Finite Group Databasebeta

← C22⋊C9×#239 #13164619

Subgroups — C22⋊C9×#239 #13164619

Order 1728.

826 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C3×Q8×C22⋊C9C211yesmaximal, characteristic, complemented
2D12×C22⋊C9C213yesmaximal, complemented
3C4×D3×C22⋊C9C213yesmaximal, complemented
4C22×C3×#239C311yesmaximal, characteristic
5#41×C22⋊C9—31nomaximal
6C2×D3×C22⋊C9C2213yes—
7C9×#239—41nomaximal
8C3×C4×C22⋊C9C2213yes—
9Dic3×C22⋊C9C2211yescharacteristic
10C22×C32×Q8C2×C311yescharacteristic
11C2×C3×#239—31no—
12C2×C3×#239—33no—
13C2×C3×#239—33no—
14C2×C3×#239—31no—
15C22×C3×D12C2×C313yes—
16C22×C3×C4×D3C2×C313yes—
17Q8×C22⋊C9—21no—
18Q8×C22⋊C9D311yescharacteristic, complemented
19D4×C22⋊C9—33no—
20C2×C4×C22⋊C9—33no—
21C2×C3×C22⋊C9C2311yescharacteristic
22D3×C22⋊C9—23no—
23C3×Q8×C9—41no—
24C9×D12—43no—
25C4×D3×C9—43no—
26C22×C3×#41—31no—
27C22×#239C911yescharacteristic, complemented
28C23×C3×D3C22×C313yes—
29C2×C32×Q8—31no—
30C2×C32×Q8—33no—
31C22×C32×C4C22×C313yes—
32C22×C3×Dic3C22×C311yescharacteristic
33C3×#239—36no—
34C3×#239—36no—
35C3×#239—32no—
36C3×#239—31no—
37C3×#239—33no—
38C3×#239—33no—
39C3×#239A411yescharacteristic
40C2×C3×D12—33no—
41C2×C3×D12—36no—
42C2×C3×D12—33no—
43C2×C3×C4×D3—33no—
44C2×C3×C4×D3—33no—
45C2×C3×C4×D3—33no—
46C2×C3×C4×D3—33no—
47C22×C22⋊C9—33no—
48C9×#41—121no—
49C4×C22⋊C9—23no—
50C4×C22⋊C9—31no—
51C4×C22⋊C9C2×D313yes—
52C2×D3×C9—43no—
53C3×C22⋊C9#4111yescharacteristic, complemented
54C3×C4×C9—43no—
55C9×Dic3—41no—
56C2×C3×#41—91no—
57C2×C3×#41—93no—
58C2×C3×#41—93no—
59C2×C3×#41—91no—
60C22×C3×Q8—21no—
61C22×C3×Q8C2×C911yescharacteristic, complemented
62C22×C3×Q8C3×D311yescharacteristic
63C22×C3×D4—33no—
64C23×C3×C4—33no—
65C2×#239—31no—
66C2×#239—33no—
67C2×#239—33no—
68C2×#239—31no—
69C22×D12C2×C913yescomplemented
70C22×C4×D3C2×C913yescomplemented
71C23×C32C23×C311yescharacteristic
72C22×C3×D3—23no—
73C22×C3×D3—33no—
74C22×C3×D3—33no—
75C22×C3×D3—63no—
76C32×Q8—32no—
77C32×Q8—33no—
78C32×Q8C2×A411yescharacteristic
79C2×C32×C4—33no—
80C2×C32×C4—33no—
81C2×C3×Dic3—31no—
82C2×C3×Dic3—31no—
83C3×D12—36no—
84C3×D12—36no—
85C3×D12—33no—
86C3×D12C2×A413yes—
87C3×C4×D3—36no—
88C3×C4×D3—33no—
89C3×C4×D3—33no—
90C3×C4×D3—33no—
91C3×C4×D3C2×A413yes—
92C2×C22⋊C9—21no—
93C2×C22⋊C9—63no—
94C2×C22⋊C9C22×D311yescharacteristic
95Q8×C9—41no—
96Q8×C9—81no—
97D4×C9—123no—
98C2×C4×C9—123no—
99C22×#41—31noSylow
100C2×C3×C9—41no—
101D3×C9—83no—
102C24×C3—33no—
103C23×D3C22×C913yes—
104C3×#41—31no—
105C3×#41—93no—
106C3×#41—93no—
107C3×#41—92no—
108C3×#41—96no—
109C3×#41—96no—
110C3×#41—91no—
111C2×C3×Q8—33no—
112C2×C3×Q8—33no—
113C2×C3×Q8—31no—
114C2×C3×Q8—31no—
115C2×C3×Q8—61no—
116C2×C3×Q8—63no—
117C2×C3×D4—93no—
118C2×C3×D4—93no—
119C2×C3×D4—96no—
120C22×C3×C4—23no—
121C22×C3×C4—31no—
122C22×C3×C4—93no—
123C22×C3×C4—93no—
124C22×C3×C4—93no—
125C22×C3×C4—93no—
126C22×C3×C4C22×C913yes—
127C22×C3×C4C2×C3×D313yes—
128C22×Dic3C22×C911yescharacteristic
129#239—36no—
130#239—36no—
131#239—32no—
132#239—31no—
133#239—33no—
134#239—33no—
135#239C22⋊C911yescharacteristic, direct product factor
136C2×D12—33no—
137C2×D12—36no—
138C2×D12—33no—
139C2×C4×D3—33no—
140C2×C4×D3—33no—
141C2×C4×D3—33no—
142C2×C4×D3—33no—
143C22×C32—31no—
144C22×C32—31no—
145C22×C32C3×#4111yescharacteristic
146C2×C3×D3—33no—
147C2×C3×D3—63no—
148C2×C3×D3—63no—
149C2×C3×D3—63no—
150C2×C3×D3—63no—
151C2×C3×D3C22×A413yes—
152C32×C4—33no—
153C32×C4C22×A413yes—
154C3×Dic3—31no—
155C3×Dic3C22×A411yescharacteristic
156C22×C9—123no—
157C22⋊C9—21no—
158C22⋊C9#23911yescharacteristic, direct product factor
159C4×C9—43no—
160C4×C9—83no—
161C4×C9—121no—
162C2×#41—91no—
163C2×#41—93no—
164C2×#41—93no—
165C2×#41—91no—
166C22×Q8D3×C911yescharacteristic, complemented
167C22×D4—33no—
168C23×C4—33no—
169C3×C9—41noSylow
170C23×C3—21no—
171C23×C3—63no—
172C23×C3—93no—
173C23×C3—93no—
174C23×C3—183no—
175C23×C3C23×C911yescharacteristic
176C23×C3C22×C3×D311yescharacteristic
177C22×D3—23no—
178C22×D3—33no—
179C22×D3—33no—
180C22×D3—63no—
181C3×Q8—21no—
182C3×Q8—32no—
183C3×Q8—32no—
184C3×Q8—33no—
185C3×Q8—33no—
186C3×Q8—63no—
187C3×Q8—62no—
188C3×Q8C2×C22⋊C911yescharacteristic, complemented
189C3×Q8D3×A411yescharacteristic
190C3×D4—33no—
191C3×D4—96no—
192C3×D4—96no—
193C3×D4—93no—
194C2×C3×C4—33no—
195C2×C3×C4—33no—
196C2×C3×C4—33no—
197C2×C3×C4—33no—
198C2×C3×C4—33no—
199C2×C3×C4—63no—
200C2×C3×C4—63no—
201C2×C3×C4—93no—
202C2×C3×C4—93no—
203C2×C3×C4—91no—
204C2×C3×C4—96no—
205C2×C3×C4—93no—
206C2×C3×C4—91no—
207C2×Dic3—31no—
208C2×Dic3—31no—
209D12—36no—
210D12—36no—
211D12—33no—
212D12C2×C22⋊C913yescomplemented
213C4×D3—36no—
214C4×D3—33no—
215C4×D3—33no—
216C4×D3—33no—
217C4×D3C2×C22⋊C913yescomplemented
218C2×C32—31no—
219C2×C32—31no—
220C2×C32C23×A411yescharacteristic
221C3×D3—23no—
222C3×D3—63no—
223C2×C9—41no—
224C2×C9—81no—
225C2×C9—243no—
226C24—33no—
227#41—31no—
228#41—93no—
229#41—93no—
230#41—92no—
231#41—96no—
232#41—96no—
233#41—91no—
234C2×Q8—33no—
235C2×Q8—31no—
236C2×D4—93no—
237C2×D4—93no—
238C2×D4—96no—
239C22×C4—31no—
240C22×C4—93no—
241C22×C4—93no—
242C22×C4—93no—
243C22×C4—93no—
244C22×C4C2×D3×C913yes—
245C22×C3—21no—
246C22×C3—31no—
247C22×C3—31no—
248C22×C3—31no—
249C22×C3—31no—
250C22×C3—33no—
251C22×C3—61no—
252C22×C3—61no—
253C22×C3—93no—
254C22×C3—183no—
255C22×C3—183no—
256C22×C3—183no—
257C22×C3—183no—
258C22×C3C9×#4111yescharacteristic, complemented
259C22×C3C3×#23911yescharacteristic
260C2×D3—33no—
261C2×D3—63no—
262C2×D3—63no—
263C2×D3—63no—
264C2×D3—63no—
265C2×D3C22×C22⋊C913yes—
266C3×C4—23no—
267C3×C4—31no—
268C3×C4—33no—
269C3×C4—33no—
270C3×C4—63no—
271C3×C4—91no—
272C3×C4C22×C22⋊C913yes—
273C3×C4C2×D3×A413yes—
274Dic3—31no—
275Dic3C22×C22⋊C911yescharacteristic
276C32A4×#4111yescharacteristic
277C9—41no—
278C9—81no—
279C23—63no—
280C23—93no—
281C23—93no—
282C23—183no—
283C23C22×D3×C911yescharacteristic
284Q8—32no—
285Q8—33no—
286Q8D3×C22⋊C911yescharacteristic, complemented
287D4—33no—
288D4—96no—
289D4—96no—
290D4—93no—
291C2×C4—33no—
292C2×C4—33no—
293C2×C4—33no—
294C2×C4—93no—
295C2×C4—91no—
296C2×C4—93no—
297C2×C4—96no—
298C2×C4—93no—
299C2×C4—91no—
300C2×C3—21no—
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—61no—
306C2×C3—61no—
307C2×C3—63no—
308C2×C3—183no—
309C2×C3C23×C22⋊C911yescharacteristic
310C2×C3C22×D3×A411yescharacteristic
311D3—23no—
312D3—63no—
313C22—31no—
314C22—31no—
315C22—33no—
316C22—93no—
317C22—183no—
318C22—183no—
319C22—183no—
320C22—183no—
321C22C9×#23911yescharacteristic, complemented
322C4—31no—
323C4—33no—
324C4—91no—
325C4C2×D3×C22⋊C913yes—
326C3—21no—
327C3#41×C22⋊C911yescharacteristic, complemented
328C3A4×#23911yescharacteristic
329C2—31no—
330C2—31no—
331C2—63no—
332C2—183no—
333C2C22×D3×C22⋊C911yescharacteristic
subgroupG/H
center, FrattiniC2×C3C22×D3×A4
commutatorC23×C3C23×C9
FittingC22×C32×Q8C2×C3
socle, cosocleC23×C32C23×C3
prime radicalC22×C3×#239C3
hypercenterC3×Q8D3×A4
hypocenterC22×C3C9×#41
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C2×C3 ◁ C3×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×C22⋊C9C2semidirect
2. D12×C22⋊C9C2semidirect
3. C4×D3×C22⋊C9C2semidirect
18. Q8×C22⋊C9D3semidirect
27. C22×#239C9semidirect
53. C3×C22⋊C9#41semidirect
61. C22×C3×Q8C2×C9semidirect
69. C22×D12C2×C9semidirect
70. C22×C4×D3C2×C9semidirect
135. #239C22⋊C9direct (N × H)
158. C22⋊C9#239direct (N × H)
166. C22×Q8D3×C9semidirect
188. C3×Q8C2×C22⋊C9semidirect
212. D12C2×C22⋊C9semidirect
217. C4×D3C2×C22⋊C9semidirect
258. C22×C3C9×#41semidirect
286. Q8D3×C22⋊C9semidirect
321. C22C9×#239semidirect
327. C3#41×C22⋊C9semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C22⋊C92
D12×C22⋊C92
C4×D3×C22⋊C92
C22×C3×#2393
#41×C22⋊C93
C9×#2394