Finite Group Databasebeta

← #3522⋊C9 #11919765

Subgroups — #3522⋊C9 #11919765

Order 1152.

501 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 99 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
1Q8×Q8⋊C9C211yesmaximal, characteristic, complemented
2C4⋊(#41⋊C9)C211yesmaximal, characteristic, complemented
3#548⋊C9C212yesmaximal, complemented
4#520⋊C9C212yesmaximal, complemented
5C2×C4×Q8⋊C9C211yesmaximal, characteristic, complemented
6C3×#3522C311yesmaximal, characteristic
7C2×#41⋊C9C2212yes—
8C22×Q8⋊C9C2211yescharacteristic
9C9×C22⋊Q8—41nomaximal
10C4×Q8⋊C9C2211yescharacteristic
11C4×Q8⋊C9C2211yescharacteristic
12C4×Q8⋊C9C2212yes—
13C3×#559—31no—
14C3×Q82C2×C311yescharacteristic
15C3×#554—31no—
16C3×#550C2×C311yescharacteristic
17C3×#548C2×C312yes—
18C3×#547—32no—
19C3×#542—32no—
20C3×#532—31no—
21C3×#529—31no—
22C3×#520C2×C312yes—
23C2×C3×C4×Q8C2×C311yescharacteristic
24C2×Q8×C9—41no—
25C22×C4×C9—41no—
26#41⋊C9—22no—
27C2×Q8⋊C9—21no—
28C2×Q8⋊C9C2311yescharacteristic
29C9×C4⋊C4—42no—
30C9×C4⋊C4—41no—
31C9×C22⋊C4—42no—
32#3522C911yescharacteristic, complemented, Sylow
33C2×C3×#41—32no—
34C2×C3×#41C22×C312yes—
35C22×C3×Q8C22×C311yescharacteristic
36C3×C4⋊Q8—31no—
37C3×C4⋊Q8—32no—
38C3×#127—31no—
39C3×#126—32no—
40C3×#125—31no—
41C3×#124—32no—
42C3×#124—31no—
43C3×#124—32no—
44C3×#123—34no—
45C3×#123—32no—
46C3×C22⋊Q8—34no—
47C3×C22⋊Q8—32no—
48C3×C22⋊Q8A411yescharacteristic
49C3×#121—32no—
50C3×#121—31no—
51C3×#121—32no—
52C3×C4×Q8—31no—
53C3×C4×Q8—31no—
54C3×C4×Q8—31no—
55C3×C4×Q8C22×C311yescharacteristic
56C3×C4×Q8C22×C312yes—
57C3×C4×Q8C22×C311yescharacteristic
58C3×C4×D4—32no—
59C3×C4×D4—31no—
60C3×C4×D4—32no—
61C3×C4×D4—32no—
62C3×C4×D4—32no—
63C3×#117—32no—
64C2×C3×C4⋊C4—31no—
65C2×C3×C4⋊C4—32no—
66C2×C3×C42—31no—
67C23×C9—41no—
68Q8×C9—82no—
69C2×C4×C9—41no—
70C2×C4×C9—42no—
71C2×C4×C9—42no—
72C2×C4×C9—41no—
73C2×C4×C9—82no—
74Q8⋊C9#4111yescharacteristic, complemented
75#559—31no—
76Q82C2×C911yescharacteristic, complemented
77#554—31no—
78#550C2×C911yescharacteristic, complemented
79#548C2×C912yes—
80#547—32no—
81#542—32no—
82#532—31no—
83#529—31no—
84#520C2×C912yescomplemented
85C2×C4×Q8C2×C911yescharacteristic
86C3×#41—22no—
87C3×#41—62no—
88C3×#41—124no—
89C2×C3×Q8—21no—
90C2×C3×Q8—32no—
91C2×C3×Q8—31no—
92C2×C3×Q8—61no—
93C2×C3×Q8C2×A411yescharacteristic
94C2×C3×Q8C23×C311yescharacteristic
95C2×C3×D4—32no—
96C2×C3×D4—32no—
97C2×C3×D4—32no—
98C2×C3×D4—31no—
99C22×C3×C4—31no—
100C22×C3×C4—31no—
101C22×C3×C4—34no—
102C22×C3×C4—32no—
103C22×C3×C4C2×A411yescharacteristic
104C3×C4⋊C4—31no—
105C3×C4⋊C4—32no—
106C3×C4⋊C4—31no—
107C3×C4⋊C4—32no—
108C3×C4⋊C4—32no—
109C3×C4⋊C4—31no—
110C3×C4⋊C4—32no—
111C3×C4⋊C4—32no—
112C3×C4⋊C4C2×A412yes—
113C3×C4⋊C4C2×A411yescharacteristic
114C3×C22⋊C4—34no—
115C3×C22⋊C4—32no—
116C3×C22⋊C4—32no—
117C3×C22⋊C4—32no—
118C3×C22⋊C4—32no—
119C3×C22⋊C4C2×A412yes—
120C3×C42—31no—
121C3×C42—31no—
122C3×C42—31no—
123C3×C42—31no—
124C3×C42—32no—
125C22×C9—42no—
126C22×C9—41no—
127C22×C9—81no—
128C22×C9—81no—
129C4×C9—42no—
130C4×C9—81no—
131C4×C9—82no—
132C4×C9—82no—
133C2×#41—32no—
134C2×#41C22×C912yes—
135C22×Q8C22×C911yescharacteristic
136C4⋊Q8—31no—
137C4⋊Q8—32no—
138#127—31no—
139#126—32no—
140#125—31no—
141#124—32no—
142#124—31no—
143#124—32no—
144#123—34no—
145#123—32no—
146C22⋊Q8—34no—
147C22⋊Q8—32no—
148C22⋊Q8C22⋊C911yescharacteristic
149#121—32no—
150#121—31no—
151#121—32no—
152C4×Q8—31no—
153C4×Q8—31no—
154C4×Q8—31no—
155C4×Q8C22×C911yescharacteristic
156C4×Q8C22×C911yescharacteristic
157C4×Q8C22×C912yes—
158C4×D4—32no—
159C4×D4—31no—
160C4×D4—32no—
161C4×D4—32no—
162C4×D4—32no—
163#117—32no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—32no—
166C2×C42—31no—
167C23×C3—32no—
168C23×C3C22×A411yescharacteristic
169C3×Q8—22no—
170C3×Q8—31no—
171C3×Q8—61no—
172C3×Q8—61no—
173C3×Q8—122no—
174C3×Q8C3×#4111yescharacteristic
175C3×D4—62no—
176C3×D4—62no—
177C3×D4—64no—
178C3×D4—62no—
179C3×D4—122no—
180C2×C3×C4—22no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—32no—
186C2×C3×C4—62no—
187C2×C3×C4—62no—
188C2×C3×C4—61no—
189C2×C3×C4—61no—
190C2×C3×C4—62no—
191C2×C3×C4—124no—
192C2×C3×C4C22×A411yescharacteristic
193C2×C3×C4C22×A412yes—
194C2×C3×C4C22×A412yes—
195C2×C3×C4C22×A411yescharacteristic
196C2×C9—41no—
197C2×C9—41no—
198C2×C9—41no—
199C2×C9—82no—
200#41—22no—
201#41—62no—
202#41—124no—
203C2×Q8—21no—
204C2×Q8—32no—
205C2×Q8—31no—
206C2×Q8—61no—
207C2×Q8C2×C22⋊C911yescharacteristic
208C2×Q8C23×C911yescharacteristic
209C2×D4—32no—
210C2×D4—32no—
211C2×D4—32no—
212C2×D4—31no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—34no—
216C22×C4—32no—
217C22×C4C2×C22⋊C911yescharacteristic
218C4⋊C4—31no—
219C4⋊C4—32no—
220C4⋊C4—31no—
221C4⋊C4—32no—
222C4⋊C4—32no—
223C4⋊C4—31no—
224C4⋊C4—32no—
225C4⋊C4—32no—
226C4⋊C4C2×C22⋊C912yes—
227C4⋊C4C2×C22⋊C911yescharacteristic
228C22⋊C4—34no—
229C22⋊C4—32no—
230C22⋊C4—32no—
231C22⋊C4—32no—
232C22⋊C4—32no—
233C22⋊C4C2×C22⋊C912yes—
234C42—31no—
235C42—31no—
236C42—31no—
237C42—31no—
238C42—32no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—62no—
242C22×C3—62no—
243C22×C3—62no—
244C22×C3#99412yes—
245C22×C3C23×A411yescharacteristic
246C3×C4—22no—
247C3×C4—21no—
248C3×C4—22no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—61no—
252C3×C4—61no—
253C3×C4—61no—
254C3×C4—62no—
255C3×C42−1+4⋊C312yes—
256C9—41noSylow
257C23—32no—
258C23C22×C22⋊C911yescharacteristic
259Q8—22no—
260Q8—31no—
261Q8—61no—
262Q8—61no—
263Q8—122no—
264Q8C9×#4111yescharacteristic
265D4—62no—
266D4—62no—
267D4—64no—
268D4—62no—
269D4—122no—
270C2×C4—22no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—32no—
276C2×C4—62no—
277C2×C4—62no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—62no—
281C2×C4—124no—
282C2×C4C22×C22⋊C911yescharacteristic
283C2×C4C22×C22⋊C912yes—
284C2×C4C22×C22⋊C912yes—
285C2×C4C22×C22⋊C911yescharacteristic
286C2×C3—22no—
287C2×C3—122no—
288C2×C3A4×#4111yescharacteristic
289C2×C3C2×#99411yescharacteristic
290C2×C3C2×2−1+4⋊C311yescharacteristic
291C22—21no—
292C22—21no—
293C22—62no—
294C22—62no—
295C22—62no—
296C22#6370012yes—
297C22C23×C22⋊C911yescharacteristic
298C4—22no—
299C4—21no—
300C4—22no—
301C4—31no—
302C4—31no—
303C4—61no—
304C4—61no—
305C4—61no—
306C4—62no—
307C42−1+4⋊C912yes—
308C3#3522⋊C311yescharacteristic
309C2—22no—
310C2—122no—
311C2#41×C22⋊C911yescharacteristic
312C2C2×#6370011yescharacteristic
313C2C2×2−1+4⋊C911yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×A4
commutatorC2×Q8C23×C9
Fitting, prime radicalC3×#3522C3
cosocleC2×C3×Q8C23×C3
hypercenterC3×C22⋊Q8A4
hypocenterQ8C9×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. Q8×Q8⋊C9C2semidirect
2. C4⋊(#41⋊C9)C2semidirect
3. #548⋊C9C2semidirect
4. #520⋊C9C2semidirect
5. C2×C4×Q8⋊C9C2semidirect
32. #3522C9semidirect
74. Q8⋊C9#41semidirect
76. Q82C2×C9semidirect
78. #550C2×C9semidirect
84. #520C2×C9semidirect

Maximal subgroups

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

subgroupindex
Q8×Q8⋊C92
C4⋊(#41⋊C9)2
#548⋊C92
#520⋊C92
C2×C4×Q8⋊C92
C3×#35223
C9×C22⋊Q84