Finite Group Databasebeta

← #11782521

Subgroups — #11782521

Order 1152.

439 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 102 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
1#10592787C211yesmaximal, characteristic
2C3×#5080C211yesmaximal, characteristic, complemented
3C3×#5079C211yesmaximal, characteristic, complemented
4#10590373C211yesmaximal, characteristic
5#10590290C211yesmaximal, characteristic, complemented
6#10590281C211yesmaximal, characteristic
7C3⋊C82C211yesmaximal, characteristic, complemented
8#71334—31nomaximal
9#71213—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11#64083C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C2×#3717—22no—
15#63636—22no—
16C4×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×#835—22no—
18C3×#808—22no—
19C3×C4×C3⋊C8C2211yescharacteristic
20C3×C4×C3⋊C8C2211yescharacteristic
21#63565—22no—
22#63564—22no—
23C3×#498—31no—
24C3×#487—31no—
25#5093—31no—
26#5088—31no—
27#5088—31no—
28#5088—31no—
29#5088—61no—
30#5085—31no—
31#5080D311yescharacteristic, complemented
32#5079—31no—
33#5079D311yescharacteristic, complemented
34#4694—31no—
35#4693—31no—
36C8×C3⋊C8—31no—
37C8×C3⋊C8—31no—
38C2×C32×Q8—21no—
39C2×C4⋊(C3⋊D3)—21no—
40C2×C4⋊(C3⋊D3)—21no—
41C2×C4⋊(C3⋊D3)—21no—
42C2×C3×Dic6—21no—
43C2×C3×Dic6—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C42C2311yescharacteristic
47C2×C3⋊(C3⋊C8)D412yes—
48C3×C4⋊Dic3—21no—
49C3×C4⋊Dic3—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C2×C3×C3⋊C8D412yes—
53C2×C3×C3⋊C8D412yes—
54#3717—44no—
55#1708—91noSylow
56C3×C4⋊Q8—21no—
57C3×C4⋊Q8—31no—
58C3×C4⋊Q8—31no—
59C3×C4⋊Q8C2×D311yescharacteristic
60C3×C4⋊Q8C2×D311yescharacteristic
61C2×#215—62no—
62C2×#204—62no—
63#874—31no—
64#874—31no—
65#874—61no—
66#869C2×D311yescharacteristic
67#869C2×D311yescharacteristic
68C3×#106—62no—
69C3×#103—62no—
70C3×C4×C8—31no—
71C3×C4×C8—31no—
72#835—22no—
73#835—62no—
74#821—62no—
75#810—62no—
76#809—62no—
77#809—62no—
78#809—64no—
79#809—62no—
80#808—22no—
81C4×C3⋊C8—31no—
82C4×C3⋊C8—31no—
83C4×C3⋊C8—61no—
84C4×C3⋊C8C2×D311yescharacteristic
85C4×C3⋊C8C2×D311yescharacteristic
86C22×C3⋊D3—41no—
87C32×Q8—42no—
88C2×C32×C4—41no—
89C2×C32×C4C2×D411yescharacteristic
90C2×C32×C4C2×D411yescharacteristic
91C2×C32×C4C2×D411yescharacteristic
92C4⋊(C3⋊D3)—42no—
93C4⋊(C3⋊D3)—42no—
94C4⋊(C3⋊D3)—42no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3—41no—
97C3×Dic6—42no—
98C3×Dic6—42no—
99C3⋊(C3⋊C8)—22no—
100C3×C3⋊C8—22no—
101C3×C3⋊C8SD1614yes—
102#498—31no—
103#492—91no—
104#487—31no—
105#487—91no—
106#486—91no—
107#486—91no—
108C82—91no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—41no—
112C2×C3×Q8—61no—
113C2×C3×Q8—61no—
114C2×D12—62no—
115C2×D12—61no—
116C2×D12—61no—
117C2×D12—61no—
118C2×D12—61no—
119C2×D12—61no—
120C2×D12—61no—
121C2×D12—121no—
122C2×D12—121no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×C3×C8—32no—
126C2×C3×C8—32no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—61no—
134C3×C4⋊C4—61no—
135C3×C4⋊C4—61no—
136C3×C4⋊C4—61no—
137C3×C42—21no—
138C3×C42C22×D311yescharacteristic
139C3×C42C22×D311yescharacteristic
140#215—124no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C2×C3⋊C8—32no—
146C2×C3⋊C8—32no—
147C2×C3⋊C8—62no—
148C2×C3⋊C8#6712yes—
149C2×C3⋊C8#6712yes—
150#204—124no—
151C22×C32#12711yescharacteristic
152C2×C3⋊D3—81no—
153C2×C3⋊D3—81no—
154C2×C3⋊D3—81no—
155C32×C4—81no—
156C32×C4C2×SD1612yes—
157C32×C4#13512yes—
158C32×C4#13512yes—
159C3×Dic3—81no—
160C3×Dic3—81no—
161C2×SD16—182no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164C4⋊Q8D3211yescharacteristic
165#127—91no—
166#106—62no—
167#103—62no—
168#103—182no—
169#102—182no—
170#102—182no—
171C4×C8—31no—
172C4×C8—31no—
173C4×C8—91no—
174C22×D3—122no—
175C22×D3—121no—
176C22×D3—121no—
177C3×Q8—42no—
178C3×Q8—42no—
179C3×Q8—82no—
180C3×Q8—122no—
181C3×Q8—122no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—42no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—121no—
189C2×C3×C4—121no—
190C2×C3×C4D3×D411yescharacteristic
191C2×C3×C4D3×D411yescharacteristic
192C2×C3×C4D3×D411yescharacteristic
193C2×C3×C4D3×D411yescharacteristic
194C2×C3×C4C2×#6711yescharacteristic
195C2×C3×C4C2×#6711yescharacteristic
196C2×Dic3—41no—
197C2×Dic3—41no—
198D12—124no—
199D12—122no—
200D12—122no—
201D12—122no—
202D12—122no—
203D12—122no—
204D12—122no—
205D12—242no—
206D12—242no—
207Dic6—42no—
208Dic6—42no—
209C3×C8—34no—
210C3×C8—62no—
211C3⋊C8—22no—
212C3⋊C8—62no—
213C3⋊C8—62no—
214C3⋊C8—122no—
215C3⋊C8#21514yes—
216C2×C32#49211yescharacteristic
217C2×C32#49511yescharacteristic
218C2×C32#49511yescharacteristic
219C3⋊D3—161no—
220C2×Q8—21no—
221C2×Q8—61no—
222C2×Q8—61no—
223C2×D4—181no—
224C2×D4—181no—
225C2×D4—181no—
226SD16—364no—
227C2×C8—32no—
228C2×C8—32no—
229C2×C8—92no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C42C2×D3211yescharacteristic
237C22×C3—21no—
238C22×C3#94011yescharacteristic
239C22×C3#94011yescharacteristic
240C2×D3—242no—
241C2×D3—242no—
242C2×D3—242no—
243C2×D3—241no—
244C2×D3—241no—
245C2×D3—241no—
246C2×D3—241no—
247C2×D3—241no—
248C2×D3—241no—
249C3×C4—22no—
250C3×C4—22no—
251C3×C4—22no—
252C3×C4—82no—
253C3×C4—81no—
254C3×C4—81no—
255C3×C4—241no—
256C3×C4—241no—
257C3×C4D3×SD1612yes—
258C3×C4#91612yes—
259C3×C4#91612yes—
260C3×C4#91912yes—
261C3×C4C2×#21512yes—
262C3×C4#95012yes—
263Dic3—81no—
264Dic3—81no—
265C32#170811yescharacteristic, complemented, Sylow
266C23—361no—
267Q8—42no—
268Q8—122no—
269Q8—122no—
270D4—362no—
271D4—362no—
272D4—362no—
273C2×C4—41no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4D3×#6711yescharacteristic
277C2×C4D3×#6711yescharacteristic
278C2×C4#381111yescharacteristic
279C8—34no—
280C8—62no—
281C8—182no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3#505911yescharacteristic
286C2×C3#508511yescharacteristic
287C2×C3#516811yescharacteristic
288C2×C3#516811yescharacteristic
289C2×C3#517511yescharacteristic
290C2×C3#519111yescharacteristic
291D3—482no—
292D3—481no—
293D3—481no—
294C22—721no—
295C22—721no—
296C22—721no—
297C22#6397911yescharacteristic
298C4—81no—
299C4—241no—
300C4—241no—
301C4#6393512yes—
302C4D3×#21512yes—
303C4#6395012yes—
304C3—21no—
305C3#7121311yescharacteristic, complemented
306C3#7133411yescharacteristic, complemented
307C2—1441no—
308C2#1059147311yescharacteristic
309C2#1059161011yescharacteristic
310C2#1059161811yescharacteristic
subgroupG/H
centerC22#63979
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#127
prime radical#64083C22
hypercenterC4⋊Q8D32
hypocenterC32#1708
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5080C2semidirect
3. C3×#5079C2semidirect
5. #10590290C2semidirect
7. C3⋊C82C2semidirect
31. #5080D3semidirect
33. #5079D3semidirect
265. C32#1708semidirect
305. C3#71213semidirect
306. C3#71334semidirect

Maximal subgroups

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

subgroupindex
#105927872
C3×#50802
C3×#50792
#105903732
#105902902
#105902812
C3⋊C822
#713343
#712133