Finite Group Databasebeta

← #11922874

Subgroups — #11922874

Order 1152.

760 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#10593655C212yesmaximal, complemented
3C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
4C4×C3⋊GL(2,𝔽3)C212yesmaximal, complemented
5#10593620C211yesmaximal, characteristic, complemented
6#85325—33nomaximal
7#82442—31nomaximal
8C2×#3828—41nomaximal
9C22×C3×SL(2,𝔽3)C2211yescharacteristic
10C2×C3⋊GL(2,𝔽3)C2212yes—
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3⋊#859C2212yes—
14C2×C3×C4×Q8D311yescharacteristic, complemented
15#5477—31no—
16C2×C4×SL(2,𝔽3)D313yescomplemented
17#5421—36no—
18C4⋊GL(2,𝔽3)—33no—
19C4×GL(2,𝔽3)—36no—
20#5386—33no—
21#5227—32no—
22#5027—31no—
23#5027—31no—
24C4×#215—32no—
25C4×#215—32no—
26#5020—31no—
27#5020—31no—
28#4968—32no—
29#4925—31no—
30C22×C32×C4—41no—
31C2×#638—42no—
32#3828—48no—
33C2×C4⋊(C3⋊D3)—41no—
34C2×C4×C3⋊D3—42no—
35C2×#634—41no—
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)C2311yescharacteristic
38C3⋊GL(2,𝔽3)—22no—
39#3103—91noSylow
40C22×C3×Q8C2×D311yescharacteristic
41C2×#235—61no—
42C2×#235—123no—
43C22×SL(2,𝔽3)C2×D313yes—
44C2×GL(2,𝔽3)—36no—
45C3×C4×Q8—31no—
46C3×C4×Q8—31no—
47C3×C4×Q8C2×D311yescharacteristic
48C3×C4×Q8C2×D311yescharacteristic
49C2×C3×C4⋊C4—31no—
50C2×C3×C42—31no—
51C2×#215—32no—
52C2×#215—32no—
53#930—32no—
54#930—61no—
55#875—61no—
56#874—31no—
57C4×D12—32no—
58C4×D12—61no—
59#869—31no—
60C4×SL(2,𝔽3)C2×D313yes—
61C4×SL(2,𝔽3)C2×D313yes—
62#859—36no—
63#835—32no—
64#835—32no—
65#830—32no—
66#809—32no—
67#809—32no—
68#808—32no—
69#808—32no—
70C4⋊(C3⋊C8)—31no—
71C4⋊(C3⋊C8)—31no—
72C4×C3⋊C8—32no—
73C23×C32—41no—
74C22×C3⋊D3—42no—
75C2×C32×C4—44no—
76C2×C32×C4—41no—
77C2×C32×C4—41no—
78#638—48no—
79C2×C3⋊Dic3—42no—
80C4⋊(C3⋊D3)—44no—
81C4×C3⋊D3—48no—
82#634—44no—
83C3×SL(2,𝔽3)#4111yescharacteristic, complemented
84#532—91no—
85C2×C4×Q8C3⋊D311yescharacteristic, complemented
86#481—92no—
87#475—91no—
88#475—91no—
89#460—91no—
90#460—91no—
91#448—92no—
92C4×SD16—92no—
93C4×SD16—92no—
94#431—91no—
95C2×C3×Q8—21no—
96C2×C3×Q8—31no—
97C2×C3×Q8—61no—
98C2×C3×Q8C22×D311yescharacteristic
99C22×C3×C4—31no—
100C22×C3×C4—31no—
101C22×C3×C4—43no—
102C22×C3×C4S411yescharacteristic
103C2×#67—62no—
104C2×#67—126no—
105#235—1224no—
106#235—124no—
107C2×D12—32no—
108C2×D12—61no—
109C2×D12—61no—
110C2×D12—123no—
111C2×C4×D3—62no—
112C2×C4×D3—126no—
113C2×Dic6—61no—
114C2×Dic6—123no—
115C2×SL(2,𝔽3)—23no—
116C2×SL(2,𝔽3)C22×D313yes—
117GL(2,𝔽3)—66no—
118C3×C4⋊C4—31no—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—61no—
121C3×C42—31no—
122C3×C42—31no—
123C3×C42—31no—
124C3×C42—31no—
125#215—62no—
126#215—64no—
127#215—62no—
128(C2×D3)⋊C4—62no—
129(C2×D3)⋊C4—62no—
130C4⋊Dic3—32no—
131C4⋊Dic3—61no—
132C2×C3⋊C8—32no—
133C2×C3⋊C8—32no—
134C22×C32—42no—
135C22×C32—41no—
136C22×C32—81no—
137C22×C32—81no—
138C2×C3⋊D3—44no—
139C2×C3⋊D3—82no—
140C2×C3⋊D3—82no—
141C32×C4—42no—
142C32×C4—42no—
143C3⋊Dic3—44no—
144C2×#41—181no—
145C22×Q8C2×C3⋊D311yescharacteristic
146C2×SD16—92no—
147C2×SD16—92no—
148C4⋊Q8—91no—
149#127—91no—
150#124—181no—
151#121—92no—
152#121—181no—
153C4×Q8—31no—
154C4×Q8—31no—
155C4×Q8C2×C3⋊D311yescharacteristic
156C4×Q8C2×C3⋊D311yescharacteristic
157C4×D4—92no—
158C4×D4—181no—
159C2×C4⋊C4—31no—
160C2×C42—31no—
161#106—92no—
162#106—92no—
163C4⋊C8—91no—
164C4⋊C8—91no—
165#103—92no—
166#103—92no—
167#102—92no—
168#102—92no—
169C22⋊C8—92no—
170C4×C8—92no—
171C23×C3—43no—
172C23×C3C2×S411yescharacteristic
173C22×D3—62no—
174C22×D3—126no—
175C3×Q8—31no—
176C3×Q8—32no—
177C3×Q8—61no—
178C3×Q8#23511yescharacteristic, complemented
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—43no—
184C2×C3×C4—43no—
185C2×C3×C4—412no—
186C2×C3×C4—61no—
187C2×C3×C4—61no—
188C2×C3×C4—121no—
189C2×C3×C4GL(2,𝔽3)14yescomplemented
190C2×C3×C4C2×S411yescharacteristic
191C2×C3×C4C2×S411yescharacteristic
192#67—1224no—
193#67—124no—
194C2×Dic3—62no—
195C2×Dic3—126no—
196D12—62no—
197D12—62no—
198D12—1212no—
199D12—122no—
200D12—122no—
201C4×D3—1224no—
202C4×D3—124no—
203Dic6—1212no—
204Dic6—122no—
205SL(2,𝔽3)#23513yescomplemented
206C3⋊C8—62no—
207C3⋊C8—62no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32—82no—
212C3⋊D3—84no—
213#41—364no—
214C2×Q8—21no—
215C2×Q8—31no—
216C2×Q8—61no—
217C2×Q8—181no—
218C2×Q8C22×C3⋊D311yescharacteristic
219C2×D4—92no—
220C2×D4—181no—
221C2×D4—182no—
222C2×D4—181no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—182no—
226C22×C4C3⋊S411yescharacteristic
227SD16—182no—
228SD16—184no—
229SD16—182no—
230C2×C8—92no—
231C2×C8—92no—
232C4⋊C4—31no—
233C4⋊C4—31no—
234C4⋊C4—61no—
235C4⋊C4—92no—
236C4⋊C4—181no—
237C22⋊C4—182no—
238C22⋊C4—182no—
239C42—31no—
240C42—31no—
241C42—31no—
242C42—31no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—46no—
246C22×C3—43no—
247C22×C3—83no—
248C22×C3—83no—
249C22×C3C2×GL(2,𝔽3)12yes—
250C22×C3C22×S411yescharacteristic
251C2×D3—1212no—
252C2×D3—122no—
253C2×D3—122no—
254C2×D3—122no—
255C2×D3—246no—
256C2×D3—246no—
257C3×C4—31no—
258C3×C4—31no—
259C3×C4—46no—
260C3×C4—46no—
261C3×C4—61no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4C2×GL(2,𝔽3)12yes—
265C3×C4C2×GL(2,𝔽3)12yes—
266Dic3—1212no—
267Dic3—122no—
268C32—41noSylow
269C23—182no—
270C23C2×C3⋊S411yescharacteristic
271Q8—31no—
272Q8—32no—
273Q8—61no—
274Q8—362no—
275Q8#382811yescharacteristic, complemented
276D4—182no—
277D4—182no—
278D4—362no—
279D4—364no—
280D4—362no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—121no—
288C2×C4—182no—
289C2×C4—364no—
290C2×C4C3⋊GL(2,𝔽3)14yescomplemented
291C2×C4C2×C3⋊S411yescharacteristic
292C2×C4C2×C3⋊S411yescharacteristic
293C8—182no—
294C8—182no—
295C2×C3—22no—
296C2×C3—43no—
297C2×C3—43no—
298C2×C3—43no—
299C2×C3—86no—
300C2×C3#591211yescharacteristic
301C2×C3C22×GL(2,𝔽3)11yescharacteristic
302C2×C3#592311yescharacteristic
303D3—2412no—
304D3—242no—
305C22—21no—
306C22—21no—
307C22—362no—
308C22—362no—
309C22—362no—
310C22C2×C3⋊GL(2,𝔽3)12yes—
311C22C22×C3⋊S411yescharacteristic
312C4—31no—
313C4—31no—
314C4—61no—
315C4—61no—
316C4—61no—
317C4—362no—
318C4C2×C3⋊GL(2,𝔽3)12yes—
319C4C2×C3⋊GL(2,𝔽3)12yes—
320C3—43no—
321C3#8532511yescharacteristic, complemented
322C2—22no—
323C2—722no—
324C2#1059657511yescharacteristic
325C2C22×C3⋊GL(2,𝔽3)11yescharacteristic
326C2#1059658611yescharacteristic
subgroupG/H
centerC2×C4C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FrattiniC22C22×C3⋊S4
FittingC2×C3×C4×Q8D3
socleC22×C3C22×S4
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×SL(2,𝔽3)C2semidirect
2. #10593655C2semidirect
3. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
4. C4×C3⋊GL(2,𝔽3)C2semidirect
5. #10593620C2semidirect
14. C2×C3×C4×Q8D3semidirect
16. C2×C4×SL(2,𝔽3)D3semidirect
83. C3×SL(2,𝔽3)#41semidirect
85. C2×C4×Q8C3⋊D3semidirect
178. C3×Q8#235semidirect
189. C2×C3×C4GL(2,𝔽3)semidirect
205. SL(2,𝔽3)#235semidirect
275. Q8#3828semidirect
290. C2×C4C3⋊GL(2,𝔽3)semidirect
321. C3#85325semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
#105936552
C4⋊(C3⋊GL(2,𝔽3))2
C4×C3⋊GL(2,𝔽3)2
#105936202
#853253
#824423
C2×#38284