Finite Group Databasebeta

← #11782428

Subgroups — #11782428

Order 1152.

477 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 477 conjugacy classes of subgroups — too many to draw. Only its 114 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#10592779C211yesmaximal, characteristic, complemented
2C3×#5085C211yesmaximal, characteristic
3C3×#5080C211yesmaximal, characteristic, complemented
4#10590375C211yesmaximal, characteristic
5#10590369C211yesmaximal, characteristic
6#10590303C211yesmaximal, characteristic, complemented
7C3⋊C82C211yesmaximal, characteristic, complemented
8#71332—31nomaximal
9#71254—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11#64078C2211yescharacteristic
12C2×C3×#215—22no—
13C3×#874C2211yescharacteristic
14C3×#869C2211yescharacteristic
15C2×#3716—22no—
16C2×#3715—22no—
17#63635—22no—
18C4×C3⋊(C3⋊C8)C2211yescharacteristic
19C3×#808—22no—
20C3×C4×C3⋊C8C2211yescharacteristic
21C3×C4×C3⋊C8C2211yescharacteristic
22#63573—22no—
23C3×#498—31no—
24C3×#492—31no—
25#5085—31no—
26#5085D311yescharacteristic
27#5080—31no—
28#5080—31no—
29#5080—31no—
30#5080—61no—
31#5080D311yescharacteristic, complemented
32#5076—31no—
33#5076—31no—
34#4693—31no—
35#4680—31no—
36C8×C3⋊C8—31no—
37C8×C3⋊C8—31no—
38C2×C32×Q8—21no—
39C2×#634—21no—
40C2×C3×D12—21no—
41C2×C3×D12—21no—
42C2×C3×D12—21no—
43C2×C3×Dic6—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C42C2311yescharacteristic
47C4⋊(C3⋊Dic3)—21no—
48C4⋊(C3⋊Dic3)—21no—
49C2×C3⋊(C3⋊C8)D412yes—
50C3×#215—44no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C2×C3×C3⋊C8D412yes—
54C2×C3×C3⋊C8D412yes—
55#3716—44no—
56#3715—44no—
57#1705—91noSylow
58C2×C3×SD16—62no—
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—31no—
61C3×C4⋊Q8C2×D311yescharacteristic
62C3×C4⋊Q8C2×D311yescharacteristic
63C3×#127—31no—
64C2×#215—22no—
65C2×#215—62no—
66C2×#214—62no—
67C2×#214—62no—
68C2×#204—62no—
69#874C2×D311yescharacteristic
70#869—31no—
71#869—31no—
72#869—61no—
73#869C2×D311yescharacteristic
74C3×#106—62no—
75C3×C4×C8—31no—
76C3×C4×C8—31no—
77#817—62no—
78#808—22no—
79#808—62no—
80#808—62no—
81#808—64no—
82#808—62no—
83C4×C3⋊C8—31no—
84C4×C3⋊C8—31no—
85C4×C3⋊C8—61no—
86C4×C3⋊C8C2×D311yescharacteristic
87C4×C3⋊C8C2×D311yescharacteristic
88C22×C3×D3—41no—
89C32×Q8—42no—
90C2×C32×C4—41no—
91C2×C32×C4C2×D411yescharacteristic
92C2×C32×C4C2×D411yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94C2×C3⋊Dic3—41no—
95#634—42no—
96C2×C3×Dic3—41no—
97C3×D12—42no—
98C3×D12—42no—
99C3×D12—42no—
100C3×Dic6—42no—
101C3⋊(C3⋊C8)SD1614yescomplemented
102C3×C3⋊C8SD1614yescomplemented
103C3×C3⋊C8SD1614yescomplemented
104#498—31no—
105#498—91no—
106#498—91no—
107#492—31no—
108#492—91no—
109#492—91no—
110C82—91no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—41no—
114C2×C3×Q8—61no—
115C2×C3×D4—61no—
116C2×C3×D4—61no—
117C2×C3×D4—61no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×Dic6—21no—
122C2×Dic6—61no—
123C2×Dic6—61no—
124C2×Dic6—121no—
125C3×SD16—124no—
126C2×C3×C8—32no—
127C2×C3×C8—32no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—61no—
135C3×C4⋊C4—61no—
136C3×C42—21no—
137C3×C42C22×D311yescharacteristic
138C3×C42C22×D311yescharacteristic
139#215—44no—
140#215—124no—
141#214—124no—
142#214—124no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—62no—
146C4⋊Dic3—61no—
147C4⋊Dic3—61no—
148C4⋊Dic3—61no—
149C4⋊Dic3—61no—
150C4⋊Dic3—121no—
151C2×C3⋊C8—32no—
152C2×C3⋊C8—32no—
153C2×C3⋊C8—62no—
154C2×C3⋊C8#6712yes—
155C2×C3⋊C8#6712yes—
156#204—124no—
157C22×C32#12711yescharacteristic
158C2×C3×D3—81no—
159C2×C3×D3—81no—
160C2×C3×D3—81no—
161C32×C4—81no—
162C32×C4C2×SD1612yes—
163C32×C4C2×SD1612yes—
164C32×C4C2×SD1612yes—
165C3⋊Dic3—81no—
166C3×Dic3—81no—
167C2×SD16—62no—
168C2×SD16—182no—
169C2×SD16—182no—
170C4⋊Q8—31no—
171C4⋊Q8—91no—
172C4⋊Q8D3211yescharacteristic
173#127—31no—
174#106—62no—
175#106—182no—
176#106—182no—
177C4×C8—31no—
178C4×C8—31no—
179C4×C8—91no—
180C23×C3—121no—
181C22×D3—41no—
182C3×Q8—42no—
183C3×Q8—42no—
184C3×Q8—82no—
185C3×Q8—122no—
186C3×D4—122no—
187C3×D4—122no—
188C3×D4—122no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—42no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—121no—
196C2×C3×C4D3×D411yescharacteristic
197C2×C3×C4D3×D411yescharacteristic
198C2×C3×C4D3×D411yescharacteristic
199C2×C3×C4D3×D411yescharacteristic
200C2×C3×C4C2×#6711yescharacteristic
201C2×C3×C4C2×#6711yescharacteristic
202C2×Dic3—41no—
203C2×Dic3—122no—
204C2×Dic3—121no—
205C2×Dic3—121no—
206D12—42no—
207D12—42no—
208D12—42no—
209Dic6—42no—
210Dic6—122no—
211Dic6—122no—
212Dic6—242no—
213C3×C8—34no—
214C3×C8—34no—
215C3⋊C8—34no—
216C3⋊C8—34no—
217C3⋊C8—64no—
218C3⋊C8#21414yescomplemented
219C3⋊C8#21514yescomplemented
220C2×C32#49211yescharacteristic
221C2×C32#49211yescharacteristic
222C2×C32#49211yescharacteristic
223C3×D3—161no—
224C2×Q8—21no—
225C2×Q8—61no—
226C2×Q8—181no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—61no—
230SD16—124no—
231SD16—364no—
232SD16—364no—
233C2×C8—32no—
234C2×C8—32no—
235C2×C8—92no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—181no—
241C4⋊C4—181no—
242C42C2×D3211yescharacteristic
243C22×C3—21no—
244C22×C3—241no—
245C22×C3—241no—
246C22×C3—241no—
247C22×C3#94011yescharacteristic
248C22×C3#94011yescharacteristic
249C2×D3—81no—
250C2×D3—81no—
251C2×D3—81no—
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—22no—
255C3×C4—82no—
256C3×C4—81no—
257C3×C4—81no—
258C3×C4—241no—
259C3×C4D3×SD1612yes—
260C3×C4D3×SD1612yes—
261C3×C4D3×SD1612yes—
262C3×C4D3×SD1612yes—
263C3×C4C2×#21412yes—
264C3×C4C2×#21512yes—
265Dic3—81no—
266Dic3—242no—
267Dic3—241no—
268Dic3—241no—
269C32#170511yescharacteristic, complemented, Sylow
270C23—121no—
271Q8—42no—
272Q8—122no—
273Q8—362no—
274D4—122no—
275D4—122no—
276D4—122no—
277C2×C4—41no—
278C2×C4—121no—
279C2×C4—361no—
280C2×C4D3×#6711yescharacteristic
281C2×C4D3×#6711yescharacteristic
282C2×C4#381111yescharacteristic
283C8—34no—
284C8—34no—
285C8—94no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—481no—
290C2×C3#507611yescharacteristic
291C2×C3#508511yescharacteristic
292C2×C3#517511yescharacteristic
293C2×C3#517511yescharacteristic
294C2×C3#517511yescharacteristic
295C2×C3#517511yescharacteristic
296D3—161no—
297C22—241no—
298C22—241no—
299C22—241no—
300C22#6397911yescharacteristic
301C4—81no—
302C4—241no—
303C4—721no—
304C4D3×#21412yes—
305C4D3×#21512yes—
306C4#6394012yes—
307C3—21no—
308C3#7125411yescharacteristic, complemented
309C3#7133211yescharacteristic, complemented
310C2—481no—
311C2#1059149011yescharacteristic
312C2#1059161811yescharacteristic
313C2#1059162011yescharacteristic
subgroupG/H
centerC22#63979
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#127
prime radicalC3×#874C22
hypercenterC4⋊Q8D32
hypocenterC32#1705
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
1. #10592779C2semidirect
3. C3×#5080C2semidirect
6. #10590303C2semidirect
7. C3⋊C82C2semidirect
31. #5080D3semidirect
101. C3⋊(C3⋊C8)SD16semidirect
102. C3×C3⋊C8SD16semidirect
103. C3×C3⋊C8SD16semidirect
218. C3⋊C8#214semidirect
219. C3⋊C8#215semidirect
269. C32#1705semidirect
308. C3#71254semidirect
309. C3#71332semidirect

Maximal subgroups

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

subgroupindex
#105927792
C3×#50852
C3×#50802
#105903752
#105903692
#105903032
C3⋊C822
#713323
#712543