Finite Group Databasebeta

← #11922903

Subgroups — #11922903

Order 1152.

792 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 89 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
2C3×#519⋊C3C211yesmaximal, characteristic, complemented
3#10593654C211yesmaximal, characteristic, complemented
4#10593628C212yesmaximal, complemented
5C4×C3⋊GL(2,𝔽3)C212yesmaximal, complemented
6#85353—33nomaximal
7#82548—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3⋊GL(2,𝔽3)C414yescomplemented
10C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
11C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
12C22⋊C4×C3⋊D3—41nomaximal
13C3×C4×SL(2,𝔽3)C2212yescomplemented
14C3⋊#859C2212yescomplemented
15C22×GL(2,𝔽3)—33no—
16C3×#519D311yescharacteristic, complemented
17C22×#215—31no—
18#5521—31no—
19#519⋊C3D313yescomplemented
20#5420—33no—
21#5394—36no—
22C4×GL(2,𝔽3)—36no—
23#5225—31no—
24#5026—32no—
25#5026—32no—
26C4×#215—32no—
27C4×#215—32no—
28#5001—31no—
29#4998—31no—
30#4997—31no—
31C23×C3⋊D3—41no—
32C2×C4×C3⋊D3—42no—
33C2×C3×SL(2,𝔽3)C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C3⋊GL(2,𝔽3)C2×C412yescomplemented
36C3⋊GL(2,𝔽3)C2×C412yescomplemented
37C32×C22⋊C4—41no—
38#3757—41no—
39#3752—42no—
40#2949—91noSylow
41C22×C3×Q8C2×D311yescharacteristic
42C22×D12—31no—
43C22×SL(2,𝔽3)C2×D313yes—
44C2×GL(2,𝔽3)—312no—
45C2×GL(2,𝔽3)—33no—
46C2×GL(2,𝔽3)—33no—
47C3×C4×Q8—32no—
48C3×C4×Q8C2×D312yescomplemented
49C3×#117—31no—
50C3×#117—31no—
51C2×#215—31no—
52C2×#215—31no—
53C2×#215—34no—
54C2×#215—34no—
55C2×#215—31no—
56C2×#215—31no—
57#926—31no—
58C2×#208—31no—
59C22×C3⋊C8—31no—
60#893—62no—
61D3×C22⋊C4—61no—
62D3×C22⋊C4—61no—
63D3×C22⋊C4—123no—
64C4×D12—32no—
65C4×D12—32no—
66C4×SL(2,𝔽3)C2×D316yescomplemented
67#859—36no—
68#835—32no—
69#835—32no—
70#809—32no—
71#809—32no—
72#808—32no—
73#807—32no—
74#803—32no—
75C4×C3⋊C8—32no—
76C23×C32—41no—
77C22×C3⋊D3—41no—
78C22×C3⋊D3—44no—
79C22×C3⋊D3—41no—
80C22×C3⋊D3—84no—
81C2×C32×C4—42no—
82C2×C3⋊Dic3—42no—
83C4×C3⋊D3—84no—
84C3×SL(2,𝔽3)C22×C411yescharacteristic
85C22×SD16—91no—
86#519C3⋊D311yescharacteristic, complemented
87#518—91no—
88#440—92no—
89#440—92no—
90C4×SD16—92no—
91C4×SD16—92no—
92#427—91no—
93#419—91no—
94#418—91no—
95#405—91no—
96C23×D3—61no—
97C23×D3—123no—
98C2×C3×Q8—31no—
99C2×C3×Q8—32no—
100C2×C3×Q8C4×D312yescomplemented
101C2×C3×Q8C22×D311yescharacteristic
102C22×C3×C4—31no—
103C2×D12—31no—
104C2×D12—31no—
105C2×D12—34no—
106C2×D12—61no—
107C2×D12—62no—
108C2×C4×D3—62no—
109C2×C4×D3—62no—
110C2×C4×D3—126no—
111C2×SL(2,𝔽3)C4×D316yescomplemented
112C2×SL(2,𝔽3)C22×D313yes—
113GL(2,𝔽3)—36no—
114GL(2,𝔽3)—36no—
115C3×C4⋊C4—32no—
116C3×C4⋊C4—32no—
117C3×C22⋊C4—31no—
118C3×C22⋊C4—43no—
119C3×C22⋊C4S411yescharacteristic
120C3×C42—32no—
121C3×C42—61no—
122#217—61no—
123#217—123no—
124#215—34no—
125#215—34no—
126#215—32no—
127#215—32no—
128#215—32no—
129#215—32no—
130(C2×D3)⋊C4—62no—
131(C2×D3)⋊C4—62no—
132(C2×D3)⋊C4—126no—
133C4⋊Dic3—32no—
134C4×Dic3—61no—
135#208—62no—
136C2×C3⋊C8—31no—
137C2×C3⋊C8—31no—
138C2×C3⋊C8—34no—
139C2×C3⋊C8—32no—
140C22×C32—42no—
141C22×C32—41no—
142C22×C32—82no—
143C2×C3⋊D3—42no—
144C2×C3⋊D3—42no—
145C2×C3⋊D3—44no—
146C2×C3⋊D3—82no—
147C2×C3⋊D3—88no—
148C32×C4—82no—
149C3⋊Dic3—82no—
150C22×Q8C2×C3⋊D311yescharacteristic
151C22×D4—91no—
152C2×SD16—94no—
153C2×SD16—91no—
154C2×SD16—91no—
155C2×SD16—94no—
156C2×SD16—91no—
157C2×SD16—91no—
158C2×M16—91no—
159C22×C8—91no—
160C4×Q8—32no—
161C4×Q8C2×C3⋊D312yescomplemented
162C4×D4—92no—
163C4×D4—92no—
164C4×D4—182no—
165#117—31no—
166#117—31no—
167#117—91no—
168C2×C22⋊C4—181no—
169C2×C22⋊C4—181no—
170#107—92no—
171#106—92no—
172#103—92no—
173#103—92no—
174#102—92no—
175#102—92no—
176#97—92no—
177C4×C8—92no—
178C23×C3—43no—
179C23×C3C2×S411yescharacteristic
180C22×D3—61no—
181C22×D3—61no—
182C22×D3—64no—
183C22×D3—123no—
184C22×D3—123no—
185C22×D3—1212no—
186C22×D3—242no—
187C22×D3—2412no—
188C3×Q8—32no—
189C3×Q8—31no—
190C3×Q8—61no—
191C3×Q8C2×C4×D311yescharacteristic
192C2×C3×C4—32no—
193C2×C3×C4—32no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—32no—
197C2×C3×C4—46no—
198C2×C3×C4C2×S412yes—
199C2×Dic3—62no—
200C2×Dic3—126no—
201D12—32no—
202D12—32no—
203D12—61no—
204D12—61no—
205D12—64no—
206C4×D3—242no—
207C4×D3—242no—
208C4×D3—2412no—
209SL(2,𝔽3)C2×C4×D313yes—
210C3⋊C8—32no—
211C3⋊C8—32no—
212C3⋊C8—62no—
213C2×C32—41no—
214C2×C32—42no—
215C2×C32—82no—
216C3⋊D3—44no—
217C3⋊D3—82no—
218C24—181no—
219C2×Q8—31no—
220C2×Q8—32no—
221C2×Q8C4×C3⋊D312yescomplemented
222C2×Q8C22×C3⋊D311yescharacteristic
223C2×D4—91no—
224C2×D4—91no—
225C2×D4—94no—
226C2×D4—181no—
227C2×D4—182no—
228C22×C4—31no—
229C22×C4—182no—
230C22×C4—182no—
231SD16—94no—
232SD16—94no—
233SD16—92no—
234SD16—92no—
235SD16—92no—
236SD16—92no—
237M16—182no—
238C2×C8—91no—
239C2×C8—91no—
240C2×C8—94no—
241C2×C8—92no—
242C4⋊C4—32no—
243C4⋊C4—32no—
244C4⋊C4—92no—
245C22⋊C4—31no—
246C22⋊C4—182no—
247C22⋊C4—181no—
248C22⋊C4—182no—
249C22⋊C4C3⋊S411yescharacteristic
250C42—32no—
251C42—61no—
252C42—181no—
253C22×C3—22no—
254C22×C3—46no—
255C22×C3—43no—
256C22×C3—86no—
257C22×C3C4×S412yes—
258C22×C3C22×S411yescharacteristic
259C2×D3—62no—
260C2×D3—62no—
261C2×D3—126no—
262C2×D3—126no—
263C2×D3—1212no—
264C2×D3—122no—
265C2×D3—122no—
266C2×D3—246no—
267C2×D3—244no—
268C2×D3—2424no—
269C3×C4—22no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—32no—
273C3×C4—62no—
274C3×C4—86no—
275Dic3—122no—
276Dic3—246no—
277C32—41noSylow
278C23—181no—
279C23—181no—
280C23—184no—
281C23—722no—
282C23C2×C3⋊S411yescharacteristic
283Q8—32no—
284Q8—31no—
285Q8—61no—
286Q8C2×C4×C3⋊D311yescharacteristic
287D4—92no—
288D4—92no—
289D4—181no—
290D4—181no—
291D4—184no—
292C2×C4—32no—
293C2×C4—32no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—32no—
297C2×C4—182no—
298C2×C4—722no—
299C2×C4—722no—
300C2×C4C2×C3⋊S412yes—
301C8—92no—
302C8—92no—
303C8—182no—
304C2×C3—22no—
305C2×C3—43no—
306C2×C3—46no—
307C2×C3—86no—
308C2×C3C2×C4×S411yescharacteristic
309C2×C3#592612yes—
310D3—1212no—
311D3—122no—
312D3—122no—
313D3—246no—
314C22—22no—
315C22—182no—
316C22—182no—
317C22—362no—
318C22—362no—
319C22—724no—
320C22C4×C3⋊S412yes—
321C22C22×C3⋊S411yescharacteristic
322C4—22no—
323C4—31no—
324C4—31no—
325C4—32no—
326C4—62no—
327C4—362no—
328C3—43no—
329C3#8535311yescharacteristic, complemented
330C2—22no—
331C2—362no—
332C2—362no—
333C2C2×C4×C3⋊S411yescharacteristic
334C2#1059658912yes—
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC3×#519D3
socleC22×C3C22×S4
prime radicalC22×C3⋊GL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C23
hypercenterC22⋊C4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#519⋊C3C2semidirect
3. #10593654C2semidirect
4. #10593628C2semidirect
5. C4×C3⋊GL(2,𝔽3)C2semidirect
9. C2×C3⋊GL(2,𝔽3)C4semidirect
13. C3×C4×SL(2,𝔽3)C22semidirect
14. C3⋊#859C22semidirect
16. C3×#519D3semidirect
19. #519⋊C3D3semidirect
33. C2×C3×SL(2,𝔽3)C2×C4semidirect
35. C3⋊GL(2,𝔽3)C2×C4semidirect
36. C3⋊GL(2,𝔽3)C2×C4semidirect
48. C3×C4×Q8C2×D3semidirect
66. C4×SL(2,𝔽3)C2×D3semidirect
86. #519C3⋊D3semidirect
100. C2×C3×Q8C4×D3semidirect
111. C2×SL(2,𝔽3)C4×D3semidirect
161. C4×Q8C2×C3⋊D3semidirect
221. C2×Q8C4×C3⋊D3semidirect
329. C3#85353semidirect

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
C3×#519⋊C32
#105936542
#105936282
C4×C3⋊GL(2,𝔽3)2
#853533
#825483
C22⋊C4×C3⋊D34