Finite Group Databasebeta

← #3523⋊C9 #11919767

Subgroups — #3523⋊C9 #11919767

Order 1152.

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

normal subgroups only

This group has 487 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
1#557⋊C9C211yesmaximal, characteristic, complemented
2D4×Q8⋊C9C211yesmaximal, characteristic, complemented
3D4×Q8⋊C9C212yesmaximal, complemented
4#519⋊C9C212yesmaximal, complemented
5C2×C4×Q8⋊C9C211yesmaximal, characteristic, complemented
6C3×#3523C311yesmaximal, characteristic
7C22×Q8⋊C9C2211yescharacteristic
8C22×Q8⋊C9C2212yes—
9C9×#121—41nomaximal
10C4×Q8⋊C9C2211yescharacteristic
11C4×Q8⋊C9C2211yescharacteristic
12C4×Q8⋊C9C2212yes—
13C3×#557—31no—
14C3×#557C2×C311yescharacteristic
15C3×#556—31no—
16C3×#552—32no—
17C3×D4×Q8C2×C312yes—
18C3×D4×Q8C2×C311yescharacteristic
19C3×#541—32no—
20C3×#533—31no—
21C3×#529—31no—
22C3×#519C2×C312yes—
23C2×C3×C4×Q8C2×C311yescharacteristic
24C2×D4×C9—41no—
25C2×D4×C9—42no—
26C22×C4×C9—41no—
27C2×Q8⋊C9—21no—
28C2×Q8⋊C9—22no—
29C2×Q8⋊C9C2311yescharacteristic
30C9×C4⋊C4—41no—
31C9×C22⋊C4—42no—
32#3523C911yescharacteristic, complemented, Sylow
33C22×C3×Q8C22×C311yescharacteristic
34C22×C3×Q8C22×C312yes—
35C3×C4⋊Q8—31no—
36C3×C4⋊Q8—31no—
37C3×C4⋊Q8—32no—
38C3×#126—34no—
39C3×#126—32no—
40C3×#125—32no—
41C3×#125—32no—
42C3×#125—34no—
43C3×#125—31no—
44C3×#124—31no—
45C3×#123—32no—
46C3×C22⋊Q8—31no—
47C3×C22⋊Q8—32no—
48C3×C22⋊Q8—32no—
49C3×C22⋊Q8—32no—
50C3×#121A411yescharacteristic
51C3×C4×Q8—32no—
52C3×C4×Q8—31no—
53C3×C4×Q8—32no—
54C3×C4×Q8—32no—
55C3×C4×Q8—31no—
56C3×C4×Q8—31no—
57C3×C4×Q8C22×C311yescharacteristic
58C3×C4×Q8C22×C311yescharacteristic
59C3×C4×Q8C22×C312yes—
60C3×C4×D4—32no—
61C3×C4×D4—31no—
62C3×#117—32no—
63C3×#117—32no—
64C3×#117—32no—
65C2×C3×C4⋊C4—31no—
66C2×C3×C42—31no—
67C23×C9—41no—
68C23×C9—42no—
69D4×C9—82no—
70D4×C9—84no—
71C2×C4×C9—41no—
72C2×C4×C9—41no—
73C2×C4×C9—42no—
74C2×C4×C9—82no—
75Q8⋊C9#4111yescharacteristic
76#557—31no—
77#557C2×C911yescharacteristic, complemented
78#556—31no—
79#552—32no—
80D4×Q8C2×C912yescomplemented
81D4×Q8C2×C911yescharacteristic, complemented
82#541—32no—
83#533—31no—
84#529—31no—
85#519C2×C912yescomplemented
86C2×C4×Q8C2×C911yescharacteristic, complemented
87C2×C3×Q8—21no—
88C2×C3×Q8—22no—
89C2×C3×Q8—31no—
90C2×C3×Q8—32no—
91C2×C3×Q8—31no—
92C2×C3×Q8—61no—
93C2×C3×Q8—62no—
94C2×C3×Q8C23×C311yescharacteristic
95C2×C3×D4C2×A412yes—
96C2×C3×D4C2×A411yescharacteristic
97C22×C3×C4—31no—
98C22×C3×C4—31no—
99C22×C3×C4—32no—
100C22×C3×C4C2×A411yescharacteristic
101C3×C4⋊C4—31no—
102C3×C4⋊C4—31no—
103C3×C4⋊C4—32no—
104C3×C4⋊C4—32no—
105C3×C4⋊C4—32no—
106C3×C4⋊C4—32no—
107C3×C4⋊C4—34no—
108C3×C4⋊C4—32no—
109C3×C4⋊C4—32no—
110C3×C4⋊C4—31no—
111C3×C4⋊C4—32no—
112C3×C4⋊C4—32no—
113C3×C4⋊C4C2×A411yescharacteristic
114C3×C22⋊C4—32no—
115C3×C22⋊C4—32no—
116C3×C22⋊C4—32no—
117C3×C22⋊C4—32no—
118C3×C22⋊C4C2×A412yes—
119C3×C42—34no—
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—
129C22×C9—82no—
130C22×C9—82no—
131C22×C9—82no—
132C4×C9—42no—
133C4×C9—81no—
134C4×C9—82no—
135C22×Q8C22×C911yescharacteristic
136C22×Q8C22×C912yes—
137C4⋊Q8—31no—
138C4⋊Q8—31no—
139C4⋊Q8—32no—
140#126—34no—
141#126—32no—
142#125—32no—
143#125—32no—
144#125—34no—
145#125—31no—
146#124—31no—
147#123—32no—
148C22⋊Q8—31no—
149C22⋊Q8—32no—
150C22⋊Q8—32no—
151C22⋊Q8—32no—
152#121C22⋊C911yescharacteristic
153C4×Q8—32no—
154C4×Q8—31no—
155C4×Q8—32no—
156C4×Q8—32no—
157C4×Q8—31no—
158C4×Q8—31no—
159C4×Q8C22×C911yescharacteristic
160C4×Q8C22×C911yescharacteristic
161C4×Q8C22×C912yes—
162C4×D4—32no—
163C4×D4—31no—
164#117—32no—
165#117—32no—
166#117—32no—
167C2×C4⋊C4—31no—
168C2×C42—31no—
169C23×C3C22×A412yes—
170C23×C3C22×A411yescharacteristic
171C3×Q8—31no—
172C3×Q8—61no—
173C3×Q8—62no—
174C3×Q8—62no—
175C3×Q8—61no—
176C3×Q8—62no—
177C3×Q8C3×#4111yescharacteristic
178C3×D4—24no—
179C3×D4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—32no—
184C2×C3×C4—31no—
185C2×C3×C4—32no—
186C2×C3×C4—31no—
187C2×C3×C4—61no—
188C2×C3×C4—62no—
189C2×C3×C4—61no—
190C2×C3×C4—62no—
191C2×C3×C4—62no—
192C2×C3×C4C22×A411yescharacteristic
193C2×C3×C4C22×A412yes—
194C2×C3×C4C22×A411yescharacteristic
195C2×C9—41no—
196C2×C9—41no—
197C2×C9—41no—
198C2×C9—82no—
199C2×C9—162no—
200C2×Q8—21no—
201C2×Q8—22no—
202C2×Q8—31no—
203C2×Q8—32no—
204C2×Q8—31no—
205C2×Q8—61no—
206C2×Q8—62no—
207C2×Q8C23×C911yescharacteristic
208C2×D4C2×C22⋊C912yes—
209C2×D4C2×C22⋊C911yescharacteristic
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—32no—
213C22×C4C2×C22⋊C911yescharacteristic
214C4⋊C4—31no—
215C4⋊C4—31no—
216C4⋊C4—32no—
217C4⋊C4—32no—
218C4⋊C4—32no—
219C4⋊C4—32no—
220C4⋊C4—34no—
221C4⋊C4—32no—
222C4⋊C4—32no—
223C4⋊C4—31no—
224C4⋊C4—32no—
225C4⋊C4—32no—
226C4⋊C4C2×C22⋊C911yescharacteristic
227C22⋊C4—32no—
228C22⋊C4—32no—
229C22⋊C4—32no—
230C22⋊C4—32no—
231C22⋊C4C2×C22⋊C912yes—
232C42—34no—
233C42—31no—
234C42—31no—
235C42—31no—
236C42—31no—
237C42—32no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C32−1+4⋊C312yes—
244C22×C3C23×A411yescharacteristic
245C3×C4—21no—
246C3×C4—22no—
247C3×C4—31no—
248C3×C4—31no—
249C3×C4—62no—
250C3×C4—61no—
251C3×C4—61no—
252C3×C4—61no—
253C3×C4—62no—
254C3×C42−1+4⋊C312yes—
255C9—41noSylow
256C23C22×C22⋊C912yes—
257C23C22×C22⋊C911yescharacteristic
258Q8—31no—
259Q8—61no—
260Q8—62no—
261Q8—62no—
262Q8—61no—
263Q8—62no—
264Q8C9×#4111yescharacteristic
265D4—24no—
266D4—22no—
267C2×C4—22no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—32no—
271C2×C4—31no—
272C2×C4—32no—
273C2×C4—31no—
274C2×C4—61no—
275C2×C4—62no—
276C2×C4—61no—
277C2×C4—62no—
278C2×C4—62no—
279C2×C4C22×C22⋊C911yescharacteristic
280C2×C4C22×C22⋊C912yes—
281C2×C4C22×C22⋊C911yescharacteristic
282C2×C3—22no—
283C2×C3—42no—
284C2×C3A4×#4111yescharacteristic
285C2×C3C2×2−1+4⋊C311yescharacteristic
286C2×C3C2×2−1+4⋊C311yescharacteristic
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C222−1+4⋊C912yes—
293C22C23×C22⋊C911yescharacteristic
294C4—21no—
295C4—22no—
296C4—31no—
297C4—31no—
298C4—62no—
299C4—61no—
300C4—61no—
301C4—61no—
302C4—62no—
303C42−1+4⋊C912yes—
304C3#3523⋊C311yescharacteristic
305C2—22no—
306C2—42no—
307C2#41×C22⋊C911yescharacteristic
308C2C2×2−1+4⋊C911yescharacteristic
309C2C2×2−1+4⋊C911yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×A4
commutatorC2×Q8C23×C9
FittingC3×#3523C3
prime radical, hypercenterC3×#121A4
cosocleC2×C3×Q8C23×C3
hypocenterQ8C9×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. #557⋊C9C2semidirect
2. D4×Q8⋊C9C2semidirect
3. D4×Q8⋊C9C2semidirect
4. #519⋊C9C2semidirect
5. C2×C4×Q8⋊C9C2semidirect
32. #3523C9semidirect
77. #557C2×C9semidirect
80. D4×Q8C2×C9semidirect
81. D4×Q8C2×C9semidirect
85. #519C2×C9semidirect
86. C2×C4×Q8C2×C9semidirect

Maximal subgroups

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

subgroupindex
#557⋊C92
D4×Q8⋊C92
#519⋊C92
C2×C4×Q8⋊C92
C3×#35233
C9×#1214