Finite Group Databasebeta

← #11782454

Subgroups — #11782454

Order 1152.

497 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 106 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#10592777C211yesmaximal, characteristic, complemented
2C3×#5088C211yesmaximal, characteristic, complemented
3C3×#5085C211yesmaximal, characteristic, complemented
4#10590369C211yesmaximal, characteristic, complemented
5#10590352C211yesmaximal, characteristic
6#10590273C211yesmaximal, characteristic, complemented
7C3⋊C82C211yesmaximal, characteristic, complemented
8#71255—31nomaximal
9#71180—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11#64078C2211yescharacteristic
12C2×C3×#215—22no—
13C3×#874C2211yescharacteristic
14C3×#874C2211yescharacteristic
15C2×#3716—22no—
16C2×#3713—22no—
17#63634—22no—
18C4×C3⋊(C3⋊C8)C2211yescharacteristic
19C3×#809—22no—
20C3×C4×C3⋊C8C2211yescharacteristic
21C3×C4×C3⋊C8C2211yescharacteristic
22#63563—22no—
23C3×#492—31no—
24C3×#486—31no—
25#5088D311yescharacteristic, complemented
26#5085D311yescharacteristic, complemented
27#5078—31no—
28#5078—31no—
29#5078—61no—
30#5076—31no—
31#5073—31no—
32#5073—31no—
33#5070—31no—
34#4694—31no—
35#4693—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×D12—21no—
44C2×C3×D12—21no—
45C2×C3×D12—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4—21no—
48C32×C42C2311yescharacteristic
49C4⋊(C3⋊Dic3)—21no—
50C4⋊(C3⋊Dic3)—21no—
51C2×C3⋊(C3⋊C8)D412yes—
52C3×#215—44no—
53C2×C3×C3⋊C8D412yes—
54C2×C3×C3⋊C8D412yes—
55#3716—44no—
56#3713—44no—
57#1706—91noSylow
58C2×C3×SD16—62no—
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8C2×D311yescharacteristic
61C3×C4⋊Q8C2×D311yescharacteristic
62C3×#127—31no—
63C3×#127—31no—
64C2×#215—22no—
65C2×#214—62no—
66C2×#213—62no—
67C2×#213—62no—
68C2×#204—62no—
69#874C2×D311yescharacteristic
70#874C2×D311yescharacteristic
71#869—31no—
72#869—31no—
73#869—61no—
74C3×#102—62no—
75C3×C4×C8—31no—
76C3×C4×C8—31no—
77#832—62no—
78#821—62no—
79#809—22no—
80#807—62no—
81#807—62no—
82#807—64no—
83C4×C3⋊C8—31no—
84C4×C3⋊C8—31no—
85C4×C3⋊C8—61no—
86C4×C3⋊C8C2×D311yescharacteristic
87C4×C3⋊C8C2×D311yescharacteristic
88C22×C3×D3—41no—
89C22×C3×D3—41no—
90C32×Q8—42no—
91C2×C32×C4—41no—
92C2×C32×C4C2×D411yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C2×C3⋊Dic3—41no—
96#634—42no—
97C3×D12—42no—
98C3×D12—42no—
99C3×D12—42no—
100C3×D12—42no—
101C3×D12—42no—
102C3×D12—42no—
103C3⋊(C3⋊C8)SD1614yescomplemented
104C3×C3⋊C8—22no—
105C3×C3⋊C8D814yescomplemented
106#500—91no—
107#493—91no—
108#492—31no—
109#492—91no—
110#486—31no—
111#486—91no—
112C82—91no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—41no—
116C2×C3×D4—61no—
117C2×C3×D4—61no—
118C2×C3×D4—61no—
119C2×C3×D4—61no—
120C2×C3×D4—61no—
121C2×C3×D4—61no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×Dic6—61no—
129C2×Dic6—61no—
130C2×Dic6—121no—
131C3×SD16—124no—
132C2×C3×C8—32no—
133C2×C3×C8—32no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C42—21no—
141C3×C42C22×D311yescharacteristic
142C3×C42C22×D311yescharacteristic
143#215—44no—
144#214—124no—
145#213—124no—
146#213—124no—
147C4⋊Dic3—62no—
148C4⋊Dic3—61no—
149C4⋊Dic3—61no—
150C4⋊Dic3—61no—
151C4⋊Dic3—61no—
152C4⋊Dic3—121no—
153C2×C3⋊C8—32no—
154C2×C3⋊C8—32no—
155C2×C3⋊C8—62no—
156C2×C3⋊C8#6712yes—
157C2×C3⋊C8#6712yes—
158#204—124no—
159C22×C32#12711yescharacteristic
160C2×C3×D3—81no—
161C2×C3×D3—81no—
162C2×C3×D3—81no—
163C2×C3×D3—81no—
164C2×C3×D3—81no—
165C2×C3×D3—81no—
166C32×C4—81no—
167C32×C4C2×D812yes—
168C32×C4C2×SD1612yes—
169C32×C4#13512yes—
170C3⋊Dic3—81no—
171C2×SD16—62no—
172C2×SD16—182no—
173C2×D8—182no—
174C4⋊Q8—91no—
175C4⋊Q8D3211yescharacteristic
176#127—31no—
177#127—31no—
178#107—182no—
179#102—62no—
180#102—182no—
181C4×C8—31no—
182C4×C8—31no—
183C4×C8—91no—
184C23×C3—121no—
185C23×C3—121no—
186C22×D3—41no—
187C22×D3—41no—
188C3×Q8—42no—
189C3×Q8—42no—
190C3×Q8—82no—
191C3×D4—122no—
192C3×D4—122no—
193C3×D4—122no—
194C3×D4—122no—
195C3×D4—122no—
196C3×D4—122no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—42no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4D3×D411yescharacteristic
204C2×C3×C4D3×D411yescharacteristic
205C2×C3×C4D3×D411yescharacteristic
206C2×C3×C4D3×D411yescharacteristic
207C2×C3×C4C2×#6711yescharacteristic
208C2×C3×C4C2×#6711yescharacteristic
209C2×Dic3—122no—
210C2×Dic3—121no—
211C2×Dic3—121no—
212D12—42no—
213D12—42no—
214D12—42no—
215D12—42no—
216D12—42no—
217D12—42no—
218Dic6—122no—
219Dic6—122no—
220Dic6—242no—
221C3×C8—34no—
222C3×C8—62no—
223C3⋊C8—22no—
224C3⋊C8—34no—
225C3⋊C8—34no—
226C3⋊C8—64no—
227C3⋊C8#21314yescomplemented
228C2×C32#49211yescharacteristic
229C2×C32#49311yescharacteristic
230C2×C32#49511yescharacteristic
231C3×D3—161no—
232C3×D3—161no—
233C2×Q8—21no—
234C2×Q8—181no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241SD16—124no—
242SD16—364no—
243D8—364no—
244C2×C8—32no—
245C2×C8—32no—
246C2×C8—92no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—181no—
250C4⋊C4—181no—
251C42C2×D3211yescharacteristic
252C22×C3—21no—
253C22×C3—241no—
254C22×C3—241no—
255C22×C3—241no—
256C22×C3—241no—
257C22×C3—241no—
258C22×C3—241no—
259C22×C3#94011yescharacteristic
260C22×C3#94011yescharacteristic
261C2×D3—81no—
262C2×D3—81no—
263C2×D3—81no—
264C2×D3—81no—
265C2×D3—81no—
266C2×D3—81no—
267C3×C4—22no—
268C3×C4—22no—
269C3×C4—22no—
270C3×C4—82no—
271C3×C4—81no—
272C3×C4—81no—
273C3×C4D3×D812yes—
274C3×C4D3×SD1612yes—
275C3×C4D3×SD1612yes—
276C3×C4#91612yes—
277C3×C4C2×#21312yes—
278C3×C4#95012yes—
279Dic3—242no—
280Dic3—241no—
281Dic3—241no—
282C32#170611yescharacteristic, complemented, Sylow
283C23—121no—
284C23—121no—
285Q8—42no—
286Q8—362no—
287D4—122no—
288D4—122no—
289D4—122no—
290D4—122no—
291D4—122no—
292D4—122no—
293C2×C4—41no—
294C2×C4—361no—
295C2×C4D3×#6711yescharacteristic
296C2×C4D3×#6711yescharacteristic
297C2×C4#381111yescharacteristic
298C8—34no—
299C8—62no—
300C8—94no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—481no—
305C2×C3—481no—
306C2×C3#505911yescharacteristic
307C2×C3#507311yescharacteristic
308C2×C3#515111yescharacteristic
309C2×C3#516811yescharacteristic
310C2×C3#517511yescharacteristic
311C2×C3#517511yescharacteristic
312D3—161no—
313D3—161no—
314C22—241no—
315C22—241no—
316C22—241no—
317C22—241no—
318C22—241no—
319C22—241no—
320C22#6397911yescharacteristic
321C4—81no—
322C4—721no—
323C4D3×#21312yes—
324C4#6394012yes—
325C4#6394712yes—
326C3—21no—
327C3#7118011yescharacteristic, complemented
328C3#7125511yescharacteristic, complemented
329C2—481no—
330C2—481no—
331C2#1059148211yescharacteristic
332C2#1059160611yescharacteristic
333C2#1059162011yescharacteristic
subgroupG/H
centerC22#63979
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#127
prime radical#10590352C2
hypercenterC4⋊Q8D32
hypocenterC32#1706
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. #10592777C2semidirect
2. C3×#5088C2semidirect
3. C3×#5085C2semidirect
4. #10590369C2semidirect
6. #10590273C2semidirect
7. C3⋊C82C2semidirect
25. #5088D3semidirect
26. #5085D3semidirect
103. C3⋊(C3⋊C8)SD16semidirect
105. C3×C3⋊C8D8semidirect
227. C3⋊C8#213semidirect
282. C32#1706semidirect
327. C3#71180semidirect
328. C3#71255semidirect

Maximal subgroups

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

subgroupindex
#105927772
C3×#50882
C3×#50852
#105903692
#105903522
#105902732
C3⋊C822
#712553
#711803