Finite Group Databasebeta

← #11782519

Subgroups — #11782519

Order 1152.

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

normal subgroups only

This group has 429 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#10592792C211yesmaximal, characteristic, complemented
2C3×#5093C211yesmaximal, characteristic, complemented
3C3×#5085C211yesmaximal, characteristic
4#10590303C211yesmaximal, characteristic, complemented
5#10590277C211yesmaximal, characteristic
6#10590273C211yesmaximal, characteristic
7C3⋊C82C211yesmaximal, characteristic, complemented
8#71256—31nomaximal
9#71213—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11#64078C2211yescharacteristic
12C2×C3×#215—22no—
13C3×#874C2211yescharacteristic
14C3×#869C2211yescharacteristic
15#63637—22no—
16C4×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×#810—22no—
18C3×C4×C3⋊C8C2211yescharacteristic
19C3×C4×C3⋊C8C2211yescharacteristic
20#63573—22no—
21#63563—22no—
22#63562—22no—
23C3×#492—31no—
24C3×#487—31no—
25#5093—31no—
26#5093—31no—
27#5093—61no—
28#5093D311yescharacteristic, complemented
29#5088—31no—
30#5085D311yescharacteristic
31#5080—31no—
32#5070—31no—
33#5070—31no—
34#4694—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#1708—91noSylow
56C2×C3×SD16—62no—
57C3×C4⋊Q8—21no—
58C3×C4⋊Q8—31no—
59C3×C4⋊Q8C2×D311yescharacteristic
60C3×C4⋊Q8C2×D311yescharacteristic
61C3×#127—31no—
62C2×#215—22no—
63#874C2×D311yescharacteristic
64#869—31no—
65#869—31no—
66#869—61no—
67#869C2×D311yescharacteristic
68C3×#103—62no—
69C3×C4×C8—31no—
70C3×C4×C8—31no—
71#832—62no—
72#832—62no—
73#821—62no—
74#817—62no—
75#810—22no—
76#810—62no—
77#810—62no—
78#810—64no—
79#809—62no—
80#808—62no—
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
92C2×C3⋊Dic3—41no—
93#634—42no—
94C2×C3×Dic3—41no—
95C3×D12—42no—
96C3×D12—42no—
97C3×D12—42no—
98C3×Dic6—42no—
99C3⋊(C3⋊C8)—22no—
100C3×C3⋊C8—22no—
101C3×C3⋊C8SD1614yes—
102#498—91no—
103#492—31no—
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×D4—61no—
114C2×C3×D4—61no—
115C2×C3×D4—61no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×Dic6—21no—
120C2×Dic6—62no—
121C2×Dic6—61no—
122C2×Dic6—61no—
123C3×SD16—124no—
124C2×C3×C8—32no—
125C2×C3×C8—32no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—61no—
133C3×C4⋊C4—61no—
134C3×C42—21no—
135C3×C42C22×D311yescharacteristic
136C3×C42C22×D311yescharacteristic
137#215—44no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—61no—
141C4⋊Dic3—61no—
142C4⋊Dic3—61no—
143C4⋊Dic3—61no—
144C4⋊Dic3—121no—
145C4⋊Dic3—121no—
146C2×C3⋊C8—32no—
147C2×C3⋊C8—32no—
148C2×C3⋊C8—62no—
149C2×C3⋊C8#6712yes—
150C2×C3⋊C8#6712yes—
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—62no—
162C4⋊Q8—31no—
163C4⋊Q8—91no—
164C4⋊Q8D3211yescharacteristic
165#127—31no—
166#106—182no—
167#103—62no—
168#103—182no—
169#102—182no—
170#102—182no—
171C4×C8—31no—
172C4×C8—31no—
173C4×C8—91no—
174C23×C3—121no—
175C22×D3—41no—
176C3×Q8—42no—
177C3×Q8—42no—
178C3×Q8—82no—
179C3×Q8—122no—
180C3×D4—122no—
181C3×D4—122no—
182C3×D4—122no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
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—122no—
198C2×Dic3—121no—
199C2×Dic3—121no—
200D12—42no—
201D12—42no—
202D12—42no—
203Dic6—42no—
204Dic6—124no—
205Dic6—122no—
206Dic6—122no—
207C3×C8—34no—
208C3×C8—62no—
209C3⋊C8—22no—
210C3⋊C8—62no—
211C3⋊C8—62no—
212C3⋊C8—122no—
213C3⋊C8#21414yes—
214C2×C32#49211yescharacteristic
215C2×C32#49511yescharacteristic
216C2×C32#49511yescharacteristic
217C3×D3—161no—
218C2×Q8—21no—
219C2×Q8—61no—
220C2×Q8—181no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4—61no—
224SD16—124no—
225C2×C8—32no—
226C2×C8—32no—
227C2×C8—92no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—181no—
233C4⋊C4—181no—
234C42C2×D3211yescharacteristic
235C22×C3—21no—
236C22×C3—241no—
237C22×C3—241no—
238C22×C3—241no—
239C22×C3#94011yescharacteristic
240C22×C3#94011yescharacteristic
241C2×D3—81no—
242C2×D3—81no—
243C2×D3—81no—
244C3×C4—22no—
245C3×C4—22no—
246C3×C4—22no—
247C3×C4—82no—
248C3×C4—81no—
249C3×C4—81no—
250C3×C4—241no—
251C3×C4D3×SD1612yes—
252C3×C4#91612yes—
253C3×C4#91612yes—
254C3×C4#91912yes—
255C3×C4C2×#21412yes—
256C3×C4#95012yes—
257Dic3—81no—
258Dic3—242no—
259Dic3—241no—
260Dic3—241no—
261C32#170811yescharacteristic, complemented, Sylow
262C23—121no—
263Q8—42no—
264Q8—122no—
265Q8—362no—
266D4—122no—
267D4—122no—
268D4—122no—
269C2×C4—41no—
270C2×C4—121no—
271C2×C4—361no—
272C2×C4D3×#6711yescharacteristic
273C2×C4D3×#6711yescharacteristic
274C2×C4#381111yescharacteristic
275C8—34no—
276C8—62no—
277C8—182no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—481no—
282C2×C3#505911yescharacteristic
283C2×C3#507611yescharacteristic
284C2×C3#516811yescharacteristic
285C2×C3#516811yescharacteristic
286C2×C3#517511yescharacteristic
287C2×C3#519111yescharacteristic
288D3—161no—
289C22—241no—
290C22—241no—
291C22—241no—
292C22#6397911yescharacteristic
293C4—81no—
294C4—241no—
295C4—721no—
296C4D3×#21412yes—
297C4#6394712yes—
298C4#6395012yes—
299C3—21no—
300C3#7121311yescharacteristic, complemented
301C3#7125611yescharacteristic, complemented
302C2—481no—
303C2#1059149011yescharacteristic
304C2#1059160611yescharacteristic
305C2#1059161011yescharacteristic
subgroupG/H
centerC22#63979
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#127
prime radicalC3×#874C22
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
1. #10592792C2semidirect
2. C3×#5093C2semidirect
4. #10590303C2semidirect
7. C3⋊C82C2semidirect
28. #5093D3semidirect
261. C32#1708semidirect
300. C3#71213semidirect
301. C3#71256semidirect

Maximal subgroups

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

subgroupindex
#105927922
C3×#50932
C3×#50852
#105903032
#105902772
#105902732
C3⋊C822
#712563
#712133