Finite Group Databasebeta

← C2×#41×C22⋊C9 #11919871

Subgroups — C2×#41×C22⋊C9 #11919871

Order 1152.

2556 conjugacy classes of subgroups; only the 350 normal subgroups are shown.

normal subgroups only

This group has 2556 conjugacy classes of subgroups — too many to draw. Only its 350 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#41×C22⋊C9C218yesmaximal, direct product factor
2C2×Q8×C22⋊C9C211yesmaximal, characteristic, complemented
3C2×D4×C22⋊C9C213yesmaximal, complemented
4C22×C4×C22⋊C9C213yesmaximal, complemented
5C23×C3×#41C311yesmaximal, characteristic
6C23×C22⋊C9C2213yes—
7C2×C9×#41—41nomaximal
8Q8×C22⋊C9C2214yescomplemented
9D4×C22⋊C9C22112yescomplemented
10C2×C4×C22⋊C9C2211yescharacteristic
11C2×C4×C22⋊C9C2213yes—
12C2×C4×C22⋊C9C22112yescomplemented
13C22×C3×#41—33no—
14C22×C3×#41—33no—
15C22×C3×#41—38no—
16C22×C3×#41—31no—
17C22×C3×#41—31no—
18C22×C3×#41C2×C318yes—
19C23×C3×Q8C2×C311yescharacteristic
20C23×C3×D4C2×C313yes—
21C24×C3×C4C2×C313yes—
22C22×C22⋊C9—26no—
23C22×C22⋊C9C2316yes—
24C22×C22⋊C9C2311yescharacteristic
25C9×#41—48no—
26C2×Q8×C9—41no—
27C2×D4×C9—43no—
28C22×C4×C9—43no—
29C4×C22⋊C9C2312yes—
30C4×C22⋊C9C2316yes—
31C23×#41C911yescharacteristic, complemented, Sylow
32C25×C3C22×C313yes—
33C2×C3×#41—324no—
34C2×C3×#41—324no—
35C2×C3×#41—38no—
36C2×C3×#41—38no—
37C2×C3×#41—324no—
38C2×C3×#41—36no—
39C2×C3×#41—324no—
40C2×C3×#41—36no—
41C2×C3×#41—32no—
42C2×C3×#41—38no—
43C2×C3×#41—38no—
44C2×C3×#41—31no—
45C2×C3×#41—33no—
46C2×C3×#41—33no—
47C2×C3×#41A411yescharacteristic
48C22×C3×Q8—33no—
49C22×C3×Q8—34no—
50C22×C3×Q8—31no—
51C22×C3×Q8C22×C314yes—
52C22×C3×D4—33no—
53C22×C3×D4—33no—
54C22×C3×D4—36no—
55C22×C3×D4—312no—
56C22×C3×D4C22×C3112yes—
57C23×C3×C4—33no—
58C23×C3×C4—33no—
59C23×C3×C4—33no—
60C23×C3×C4—312no—
61C23×C3×C4—33no—
62C23×C3×C4C22×C311yescharacteristic
63C23×C3×C4C22×C313yes—
64C23×C3×C4C22×C3112yes—
65C23×C9—43no—
66C2×C22⋊C9—26no—
67C2×C22⋊C9#4112yesdirect product factor
68C2×C22⋊C9C2411yescharacteristic
69Q8×C9—44no—
70D4×C9—412no—
71C2×C4×C9—43no—
72C2×C4×C9—412no—
73C2×C4×C9—41no—
74C22×#41—33no—
75C22×#41—33no—
76C22×#41—38no—
77C22×#41—31no—
78C22×#41—31no—
79C22×#41C2×C918yescomplemented
80C23×Q8C2×C911yescharacteristic, complemented
81C23×D4C2×C913yescomplemented
82C24×C4C2×C913yescomplemented
83C24×C3—26no—
84C24×C3—36no—
85C24×C3—33no—
86C24×C3—33no—
87C24×C3—66no—
88C24×C3C23×C316yes—
89C24×C3C23×C311yescharacteristic
90C3×#41—348no—
91C3×#41—348no—
92C3×#41—316no—
93C3×#41—38no—
94C3×#41—324no—
95C3×#41—324no—
96C3×#41C2×A418yes—
97C2×C3×Q8—312no—
98C2×C3×Q8—312no—
99C2×C3×Q8—32no—
100C2×C3×Q8—34no—
101C2×C3×Q8—34no—
102C2×C3×Q8—33no—
103C2×C3×Q8C2×A411yescharacteristic
104C2×C3×D4—324no—
105C2×C3×D4—36no—
106C2×C3×D4—312no—
107C2×C3×D4—36no—
108C2×C3×D4—312no—
109C2×C3×D4—312no—
110C2×C3×D4—312no—
111C2×C3×D4—324no—
112C2×C3×D4—33no—
113C2×C3×D4C2×A413yes—
114C22×C3×C4—312no—
115C22×C3×C4—312no—
116C22×C3×C4—312no—
117C22×C3×C4—36no—
118C22×C3×C4—33no—
119C22×C3×C4—33no—
120C22×C3×C4—312no—
121C22×C3×C4—33no—
122C22×C3×C4—312no—
123C22×C3×C4—31no—
124C22×C3×C4—33no—
125C22×C3×C4—31no—
126C22×C3×C4—32no—
127C22×C3×C4—33no—
128C22×C3×C4—36no—
129C22×C3×C4—312no—
130C22×C3×C4—312no—
131C22×C3×C4—312no—
132C22×C3×C4C2×A413yes—
133C22×C3×C4C23×C312yes—
134C22×C3×C4C23×C316yes—
135C22×C9—41no—
136C22×C9—46no—
137C22×C9—86no—
138C22⋊C9C2×#4111yescharacteristic, direct product factor
139C4×C9—42no—
140C4×C9—46no—
141C25C22×C913yes—
142C2×#41—324no—
143C2×#41—324no—
144C2×#41—38no—
145C2×#41—38no—
146C2×#41—324no—
147C2×#41—36no—
148C2×#41—36no—
149C2×#41—324no—
150C2×#41—32no—
151C2×#41—38no—
152C2×#41—38no—
153C2×#41—31no—
154C2×#41—33no—
155C2×#41—33no—
156C2×#41C22⋊C911yescharacteristic, direct product factor
157C22×Q8—33no—
158C22×Q8—34no—
159C22×Q8—31no—
160C22×Q8C22×C914yescomplemented
161C22×D4—33no—
162C22×D4—33no—
163C22×D4—36no—
164C22×D4—312no—
165C22×D4C22×C9112yescomplemented
166C23×C4—33no—
167C23×C4—33no—
168C23×C4—33no—
169C23×C4—33no—
170C23×C4—312no—
171C23×C4C22×C911yescharacteristic
172C23×C4C22×C913yes—
173C23×C4C22×C9112yescomplemented
174C23×C3—26no—
175C23×C3—31no—
176C23×C3—32no—
177C23×C3—31no—
178C23×C3—36no—
179C23×C3—36no—
180C23×C3—36no—
181C23×C3—33no—
182C23×C3—36no—
183C23×C3—66no—
184C23×C3—66no—
185C23×C3—66no—
186C23×C3—612no—
187C23×C3—66no—
188C23×C3—66no—
189C23×C3—612no—
190C23×C3C3×#4112yes—
191C23×C3C22×A413yes—
192C23×C3C24×C311yescharacteristic
193C3×Q8—38no—
194C3×Q8—312no—
195C3×Q8C22×A414yes—
196C3×D4—324no—
197C3×D4—324no—
198C3×D4—312no—
199C3×D4C22×A4112yes—
200C2×C3×C4—32no—
201C2×C3×C4—36no—
202C2×C3×C4—324no—
203C2×C3×C4—31no—
204C2×C3×C4—33no—
205C2×C3×C4—312no—
206C2×C3×C4—32no—
207C2×C3×C4—32no—
208C2×C3×C4—36no—
209C2×C3×C4—36no—
210C2×C3×C4—312no—
211C2×C3×C4—312no—
212C2×C3×C4—32no—
213C2×C3×C4—36no—
214C2×C3×C4C22×A413yes—
215C2×C3×C4C22×A4112yes—
216C2×C3×C4C22×A411yescharacteristic
217C2×C9—42no—
218C2×C9—41no—
219C2×C9—86no—
220C24—26no—
221C24—36no—
222C24—33no—
223C24—33no—
224C24—66no—
225C24C23×C916yes—
226C24C23×C911yescharacteristic
227#41—348no—
228#41—348no—
229#41—316no—
230#41—38no—
231#41—324no—
232#41—324no—
233#41C2×C22⋊C918yesdirect product factor
234C2×Q8—312no—
235C2×Q8—312no—
236C2×Q8—32no—
237C2×Q8—34no—
238C2×Q8—34no—
239C2×Q8—33no—
240C2×Q8C2×C22⋊C911yescharacteristic, complemented
241C2×D4—324no—
242C2×D4—36no—
243C2×D4—312no—
244C2×D4—36no—
245C2×D4—312no—
246C2×D4—312no—
247C2×D4—312no—
248C2×D4—324no—
249C2×D4—33no—
250C2×D4C2×C22⋊C913yescomplemented
251C22×C4—312no—
252C22×C4—312no—
253C22×C4—312no—
254C22×C4—36no—
255C22×C4—33no—
256C22×C4—31no—
257C22×C4—33no—
258C22×C4—33no—
259C22×C4—33no—
260C22×C4—312no—
261C22×C4—312no—
262C22×C4—31no—
263C22×C4—32no—
264C22×C4—33no—
265C22×C4—36no—
266C22×C4—312no—
267C22×C4—312no—
268C22×C4—312no—
269C22×C4C2×C22⋊C913yescomplemented
270C22×C4C23×C912yes—
271C22×C4C23×C916yes—
272C22×C3—26no—
273C22×C3—32no—
274C22×C3—31no—
275C22×C3—32no—
276C22×C3—32no—
277C22×C3—32no—
278C22×C3—31no—
279C22×C3—31no—
280C22×C3—36no—
281C22×C3—66no—
282C22×C3—612no—
283C22×C3—66no—
284C22×C3—66no—
285C22×C3—66no—
286C22×C3—612no—
287C22×C3—66no—
288C22×C3C2×C3×#4111yescharacteristic
289C22×C3C23×A416yes—
290C22×C3C23×A411yescharacteristic
291C3×C4—32no—
292C3×C4—36no—
293C3×C4C23×A412yes—
294C3×C4C23×A416yes—
295C9—41noSylow
296C23—26no—
297C23—31no—
298C23—32no—
299C23—31no—
300C23—36no—
301C23—36no—
302C23—36no—
303C23—36no—
304C23—33no—
305C23—66no—
306C23—66no—
307C23—66no—
308C23—612no—
309C23—66no—
310C23—66no—
311C23—612no—
312C23C9×#4112yescomplemented
313C23C22×C22⋊C913yes—
314C23C24×C911yescharacteristic
315Q8—38no—
316Q8—312no—
317Q8C22×C22⋊C914yescomplemented
318D4—324no—
319D4—324no—
320D4—312no—
321D4C22×C22⋊C9112yescomplemented
322C2×C4—32no—
323C2×C4—36no—
324C2×C4—324no—
325C2×C4—31no—
326C2×C4—33no—
327C2×C4—312no—
328C2×C4—32no—
329C2×C4—32no—
330C2×C4—36no—
331C2×C4—36no—
332C2×C4—312no—
333C2×C4—312no—
334C2×C4—32no—
335C2×C4—36no—
336C2×C4C22×C22⋊C911yescharacteristic
337C2×C4C22×C22⋊C913yes—
338C2×C4C22×C22⋊C9112yescomplemented
339C2×C3—26no—
340C2×C3—32no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—66no—
344C2×C3A4×#4112yes—
345C2×C3C24×A411yescharacteristic
346C22—26no—
347C22—32no—
348C22—31no—
349C22—32no—
350C22—31no—
351C22—31no—
352C22—32no—
353C22—32no—
354C22—36no—
355C22—66no—
356C22—66no—
357C22—612no—
358C22—66no—
359C22—66no—
360C22—612no—
361C22—66no—
362C22C2×C9×#4111yescharacteristic, complemented
363C22C23×C22⋊C916yes—
364C22C23×C22⋊C911yescharacteristic
365C4—32no—
366C4—36no—
367C4C23×C22⋊C912yes—
368C4C23×C22⋊C916yes—
369C3C2×A4×#4111yescharacteristic
370C2—26no—
371C2—31no—
372C2—32no—
373C2—31no—
374C2—66no—
375C2#41×C22⋊C912yesdirect product factor
376C2C24×C22⋊C911yescharacteristic
subgroupG/H
centerC2×C3×C4C22×A4
commutatorC23C24×C9
FrattiniC2×C3C24×A4
Fitting, prime radicalC23×C3×#41C3
socleC24×C3C23×C3
cosocleC23×C3C24×C3
hypercenterC2×C3×#41A4
hypocenterC22C2×C9×#41
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×#41

Complements

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

NH ≅ G/Ntype
1. #41×C22⋊C9C2direct (N × H)
2. C2×Q8×C22⋊C9C2semidirect
3. C2×D4×C22⋊C9C2semidirect
4. C22×C4×C22⋊C9C2semidirect
8. Q8×C22⋊C9C22semidirect
9. D4×C22⋊C9C22semidirect
12. C2×C4×C22⋊C9C22semidirect
31. C23×#41C9semidirect
67. C2×C22⋊C9#41direct (N × H)
79. C22×#41C2×C9semidirect
80. C23×Q8C2×C9semidirect
81. C23×D4C2×C9semidirect
82. C24×C4C2×C9semidirect
138. C22⋊C9C2×#41direct (N × H)
156. C2×#41C22⋊C9direct (N × H)
160. C22×Q8C22×C9semidirect
165. C22×D4C22×C9semidirect
173. C23×C4C22×C9semidirect
233. #41C2×C22⋊C9direct (N × H)
240. C2×Q8C2×C22⋊C9semidirect
250. C2×D4C2×C22⋊C9semidirect
269. C22×C4C2×C22⋊C9semidirect
312. C23C9×#41semidirect
317. Q8C22×C22⋊C9semidirect
321. D4C22×C22⋊C9semidirect
338. C2×C4C22×C22⋊C9semidirect
362. C22C2×C9×#41semidirect
375. C2#41×C22⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
#41×C22⋊C92
C2×Q8×C22⋊C92
C2×D4×C22⋊C92
C22×C4×C22⋊C92
C23×C3×#413
C2×C9×#414