Finite Group Databasebeta

← #11922880

Subgroups — #11922880

Order 1152.

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

normal subgroups only

This group has 704 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×#64264C212yesmaximal, complemented
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#10593629C212yesmaximal, complemented
4C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
5C4⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
6#85331—33nomaximal
7#82441—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9#64264—24no—
10C2×C3⋊GL(2,𝔽3)C2212yes—
11C2×C3⋊#226C2212yes—
12#64139—41nomaximal
13C3×C4×SL(2,𝔽3)C2211yescharacteristic
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C2×#983—36no—
16C2×C3×C4×Q8D311yescharacteristic, complemented
17C2×#942—32no—
18#5477—31no—
19C2×C4×SL(2,𝔽3)D313yescomplemented
20#5395—36no—
21C4⋊GL(2,𝔽3)—33no—
22C4⋊#226—33no—
23#5031—31no—
24#5031—31no—
25#5028—32no—
26#5028—32no—
27#5027—31no—
28#5027—31no—
29#4966—32no—
30#4924—31no—
31C22×C32×C4—41no—
32C2×#638—42no—
33C2×C4⋊(C3⋊D3)—41no—
34C2×C3×SL(2,𝔽3)D412yescomplemented
35C2×C3×SL(2,𝔽3)C2311yescharacteristic
36C3⋊GL(2,𝔽3)—22no—
37C3⋊#226—22no—
38#3752—42no—
39C4⋊(C3⋊Dic3)—41no—
40#3037—91noSylow
41C22×C3×Q8C2×D311yescharacteristic
42C2×#235—32no—
43C22×SL(2,𝔽3)C2×D313yes—
44#983—612no—
45C2×GL(2,𝔽3)—36no—
46C2×#226—36no—
47C3×C4×Q8—31no—
48C3×C4×Q8—31no—
49C3×C4×Q8C2×D311yescharacteristic
50C3×C4×Q8C2×D311yescharacteristic
51C2×C3×C4⋊C4—31no—
52C2×C3×C42—31no—
53C2×#216—32no—
54C2×#216—32no—
55#942—64no—
56#942—64no—
57C2×#215—32no—
58C2×#215—32no—
59#930—61no—
60#930—61no—
61#930—123no—
62C2×#208—32no—
63#875—32no—
64#874—31no—
65C4×D12—61no—
66C4×D12—61no—
67#869—31no—
68C4×SL(2,𝔽3)C2×D313yes—
69C4×SL(2,𝔽3)C2×D313yes—
70#810—32no—
71#810—32no—
72#809—32no—
73#809—32no—
74C4⋊(C3⋊C8)—31no—
75C4⋊(C3⋊C8)—31no—
76C4⋊(C3⋊C8)—32no—
77C23×C32—41no—
78C22×C3⋊D3—42no—
79C2×C32×C4—41no—
80C2×C32×C4—41no—
81C2×C32×C4—82no—
82#638—84no—
83C2×C3⋊Dic3—42no—
84C4⋊(C3⋊D3)—82no—
85C3×SL(2,𝔽3)C2×D411yescharacteristic
86C2×#137—92no—
87#532—91no—
88C2×C4×Q8C3⋊D311yescharacteristic, complemented
89#464—92no—
90#464—92no—
91#462—91no—
92#462—91no—
93#460—91no—
94#460—91no—
95#423—91no—
96#417—92no—
97C2×C3×Q8—31no—
98C2×C3×Q8—32no—
99C2×C3×Q8D1212yescomplemented
100C2×C3×Q8C22×D311yescharacteristic
101C22×C3×C4—31no—
102C22×C3×C4—31no—
103C22×C3×C4—43no—
104C22×C3×C4S411yescharacteristic
105C2×#67—62no—
106C2×#67—126no—
107#235—64no—
108#235—64no—
109C2×D12—32no—
110C2×D12—61no—
111C2×D12—61no—
112C2×D12—123no—
113C2×C4×D3—62no—
114C2×Dic6—32no—
115C2×SL(2,𝔽3)D1216yescomplemented
116C2×SL(2,𝔽3)C22×D313yes—
117GL(2,𝔽3)—66no—
118#226—66no—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—31no—
121C3×C4⋊C4—61no—
122C3×C42—31no—
123C3×C42—31no—
124C3×C42—31no—
125C3×C42—31no—
126#216—64no—
127#216—62no—
128#216—62no—
129#215—64no—
130#215—62no—
131#215—62no—
132(C2×D3)⋊C4—62no—
133(C2×D3)⋊C4—62no—
134(C2×D3)⋊C4—126no—
135C4⋊Dic3—61no—
136C4⋊Dic3—61no—
137C4⋊Dic3—123no—
138#208—64no—
139C2×C3⋊C8—32no—
140C2×C3⋊C8—32no—
141C22×C32—42no—
142C22×C32—41no—
143C22×C32—81no—
144C22×C32—81no—
145C2×C3⋊D3—82no—
146C2×C3⋊D3—82no—
147C2×C3⋊D3—82no—
148C32×C4—42no—
149C32×C4—81no—
150C3⋊Dic3—82no—
151C2×#41—92no—
152C22×Q8C2×C3⋊D311yescharacteristic
153#137—184no—
154#137—184no—
155C2×Q16—92no—
156C2×Q16—92no—
157C2×SD16—92no—
158C2×SD16—92no—
159C2×M16—92no—
160C4⋊Q8—91no—
161#127—91no—
162#124—92no—
163#121—181no—
164#121—181no—
165C4×Q8—31no—
166C4×Q8—31no—
167C4×Q8C2×C3⋊D311yescharacteristic
168C4×Q8C2×C3⋊D311yescharacteristic
169C4×D4—181no—
170C4×D4—181no—
171C2×C4⋊C4—31no—
172C2×C42—31no—
173C4⋊C8—91no—
174C4⋊C8—91no—
175C4⋊C8—92no—
176#103—92no—
177#103—92no—
178#102—92no—
179#102—92no—
180C23×C3—43no—
181C23×C3C2×S411yescharacteristic
182C22×D3—62no—
183C22×D3—126no—
184C3×Q8—32no—
185C3×Q8—31no—
186C3×Q8—61no—
187C3×Q8C2×D1211yescharacteristic
188C2×C3×C4—22no—
189C2×C3×C4—32no—
190C2×C3×C4—32no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—43no—
196C2×C3×C4—43no—
197C2×C3×C4—86no—
198C2×C3×C4—121no—
199C2×C3×C4C2×S411yescharacteristic
200C2×C3×C4C2×S411yescharacteristic
201#67—124no—
202#67—2412no—
203C2×Dic3—62no—
204C2×Dic3—126no—
205D12—62no—
206D12—62no—
207D12—122no—
208D12—122no—
209D12—246no—
210C4×D3—124no—
211Dic6—62no—
212Dic6—62no—
213SL(2,𝔽3)C2×D1213yes—
214C3⋊C8—62no—
215C3⋊C8—62no—
216C2×C32—41no—
217C2×C32—41no—
218C2×C32—41no—
219C2×C32—82no—
220C3⋊D3—162no—
221#41—184no—
222#41—184no—
223C2×Q8—31no—
224C2×Q8—32no—
225C2×Q8—92no—
226C2×Q8C4⋊(C3⋊D3)12yescomplemented
227C2×Q8C22×C3⋊D311yescharacteristic
228C2×D4—92no—
229C2×D4—181no—
230C2×D4—182no—
231C2×D4—181no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—182no—
235C22×C4C3⋊S411yescharacteristic
236Q16—184no—
237Q16—182no—
238Q16—182no—
239SD16—184no—
240SD16—182no—
241SD16—182no—
242M16—184no—
243C2×C8—92no—
244C2×C8—92no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—61no—
248C4⋊C4—181no—
249C4⋊C4—181no—
250C22⋊C4—182no—
251C22⋊C4—182no—
252C42—31no—
253C42—31no—
254C42—31no—
255C42—31no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—46no—
259C22×C3—43no—
260C22×C3—83no—
261C22×C3—83no—
262C22×C3C4⋊S412yes—
263C22×C3C22×S411yescharacteristic
264C2×D3—122no—
265C2×D3—122no—
266C2×D3—122no—
267C2×D3—246no—
268C2×D3—246no—
269C2×D3—246no—
270C3×C4—21no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—32no—
274C3×C4—46no—
275C3×C4—61no—
276C3×C4—61no—
277C3×C4—83no—
278C3×C4#98312yescomplemented
279Dic3—122no—
280Dic3—246no—
281C32—41noSylow
282C23—182no—
283C23C2×C3⋊S411yescharacteristic
284Q8—32no—
285Q8—31no—
286Q8—61no—
287Q8—182no—
288Q8—182no—
289Q8C2×C4⋊(C3⋊D3)11yescharacteristic
290D4—182no—
291D4—182no—
292D4—362no—
293D4—364no—
294D4—362no—
295C2×C4—22no—
296C2×C4—32no—
297C2×C4—32no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—121no—
303C2×C4—182no—
304C2×C4—364no—
305C2×C4C2×C3⋊S411yescharacteristic
306C2×C4C2×C3⋊S411yescharacteristic
307C8—182no—
308C8—182no—
309C2×C3—22no—
310C2×C3—43no—
311C2×C3—43no—
312C2×C3—43no—
313C2×C3—86no—
314C2×C3C2×C4⋊S411yescharacteristic
315C2×C3C2×#98311yescharacteristic
316C2×C3#592311yescharacteristic
317D3—242no—
318D3—486no—
319C22—21no—
320C22—21no—
321C22—362no—
322C22—362no—
323C22—362no—
324C22C4⋊(C3⋊S4)12yes—
325C22C22×C3⋊S411yescharacteristic
326C4—21no—
327C4—31no—
328C4—32no—
329C4—31no—
330C4—61no—
331C4—61no—
332C4—362no—
333C4#6426412yescomplemented
334C3—43no—
335C3#8533111yescharacteristic, complemented
336C2—22no—
337C2—722no—
338C2C2×C4⋊(C3⋊S4)11yescharacteristic
339C2C2×#6426411yescharacteristic
340C2#1059658611yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×C4×Q8D3
socleC22×C3C22×S4
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#64264C2semidirect
2. C2×C3×C4×SL(2,𝔽3)C2semidirect
3. #10593629C2semidirect
4. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
5. C4⋊(C3⋊#226)C2semidirect
16. C2×C3×C4×Q8D3semidirect
19. C2×C4×SL(2,𝔽3)D3semidirect
34. C2×C3×SL(2,𝔽3)D4semidirect
88. C2×C4×Q8C3⋊D3semidirect
99. C2×C3×Q8D12semidirect
115. C2×SL(2,𝔽3)D12semidirect
226. C2×Q8C4⋊(C3⋊D3)semidirect
278. C3×C4#983semidirect
333. C4#64264semidirect
335. C3#85331semidirect

Maximal subgroups

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

subgroupindex
C2×#642642
C2×C3×C4×SL(2,𝔽3)2
#105936292
C4⋊(C3⋊GL(2,𝔽3))2
C4⋊(C3⋊#226)2
#853313
#824413
#641394