Finite Group Databasebeta

← C22×#88936 #13181118

Subgroups — C22×#88936 #13181118

Order 1728.

1176 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1176 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
2C22×C3×#627C211yesmaximal, characteristic, complemented
3C22×C3⋊#622C211yesmaximal, characteristic, complemented
4C2×#88936C2112yesmaximal, direct product factor
5C22×#3775—31nomaximal
6C23×C3×C3⋊D3—21no—
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C22×C32×Dic3—21no—
9C2×#6295—23no—
10C2×#6295C2216yes—
11C2×C3×#627—23no—
12C2×C3×#627C2216yescomplemented
13C2×C3⋊#622C2216yescomplemented
14#88936C22116yesdirect product factor
15C22×C3×Dic6—61no—
16C22×#627D311yescharacteristic, complemented
17C22×#626—61no—
18C22×#625—31no—
19C22×#622—31no—
20C2×#3775—312no—
21C23×C33D411yescharacteristic
22C22×C3×C3⋊D3—26no—
23C22×C3×C3⋊D3—44no—
24C2×C3×C3⋊Dic3D413yes—
25C2×C3×C3⋊Dic3C2313yes—
26C2×C32×Dic3—26no—
27#6295—26no—
28#6295C2314yes—
29C3×#627—26no—
30C3×#627C2314yescomplemented
31C3⋊#622C2314yescomplemented
32C22×#215—91nomaximal
33C23×C3⋊D3—21no—
34C23×C3×D3—61no—
35C23×C3×D3—61no—
36C23×C3×D3—121no—
37C22×C32×C4—61no—
38C22×C3⋊Dic3C2×D311yescharacteristic
39C22×C3×Dic3—21no—
40C22×C3×Dic3—21no—
41C22×C3×Dic3—41no—
42C22×C3×Dic3—61no—
43C22×C3×Dic3—61no—
44C2×C3×Dic6—66no—
45C2×C3×Dic6—66no—
46C2×#627—23no—
47C2×#627C2×D316yescomplemented
48C2×#626—66no—
49C2×#626—66no—
50C2×#625—36no—
51C2×#625—63no—
52C2×#622—36no—
53#3775—316no—
54C22×C33SD1614yescomplemented
55C22×C33C2×D413yes—
56C2×C3×C3⋊D3—24no—
57C2×C3×C3⋊D3—412no—
58C3×C3⋊Dic3C2×D413yes—
59C3×C3⋊Dic3C2411yescharacteristic
60C32×Dic3—24no—
61C22×C3×Q8—91no—
62C22×#67—181no—
63C22×D12—91no—
64C22×Dic6—61no—
65C2×#215—912no—
66C22×C3⋊C8—91no—
67C23×C32—21no—
68C23×C32—21no—
69C23×C32—41no—
70C23×C32—41no—
71C23×C32#6711yescharacteristic
72C22×C3⋊D3—26no—
73C22×C3⋊D3—44no—
74C22×C3×D3—66no—
75C22×C3×D3—66no—
76C22×C3×D3—126no—
77C22×C3×D3—124no—
78C22×C3×D3—128no—
79C22×C3×D3—124no—
80C2×C32×C4—66no—
81C2×C3⋊Dic3#6713yes—
82C2×C3⋊Dic3C22×D313yes—
83C2×C3×Dic3—26no—
84C2×C3×Dic3—26no—
85C2×C3×Dic3—46no—
86C2×C3×Dic3—63no—
87C2×C3×Dic3—63no—
88C2×C3×Dic3—63no—
89C2×C3×Dic3—63no—
90C3×Dic6—64no—
91C3×Dic6—612no—
92#627—26no—
93#627C22×D314yescomplemented
94#626—64no—
95#626—612no—
96#625—34no—
97#625—66no—
98#622—34no—
99C22×SD16—271noSylow
100C2×C33C2×SD1616yescomplemented
101C2×C33C22×D411yescharacteristic
102C3×C3⋊D3—44no—
103C24×C3—181no—
104C23×D3—61no—
105C23×D3—61no—
106C23×D3—121no—
107C2×C3×Q8—96no—
108C2×C3×Q8—183no—
109C22×C3×C4—61no—
110C22×C3×C4—61no—
111C22×C3×C4—91no—
112C22×C3×C4—121no—
113C2×#67—186no—
114C2×#67—186no—
115C22×Dic3—21no—
116C22×Dic3—61no—
117C22×Dic3—61no—
118C2×D12—96no—
119C2×D12—183no—
120C2×Dic6—66no—
121C2×Dic6—66no—
122#215—916no—
123C2×C3⋊C8—96no—
124C22×C32—23no—
125C22×C32—23no—
126C22×C32—24no—
127C22×C32—24no—
128C22×C32—44no—
129C22×C32—44no—
130C22×C32—43no—
131C22×C32—43no—
132C22×C32#21514yescomplemented
133C22×C32C2×#6713yes—
134C2×C3⋊D3—24no—
135C2×C3⋊D3—412no—
136C2×C3×D3—64no—
137C2×C3×D3—64no—
138C2×C3×D3—124no—
139C2×C3×D3—1212no—
140C2×C3×D3—1224no—
141C2×C3×D3—1212no—
142C32×C4—64no—
143C3⋊Dic3C2×#6713yes—
144C3⋊Dic3C23×D311yescharacteristic
145C3×Dic3—24no—
146C3×Dic3—24no—
147C3×Dic3—44no—
148C3×Dic3—63no—
149C3×Dic3—63no—
150C3×Dic3—61no—
151C3×Dic3—61no—
152C22×Q8—91no—
153C22×D4—271no—
154C2×SD16—2712no—
155C22×C8—271no—
156C33C22×SD1611yescharacteristic, complemented, Sylow
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3—186no—
162C23×C3—364no—
163C23×C3GO+(4,𝔽2)11yescharacteristic
164C22×D3—66no—
165C22×D3—66no—
166C22×D3—126no—
167C22×D3—124no—
168C22×D3—128no—
169C22×D3—124no—
170C3×Q8—94no—
171C3×Q8—186no—
172C2×C3×C4—66no—
173C2×C3×C4—66no—
174C2×C3×C4—93no—
175C2×C3×C4—93no—
176C2×C3×C4—126no—
177#67—184no—
178#67—1812no—
179C2×Dic3—26no—
180C2×Dic3—63no—
181C2×Dic3—63no—
182C2×Dic3—63no—
183C2×Dic3—63no—
184D12—94no—
185D12—186no—
186Dic6—64no—
187Dic6—612no—
188C3⋊C8—94no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32—26no—
192C2×C32—26no—
193C2×C32—46no—
194C2×C32—46no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32C2×#21516yescomplemented
198C2×C32C22×#6711yescharacteristic
199C3⋊D3—44no—
200C3×D3—124no—
201C3×D3—128no—
202C3×D3—124no—
203C24—181no—
204C2×Q8—96no—
205C2×Q8—183no—
206C2×D4—276no—
207C2×D4—543no—
208C22×C4—61no—
209C22×C4—91no—
210SD16—2716no—
211C2×C8—276no—
212C22×C3—23no—
213C22×C3—23no—
214C22×C3—24no—
215C22×C3—24no—
216C22×C3—43no—
217C22×C3—43no—
218C22×C3—44no—
219C22×C3—44no—
220C22×C3—184no—
221C22×C3—3612no—
222C22×C3#377514yescomplemented
223C22×C3C2×GO+(4,𝔽2)13yes—
224C2×D3—64no—
225C2×D3—64no—
226C2×D3—124no—
227C2×D3—1212no—
228C2×D3—1224no—
229C2×D3—1212no—
230C3×C4—64no—
231C3×C4—64no—
232C3×C4—91no—
233C3×C4—93no—
234C3×C4—124no—
235Dic3—24no—
236Dic3—63no—
237Dic3—61no—
238Dic3—63no—
239Dic3—61no—
240C32—21no—
241C32—21no—
242C32—41no—
243C32—41no—
244C32C22×#21511yescharacteristic, complemented
245C23—186no—
246C23—364no—
247C23D32⋊D311yescharacteristic
248Q8—94no—
249Q8—186no—
250D4—274no—
251D4—546no—
252C2×C4—66no—
253C2×C4—93no—
254C2×C4—93no—
255C8—274no—
256C2×C3—26no—
257C2×C3—26no—
258C2×C3—21no—
259C2×C3—21no—
260C2×C3—46no—
261C2×C3—46no—
262C2×C3—41no—
263C2×C3—41no—
264C2×C3—364no—
265C2×C3C2×#377516yescomplemented
266C2×C3C22×GO+(4,𝔽2)11yescharacteristic
267D3—124no—
268D3—128no—
269D3—124no—
270C22—184no—
271C22—3612no—
272C22#8893614yesdirect product factor
273C22C2×D32⋊D313yes—
274C4—64no—
275C4—91no—
276C4—93no—
277C3—21no—
278C3—21no—
279C3—41no—
280C3—41no—
281C3C22×#377511yescharacteristic, complemented
282C2—364no—
283C2C2×#8893616yesdirect product factor
284C2C22×D32⋊D311yescharacteristic
subgroupG/H
center, hypercenterC23D32⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×D32⋊D3
Fitting, socleC23×C33D4
prime radicalC22×#6295C2
hypocenterC33C22×SD16
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3×#627C2semidirect
3. C22×C3⋊#622C2semidirect
4. C2×#88936C2direct (N × H)
12. C2×C3×#627C22semidirect
13. C2×C3⋊#622C22semidirect
14. #88936C22direct (N × H)
16. C22×#627D3semidirect
30. C3×#627C23semidirect
31. C3⋊#622C23semidirect
47. C2×#627C2×D3semidirect
54. C22×C33SD16semidirect
93. #627C22×D3semidirect
100. C2×C33C2×SD16semidirect
132. C22×C32#215semidirect
156. C33C22×SD16semidirect
197. C2×C32C2×#215semidirect
222. C22×C3#3775semidirect
244. C32C22×#215semidirect
265. C2×C3C2×#3775semidirect
272. C22#88936direct (N × H)
281. C3C22×#3775semidirect
283. C2C2×#88936direct (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×#6272
C22×C3⋊#6222
C2×#889362
C22×#37753
C22×#2159