Finite Group Databasebeta

← #11922893

Subgroups — #11922893

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×#64266C212yesmaximal, complemented
2C3×C4×#41⋊C3C211yesmaximal, characteristic, complemented
3#10593629C212yesmaximal, complemented
4C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
5C4⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
6#85344—33nomaximal
7#82525—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#64266—24no—
10C2×C3⋊GL(2,𝔽3)C2212yes—
11C2×C3⋊#226C2212yes—
12C4×C4⋊(C3⋊D3)—41nomaximal
13C3×C4×SL(2,𝔽3)C2211yescharacteristic
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C2×#985—36no—
16C3×C4×#41D311yescharacteristic, complemented
17C2×#950—32no—
18#5477—31no—
19C4×#41⋊C3D313yescomplemented
20#5395—36no—
21C4⋊GL(2,𝔽3)—33no—
22C4⋊#226—33no—
23#5031—31no—
24#5028—32no—
25#5027—31no—
26#5019—31no—
27#5016—32no—
28#5015—31no—
29#5002—32no—
30C2×C4⋊(C3⋊C8)—31no—
31C2×C4⋊(C3⋊D3)—41no—
32C2×C4×C3⋊D3—42no—
33C3×#41⋊C3D412yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C3⋊GL(2,𝔽3)—22no—
36C3⋊#226—22no—
37C32×C42—41no—
38#3752—42no—
39C4⋊(C3⋊Dic3)—41no—
40#3035—91noSylow
41C2×C3×#41C2×D311yescharacteristic
42C2×#235—32no—
43C2×#41⋊C3C2×D313yes—
44#985—612no—
45C2×GL(2,𝔽3)—36no—
46C2×#226—36no—
47C3×C4×Q8C2×D311yescharacteristic
48C3×C4×Q8C2×D311yescharacteristic
49C3×C4×D4—31no—
50C3×C4×D4—31no—
51C3×#117—31no—
52C2×C3×C42—31no—
53#950—64no—
54#950—64no—
55C2×#216—32no—
56C2×#215—32no—
57C2×#214—32no—
58C2×#213—32no—
59#930—61no—
60#930—61no—
61C22×C3⋊C8—32no—
62#875—32no—
63#874—31no—
64C4×D12—61no—
65C4×D12—61no—
66C4×D12—123no—
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)—32no—
76C4⋊(C3⋊C8)—31no—
77C22×C3⋊D3—42no—
78C2×C32×C4—41no—
79C2×C32×C4—41no—
80C2×C32×C4—41no—
81C2×C3⋊Dic3—42no—
82C4⋊(C3⋊D3)—44no—
83C4×C3⋊D3—84no—
84C3×SL(2,𝔽3)C2×D411yescharacteristic
85C2×#135—92no—
86#532—91no—
87C4×#41C3⋊D311yescharacteristic, complemented
88#464—92no—
89#463—92no—
90#462—91no—
91#461—91no—
92#460—91no—
93#459—91no—
94C2×C4⋊C8—91no—
95#416—92no—
96C3×#41—32no—
97C3×#41D1212yescomplemented
98C2×C3×Q8C22×D311yescharacteristic
99C2×C3×D4—31no—
100C22×C3×C4—31no—
101C22×C3×C4—31no—
102C22×C3×C4—31no—
103C2×#67—62no—
104#235—64no—
105#235—64no—
106C2×D12—32no—
107C2×D12—61no—
108C2×D12—61no—
109C2×D12—123no—
110C2×C4×D3—62no—
111C2×C4×D3—126no—
112C2×Dic6—32no—
113#41⋊C3D1216yescomplemented
114C2×SL(2,𝔽3)C22×D313yes—
115GL(2,𝔽3)—66no—
116#226—66no—
117C3×C4⋊C4—31no—
118C3×C4⋊C4—31no—
119C3×C22⋊C4—61no—
120C3×C42—31no—
121C3×C42—31no—
122C3×C42—31no—
123C3×C42—43no—
124C3×C42S411yescharacteristic
125#216—62no—
126#216—62no—
127#215—62no—
128#215—62no—
129#214—64no—
130#213—64no—
131(C2×D3)⋊C4—62no—
132(C2×D3)⋊C4—62no—
133(C2×D3)⋊C4—126no—
134C4⋊Dic3—61no—
135C4⋊Dic3—61no—
136C4⋊Dic3—123no—
137C2×C3⋊C8—32no—
138C2×C3⋊C8—32no—
139C2×C3⋊C8—64no—
140C22×C32—41no—
141C2×C3⋊D3—44no—
142C2×C3⋊D3—82no—
143C2×C3⋊D3—82no—
144C32×C4—42no—
145C32×C4—42no—
146C32×C4—81no—
147C3⋊Dic3—82no—
148C2×#41—92no—
149C2×#41C2×C3⋊D311yescharacteristic
150#135—184no—
151#135—184no—
152C2×Q16—92no—
153C2×SD16—92no—
154C2×SD16—92no—
155C2×D8—92no—
156C22×C8—92no—
157C4⋊Q8—91no—
158#127—91no—
159#124—92no—
160#121—181no—
161#121—181no—
162C4×Q8C2×C3⋊D311yescharacteristic
163C4×Q8C2×C3⋊D311yescharacteristic
164C4×D4—31no—
165C4×D4—31no—
166C4×D4—181no—
167C4×D4—181no—
168#117—31no—
169C2×C42—31no—
170C4⋊C8—91no—
171C4⋊C8—91no—
172C4⋊C8—92no—
173#103—92no—
174#103—92no—
175#102—92no—
176#102—92no—
177C23×C3—31no—
178C22×D3—62no—
179C22×D3—126no—
180C3×Q8—31no—
181C3×Q8C2×D1211yescharacteristic
182C3×D4—32no—
183C3×D4—61no—
184C2×C3×C4—31no—
185C2×C3×C4—32no—
186C2×C3×C4—32no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—43no—
190C2×C3×C4—43no—
191C2×C3×C4—43no—
192C2×C3×C4—62no—
193C2×C3×C4—121no—
194C2×C3×C4C2×S411yescharacteristic
195C2×C3×C4C2×S411yescharacteristic
196C2×C3×C4C2×S411yescharacteristic
197#67—124no—
198C2×Dic3—62no—
199C2×Dic3—126no—
200D12—62no—
201D12—62no—
202D12—1212no—
203D12—122no—
204D12—122no—
205C4×D3—124no—
206C4×D3—2412no—
207Dic6—62no—
208Dic6—62no—
209SL(2,𝔽3)C2×D1213yes—
210C3⋊C8—62no—
211C3⋊C8—62no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C3⋊D3—84no—
216#41—32no—
217#41—184no—
218#41—184no—
219#41C4⋊(C3⋊D3)12yescomplemented
220C2×Q8—92no—
221C2×Q8C22×C3⋊D311yescharacteristic
222C2×D4—31no—
223C2×D4—92no—
224C2×D4—181no—
225C2×D4—182no—
226C2×D4—181no—
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—182no—
231Q16—182no—
232Q16—182no—
233SD16—184no—
234SD16—182no—
235SD16—182no—
236D8—184no—
237C2×C8—92no—
238C2×C8—92no—
239C2×C8—184no—
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—181no—
243C4⋊C4—181no—
244C22⋊C4—61no—
245C22⋊C4—182no—
246C22⋊C4—182no—
247C42—31no—
248C42—31no—
249C42—31no—
250C42C3⋊S411yescharacteristic
251C22×C3—32no—
252C22×C3—43no—
253C22×C3—61no—
254C22×C3—61no—
255C22×C3C22×S411yescharacteristic
256C2×D3—1212no—
257C2×D3—122no—
258C2×D3—122no—
259C2×D3—122no—
260C2×D3—246no—
261C2×D3—246no—
262C3×C4—21no—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—46no—
266C3×C4—46no—
267C3×C4—61no—
268C3×C4—61no—
269C3×C4—83no—
270C3×C4C4⋊S412yes—
271C3×C4#98512yescomplemented
272Dic3—122no—
273Dic3—246no—
274C32—41noSylow
275C23—31no—
276C23—182no—
277Q8—31no—
278Q8—182no—
279Q8—182no—
280Q8C2×C4⋊(C3⋊D3)11yescharacteristic
281D4—32no—
282D4—61no—
283D4—182no—
284D4—182no—
285D4—362no—
286D4—364no—
287D4—362no—
288C2×C4—31no—
289C2×C4—32no—
290C2×C4—32no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—62no—
294C2×C4—121no—
295C2×C4—182no—
296C2×C4—364no—
297C2×C4C2×C3⋊S411yescharacteristic
298C2×C4C2×C3⋊S411yescharacteristic
299C2×C4C2×C3⋊S411yescharacteristic
300C8—182no—
301C8—182no—
302C2×C3—43no—
303C2×C3—43no—
304C2×C3—43no—
305C2×C3—62no—
306C2×C3C2×C4⋊S411yescharacteristic
307C2×C3C2×#98511yescharacteristic
308C2×C3#592311yescharacteristic
309D3—2412no—
310D3—242no—
311C22—32no—
312C22—61no—
313C22—61no—
314C22—362no—
315C22—362no—
316C22—362no—
317C22C22×C3⋊S411yescharacteristic
318C4—21no—
319C4—31no—
320C4—31no—
321C4—61no—
322C4—61no—
323C4—362no—
324C4C4⋊(C3⋊S4)12yes—
325C4#6426612yescomplemented
326C3—43no—
327C3#8534411yescharacteristic, complemented
328C2—62no—
329C2—722no—
330C2C2×C4⋊(C3⋊S4)11yescharacteristic
331C2C2×#6426611yescharacteristic
332C2#1059658611yescharacteristic
subgroupG/H
centerC2×C4C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FrattiniC22C22×C3⋊S4
FittingC3×C4×#41D3
socleC22×C3C22×S4
hypercenterC42C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#64266C2semidirect
2. C3×C4×#41⋊C3C2semidirect
3. #10593629C2semidirect
4. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
5. C4⋊(C3⋊#226)C2semidirect
16. C3×C4×#41D3semidirect
19. C4×#41⋊C3D3semidirect
33. C3×#41⋊C3D4semidirect
87. C4×#41C3⋊D3semidirect
97. C3×#41D12semidirect
113. #41⋊C3D12semidirect
219. #41C4⋊(C3⋊D3)semidirect
271. C3×C4#985semidirect
325. C4#64266semidirect
327. C3#85344semidirect

Maximal subgroups

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

subgroupindex
C2×#642662
C3×C4×#41⋊C32
#105936292
C4⋊(C3⋊GL(2,𝔽3))2
C4⋊(C3⋊#226)2
#853443
#825253
C4×C4⋊(C3⋊D3)4