Finite Group Databasebeta

← #3518⋊C9 #11919763

Subgroups — #3518⋊C9 #11919763

Order 1152.

472 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 95 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
2D4×Q8⋊C9C211yesmaximal, characteristic, complemented
3#548⋊C9C212yesmaximal, complemented
4#519⋊C9C212yesmaximal, complemented
5C4×#41⋊C9C211yesmaximal, characteristic, complemented
6C3×#3518C311yesmaximal, characteristic
7C2×#41⋊C9C2211yescharacteristic
8C22×Q8⋊C9C2212yes—
9C9×#124—41nomaximal
10C4×Q8⋊C9C2211yescharacteristic
11C4×Q8⋊C9C2211yescharacteristic
12C4×Q8⋊C9C2212yes—
13C3×Q82C2×C311yescharacteristic
14C3×#556—32no—
15C3×#556—31no—
16C3×#555—31no—
17C3×D4×Q8C2×C311yescharacteristic
18C3×#548C2×C312yes—
19C3×#542—32no—
20C3×#533—31no—
21C3×#529—31no—
22C3×#519C2×C312yes—
23C3×C4×#41C2×C311yescharacteristic
24C2×Q8×C9—41no—
25C2×D4×C9—41no—
26#41⋊C9—21no—
27C2×Q8⋊C9—22no—
28C2×Q8⋊C9C2311yescharacteristic
29C9×C22⋊C4—42no—
30C9×C22⋊C4—42no—
31C42×C9—41no—
32#3518C911yescharacteristic, complemented, Sylow
33C2×C3×#41C22×C311yescharacteristic
34C22×C3×Q8C22×C312yes—
35C3×C4⋊Q8—31no—
36C3×C4⋊Q8—32no—
37C3×C4⋊Q8—31no—
38C3×#126—34no—
39C3×#126—32no—
40C3×#125—31no—
41C3×#124—34no—
42C3×#124—32no—
43C3×#124A411yescharacteristic
44C3×#123—32no—
45C3×C22⋊Q8—32no—
46C3×C22⋊Q8—31no—
47C3×C22⋊Q8—32no—
48C3×C22⋊Q8—32no—
49C3×#121—32no—
50C3×#121—31no—
51C3×C4×Q8—32no—
52C3×C4×Q8—31no—
53C3×C4×Q8—32no—
54C3×C4×Q8—32no—
55C3×C4×Q8C22×C311yescharacteristic
56C3×C4×Q8C22×C312yes—
57C3×C4×Q8C22×C311yescharacteristic
58C3×C4×D4—31no—
59C3×C4×D4—31no—
60C3×C4×D4—31no—
61C3×C4×D4—32no—
62C3×#117—32no—
63C3×#117—32no—
64C3×#117—32no—
65C3×#117—31no—
66C2×C3×C42—31no—
67C23×C9—42no—
68Q8×C9—82no—
69D4×C9—82no—
70C2×C4×C9—41no—
71C2×C4×C9—41no—
72C2×C4×C9—41no—
73C2×C4×C9—42no—
74Q8⋊C9#4111yescharacteristic
75Q82C2×C911yescharacteristic, complemented
76#556—32no—
77#556—31no—
78#555—31no—
79D4×Q8C2×C911yescharacteristic
80#548C2×C912yescomplemented
81#542—32no—
82#533—31no—
83#529—31no—
84#519C2×C912yescomplemented
85C4×#41C2×C911yescharacteristic, complemented
86C3×#41—21no—
87C3×#41—61no—
88C2×C3×Q8—22no—
89C2×C3×Q8—32no—
90C2×C3×Q8—32no—
91C2×C3×Q8—62no—
92C2×C3×Q8C2×A411yescharacteristic
93C2×C3×Q8C23×C311yescharacteristic
94C2×C3×D4—31no—
95C2×C3×D4—32no—
96C2×C3×D4C2×A411yescharacteristic
97C22×C3×C4—32no—
98C22×C3×C4—31no—
99C22×C3×C4—31no—
100C22×C3×C4—31no—
101C3×C4⋊C4—31no—
102C3×C4⋊C4—31no—
103C3×C4⋊C4—32no—
104C3×C4⋊C4—32no—
105C3×C4⋊C4—31no—
106C3×C4⋊C4—31no—
107C3×C4⋊C4—32no—
108C3×C4⋊C4—32no—
109C3×C4⋊C4—32no—
110C3×C4⋊C4—32no—
111C3×C22⋊C4—32no—
112C3×C22⋊C4—34no—
113C3×C22⋊C4—32no—
114C3×C22⋊C4—32no—
115C3×C22⋊C4—32no—
116C3×C22⋊C4C2×A412yes—
117C3×C22⋊C4C2×A412yes—
118C3×C42—34no—
119C3×C42—31no—
120C3×C42—31no—
121C3×C42—31no—
122C3×C42—32no—
123C3×C42C2×A411yescharacteristic
124C22×C9—41no—
125C22×C9—82no—
126C22×C9—82no—
127C22×C9—82no—
128C4×C9—42no—
129C4×C9—81no—
130C4×C9—82no—
131C4×C9—81no—
132C2×#41C22×C911yescharacteristic
133C22×Q8C22×C912yes—
134C4⋊Q8—31no—
135C4⋊Q8—31no—
136C4⋊Q8—32no—
137#126—34no—
138#126—32no—
139#125—31no—
140#124—34no—
141#124—32no—
142#124C22⋊C911yescharacteristic
143#123—32no—
144C22⋊Q8—32no—
145C22⋊Q8—31no—
146C22⋊Q8—32no—
147C22⋊Q8—32no—
148#121—32no—
149#121—31no—
150C4×Q8—32no—
151C4×Q8—31no—
152C4×Q8—32no—
153C4×Q8—32no—
154C4×Q8C22×C911yescharacteristic
155C4×Q8C22×C911yescharacteristic
156C4×Q8C22×C912yes—
157C4×D4—31no—
158C4×D4—31no—
159C4×D4—31no—
160C4×D4—32no—
161#117—32no—
162#117—32no—
163#117—32no—
164#117—31no—
165C2×C42—31no—
166C23×C3—31no—
167C23×C3C22×A412yes—
168C3×Q8—22no—
169C3×Q8—31no—
170C3×Q8—62no—
171C3×Q8—62no—
172C3×Q8—122no—
173C3×Q8C3×#4111yescharacteristic
174C3×D4—22no—
175C3×D4—61no—
176C3×D4—61no—
177C3×D4—122no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—32no—
181C2×C3×C4—32no—
182C2×C3×C4—31no—
183C2×C3×C4—61no—
184C2×C3×C4—61no—
185C2×C3×C4—62no—
186C2×C3×C4—62no—
187C2×C3×C4—62no—
188C2×C3×C4—121no—
189C2×C3×C4C22×A411yescharacteristic
190C2×C3×C4C22×A411yescharacteristic
191C2×C3×C4C22×A412yes—
192C2×C3×C4C22×A411yescharacteristic
193C2×C9—41no—
194C2×C9—41no—
195C2×C9—41no—
196C2×C9—162no—
197#41—21no—
198#41—61no—
199C2×Q8—22no—
200C2×Q8—32no—
201C2×Q8—32no—
202C2×Q8—62no—
203C2×Q8C2×C22⋊C911yescharacteristic
204C2×Q8C23×C911yescharacteristic
205C2×D4—31no—
206C2×D4—32no—
207C2×D4C2×C22⋊C911yescharacteristic
208C22×C4—32no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—31no—
212C4⋊C4—31no—
213C4⋊C4—31no—
214C4⋊C4—32no—
215C4⋊C4—32no—
216C4⋊C4—31no—
217C4⋊C4—31no—
218C4⋊C4—32no—
219C4⋊C4—32no—
220C4⋊C4—32no—
221C4⋊C4—32no—
222C22⋊C4—32no—
223C22⋊C4—34no—
224C22⋊C4—32no—
225C22⋊C4—32no—
226C22⋊C4—32no—
227C22⋊C4C2×C22⋊C912yes—
228C22⋊C4C2×C22⋊C912yes—
229C42—34no—
230C42—31no—
231C42—31no—
232C42—31no—
233C42—32no—
234C42C2×C22⋊C911yescharacteristic
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—22no—
238C22×C3—61no—
239C22×C3—61no—
240C22×C3—61no—
241C22×C3C23×A411yescharacteristic
242C3×C4—21no—
243C3×C4—22no—
244C3×C4—21no—
245C3×C4—31no—
246C3×C4—31no—
247C3×C4—62no—
248C3×C4—61no—
249C3×C4—61no—
250C3×C4—62no—
251C3×C42−1+4⋊C312yes—
252C9—41noSylow
253C23—31no—
254C23C22×C22⋊C912yes—
255Q8—22no—
256Q8—31no—
257Q8—62no—
258Q8—62no—
259Q8—122no—
260Q8C9×#4111yescharacteristic
261D4—22no—
262D4—61no—
263D4—61no—
264D4—122no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—32no—
268C2×C4—32no—
269C2×C4—31no—
270C2×C4—61no—
271C2×C4—61no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—62no—
275C2×C4—121no—
276C2×C4C22×C22⋊C911yescharacteristic
277C2×C4C22×C22⋊C911yescharacteristic
278C2×C4C22×C22⋊C912yes—
279C2×C4C22×C22⋊C911yescharacteristic
280C2×C3—42no—
281C2×C3—121no—
282C2×C3A4×#4111yescharacteristic
283C2×C3C2×2−1+4⋊C311yescharacteristic
284C2×C3#594511yescharacteristic
285C22—22no—
286C22—22no—
287C22—22no—
288C22—61no—
289C22—61no—
290C22—61no—
291C22C23×C22⋊C911yescharacteristic
292C4—21no—
293C4—22no—
294C4—21no—
295C4—31no—
296C4—31no—
297C4—62no—
298C4—61no—
299C4—61no—
300C4—62no—
301C42−1+4⋊C912yes—
302C3#3518⋊C311yescharacteristic
303C2—42no—
304C2—121no—
305C2#41×C22⋊C911yescharacteristic
306C2C2×2−1+4⋊C911yescharacteristic
307C2#1059312011yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×A4
commutatorC2×Q8C23×C9
Fitting, prime radicalC3×#3518C3
cosocleC2×C3×Q8C23×C3
hypercenterC3×#124A4
hypocenterQ8C9×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. Q8×Q8⋊C9C2semidirect
2. D4×Q8⋊C9C2semidirect
3. #548⋊C9C2semidirect
4. #519⋊C9C2semidirect
5. C4×#41⋊C9C2semidirect
32. #3518C9semidirect
75. Q82C2×C9semidirect
80. #548C2×C9semidirect
84. #519C2×C9semidirect
85. C4×#41C2×C9semidirect

Maximal subgroups

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

subgroupindex
Q8×Q8⋊C92
D4×Q8⋊C92
#548⋊C92
#519⋊C92
C4×#41⋊C92
C3×#35183
C9×#1244