Finite Group Databasebeta

← C22×#10620203 #72252983

Subgroups — C22×#10620203 #72252983

Order 2592.

1946 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1946 conjugacy classes of subgroups — too many to draw. Only its 168 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
1C22×C3×#1113C212yesmaximal, complemented
2C22×C3⋊(C3⋊#157)C211yesmaximal, characteristic, complemented
3C2×#10620203C2112yesmaximal, direct product factor
4C22×D32⋊D3—32nomaximal
5C22×C32×C3⋊D3—22no—
6C22×C32×C3⋊D3C2211yescharacteristic
7C2×C3×#1113—26no—
8C2×C3×#1113C22112yescomplemented
9C2×C3⋊(C3⋊#157)C2216yescomplemented
10#10620203C22116yesdirect product factor
11C22×#1113D312yescomplemented
12C22×C3×D32—32no—
13C22×C3×D32—62no—
14C22×C3⋊#157—32no—
15C22×C3⋊#157—61no—
16C2×D32⋊D3—324no—
17C22×C34D411yescharacteristic, complemented
18C2×C32×C3⋊D3—212no—
19C2×C32×C3⋊D3D413yescomplemented
20C2×C32×C3⋊D3C2313yes—
21C3×#1113—212no—
22C3×#1113C2318yescomplemented
23C3⋊(C3⋊#157)C2314yescomplemented
24C22×GO+(4,𝔽2)—91no—
25C22×#625—91nomaximal
26C22×C3×C3⋊D3—22no—
27C22×C3×C3⋊D3—21no—
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3—42no—
30C22×C3×C3⋊D3C2×D312yes—
31C22×C32×D3—62no—
32C22×C32×D3—62no—
33C22×C32×D3—62no—
34C22×C32×D3—122no—
35C2×#1113—26no—
36C2×#1113C2×D3112yescomplemented
37C2×C3×D32—312no—
38C2×C3×D32—612no—
39C2×C3×D32—612no—
40C2×C3×D32—66no—
41C2×C3⋊#157—312no—
42C2×C3⋊#157—66no—
43D32⋊D3—332no—
44C2×C34C2×D413yescomplemented
45C32×C3⋊D3—28no—
46C32×C3⋊D3C2×D413yescomplemented
47C32×C3⋊D3C2411yescharacteristic
48C23×C3×D3—92no—
49C23×C3×D3—182no—
50C22×D32—32no—
51C22×D32—62no—
52C22×#157—91no—
53C2×GO+(4,𝔽2)—912no—
54C22×C3⋊Dic3—91no—
55C2×#625—912no—
56C22×C33—22no—
57C22×C33—21no—
58C22×C33—42no—
59C22×C33—42no—
60C22×C33—82no—
61C22×C33#6712yescomplemented
62C2×C3×C3⋊D3—212no—
63C2×C3×C3⋊D3—23no—
64C2×C3×C3⋊D3—23no—
65C2×C3×C3⋊D3—212no—
66C2×C3×C3⋊D3—412no—
67C2×C3×C3⋊D3#6716yescomplemented
68C2×C3×C3⋊D3C22×D316yes—
69C2×C32×D3—66no—
70C2×C32×D3—66no—
71C2×C32×D3—612no—
72C2×C32×D3—612no—
73C2×C32×D3—1212no—
74#1113—212no—
75#1113C22×D318yescomplemented
76C3×D32—38no—
77C3×D32—68no—
78C3×D32—624no—
79C3×D32—612no—
80C3⋊#157—38no—
81C3⋊#157—64no—
82C22×#67—272no—
83C34C22×D411yescharacteristic, complemented, Sylow
84C23×C32—91no—
85C23×C32—182no—
86C22×C3⋊D3—22no—
87C22×C3⋊D3D3211yescharacteristic
88C22×C3×D3—62no—
89C22×C3×D3—62no—
90C22×C3×D3—62no—
91C22×C3×D3—62no—
92C22×C3×D3—62no—
93C22×C3×D3—62no—
94C22×C3×D3—912no—
95C22×C3×D3—122no—
96C22×C3×D3—122no—
97C22×C3×D3—122no—
98C22×C3×D3—122no—
99C22×C3×D3—122no—
100C22×C3×D3—124no—
101C22×C3×D3—1812no—
102C22×C3×D3—186no—
103C22×C3×D3—1812no—
104C2×D32—312no—
105C2×D32—612no—
106C2×D32—612no—
107C2×D32—66no—
108C2×#157—96no—
109GO+(4,𝔽2)—916no—
110C2×C3⋊Dic3—96no—
111#625—916no—
112C2×C33—26no—
113C2×C33—23no—
114C2×C33—46no—
115C2×C33—46no—
116C2×C33—86no—
117C2×C33C2×#6716yescomplemented
118C3×C3⋊D3—28no—
119C3×C3⋊D3—28no—
120C3×C3⋊D3—21no—
121C3×C3⋊D3—23no—
122C3×C3⋊D3—48no—
123C3×C3⋊D3C2×#6716yescomplemented
124C3×C3⋊D3C23×D312yes—
125C32×D3—66no—
126C32×D3—62no—
127C32×D3—68no—
128C32×D3—68no—
129C32×D3—128no—
130C24×C3—272no—
131C23×D3—92no—
132C23×D3—182no—
133C2×#67—2724no—
134C22×Dic3—272no—
135C22×Dic3—541no—
136C22×C32—22no—
137C22×C32—22no—
138C22×C32—42no—
139C22×C32—42no—
140C22×C32—42no—
141C22×C32—44no—
142C22×C32—42no—
143C22×C32—42no—
144C22×C32—82no—
145C22×C32—82no—
146C22×C32—82no—
147C22×C32—82no—
148C22×C32—93no—
149C22×C32—93no—
150C22×C32—1812no—
151C22×C32#62511yescharacteristic, complemented
152C22×C32GO+(4,𝔽2)11yescharacteristic, complemented
153C2×C3⋊D3—212no—
154C2×C3⋊D3#62513yescomplemented
155C2×C3⋊D3C2×D3213yes—
156C2×C3×D3—66no—
157C2×C3×D3—66no—
158C2×C3×D3—66no—
159C2×C3×D3—66no—
160C2×C3×D3—612no—
161C2×C3×D3—612no—
162C2×C3×D3—612no—
163C2×C3×D3—612no—
164C2×C3×D3—98no—
165C2×C3×D3—126no—
166C2×C3×D3—126no—
167C2×C3×D3—1212no—
168C2×C3×D3—1212no—
169C2×C3×D3—1212no—
170C2×C3×D3—1212no—
171C2×C3×D3—1224no—
172C2×C3×D3—188no—
173C2×C3×D3—1812no—
174C2×C3×D3—1824no—
175D32—38no—
176D32—68no—
177D32—624no—
178D32—612no—
179#157—94no—
180C3⋊Dic3—94no—
181C22×D4—811noSylow
182C33—22no—
183C33—21no—
184C33—42no—
185C33—42no—
186C33—82no—
187C33C22×#6712yescomplemented
188C23×C3—92no—
189C23×C3—181no—
190C23×C3—182no—
191C23×C3—182no—
192C23×C3—2712no—
193C23×C3—362no—
194C23×C3—546no—
195C22×D3—62no—
196C22×D3—62no—
197C22×D3—62no—
198C22×D3—912no—
199C22×D3—122no—
200C22×D3—1812no—
201C22×D3—186no—
202C22×D3—1812no—
203#67—2732no—
204C2×Dic3—2712no—
205C2×Dic3—546no—
206C2×C32—26no—
207C2×C32—26no—
208C2×C32—46no—
209C2×C32—46no—
210C2×C32—46no—
211C2×C32—412no—
212C2×C32—46no—
213C2×C32—46no—
214C2×C32—86no—
215C2×C32—86no—
216C2×C32—86no—
217C2×C32—86no—
218C2×C32—91no—
219C2×C32—93no—
220C2×C32—188no—
221C2×C32C2×#62513yescomplemented
222C2×C32C2×GO+(4,𝔽2)13yescomplemented
223C3⋊D3—28no—
224C3⋊D3C2×#62513yescomplemented
225C3⋊D3C22×D3211yescharacteristic
226C3×D3—66no—
227C3×D3—66no—
228C3×D3—62no—
229C3×D3—62no—
230C3×D3—68no—
231C3×D3—68no—
232C3×D3—68no—
233C3×D3—68no—
234C3×D3—122no—
235C3×D3—126no—
236C3×D3—128no—
237C3×D3—128no—
238C3×D3—128no—
239C3×D3—128no—
240C3×D3—1216no—
241C24—272no—
242C2×D4—8112no—
243C22×C4—811no—
244C22×C3—22no—
245C22×C3—21no—
246C22×C3—42no—
247C22×C3—42no—
248C22×C3—82no—
249C22×C3—96no—
250C22×C3—96no—
251C22×C3—183no—
252C22×C3—183no—
253C22×C3—1812no—
254C22×C3—1812no—
255C22×C3—278no—
256C22×C3—3612no—
257C22×C3—5412no—
258C22×C3D32⋊D312yescomplemented
259C2×D3—66no—
260C2×D3—66no—
261C2×D3—612no—
262C2×D3—612no—
263C2×D3—98no—
264C2×D3—1212no—
265C2×D3—188no—
266C2×D3—1812no—
267C2×D3—1824no—
268Dic3—278no—
269Dic3—544no—
270C32—22no—
271C32—22no—
272C32—42no—
273C32—42no—
274C32—42no—
275C32—44no—
276C32—42no—
277C32—42no—
278C32—82no—
279C32—82no—
280C32—82no—
281C32—82no—
282C32C22×#62511yescharacteristic, complemented
283C32C22×GO+(4,𝔽2)11yescharacteristic, complemented
284C23—91no—
285C23—182no—
286C23—2712no—
287C23—546no—
288D4—8116no—
289C2×C4—816no—
290C2×C3—26no—
291C2×C3—23no—
292C2×C3—46no—
293C2×C3—46no—
294C2×C3—86no—
295C2×C3—92no—
296C2×C3—96no—
297C2×C3—181no—
298C2×C3—183no—
299C2×C3—188no—
300C2×C3—188no—
301C2×C3—368no—
302C2×C3C2×D32⋊D316yescomplemented
303D3—62no—
304D3—66no—
305D3—68no—
306D3—68no—
307D3—128no—
308C22—93no—
309C22—93no—
310C22—1812no—
311C22—278no—
312C22—5412no—
313C22#1062020311yescharacteristic, direct product factor
314C4—814no—
315C3—22no—
316C3—21no—
317C3—42no—
318C3—42no—
319C3—82no—
320C3C22×D32⋊D312yescomplemented
321C2—91no—
322C2—93no—
323C2—188no—
324C2C2×#1062020313yesdirect product factor
subgroupG/H
center, hypercenterC22#10620203
commutator, cosocleC32×C3⋊D3C24
FrattiniC1G
Fitting, socleC22×C34D4
hypocenterC34C22×D4
LCSG ▷ C32×C3⋊D3 ▷ C34
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×#1113C2semidirect
2. C22×C3⋊(C3⋊#157)C2semidirect
3. C2×#10620203C2direct (N × H)
8. C2×C3×#1113C22semidirect
9. C2×C3⋊(C3⋊#157)C22semidirect
10. #10620203C22direct (N × H)
11. C22×#1113D3semidirect
17. C22×C34D4semidirect
19. C2×C32×C3⋊D3D4semidirect
22. C3×#1113C23semidirect
23. C3⋊(C3⋊#157)C23semidirect
36. C2×#1113C2×D3semidirect
44. C2×C34C2×D4semidirect
46. C32×C3⋊D3C2×D4semidirect
61. C22×C33#67semidirect
67. C2×C3×C3⋊D3#67semidirect
75. #1113C22×D3semidirect
83. C34C22×D4semidirect
117. C2×C33C2×#67semidirect
123. C3×C3⋊D3C2×#67semidirect
151. C22×C32#625semidirect
152. C22×C32GO+(4,𝔽2)semidirect
154. C2×C3⋊D3#625semidirect
187. C33C22×#67semidirect
221. C2×C32C2×#625semidirect
222. C2×C32C2×GO+(4,𝔽2)semidirect
224. C3⋊D3C2×#625semidirect
258. C22×C3D32⋊D3semidirect
282. C32C22×#625semidirect
283. C32C22×GO+(4,𝔽2)semidirect
302. C2×C3C2×D32⋊D3semidirect
313. C22#10620203direct (N × H)
320. C3C22×D32⋊D3semidirect
324. C2C2×#10620203direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#11132
C22×C3⋊(C3⋊#157)2
C2×#106202032
C22×D32⋊D33
C22×#6259