Finite Group Databasebeta

← C22×#88935 #13181113

Subgroups — C22×#88935 #13181113

Order 1728.

1416 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 159 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×#6295C211yesmaximal, characteristic, complemented
2C22×C3×#625C211yesmaximal, characteristic, complemented
3C22×C3⋊#622C211yesmaximal, characteristic, complemented
4C2×#88935C2112yesmaximal, direct product factor
5C22×#3774—31nomaximal
6C23×C3×C3⋊D3—21no—
7C23×C32×D3—21no—
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×#6295—23no—
10C2×#6295C2216yescomplemented
11C2×C3×#625—23no—
12C2×C3×#625C2216yescomplemented
13C2×C3⋊#622C2216yescomplemented
14#88935C22116yesdirect product factor
15C22×C3×#67—61no—
16C22×#626—61no—
17C22×#625—31no—
18C22×#625D311yescharacteristic, complemented
19C22×#622—31no—
20C2×#3774—312no—
21C23×C33D411yescharacteristic
22C22×C3×C3⋊D3—26no—
23C22×C3×C3⋊D3—44no—
24C22×C32×D3—26no—
25C22×C32×D3—44no—
26C2×C3×C3⋊Dic3D413yes—
27C2×C3×C3⋊Dic3C2313yes—
28#6295—26no—
29#6295C2314yescomplemented
30C3×#625—26no—
31C3×#625C2314yescomplemented
32C3⋊#622C2314yescomplemented
33C22×#213—91nomaximal
34C24×C32—61no—
35C23×C3⋊D3—21no—
36C23×C3×D3—21no—
37C23×C3×D3—21no—
38C23×C3×D3—41no—
39C23×C3×D3—61no—
40C23×C3×D3—61no—
41C23×C3×D3—121no—
42C22×C3⋊Dic3C2×D311yescharacteristic
43C2×C3×#67—66no—
44C2×C3×#67—66no—
45C22×C3×Dic3—61no—
46C22×C3×Dic3—61no—
47C2×#626—66no—
48C2×#626—66no—
49C2×#625—23no—
50C2×#625—36no—
51C2×#625—63no—
52C2×#625C2×D316yescomplemented
53C2×#622—36no—
54#3774—316no—
55C22×C33D814yescomplemented
56C22×C33C2×D413yes—
57C2×C3×C3⋊D3—24no—
58C2×C3×C3⋊D3—412no—
59C2×C32×D3—24no—
60C2×C32×D3—412no—
61C3×C3⋊Dic3C2×D413yes—
62C3×C3⋊Dic3C2411yescharacteristic
63C22×C3×D4—91no—
64C22×#67—61no—
65C22×#67—181no—
66C22×D12—91no—
67C2×#213—912no—
68C22×C3⋊C8—91no—
69C23×C32—21no—
70C23×C32—21no—
71C23×C32—41no—
72C23×C32—41no—
73C23×C32—66no—
74C23×C32—124no—
75C23×C32#6711yescharacteristic
76C22×C3⋊D3—26no—
77C22×C3⋊D3—44no—
78C22×C3×D3—26no—
79C22×C3×D3—26no—
80C22×C3×D3—46no—
81C22×C3×D3—44no—
82C22×C3×D3—48no—
83C22×C3×D3—44no—
84C22×C3×D3—66no—
85C22×C3×D3—66no—
86C22×C3×D3—126no—
87C22×C3×D3—124no—
88C22×C3×D3—128no—
89C22×C3×D3—124no—
90C2×C3⋊Dic3#6713yes—
91C2×C3⋊Dic3C22×D313yes—
92C3×#67—64no—
93C3×#67—612no—
94C2×C3×Dic3—63no—
95C2×C3×Dic3—63no—
96C2×C3×Dic3—63no—
97C2×C3×Dic3—63no—
98#626—64no—
99#626—612no—
100#625—26no—
101#625—34no—
102#625—66no—
103#625C22×D314yescomplemented
104#622—34no—
105C22×D8—271noSylow
106C2×C33C2×D816yescomplemented
107C2×C33C22×D411yescharacteristic
108C3×C3⋊D3—44no—
109C32×D3—44no—
110C24×C3—61no—
111C24×C3—61no—
112C24×C3—121no—
113C24×C3—181no—
114C23×D3—21no—
115C23×D3—61no—
116C23×D3—61no—
117C23×D3—121no—
118C2×C3×D4—96no—
119C2×C3×D4—183no—
120C22×C3×C4—91no—
121C2×#67—66no—
122C2×#67—66no—
123C2×#67—186no—
124C2×#67—186no—
125C22×Dic3—61no—
126C22×Dic3—61no—
127C2×D12—96no—
128C2×D12—183no—
129#213—916no—
130C2×C3⋊C8—96no—
131C22×C32—23no—
132C22×C32—23no—
133C22×C32—24no—
134C22×C32—24no—
135C22×C32—44no—
136C22×C32—44no—
137C22×C32—43no—
138C22×C32—43no—
139C22×C32—64no—
140C22×C32—1212no—
141C22×C32#21314yescomplemented
142C22×C32C2×#6713yes—
143C2×C3⋊D3—24no—
144C2×C3⋊D3—412no—
145C2×C3×D3—24no—
146C2×C3×D3—24no—
147C2×C3×D3—44no—
148C2×C3×D3—412no—
149C2×C3×D3—424no—
150C2×C3×D3—412no—
151C2×C3×D3—64no—
152C2×C3×D3—64no—
153C2×C3×D3—124no—
154C2×C3×D3—1212no—
155C2×C3×D3—1224no—
156C2×C3×D3—1212no—
157C3⋊Dic3C2×#6713yes—
158C3⋊Dic3C23×D311yescharacteristic
159C3×Dic3—63no—
160C3×Dic3—63no—
161C3×Dic3—61no—
162C3×Dic3—61no—
163C22×D4—91no—
164C22×D4—271no—
165C2×D8—2712no—
166C22×C8—271no—
167C33C22×D811yescharacteristic, complemented, Sylow
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—41no—
171C23×C3—41no—
172C23×C3—66no—
173C23×C3—66no—
174C23×C3—126no—
175C23×C3—124no—
176C23×C3—128no—
177C23×C3—124no—
178C23×C3—186no—
179C23×C3—364no—
180C23×C3GO+(4,𝔽2)11yescharacteristic
181C22×D3—26no—
182C22×D3—44no—
183C22×D3—66no—
184C22×D3—66no—
185C22×D3—126no—
186C22×D3—124no—
187C22×D3—128no—
188C22×D3—124no—
189C3×D4—94no—
190C3×D4—186no—
191C2×C3×C4—93no—
192C2×C3×C4—93no—
193#67—64no—
194#67—612no—
195#67—184no—
196#67—1812no—
197C2×Dic3—63no—
198C2×Dic3—63no—
199C2×Dic3—63no—
200C2×Dic3—63no—
201D12—94no—
202D12—186no—
203C3⋊C8—94no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—26no—
207C2×C32—26no—
208C2×C32—46no—
209C2×C32—46no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32—124no—
213C2×C32C2×#21316yescomplemented
214C2×C32C22×#6711yescharacteristic
215C3⋊D3—44no—
216C3×D3—44no—
217C3×D3—48no—
218C3×D3—44no—
219C3×D3—124no—
220C3×D3—128no—
221C3×D3—124no—
222C24—61no—
223C24—181no—
224C2×D4—96no—
225C2×D4—183no—
226C2×D4—276no—
227C2×D4—543no—
228C22×C4—91no—
229D8—2716no—
230C2×C8—276no—
231C22×C3—23no—
232C22×C3—23no—
233C22×C3—24no—
234C22×C3—24no—
235C22×C3—43no—
236C22×C3—43no—
237C22×C3—44no—
238C22×C3—44no—
239C22×C3—64no—
240C22×C3—64no—
241C22×C3—124no—
242C22×C3—1212no—
243C22×C3—1224no—
244C22×C3—1212no—
245C22×C3—184no—
246C22×C3—3612no—
247C22×C3#377414yescomplemented
248C22×C3C2×GO+(4,𝔽2)13yes—
249C2×D3—24no—
250C2×D3—412no—
251C2×D3—64no—
252C2×D3—64no—
253C2×D3—124no—
254C2×D3—1212no—
255C2×D3—1224no—
256C2×D3—1212no—
257C3×C4—91no—
258C3×C4—93no—
259Dic3—63no—
260Dic3—61no—
261Dic3—63no—
262Dic3—61no—
263C32—21no—
264C32—21no—
265C32—41no—
266C32—41no—
267C32C22×#21311yescharacteristic, complemented
268C23—66no—
269C23—124no—
270C23—186no—
271C23—364no—
272C23D32⋊D311yescharacteristic
273D4—94no—
274D4—186no—
275D4—274no—
276D4—546no—
277C2×C4—93no—
278C2×C4—93no—
279C8—274no—
280C2×C3—26no—
281C2×C3—26no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—46no—
285C2×C3—46no—
286C2×C3—41no—
287C2×C3—41no—
288C2×C3—124no—
289C2×C3—128no—
290C2×C3—124no—
291C2×C3—364no—
292C2×C3C2×#377416yescomplemented
293C2×C3C22×GO+(4,𝔽2)11yescharacteristic
294D3—44no—
295D3—124no—
296D3—128no—
297D3—124no—
298C22—64no—
299C22—1212no—
300C22—184no—
301C22—3612no—
302C22#8893514yesdirect product factor
303C22C2×D32⋊D313yes—
304C4—91no—
305C4—93no—
306C3—21no—
307C3—21no—
308C3—41no—
309C3—41no—
310C3C22×#377411yescharacteristic, complemented
311C2—124no—
312C2—364no—
313C2C2×#8893516yesdirect product factor
314C2C22×D32⋊D311yescharacteristic
subgroupG/H
center, hypercenterC23D32⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×D32⋊D3
Fitting, socleC23×C33D4
hypocenterC33C22×D8
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×#6295C2semidirect
2. C22×C3×#625C2semidirect
3. C22×C3⋊#622C2semidirect
4. C2×#88935C2direct (N × H)
10. C2×#6295C22semidirect
12. C2×C3×#625C22semidirect
13. C2×C3⋊#622C22semidirect
14. #88935C22direct (N × H)
18. C22×#625D3semidirect
29. #6295C23semidirect
31. C3×#625C23semidirect
32. C3⋊#622C23semidirect
52. C2×#625C2×D3semidirect
55. C22×C33D8semidirect
103. #625C22×D3semidirect
106. C2×C33C2×D8semidirect
141. C22×C32#213semidirect
167. C33C22×D8semidirect
213. C2×C32C2×#213semidirect
247. C22×C3#3774semidirect
267. C32C22×#213semidirect
292. C2×C3C2×#3774semidirect
302. C22#88935direct (N × H)
310. C3C22×#3774semidirect
313. C2C2×#88935direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62952
C22×C3×#6252
C22×C3⋊#6222
C2×#889352
C22×#37743
C22×#2139