Finite Group Databasebeta

← #3502⋊C9 #11919753

Subgroups — #3502⋊C9 #11919753

Order 1152.

452 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 105 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
2#557⋊C9C212yesmaximal, complemented
3#557⋊C9C211yesmaximal, characteristic, complemented
4#519⋊C9C212yesmaximal
5C2×C4×Q8⋊C9C211yesmaximal, characteristic
6C3×#3502C311yesmaximal, characteristic
7C22×Q8⋊C9C2211yescharacteristic
8C9×C22⋊Q8—41nomaximal
9C4×Q8⋊C9C2211yescharacteristic
10C4×Q8⋊C9C2211yescharacteristic
11C4×Q8⋊C9C2212yes—
12C4×Q8⋊C9C2212yes—
13C3×Q82—31no—
14C3×Q82C2×C311yescharacteristic
15C3×#557—31no—
16C3×#557—32no—
17C3×#557C2×C311yescharacteristic
18C3×#557C2×C312yes—
19C3×#544—32no—
20C3×#533—31no—
21C3×#533—31no—
22C3×#519C2×C312yes—
23C2×C3×C4×Q8C2×C311yescharacteristic
24C2×Q8×C9—41no—
25C22×C4×C9—41no—
26C2×Q8⋊C9Q812yescomplemented
27C2×Q8⋊C9C2311yescharacteristic
28C9×C4⋊C4—41no—
29C9×C4⋊C4—42no—
30C9×C22⋊C4—42no—
31#3502C911yescharacteristic, complemented, Sylow
32C22×C3×Q8C22×C311yescharacteristic
33C3×C4⋊Q8—32no—
34C3×C4⋊Q8—31no—
35C3×C4⋊Q8—34no—
36C3×C4⋊Q8—31no—
37C3×C4⋊Q8—31no—
38C3×C4⋊Q8—31no—
39C3×C4⋊Q8—32no—
40C3×#125—32no—
41C3×#125—32no—
42C3×#125—32no—
43C3×#125—34no—
44C3×#125—32no—
45C3×C22⋊Q8—32no—
46C3×C22⋊Q8—31no—
47C3×C22⋊Q8—32no—
48C3×C22⋊Q8A411yescharacteristic
49C3×C4×Q8—32no—
50C3×C4×Q8—32no—
51C3×C4×Q8—31no—
52C3×C4×Q8—31no—
53C3×C4×Q8—32no—
54C3×C4×Q8—32no—
55C3×C4×Q8—32no—
56C3×C4×Q8—31no—
57C3×C4×Q8—31no—
58C3×C4×Q8C22×C311yescharacteristic
59C3×C4×Q8C22×C312yes—
60C3×C4×Q8C22×C311yescharacteristic
61C3×C4×Q8C22×C312yes—
62C3×#117—32no—
63C3×#117—32no—
64C2×C3×C4⋊C4—31no—
65C2×C3×C42—31no—
66C23×C9—41no—
67Q8×C9—82no—
68C2×C4×C9—41no—
69C2×C4×C9—42no—
70C2×C4×C9—42no—
71C2×C4×C9—41no—
72C2×C4×C9—82no—
73Q8⋊C9C2×Q811yescharacteristic
74Q82—31no—
75Q82C2×C911yescharacteristic, complemented
76#557—31no—
77#557—32no—
78#557C2×C912yescomplemented
79#557C2×C911yescharacteristic, complemented
80#544—32no—
81#533—31no—
82#533—31no—
83#519C2×C912yes—
84C2×C4×Q8C2×C911yescharacteristic
85C2×C3×Q8—32no—
86C2×C3×Q8—31no—
87C2×C3×Q8—32no—
88C2×C3×Q8—31no—
89C2×C3×Q8—32no—
90C2×C3×Q8C3×Q812yes—
91C2×C3×Q8C2×A411yescharacteristic
92C2×C3×Q8C23×C311yescharacteristic
93C22×C3×C4—31no—
94C22×C3×C4—31no—
95C22×C3×C4C2×A411yescharacteristic
96C3×C4⋊C4—31no—
97C3×C4⋊C4—31no—
98C3×C4⋊C4—32no—
99C3×C4⋊C4—32no—
100C3×C4⋊C4—32no—
101C3×C4⋊C4—32no—
102C3×C4⋊C4—34no—
103C3×C4⋊C4—32no—
104C3×C4⋊C4—32no—
105C3×C4⋊C4—32no—
106C3×C4⋊C4—31no—
107C3×C4⋊C4—32no—
108C3×C4⋊C4—32no—
109C3×C4⋊C4—32no—
110C3×C4⋊C4C2×A412yes—
111C3×C4⋊C4C2×A411yescharacteristic
112C3×C22⋊C4—32no—
113C3×C22⋊C4C2×A412yes—
114C3×C42—34no—
115C3×C42—31no—
116C3×C42—31no—
117C3×C42—31no—
118C3×C42—31no—
119C3×C42—32no—
120C3×C42—32no—
121C22×C9—42no—
122C22×C9—41no—
123C22×C9—81no—
124C22×C9—81no—
125C4×C9—42no—
126C4×C9—82no—
127C4×C9—81no—
128C4×C9—82no—
129C22×Q8C22×C911yescharacteristic
130C4⋊Q8—31no—
131C4⋊Q8—32no—
132C4⋊Q8—34no—
133C4⋊Q8—31no—
134C4⋊Q8—31no—
135C4⋊Q8—31no—
136C4⋊Q8—32no—
137#125—32no—
138#125—32no—
139#125—32no—
140#125—34no—
141#125—32no—
142C22⋊Q8—32no—
143C22⋊Q8—31no—
144C22⋊Q8—32no—
145C22⋊Q8C22⋊C911yescharacteristic
146C4×Q8—32no—
147C4×Q8—32no—
148C4×Q8—31no—
149C4×Q8—32no—
150C4×Q8—31no—
151C4×Q8—32no—
152C4×Q8—32no—
153C4×Q8—31no—
154C4×Q8—31no—
155C4×Q8C22×C911yescharacteristic
156C4×Q8C22×C911yescharacteristic
157C4×Q8C22×C912yes—
158C4×Q8C22×C912yes—
159#117—32no—
160#117—32no—
161C2×C4⋊C4—31no—
162C2×C42—31no—
163C23×C3C22×A411yescharacteristic
164C3×Q8—22no—
165C3×Q8—32no—
166C3×Q8—31no—
167C3×Q8—32no—
168C3×Q8—62no—
169C3×Q8—122no—
170C3×Q8—122no—
171C3×Q8C2×C3×Q811yescharacteristic
172C2×C3×C4—22no—
173C2×C3×C4—32no—
174C2×C3×C4—32no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—32no—
178C2×C3×C4—32no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—121no—
182C2×C3×C4C22×A411yescharacteristic
183C2×C3×C4C22×A412yes—
184C2×C3×C4C22×A412yes—
185C2×C3×C4C22×A411yescharacteristic
186C2×C9—41no—
187C2×C9—41no—
188C2×C9—41no—
189C2×C9—82no—
190C2×Q8—32no—
191C2×Q8—31no—
192C2×Q8—32no—
193C2×Q8—31no—
194C2×Q8—32no—
195C2×Q8Q8×C912yescomplemented
196C2×Q8C2×C22⋊C911yescharacteristic
197C2×Q8C23×C911yescharacteristic
198C22×C4—31no—
199C22×C4—31no—
200C22×C4C2×C22⋊C911yescharacteristic
201C4⋊C4—31no—
202C4⋊C4—31no—
203C4⋊C4—32no—
204C4⋊C4—32no—
205C4⋊C4—32no—
206C4⋊C4—32no—
207C4⋊C4—34no—
208C4⋊C4—32no—
209C4⋊C4—32no—
210C4⋊C4—32no—
211C4⋊C4—31no—
212C4⋊C4—32no—
213C4⋊C4—32no—
214C4⋊C4—32no—
215C4⋊C4C2×C22⋊C912yes—
216C4⋊C4C2×C22⋊C911yescharacteristic
217C22⋊C4—32no—
218C22⋊C4C2×C22⋊C912yes—
219C42—34no—
220C42—31no—
221C42—31no—
222C42—31no—
223C42—31no—
224C42—32no—
225C42—32no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3Q8×A412yes—
229C22×C3C23×A411yescharacteristic
230C3×C4—21no—
231C3×C4—22no—
232C3×C4—22no—
233C3×C4—31no—
234C3×C4—32no—
235C3×C4—31no—
236C3×C4—62no—
237C3×C4—61no—
238C3×C4—61no—
239C3×C4—62no—
240C3×C42−1+4⋊C312yes—
241C9—41noSylow
242C23C22×C22⋊C911yescharacteristic
243Q8—22no—
244Q8—32no—
245Q8—31no—
246Q8—32no—
247Q8—62no—
248Q8—122no—
249Q8—122no—
250Q8C2×Q8×C911yescharacteristic
251C2×C4—22no—
252C2×C4—31no—
253C2×C4—31no—
254C2×C4—32no—
255C2×C4—32no—
256C2×C4—32no—
257C2×C4—32no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—121no—
261C2×C4C22×C22⋊C911yescharacteristic
262C2×C4C22×C22⋊C912yes—
263C2×C4C22×C22⋊C912yes—
264C2×C4C22×C22⋊C911yescharacteristic
265C2×C3—22no—
266C2×C3C2×Q8×A411yescharacteristic
267C2×C3C2×2−1+4⋊C311yescharacteristic
268C2×C3#594511yescharacteristic
269C22—21no—
270C22—21no—
271C22Q8×C22⋊C912yes—
272C22C23×C22⋊C911yescharacteristic
273C4—21no—
274C4—22no—
275C4—22no—
276C4—31no—
277C4—31no—
278C4—32no—
279C4—62no—
280C4—61no—
281C4—61no—
282C4—62no—
283C42−1+4⋊C912yes—
284C3#3502⋊C311yescharacteristic
285C2—22no—
286C2C2×Q8×C22⋊C911yescharacteristic
287C2C2×2−1+4⋊C911yescharacteristic
288C2#1059312011yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×A4
commutatorC2×Q8C23×C9
FittingC3×#3502C3
prime radicalC23×C3C22×A4
cosocleC2×C3×Q8C23×C3
hypercenterC3×C22⋊Q8A4
hypocenterQ8C2×Q8×C9
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. #557⋊C9C2semidirect
3. #557⋊C9C2semidirect
26. C2×Q8⋊C9Q8semidirect
31. #3502C9semidirect
75. Q82C2×C9semidirect
78. #557C2×C9semidirect
79. #557C2×C9semidirect
195. C2×Q8Q8×C9semidirect

Maximal subgroups

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

subgroupindex
Q8×Q8⋊C92
#557⋊C92
#519⋊C92
C2×C4×Q8⋊C92
C3×#35023
C9×C22⋊Q84