Finite Group Databasebeta

← C2×#10593214 #11922145

Subgroups — C2×#10593214 #11922145

Order 1152.

849 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 849 conjugacy classes of subgroups — too many to draw. Only its 91 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×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×C3×#859C211yesmaximal, characteristic, complemented
4#10593214C214yesmaximal, direct product factor
5C2×C4×GL(2,𝔽3)—31nomaximal
6C2×#4788—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C22×#624—41nomaximal
9C2×C3⋊GL(2,𝔽3)C414yescomplemented
10C2×C3⋊GL(2,𝔽3)C2212yes—
11Dic3×SL(2,𝔽3)C2212yescomplemented
12C3×#859C2212yescomplemented
13C22×GL(2,𝔽3)—31no—
14C22×GL(2,𝔽3)—61no—
15C2×Q8×Dic3D311yescharacteristic, complemented
16C22×#215—31no—
17C2×#893—31no—
18C2×C4×SL(2,𝔽3)—31no—
19C2×#859D311yescharacteristic, complemented
20C4×GL(2,𝔽3)—34no—
21C2×C3×#103—31no—
22C2×#821—31no—
23C2×C8×Dic3—31no—
24C2×#808—31no—
25#4788—34no—
26#4788—34no—
27C23×C3⋊D3—41no—
28C22×C3×Dic3—41no—
29C22×C3×Dic3—41no—
30C2×C3×SL(2,𝔽3)C2×C412yescomplemented
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C2×#624—44no—
33C2×#624—48no—
34C3⋊GL(2,𝔽3)C2×C414yescomplemented
35C2×C4×SD16—91noSylow
36C22×C3×Q8C2×D311yescharacteristic
37C22×D12—31no—
38C22×C4×D3—61no—
39C22×C4×D3—121no—
40C22×SL(2,𝔽3)—21no—
41C22×SL(2,𝔽3)C2×D311yescharacteristic
42C2×GL(2,𝔽3)—34no—
43C2×GL(2,𝔽3)—32no—
44C2×GL(2,𝔽3)—64no—
45C2×GL(2,𝔽3)—62no—
46C22×C3×C8—31no—
47C2×C3×C4⋊C4—31no—
48Q8×Dic3—32no—
49Q8×Dic3C2×D312yescomplemented
50C2×#215—34no—
51C2×#215—32no—
52C2×#215—34no—
53C2×#215—32no—
54C2×(C2×D3)⋊C4—61no—
55C2×C4⋊Dic3—31no—
56C2×C4×Dic3—31no—
57C22×C3⋊C8—31no—
58#893—34no—
59#893—62no—
60C4×SL(2,𝔽3)—32no—
61#859C2×D312yescomplemented
62C3×#103—32no—
63C3×#103—32no—
64#821—34no—
65C8×Dic3—34no—
66#808—34no—
67C23×C32—41no—
68C22×C3⋊D3—44no—
69C22×C3⋊D3—42no—
70C22×C3⋊D3—44no—
71C22×C3⋊D3—44no—
72C2×C3×Dic3—42no—
73C2×C3×Dic3—44no—
74C2×C3×Dic3—44no—
75C2×C3×Dic3—42no—
76C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
77#624—416no—
78C22×SD16—91no—
79C2×C4×Q8—31no—
80C2×C4×D4—91no—
81C4×SD16—94no—
82C4×SD16—94no—
83C2×#106—91no—
84C2×#103—31no—
85C2×#102—91no—
86C2×C4×C8—91no—
87C23×D3—61no—
88C23×D3—121no—
89C23×D3—241no—
90C2×C3×Q8—31no—
91C2×C3×Q8—32no—
92C2×C3×Q8C4×D312yescomplemented
93C2×C3×Q8C22×D311yescharacteristic
94C22×C3×C4—31no—
95C22×C3×C4—61no—
96C22×C3×C4—121no—
97C22×Dic3—31no—
98C22×Dic3—41no—
99C22×Dic3S411yescharacteristic
100C2×D12—32no—
101C2×D12—34no—
102C2×D12—61no—
103C2×D12—62no—
104C2×C4×D3—64no—
105C2×C4×D3—124no—
106C2×C4×D3—128no—
107C2×C4×D3—124no—
108C2×SL(2,𝔽3)—22no—
109C2×SL(2,𝔽3)—21no—
110C2×SL(2,𝔽3)C4×D312yescomplemented
111C2×SL(2,𝔽3)C22×D311yescharacteristic
112GL(2,𝔽3)—34no—
113GL(2,𝔽3)—64no—
114C2×C3×C8—32no—
115C2×C3×C8—62no—
116C3×C4⋊C4—32no—
117C3×C4⋊C4—61no—
118#215—38no—
119#215—34no—
120#215—34no—
121(C2×D3)⋊C4—64no—
122C4⋊Dic3—32no—
123C4⋊Dic3—61no—
124C4×Dic3—32no—
125C4×Dic3—32no—
126C2×C3⋊C8—32no—
127C2×C3⋊C8—34no—
128C22×C32—42no—
129C22×C32—42no—
130C22×C32—42no—
131C22×C32—41no—
132C2×C3⋊D3—44no—
133C2×C3⋊D3—44no—
134C2×C3⋊D3—416no—
135C2×C3⋊D3—44no—
136C3×Dic3—44no—
137C3×Dic3—44no—
138C22×Q8D3211yescharacteristic
139C22×D4—91no—
140C23×C4—181no—
141C2×SD16—94no—
142C2×SD16—92no—
143C2×SD16—94no—
144C2×SD16—92no—
145C22×C8—31no—
146C22×C8—91no—
147C4×Q8—32no—
148C4×Q8—92no—
149C4×D4—94no—
150C4×D4—182no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—91no—
153C2×C22⋊C4—181no—
154C2×C42—91no—
155#106—94no—
156#103—32no—
157#103—32no—
158#102—94no—
159C4×C8—94no—
160C23×C3—41no—
161C23×C3—81no—
162C23×C3C2×S411yescharacteristic
163C22×D3—64no—
164C22×D3—62no—
165C22×D3—122no—
166C22×D3—124no—
167C22×D3—124no—
168C22×D3—124no—
169C22×D3—122no—
170C22×D3—122no—
171C22×D3—242no—
172C22×D3—244no—
173C22×D3—244no—
174C22×D3—244no—
175C3×Q8—32no—
176C3×Q8—31no—
177C3×Q8—61no—
178C3×Q8C2×C4×D311yescharacteristic, complemented
179C2×C3×C4—32no—
180C2×C3×C4—32no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—62no—
184C2×C3×C4—122no—
185C2×C3×C4—124no—
186C2×C3×C4—122no—
187C2×Dic3—32no—
188C2×Dic3—42no—
189C2×Dic3—44no—
190C2×Dic3—62no—
191C2×Dic3GL(2,𝔽3)14yescomplemented
192C2×Dic3C2×S412yes—
193D12—34no—
194D12—62no—
195D12—64no—
196C4×D3—1216no—
197C4×D3—128no—
198SL(2,𝔽3)—21no—
199SL(2,𝔽3)C2×C4×D311yescharacteristic, complemented
200C3×C8—62no—
201C3⋊C8—34no—
202C2×C32—44no—
203C2×C32—41no—
204C2×C32—41no—
205C2×C32—41no—
206C3⋊D3—48no—
207C24—181no—
208C2×Q8—31no—
209C2×Q8—32no—
210C2×Q8#62412yescomplemented
211C2×Q8C2×D3211yescharacteristic
212C2×D4—92no—
213C2×D4—94no—
214C2×D4—181no—
215C2×D4—182no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—61no—
219C22×C4—91no—
220C22×C4—184no—
221C22×C4—364no—
222SD16—98no—
223SD16—94no—
224SD16—94no—
225C2×C8—32no—
226C2×C8—62no—
227C2×C8—92no—
228C2×C8—94no—
229C4⋊C4—32no—
230C4⋊C4—61no—
231C4⋊C4—92no—
232C4⋊C4—181no—
233C22⋊C4—184no—
234C42—92no—
235C42—92no—
236C22×C3—41no—
237C22×C3—42no—
238C22×C3—42no—
239C22×C3—42no—
240C22×C3—81no—
241C22×C3—82no—
242C22×C3—82no—
243C22×C3—82no—
244C22×C3C4×S412yes—
245C22×C3C2×GL(2,𝔽3)12yes—
246C22×C3#98512yes—
247C22×C3C22×S411yescharacteristic
248C2×D3—64no—
249C2×D3—124no—
250C2×D3—124no—
251C2×D3—124no—
252C2×D3—1216no—
253C2×D3—128no—
254C2×D3—122no—
255C2×D3—122no—
256C2×D3—244no—
257C2×D3—244no—
258C2×D3—244no—
259C2×D3—2416no—
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—32no—
263C3×C4—124no—
264C3×C4—122no—
265Dic3—44no—
266Dic3—62no—
267Dic3C2×GL(2,𝔽3)14yescomplemented
268C32—41noSylow
269C23—182no—
270C23—184no—
271C23—362no—
272C23—362no—
273C23D3×S411yescharacteristic
274Q8—32no—
275Q8—31no—
276Q8—61no—
277Q8C2×#62411yescharacteristic, complemented
278D4—94no—
279D4—182no—
280D4—184no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—32no—
284C2×C4—34no—
285C2×C4—32no—
286C2×C4—32no—
287C2×C4—62no—
288C2×C4—92no—
289C2×C4—122no—
290C2×C4—182no—
291C2×C4—368no—
292C8—62no—
293C8—94no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—44no—
298C2×C3—81no—
299C2×C3—81no—
300C2×C3—81no—
301C2×C3—84no—
302C2×C3C4×GL(2,𝔽3)14yescomplemented
303C2×C3C2×C4×S411yescharacteristic
304C2×C3C22×GL(2,𝔽3)11yescharacteristic
305C2×C3C2×#98511yescharacteristic
306D3—128no—
307D3—124no—
308D3—248no—
309C22—184no—
310C22—362no—
311C22—362no—
312C22—368no—
313C22#6419812yes—
314C22D3×GL(2,𝔽3)12yes—
315C22(C3⋊S4)⋊C412yes—
316C22C2×D3×S411yescharacteristic
317C4—34no—
318C4—31no—
319C4—31no—
320C4—32no—
321C4—122no—
322C4—182no—
323C3—41no—
324C3—81no—
325C3C2×C4×GL(2,𝔽3)11yescharacteristic, complemented
326C2—364no—
327C2#1059321414yesdirect product factor
328C2C2×#6419811yescharacteristic
329C2C2×D3×GL(2,𝔽3)11yescharacteristic
330C2C2×(C3⋊S4)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC23D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FrattiniC22C2×D3×S4
FittingC22×C3×Q8C2×D3
socleC23×C3C2×S4
prime radicalC22×C3⋊GL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C23
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×SL(2,𝔽3)C2semidirect
3. C2×C3×#859C2semidirect
4. #10593214C2direct (N × H)
9. C2×C3⋊GL(2,𝔽3)C4semidirect
11. Dic3×SL(2,𝔽3)C22semidirect
12. C3×#859C22semidirect
15. C2×Q8×Dic3D3semidirect
19. C2×#859D3semidirect
30. C2×C3×SL(2,𝔽3)C2×C4semidirect
34. C3⋊GL(2,𝔽3)C2×C4semidirect
49. Q8×Dic3C2×D3semidirect
61. #859C2×D3semidirect
76. C3×SL(2,𝔽3)C22×C4semidirect
92. C2×C3×Q8C4×D3semidirect
110. C2×SL(2,𝔽3)C4×D3semidirect
178. C3×Q8C2×C4×D3semidirect
191. C2×Dic3GL(2,𝔽3)semidirect
199. SL(2,𝔽3)C2×C4×D3semidirect
210. C2×Q8#624semidirect
267. Dic3C2×GL(2,𝔽3)semidirect
277. Q8C2×#624semidirect
302. C2×C3C4×GL(2,𝔽3)semidirect
325. C3C2×C4×GL(2,𝔽3)semidirect
327. C2#10593214direct (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×Dic3×SL(2,𝔽3)2
C2×C3×#8592
#105932142
C2×C4×GL(2,𝔽3)3
C2×#47883
C22×#6244