Finite Group Databasebeta

← C22×C3×#3775 #13181052

Subgroups — C22×C3×#3775 #13181052

Order 1728.

1176 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1176 conjugacy classes of subgroups — too many to draw. Only its 210 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×#627C211yesmaximal, characteristic, complemented
2C22×C3×#625C211yesmaximal, characteristic
3C22×C3×#622C211yesmaximal, characteristic, complemented
4C2×C3×#3775C2112yesmaximal, direct product factor
5C22×#3775C311yesmaximal, characteristic, direct product factor
6C23×C32×D3—21no—
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C22×C32×Dic3—21no—
9C2×C3×#627—23no—
10C2×C3×#627C2216yescomplemented
11C2×C3×#625—23no—
12C2×C3×#625C2216yes—
13C2×C3×#622C2216yescomplemented
14C3×#3775C22116yesdirect product factor
15C22×C3×#67—61no—
16C22×C3×Dic6—61no—
17C22×#627C2×C311yescharacteristic, complemented
18C22×#625C2×C311yescharacteristic
19C22×#622C2×C311yescharacteristic, complemented
20C2×#3775C2×C3112yesdirect product factor
21C23×C33D411yescharacteristic
22C22×C32×D3—26no—
23C22×C32×D3—44no—
24C2×C3×C3⋊Dic3D413yes—
25C2×C3×C3⋊Dic3C2313yes—
26C2×C32×Dic3—26no—
27C3×#627—26no—
28C3×#627C2314yescomplemented
29C3×#625—26no—
30C3×#625C2314yes—
31C3×#622C2314yescomplemented
32C22×C3×SD16—91nomaximal
33C24×C32—61no—
34C23×C3×D3—21no—
35C23×C3×D3—21no—
36C23×C3×D3—41no—
37C22×C32×C4—61no—
38C22×C3⋊Dic3C22×C311yescharacteristic
39C2×C3×#67—66no—
40C2×C3×#67—66no—
41C22×C3×Dic3—21no—
42C22×C3×Dic3—21no—
43C22×C3×Dic3—41no—
44C22×C3×Dic3—61no—
45C22×C3×Dic3—61no—
46C2×C3×Dic6—66no—
47C2×C3×Dic6—66no—
48C2×#627—23no—
49C2×#627C22×C316yescomplemented
50C2×#625—23no—
51C2×#625C22×C316yes—
52C2×#622C22×C316yescomplemented
53#3775C22×C3116yesdirect product factor
54C22×C33SD1614yescomplemented
55C22×C33C2×D413yes—
56C2×C32×D3—24no—
57C2×C32×D3—412no—
58C3×C3⋊Dic3C2×D413yes—
59C3×C3⋊Dic3C2411yescharacteristic
60C32×Dic3—24no—
61C22×C3×Q8—91no—
62C22×C3×D4—91no—
63C22×#67—61no—
64C22×Dic6—61no—
65C2×C3×SD16—912no—
66C22×C3×C8—91no—
67C23×C32—21no—
68C23×C32—21no—
69C23×C32—41no—
70C23×C32—41no—
71C23×C32—66no—
72C23×C32—124no—
73C23×C32C3×D411yescharacteristic
74C22×C3×D3—26no—
75C22×C3×D3—26no—
76C22×C3×D3—46no—
77C22×C3×D3—44no—
78C22×C3×D3—48no—
79C22×C3×D3—44no—
80C2×C32×C4—66no—
81C2×C3⋊Dic3C3×D413yes—
82C2×C3⋊Dic3C23×C313yes—
83C3×#67—64no—
84C3×#67—612no—
85C2×C3×Dic3—26no—
86C2×C3×Dic3—26no—
87C2×C3×Dic3—46no—
88C2×C3×Dic3—63no—
89C2×C3×Dic3—63no—
90C2×C3×Dic3—63no—
91C2×C3×Dic3—63no—
92C3×Dic6—64no—
93C3×Dic6—612no—
94#627—26no—
95#627C23×C314yescomplemented
96#625—26no—
97#625C23×C314yes—
98#622C23×C314yescomplemented
99C22×SD16—91noSylow
100C2×C33C2×SD1616yescomplemented
101C2×C33C22×D411yescharacteristic
102C32×D3—44no—
103C24×C3—61no—
104C24×C3—61no—
105C24×C3—121no—
106C23×D3—21no—
107C2×C3×Q8—96no—
108C2×C3×Q8—183no—
109C2×C3×D4—96no—
110C2×C3×D4—183no—
111C22×C3×C4—61no—
112C22×C3×C4—61no—
113C22×C3×C4—91no—
114C22×C3×C4—121no—
115C2×#67—66no—
116C2×#67—66no—
117C22×Dic3—21no—
118C22×Dic3—61no—
119C22×Dic3—61no—
120C2×Dic6—66no—
121C2×Dic6—66no—
122C3×SD16—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—64no—
133C22×C32—1212no—
134C22×C32C3×SD1614yescomplemented
135C22×C32C2×C3×D413yes—
136C2×C3×D3—24no—
137C2×C3×D3—24no—
138C2×C3×D3—44no—
139C2×C3×D3—412no—
140C2×C3×D3—424no—
141C2×C3×D3—412no—
142C32×C4—64no—
143C3⋊Dic3C2×C3×D413yes—
144C3⋊Dic3C24×C311yescharacteristic
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—91no—
154C2×SD16—912no—
155C22×C8—91no—
156C33C22×SD1611yescharacteristic, complemented, Sylow
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3—66no—
162C23×C3—66no—
163C23×C3—126no—
164C23×C3—124no—
165C23×C3—128no—
166C23×C3—124no—
167C23×C3GO+(4,𝔽2)11yescharacteristic
168C22×D3—26no—
169C22×D3—44no—
170C3×Q8—94no—
171C3×Q8—186no—
172C3×D4—94no—
173C3×D4—186no—
174C2×C3×C4—66no—
175C2×C3×C4—66no—
176C2×C3×C4—93no—
177C2×C3×C4—93no—
178C2×C3×C4—126no—
179#67—64no—
180#67—612no—
181C2×Dic3—26no—
182C2×Dic3—63no—
183C2×Dic3—63no—
184C2×Dic3—63no—
185C2×Dic3—63no—
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×C32—124no—
198C2×C32C2×C3×SD1616yescomplemented
199C2×C32C22×C3×D411yescharacteristic
200C3×D3—44no—
201C3×D3—48no—
202C3×D3—44no—
203C24—61no—
204C2×Q8—96no—
205C2×Q8—183no—
206C2×D4—96no—
207C2×D4—183no—
208C22×C4—61no—
209C22×C4—91no—
210SD16—916no—
211C2×C8—96no—
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—64no—
221C22×C3—64no—
222C22×C3—124no—
223C22×C3—1212no—
224C22×C3—1224no—
225C22×C3—1212no—
226C22×C3#377514yesdirect product factor
227C22×C3C2×GO+(4,𝔽2)13yes—
228C2×D3—24no—
229C2×D3—412no—
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×C3×SD1611yescharacteristic, complemented
245C23—66no—
246C23—124no—
247C23C3×GO+(4,𝔽2)11yescharacteristic
248Q8—94no—
249Q8—186no—
250D4—94no—
251D4—186no—
252C2×C4—66no—
253C2×C4—93no—
254C2×C4—93no—
255C8—94no—
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—124no—
265C2×C3—128no—
266C2×C3—124no—
267C2×C3C2×#377516yesdirect product factor
268C2×C3C22×GO+(4,𝔽2)11yescharacteristic
269D3—44no—
270C22—64no—
271C22—1212no—
272C22C3×#377514yesdirect product factor
273C22C2×C3×GO+(4,𝔽2)13yes—
274C4—64no—
275C4—91no—
276C4—93no—
277C3—21no—
278C3—21no—
279C3—41no—
280C3—41no—
281C3C22×#377511yescharacteristic, direct product factor
282C2—124no—
283C2C2×C3×#377516yesdirect product factor
284C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC23×C3GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
Fitting, socleC23×C33D4
prime radicalC22×C3×#625C2
hypocenterC32C22×C3×SD16
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×#627C2semidirect
3. C22×C3×#622C2semidirect
4. C2×C3×#3775C2direct (N × H)
5. C22×#3775C3direct (N × H)
10. C2×C3×#627C22semidirect
13. C2×C3×#622C22semidirect
14. C3×#3775C22direct (N × H)
17. C22×#627C2×C3semidirect
19. C22×#622C2×C3semidirect
20. C2×#3775C2×C3direct (N × H)
28. C3×#627C23semidirect
31. C3×#622C23semidirect
49. C2×#627C22×C3semidirect
52. C2×#622C22×C3semidirect
53. #3775C22×C3direct (N × H)
54. C22×C33SD16semidirect
95. #627C23×C3semidirect
98. #622C23×C3semidirect
100. C2×C33C2×SD16semidirect
134. C22×C32C3×SD16semidirect
156. C33C22×SD16semidirect
198. C2×C32C2×C3×SD16semidirect
226. C22×C3#3775direct (N × H)
244. C32C22×C3×SD16semidirect
267. C2×C3C2×#3775direct (N × H)
272. C22C3×#3775direct (N × H)
281. C3C22×#3775direct (N × H)
283. C2C2×C3×#3775direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6272
C22×C3×#6252
C22×C3×#6222
C2×C3×#37752
C22×#37753
C22×C3×SD169