Finite Group Databasebeta

← #3500⋊C9 #11919751

Subgroups — #3500⋊C9 #11919751

Order 1152.

665 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 665 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×2−1+4⋊C9C211yesmaximal, characteristic, complemented
2C2×#63700C211yesmaximal, characteristic, complemented
3#548⋊C9C214yesmaximal, complemented
4C4×#41⋊C9C211yesmaximal, characteristic, complemented
5C3×#3500C311yesmaximal, characteristic
62−1+4⋊C9—22no—
7#63700—22no—
8C2×#41⋊C9C2212yes—
9C2×#41⋊C9C2211yescharacteristic
10C22×Q8⋊C9C2212yescomplemented
11C9×#124—41nomaximal
12C4×Q8⋊C9C2212yescomplemented
13C2×C3×2−1+4C2×C311yescharacteristic
14C2×C3×H(5,𝔽2)C2×C311yescharacteristic
15C3×#548C2×C314yes—
16C3×#546—34no—
17C3×#536—31no—
18C3×#535—31no—
19C3×#532—31no—
20C2×C3×#124—31no—
21C3×C4×#41C2×C311yescharacteristic
22C2×Q8×C9—41no—
23C2×D4×C9—41no—
24#41⋊C9—22no—
25#41⋊C9D412yescomplemented
26C2×Q8⋊C9—22no—
27C2×Q8⋊C9C2311yescharacteristic
28C9×C22⋊C4—44no—
29C42×C9—41no—
30#3500C911yescharacteristic, complemented, Sylow
31C3×2−1+4—22no—
32C3×2−1+4—62no—
33C3×H(5,𝔽2)—22no—
34C3×H(5,𝔽2)—62no—
35C2×C3×#41—31no—
36C2×C3×#41—31no—
37C2×C3×#41—32no—
38C2×C3×#41C22×C312yes—
39C2×C3×#41C22×C311yescharacteristic
40C22×C3×Q8—31no—
41C22×C3×Q8C22×C312yes—
42C22×C3×D4—31no—
43C22×C3×D4—32no—
44C3×C4⋊Q8—31no—
45C3×#127—31no—
46C3×#124—32no—
47C3×#124—34no—
48C3×#124—31no—
49C3×#124A411yescharacteristic
50C3×#123—34no—
51C3×C22⋊Q8—34no—
52C3×#121—34no—
53C3×#121—34no—
54C3×#120—34no—
55C3×C4×Q8C22×C312yes—
56C3×C4×D4—32no—
57C3×C4×D4—34no—
58C3×#117—31no—
59C2×C3×C22⋊C4—34no—
60C2×C3×C42—31no—
61C23×C9—42no—
62Q8×C9—82no—
63D4×C9—82no—
64C2×C4×C9—41no—
65C2×C4×C9—42no—
66C2×C4×C9—42no—
67Q8⋊C9C2×D411yescharacteristic
68C2×2−1+4C2×C911yescharacteristic
69C2×H(5,𝔽2)C2×C911yescharacteristic, complemented
70#548C2×C914yescomplemented
71#546—34no—
72#536—31no—
73#535—31no—
74#532—31no—
75C2×#124—31no—
76C4×#41C2×C911yescharacteristic, complemented
77C24×C3—32no—
78C3×#41—22no—
79C3×#41—32no—
80C3×#41—62no—
81C3×#41—62no—
82C3×#41—62no—
83C3×#41—62no—
84C3×#41—64no—
85C3×#41—64no—
86C3×#41—62no—
87C3×#41C3×D412yes—
88C2×C3×Q8—22no—
89C2×C3×Q8—31no—
90C2×C3×Q8—32no—
91C2×C3×Q8—62no—
92C2×C3×Q8—62no—
93C2×C3×Q8—62no—
94C2×C3×Q8C2×A411yescharacteristic
95C2×C3×Q8C23×C311yescharacteristic
96C2×C3×D4—31no—
97C2×C3×D4—32no—
98C2×C3×D4—32no—
99C2×C3×D4—34no—
100C2×C3×D4—32no—
101C2×C3×D4—31no—
102C2×C3×D4—62no—
103C2×C3×D4—62no—
104C2×C3×D4—64no—
105C2×C3×D4—64no—
106C2×C3×D4C2×A411yescharacteristic
107C22×C3×C4—31no—
108C22×C3×C4—32no—
109C22×C3×C4—32no—
110C22×C3×C4—32no—
111C22×C3×C4—32no—
112C3×C4⋊C4—32no—
113C3×C4⋊C4—34no—
114C3×C22⋊C4—38no—
115C3×C22⋊C4—32no—
116C3×C22⋊C4—34no—
117C3×C22⋊C4C2×A414yes—
118C3×C42—31no—
119C3×C42—32no—
120C3×C42C2×A411yescharacteristic
121C22×C9—41no—
122C22×C9—82no—
123C22×C9—84no—
124C4×C9—42no—
125C4×C9—82no—
126C4×C9—82no—
1272−1+4—22no—
1282−1+4—62no—
129H(5,𝔽2)—22no—
130H(5,𝔽2)—62no—
131C2×#41—31no—
132C2×#41—31no—
133C2×#41—32no—
134C2×#41C22×C912yes—
135C2×#41C22×C911yescharacteristic
136C22×Q8—31no—
137C22×Q8C22×C912yes—
138C22×D4—31no—
139C22×D4—32no—
140C4⋊Q8—31no—
141#127—31no—
142#124—32no—
143#124—34no—
144#124—31no—
145#124C22⋊C911yescharacteristic
146#123—34no—
147C22⋊Q8—34no—
148#121—34no—
149#121—34no—
150#120—34no—
151C4×Q8C22×C912yes—
152C4×D4—32no—
153C4×D4—34no—
154#117—31no—
155C2×C22⋊C4—34no—
156C2×C42—31no—
157C23×C3—31no—
158C23×C3—32no—
159C23×C3—64no—
160C23×C3—124no—
161C23×C3C22×A412yes—
162C3×Q8—22no—
163C3×Q8—31no—
164C3×Q8—62no—
165C3×Q8—62no—
166C3×Q8—62no—
167C3×Q8C2×C3×D411yescharacteristic
168C3×D4—22no—
169C3×D4—32no—
170C3×D4—32no—
171C3×D4—64no—
172C3×D4—62no—
173C3×D4—62no—
174C3×D4—64no—
175C3×D4—62no—
176C3×D4—124no—
177C2×C3×C4—31no—
178C2×C3×C4—32no—
179C2×C3×C4—32no—
180C2×C3×C4—32no—
181C2×C3×C4—32no—
182C2×C3×C4—62no—
183C2×C3×C4—62no—
184C2×C3×C4—62no—
185C2×C3×C4—62no—
186C2×C3×C4—64no—
187C2×C3×C4—122no—
188C2×C3×C4C22×A412yes—
189C2×C3×C4C22×A412yes—
190C2×C3×C4C22×A411yescharacteristic
191C2×C9—41no—
192C2×C9—42no—
193C2×C9—162no—
194C24—32no—
195#41—22no—
196#41—32no—
197#41—62no—
198#41—62no—
199#41—62no—
200#41—62no—
201#41—64no—
202#41—64no—
203#41—62no—
204#41D4×C912yes—
205C2×Q8—22no—
206C2×Q8—31no—
207C2×Q8—32no—
208C2×Q8—62no—
209C2×Q8—62no—
210C2×Q8—62no—
211C2×Q8C2×C22⋊C911yescharacteristic
212C2×Q8C23×C911yescharacteristic
213C2×D4—31no—
214C2×D4—32no—
215C2×D4—32no—
216C2×D4—34no—
217C2×D4—32no—
218C2×D4—31no—
219C2×D4—62no—
220C2×D4—62no—
221C2×D4—64no—
222C2×D4—64no—
223C2×D4C2×C22⋊C911yescharacteristic
224C22×C4—31no—
225C22×C4—32no—
226C22×C4—32no—
227C22×C4—32no—
228C22×C4—32no—
229C4⋊C4—32no—
230C4⋊C4—34no—
231C22⋊C4—38no—
232C22⋊C4—32no—
233C22⋊C4—34no—
234C22⋊C4C2×C22⋊C914yes—
235C42—31no—
236C42—32no—
237C42C2×C22⋊C911yescharacteristic
238C22×C3—22no—
239C22×C3—24no—
240C22×C3—32no—
241C22×C3—62no—
242C22×C3—62no—
243C22×C3—64no—
244C22×C3—244no—
245C22×C3C23×A411yescharacteristic
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—62no—
251C3×C4—62no—
252C3×C4D4×A412yes—
253C9—41noSylow
254C23—31no—
255C23—32no—
256C23—64no—
257C23—124no—
258C23C22×C22⋊C912yes—
259Q8—22no—
260Q8—31no—
261Q8—62no—
262Q8—62no—
263Q8—62no—
264Q8C2×D4×C911yescharacteristic
265D4—22no—
266D4—32no—
267D4—32no—
268D4—64no—
269D4—62no—
270D4—62no—
271D4—64no—
272D4—62no—
273D4—124no—
274C2×C4—31no—
275C2×C4—32no—
276C2×C4—32no—
277C2×C4—32no—
278C2×C4—32no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—62no—
283C2×C4—64no—
284C2×C4—122no—
285C2×C4C22×C22⋊C912yes—
286C2×C4C22×C22⋊C912yes—
287C2×C4C22×C22⋊C911yescharacteristic
288C2×C3—42no—
289C2×C3—62no—
290C2×C3—122no—
291C2×C3C2×D4×A411yescharacteristic
292C2×C3#594512yes—
293C22—22no—
294C22—24no—
295C22—32no—
296C22—62no—
297C22—62no—
298C22—64no—
299C22—244no—
300C22C23×C22⋊C911yescharacteristic
301C4—22no—
302C4—22no—
303C4—31no—
304C4—31no—
305C4—62no—
306C4—62no—
307C4D4×C22⋊C912yes—
308C3#3500⋊C311yescharacteristic
309C2—42no—
310C2—62no—
311C2—122no—
312C2C2×D4×C22⋊C911yescharacteristic
313C2#1059312012yes—
subgroupG/H
center, Frattini, socleC22×C3C23×A4
commutatorC2×Q8C23×C9
Fitting, prime radicalC3×#3500C3
cosocleC2×C3×Q8C23×C3
hypercenterC3×#124A4
hypocenterQ8C2×D4×C9
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. C2×2−1+4⋊C9C2semidirect
2. C2×#63700C2semidirect
3. #548⋊C9C2semidirect
4. C4×#41⋊C9C2semidirect
10. C22×Q8⋊C9C22semidirect
12. C4×Q8⋊C9C22semidirect
25. #41⋊C9D4semidirect
30. #3500C9semidirect
69. C2×H(5,𝔽2)C2×C9semidirect
70. #548C2×C9semidirect
76. C4×#41C2×C9semidirect

Maximal subgroups

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

subgroupindex
C2×2−1+4⋊C92
C2×#637002
#548⋊C92
C4×#41⋊C92
C3×#35003
C9×#1244