Finite Group Databasebeta

← #11922960

Subgroups — #11922960

Order 1152.

762 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 71 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×#64267C212yesmaximal, complemented
2C3×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3#10593661C212yesmaximal, complemented
4C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
5C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#85410—33nomaximal
7#83578—31nomaximal
8C2×C3×#41⋊C3C2212yescomplemented
9#64267—24no—
10C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
11C2×C3⋊GL(2,𝔽3)C2212yescomplemented
12#64104—41nomaximal
13C3×C4×SL(2,𝔽3)C2211yescharacteristic, complemented
14C3⋊#859C2211yescharacteristic, complemented
15C2×#986—36no—
16C3×#550D311yescharacteristic, complemented
17C2×#949—32no—
18D4×D12—31no—
19C4⋊(#41⋊C3)D313yescomplemented
20#5427—36no—
21C4⋊GL(2,𝔽3)—33no—
22C4×GL(2,𝔽3)—33no—
23#5238—32no—
24#5073—31no—
25#5072—31no—
26#5039—32no—
27#5036—32no—
28#5027—31no—
29C4×#215—31no—
30#5012—31no—
31C2×C4⋊(C3⋊D3)—41no—
32C2×C4⋊(C3⋊D3)—42no—
33C2×C4×C3⋊D3—41no—
34C3×#41⋊C3—22no—
35C2×C3×SL(2,𝔽3)C2311yescharacteristic
36C3⋊GL(2,𝔽3)—22no—
37C3⋊GL(2,𝔽3)D412yescomplemented
38C32×C4⋊C4—41no—
39#3752—42no—
40#3274—91noSylow
41C2×C3×#41C2×D312yes—
42C2×D3×D4—61no—
43C22×D12—32no—
44C2×#41⋊C3C2×D316yescomplemented
45#986—612no—
46C2×GL(2,𝔽3)—33no—
47C2×GL(2,𝔽3)—36no—
48C3×#127—31no—
49C3×#121—32no—
50C3×C4×Q8C2×D311yescharacteristic
51C3×C4×D4—31no—
52#949—64no—
53#949—64no—
54C2×#215—31no—
55C2×#215—32no—
56C2×#213—32no—
57C2×#213—32no—
58C2×#213—32no—
59#930—32no—
60C2×#208—32no—
61#895—61no—
62#895—123no—
63#882—62no—
64#874—31no—
65C4×D12—31no—
66C4×SL(2,𝔽3)C2×D313yescomplemented
67#859—33no—
68#835—31no—
69#832—32no—
70#830—32no—
71#809—31no—
72#809—32no—
73#808—31no—
74C4⋊(C3⋊C8)—31no—
75C4×C3⋊C8—31no—
76C22×C3⋊D3—41no—
77C22×C3⋊D3—42no—
78C2×C32×C4—41no—
79C2×C32×C4—42no—
80C2×C3⋊Dic3—41no—
81C4⋊(C3⋊D3)—82no—
82C4⋊(C3⋊D3)—84no—
83C4×C3⋊D3—82no—
84C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
85C2×AGL(1,ℤ/8)—92no—
86#550C3⋊D311yescharacteristic, complemented
87D42—91no—
88#493—91no—
89#469—92no—
90#460—91no—
91#459—91no—
92#449—92no—
93#447—92no—
94C4×SD16—91no—
95#436—91no—
96C23×D3—62no—
97C3×#41—22no—
98C3×#41—62no—
99C2×C3×Q8C22×D311yescharacteristic
100C2×C3×D4—32no—
101C2×C3×D4—31no—
102C2×C3×D4—61no—
103C22×C3×C4—32no—
104C2×#67—62no—
105D3×D4—124no—
106C2×D12—31no—
107C2×D12—32no—
108C2×D12—64no—
109C2×D12—61no—
110C2×D12—62no—
111C2×D12—61no—
112C2×D12—123no—
113C2×D12—126no—
114C2×D12—122no—
115C2×C4×D3—61no—
116C2×C4×D3—123no—
117#41⋊C3—26no—
118C2×SL(2,𝔽3)C22×D313yes—
119GL(2,𝔽3)—36no—
120GL(2,𝔽3)—66no—
121C3×C4⋊C4—31no—
122C3×C4⋊C4—43no—
123C3×C4⋊C4S411yescharacteristic
124C3×C22⋊C4—32no—
125C3×C42—31no—
126#215—32no—
127#215—32no—
128#215—62no—
129#215—62no—
130#213—64no—
131#213—64no—
132#213—122no—
133(C2×D3)⋊C4—61no—
134(C2×D3)⋊C4—62no—
135(C2×D3)⋊C4—126no—
136C4⋊Dic3—31no—
137#208—64no—
138C2×C3⋊C8—31no—
139C2×C3⋊C8—31no—
140C2×C3⋊C8—32no—
141C22×C32—41no—
142C2×C3⋊D3—42no—
143C2×C3⋊D3—81no—
144C2×C3⋊D3—81no—
145C2×C3⋊D3—82no—
146C2×C3⋊D3—82no—
147C2×C3⋊D3—82no—
148C32×C4—42no—
149C32×C4—82no—
150C3⋊Dic3—81no—
151C2×#41C2×C3⋊D312yes—
152C22×D4—92no—
153C22×D4—181no—
154AGL(1,ℤ/8)—184no—
155AGL(1,ℤ/8)—184no—
156C2×SD16—91no—
157C2×SD16—92no—
158C2×D8—92no—
159C2×D8—92no—
160C2×D8—92no—
161C2×M16—92no—
162#127—31no—
163#127—91no—
164#121—32no—
165#121—92no—
166#121—181no—
167#120—182no—
168C4×Q8C2×C3⋊D311yescharacteristic
169C4×D4—31no—
170C4×D4—91no—
171#106—91no—
172C4⋊C8—91no—
173#103—91no—
174#102—91no—
175#102—92no—
176#102—92no—
177C22⋊C8—92no—
178C4×C8—91no—
179C23×C3—32no—
180C22×D3—61no—
181C22×D3—62no—
182C22×D3—123no—
183C22×D3—126no—
184C22×D3—124no—
185C22×D3—124no—
186C22×D3—242no—
187C3×Q8—31no—
188C3×Q8D3×D411yescharacteristic
189C3×D4—62no—
190C3×D4—62no—
191C3×D4—62no—
192C3×D4—241no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—43no—
196C2×C3×C4—46no—
197C2×C3×C4—62no—
198C2×C3×C4—62no—
199C2×C3×C4C2×S412yes—
200C2×C3×C4C2×S411yescharacteristic
201#67—124no—
202C2×Dic3—61no—
203C2×Dic3—123no—
204D12—32no—
205D12—61no—
206D12—62no—
207D12—62no—
208D12—124no—
209D12—122no—
210D12—246no—
211D12—2412no—
212D12—241no—
213C4×D3—122no—
214C4×D3—246no—
215SL(2,𝔽3)D3×D413yescomplemented
216C3⋊C8—32no—
217C3⋊C8—62no—
218C3⋊C8—61no—
219C2×C32—41no—
220C2×C32—41no—
221C2×C32—41no—
222C3⋊D3—82no—
223C3⋊D3—162no—
224C24—182no—
225#41—22no—
226#41—62no—
227C2×Q8C22×C3⋊D311yescharacteristic
228C2×D4—32no—
229C2×D4—31no—
230C2×D4—61no—
231C2×D4—91no—
232C2×D4—92no—
233C2×D4—181no—
234C2×D4—184no—
235C2×D4—182no—
236C2×D4—182no—
237C2×D4—181no—
238C2×D4—362no—
239C2×D4—364no—
240C22×C4—32no—
241C22×C4—181no—
242SD16—92no—
243SD16—92no—
244SD16—182no—
245SD16—182no—
246D8—184no—
247D8—184no—
248D8—362no—
249M16—184no—
250C2×C8—91no—
251C2×C8—91no—
252C2×C8—92no—
253C4⋊C4—31no—
254C4⋊C4—91no—
255C4⋊C4C3⋊S411yescharacteristic
256C22⋊C4—32no—
257C22⋊C4—181no—
258C22⋊C4—182no—
259C42—31no—
260C22×C3—43no—
261C22×C3—62no—
262C22×C3—62no—
263C22×C3—62no—
264C22×C3C22×S411yescharacteristic
265C2×D3—62no—
266C2×D3—126no—
267C2×D3—121no—
268C2×D3—121no—
269C2×D3—122no—
270C2×D3—122no—
271C2×D3—122no—
272C2×D3—243no—
273C2×D3—243no—
274C2×D3—246no—
275C2×D3—246no—
276C2×D3—246no—
277C2×D3—248no—
278C3×C4—22no—
279C3×C4—31no—
280C3×C4—31no—
281C3×C4—46no—
282C3×C4—61no—
283C3×C4—86no—
284C3×C4#98612yescomplemented
285Dic3—121no—
286Dic3—243no—
287C32—41noSylow
288C23—32no—
289C23—181no—
290C23—182no—
291C23—364no—
292C23—364no—
293C23—722no—
294Q8—31no—
295Q8D4×C3⋊D311yescharacteristic
296D4—62no—
297D4—62no—
298D4—62no—
299D4—92no—
300D4—181no—
301D4—182no—
302D4—182no—
303D4—241no—
304D4—364no—
305D4—364no—
306D4—362no—
307D4—721no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—181no—
313C2×C4—362no—
314C2×C4C2×C3⋊S412yes—
315C2×C4C2×C3⋊S411yescharacteristic
316C8—92no—
317C8—182no—
318C8—181no—
319C2×C3—43no—
320C2×C3—43no—
321C2×C3—43no—
322C2×C3—122no—
323C2×C3D4×S411yescharacteristic
324C2×C3C2×#98611yescharacteristic
325C2×C3#592511yescharacteristic
326D3—122no—
327D3—246no—
328D3—242no—
329D3—486no—
330C22—62no—
331C22—62no—
332C22—62no—
333C22—182no—
334C22—361no—
335C22—361no—
336C22—362no—
337C22—362no—
338C22—362no—
339C22—728no—
340C22C22×C3⋊S411yescharacteristic
341C4—22no—
342C4—31no—
343C4—31no—
344C4—61no—
345C4—361no—
346C4#6426712yescomplemented
347C3—43no—
348C3#8541011yescharacteristic, complemented
349C2—122no—
350C2—362no—
351C2—722no—
352C2D4×C3⋊S411yescharacteristic
353C2C2×#6426711yescharacteristic
354C2#1059658811yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#550D3
socleC22×C3C22×S4
hypercenterC4⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#64267C2semidirect
2. C3×C4⋊(#41⋊C3)C2semidirect
3. #10593661C2semidirect
4. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
5. C4×C3⋊GL(2,𝔽3)C2semidirect
8. C2×C3×#41⋊C3C22semidirect
11. C2×C3⋊GL(2,𝔽3)C22semidirect
13. C3×C4×SL(2,𝔽3)C22semidirect
14. C3⋊#859C22semidirect
16. C3×#550D3semidirect
19. C4⋊(#41⋊C3)D3semidirect
37. C3⋊GL(2,𝔽3)D4semidirect
44. C2×#41⋊C3C2×D3semidirect
66. C4×SL(2,𝔽3)C2×D3semidirect
84. C3×SL(2,𝔽3)C2×D4semidirect
86. #550C3⋊D3semidirect
215. SL(2,𝔽3)D3×D4semidirect
284. C3×C4#986semidirect
346. C4#64267semidirect
348. C3#85410semidirect

Maximal subgroups

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

subgroupindex
C2×#642672
C3×C4⋊(#41⋊C3)2
#105936612
C4⋊(C3⋊GL(2,𝔽3))2
C4×C3⋊GL(2,𝔽3)2
#854103
#835783
#641044