharacteristic
301C2C22×C3⋊GL(2,𝔽3)11yescharacteristic
302C2C2×#6426611yescharacteristic
subgroupG/H
center, hypercenterC22×C4C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FrattiniC22C22×C3⋊S4
FittingC2×C3×C4×Q8D3
socleC23×C3C2×S4
prime radicalC22×C3⋊GL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C23
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×SL(2,𝔽3)C2semidirect
3. C2×C3⋊#859C2semidirect
4. C4×C3⋊GL(2,𝔽3)C2direct (N × H)
9. C2×C3⋊GL(2,𝔽3)C4direct (N × H)
11. C3×C4×SL(2,𝔽3)C22semidirect
12. C3⋊#859C22semidirect
14. C2×C3×C4×Q8D3semidirect
17. C2×C4×SL(2,𝔽3)D3semidirect
31. C2×C3×SL(2,𝔽3)C2×C4semidirect
33. C3⋊GL(2,𝔽3)C2×C4direct (N × H)
43. C3×C4×Q8C2×D3semidirect
56. C4×SL(2,𝔽3)C2×D3semidirect
73. C3×SL(2,𝔽3)C22×C4semidirect
75. C2×C4×Q8C3⋊D3semidirect
87. C2×C3×Q8C4×D3semidirect
103. C2×SL(2,𝔽3)C4×D3semidirect
140. C4×Q8C2×C3⋊D3semidirect
165. C3×Q8C2×C4×D3semidirect
174. C2×C3×C4GL(2,𝔽3)semidirect
185. SL(2,𝔽3)C2×C4×D3semidirect
196. C2×Q8C4×C3⋊D3semidirect
243. C3×C4C2×GL(2,𝔽3)semidirect
255. Q8C2×C4×C3⋊D3semidirect
268. C2×C4C3⋊GL(2,𝔽3)direct (N × H)
276. C2×C3C4×GL(2,𝔽3)semidirect
295. C4C2×C3⋊GL(2,𝔽3)direct (N × H)
297. C3C2×C4×GL(2,𝔽3)semidirect
299. C2C4×C3⋊GL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊GL(2,𝔽3)2
C2×C3×C4×SL(2,𝔽3)2
C2×C3⋊#8592
C4×C3⋊GL(2,𝔽3)2
C2×C4×GL(2,𝔽3)3
C2×C4×#2153
C22×C4×C3⋊D34