Finite Group Databasebeta

← #3517⋊C9 #11919761

Subgroups — #3517⋊C9 #11919761

Order 1152.

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

normal subgroups only

This group has 486 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
1#557⋊C9C211yesmaximal, characteristic, complemented
2C4⋊(#41⋊C9)C211yesmaximal, characteristic, complemented
3C4⋊(#41⋊C9)C212yesmaximal, complemented
4#520⋊C9C212yesmaximal, complemented
5C4×#41⋊C9C211yesmaximal, characteristic, complemented
6C3×#3517C311yesmaximal, characteristic
7C2×#41⋊C9C2211yescharacteristic
8C2×#41⋊C9C2212yes—
9C9×#125—41nomaximal
10C4×Q8⋊C9C2211yescharacteristic
11C4×Q8⋊C9C2211yescharacteristic
12C4×Q8⋊C9C2212yes—
13C3×#557C2×C311yescharacteristic
14C3×#553—32no—
15C3×#550—31no—
16C3×#550C2×C311yescharacteristic
17C3×#550C2×C312yes—
18C3×#547—31no—
19C3×#540—32no—
20C3×#532—31no—
21C3×#529—31no—
22C3×#520C2×C312yes—
23C3×C4×#41C2×C311yescharacteristic
24#41⋊C9—21no—
25#41⋊C9—22no—
26C2×Q8⋊C9C2311yescharacteristic
27C9×C4⋊C4—42no—
28C9×C4⋊C4—41no—
29C9×C4⋊C4—42no—
30C9×C4⋊C4—41no—
31C42×C9—41no—
32#3517C911yescharacteristic, complemented, Sylow
33C2×C3×#41—32no—
34C2×C3×#41C22×C311yescharacteristic
35C2×C3×#41C22×C312yes—
36C3×C4⋊Q8—31no—
37C3×#127—31no—
38C3×#127—32no—
39C3×#126—32no—
40C3×#125—32no—
41C3×#125A411yescharacteristic
42C3×#124—31no—
43C3×#123—34no—
44C3×#123—32no—
45C3×C22⋊Q8—31no—
46C3×#121—32no—
47C3×#121—34no—
48C3×#121—31no—
49C3×#121—32no—
50C3×#121—32no—
51C3×#121—32no—
52C3×C4×Q8—31no—
53C3×C4×Q8C22×C311yescharacteristic
54C3×C4×Q8C22×C311yescharacteristic
55C3×C4×Q8C22×C312yes—
56C3×C4×D4—32no—
57C3×C4×D4—31no—
58C3×C4×D4—31no—
59C3×C4×D4—31no—
60C3×C4×D4—32no—
61C3×C4×D4—32no—
62C3×C4×D4—32no—
63C3×#117—32no—
64C3×#117—31no—
65C2×C3×C4⋊C4—32no—
66C2×C3×C42—31no—
67C2×C4×C9—41no—
68C2×C4×C9—41no—
69C2×C4×C9—41no—
70C2×C4×C9—42no—
71C2×C4×C9—42no—
72Q8⋊C9#4111yescharacteristic, complemented
73#557C2×C911yescharacteristic
74#553—32no—
75#550—31no—
76#550C2×C911yescharacteristic
77#550C2×C912yes—
78#547—31no—
79#540—32no—
80#532—31no—
81#529—31no—
82#520C2×C912yes—
83C4×#41C2×C911yescharacteristic
84C3×#41—21no—
85C3×#41—22no—
86C3×#41—61no—
87C3×#41—62no—
88C3×#41—124no—
89C2×C3×Q8—31no—
90C2×C3×Q8C23×C311yescharacteristic
91C2×C3×D4—31no—
92C2×C3×D4—32no—
93C2×C3×D4—32no—
94C2×C3×D4—34no—
95C2×C3×D4—31no—
96C2×C3×D4—32no—
97C22×C3×C4—34no—
98C22×C3×C4—31no—
99C22×C3×C4—31no—
100C22×C3×C4—32no—
101C22×C3×C4—31no—
102C3×C4⋊C4—31no—
103C3×C4⋊C4—32no—
104C3×C4⋊C4—31no—
105C3×C4⋊C4—32no—
106C3×C4⋊C4—32no—
107C3×C4⋊C4—32no—
108C3×C4⋊C4C2×A412yes—
109C3×C4⋊C4C2×A411yescharacteristic
110C3×C4⋊C4C2×A412yes—
111C3×C4⋊C4C2×A411yescharacteristic
112C3×C22⋊C4—32no—
113C3×C22⋊C4—32no—
114C3×C22⋊C4—32no—
115C3×C22⋊C4—32no—
116C3×C22⋊C4—32no—
117C3×C22⋊C4—32no—
118C3×C42—31no—
119C3×C42—31no—
120C3×C42—31no—
121C3×C42—32no—
122C3×C42C2×A411yescharacteristic
123C22×C9—41no—
124C4×C9—42no—
125C4×C9—81no—
126C4×C9—82no—
127C4×C9—81no—
128C4×C9—82no—
129C2×#41—32no—
130C2×#41C22×C911yescharacteristic
131C2×#41C22×C912yes—
132C4⋊Q8—31no—
133#127—31no—
134#127—32no—
135#126—32no—
136#125—32no—
137#125C22⋊C911yescharacteristic
138#124—31no—
139#123—34no—
140#123—32no—
141C22⋊Q8—31no—
142#121—32no—
143#121—34no—
144#121—31no—
145#121—32no—
146#121—32no—
147#121—32no—
148C4×Q8—31no—
149C4×Q8C22×C911yescharacteristic
150C4×Q8C22×C911yescharacteristic
151C4×Q8C22×C912yes—
152C4×D4—32no—
153C4×D4—31no—
154C4×D4—31no—
155C4×D4—31no—
156C4×D4—32no—
157C4×D4—32no—
158C4×D4—32no—
159#117—32no—
160#117—31no—
161C2×C4⋊C4—32no—
162C2×C42—31no—
163C23×C3—31no—
164C23×C3—32no—
165C3×Q8—31no—
166C3×Q8—62no—
167C3×Q8C3×#4111yescharacteristic
168C3×D4—61no—
169C3×D4—62no—
170C3×D4—61no—
171C3×D4—62no—
172C3×D4—62no—
173C3×D4—122no—
174C3×D4—122no—
175C3×D4—124no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—32no—
180C2×C3×C4—61no—
181C2×C3×C4—62no—
182C2×C3×C4—61no—
183C2×C3×C4—62no—
184C2×C3×C4—62no—
185C2×C3×C4—121no—
186C2×C3×C4—124no—
187C2×C3×C4C22×A411yescharacteristic
188C2×C3×C4C22×A411yescharacteristic
189C2×C3×C4C22×A412yes—
190C2×C3×C4C22×A412yes—
191C2×C3×C4C22×A411yescharacteristic
192C2×C9—41no—
193C2×C9—41no—
194C2×C9—41no—
195#41—21no—
196#41—22no—
197#41—61no—
198#41—62no—
199#41—124no—
200C2×Q8—31no—
201C2×Q8C23×C911yescharacteristic
202C2×D4—31no—
203C2×D4—32no—
204C2×D4—32no—
205C2×D4—34no—
206C2×D4—31no—
207C2×D4—32no—
208C22×C4—34no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—32no—
212C22×C4—31no—
213C4⋊C4—31no—
214C4⋊C4—32no—
215C4⋊C4—31no—
216C4⋊C4—32no—
217C4⋊C4—32no—
218C4⋊C4—32no—
219C4⋊C4C2×C22⋊C912yes—
220C4⋊C4C2×C22⋊C911yescharacteristic
221C4⋊C4C2×C22⋊C912yes—
222C4⋊C4C2×C22⋊C911yescharacteristic
223C22⋊C4—32no—
224C22⋊C4—32no—
225C22⋊C4—32no—
226C22⋊C4—32no—
227C22⋊C4—32no—
228C22⋊C4—32no—
229C42—31no—
230C42—31no—
231C42—31no—
232C42—32no—
233C42C2×C22⋊C911yescharacteristic
234C22×C3—61no—
235C22×C3—62no—
236C22×C3—61no—
237C22×C3—62no—
238C22×C3—61no—
239C22×C3—62no—
240C22×C3C23×A411yescharacteristic
241C3×C4—21no—
242C3×C4—22no—
243C3×C4—21no—
244C3×C4—22no—
245C3×C4—31no—
246C3×C4—31no—
247C3×C4—61no—
248C3×C4—62no—
249C3×C4—61no—
250C3×C4#99412yes—
251C9—41noSylow
252C23—31no—
253C23—32no—
254Q8—31no—
255Q8—62no—
256Q8C9×#4111yescharacteristic
257D4—61no—
258D4—62no—
259D4—61no—
260D4—62no—
261D4—62no—
262D4—122no—
263D4—124no—
264D4—122no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—31no—
268C2×C4—32no—
269C2×C4—61no—
270C2×C4—62no—
271C2×C4—61no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—121no—
275C2×C4—124no—
276C2×C4C22×C22⋊C911yescharacteristic
277C2×C4C22×C22⋊C911yescharacteristic
278C2×C4C22×C22⋊C912yes—
279C2×C4C22×C22⋊C912yes—
280C2×C4C22×C22⋊C911yescharacteristic
281C2×C3—121no—
282C2×C3—122no—
283C2×C3A4×#4111yescharacteristic
284C2×C3C2×#99411yescharacteristic
285C2×C3#594511yescharacteristic
286C22—61no—
287C22—62no—
288C22—61no—
289C22—62no—
290C22—61no—
291C22—62no—
292C22C23×C22⋊C911yescharacteristic
293C4—21no—
294C4—22no—
295C4—21no—
296C4—22no—
297C4—31no—
298C4—31no—
299C4—61no—
300C4—62no—
301C4—61no—
302C4#6370012yes—
303C3#3517⋊C311yescharacteristic
304C2—121no—
305C2—122no—
306C2#41×C22⋊C911yescharacteristic
307C2C2×#6370011yescharacteristic
308C2#1059312011yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×A4
commutatorC2×Q8C23×C9
Fitting, prime radicalC3×#3517C3
cosocleC2×C3×Q8C23×C3
hypercenterC3×#125A4
hypocenterQ8C9×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
1. #557⋊C9C2semidirect
2. C4⋊(#41⋊C9)C2semidirect
3. C4⋊(#41⋊C9)C2semidirect
4. #520⋊C9C2semidirect
5. C4×#41⋊C9C2semidirect
32. #3517C9semidirect
72. Q8⋊C9#41semidirect

Maximal subgroups

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

subgroupindex
#557⋊C92
C4⋊(#41⋊C9)2
#520⋊C92
C4×#41⋊C92
C3×#35173
C9×#1254