Finite Group Databasebeta

← C22×#88809 #13178509

Subgroups — C22×#88809 #13178509

Order 1728.

1634 conjugacy classes of subgroups; only the 391 normal subgroups are shown.

normal subgroups only

This group has 1634 conjugacy classes of subgroups — too many to draw. Only its 391 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×C4×C3⋊D3C211yesmaximal, characteristic
2C22×C3×C3⋊(C3⋊C8)C212yesmaximal, complemented
3C2×#88809C2112yesmaximal, direct product factor
4C22×#3712—31nomaximal
5C22×#3711—32nomaximal
6C23×C3×C3⋊D3C411yescharacteristic
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C411yescharacteristic
9C2×C3×C4×C3⋊D3C416yes—
10C2×C3×C4×C3⋊D3C2216yes—
11C2×C3×C3⋊(C3⋊C8)C22112yescomplemented
12#88809C22116yesdirect product factor
13C22×C4×C3⋊D3D311yescharacteristic
14C22×C3×C4×D3—32no—
15C22×C3×C4×D3—61no—
16C22×C3⋊(C3⋊C8)D312yescomplemented
17C22×C3×C3⋊C8—32no—
18C22×C3×C3⋊C8—32no—
19C22×C3×C3⋊C8—62no—
20C2×#3712—312no—
21C2×#3711—324no—
22C23×C33C2×C411yescharacteristic
23C22×C3×C3⋊D3—24no—
24C22×C3×C3⋊D3C2×C416yes—
25C2×C33×C4C2×C413yes—
26C2×C33×C4C2313yes—
27C2×C3×C3⋊Dic3C2×C416yes—
28C3×C4×C3⋊D3C2×C4112yes—
29C3×C4×C3⋊D3C2314yes—
30C3×C3⋊(C3⋊C8)C2318yescomplemented
31C22×#208—91no—
32C22×ΓL(1,𝔽25)—92no—
33C23×C3⋊D3Dic311yescharacteristic
34C23×C3×D3—32no—
35C23×C3×D3—61no—
36C22×C32×C4—22no—
37C22×C32×C4—21no—
38C22×C32×C4—41no—
39C22×C32×C4C2×D312yes—
40C22×C32×C4C2×D311yescharacteristic
41C22×C3⋊Dic3Dic311yescharacteristic
42C2×C4×C3⋊D3Dic316yes—
43C2×C4×C3⋊D3C2×D316yes—
44C22×C3×Dic3—32no—
45C22×C3×Dic3—61no—
46C2×C3×C4×D3—312no—
47C2×C3×C4×D3—312no—
48C2×C3×C4×D3—66no—
49C2×C3×C4×D3—66no—
50C2×C3⋊(C3⋊C8)C2×D3112yescomplemented
51C2×C3×C3⋊C8—312no—
52C2×C3×C3⋊C8—312no—
53C2×C3×C3⋊C8—612no—
54#3712—316no—
55#3711—332no—
56C22×C33M1614yescomplemented
57C22×C33C22×C413yes—
58C2×C3×C3⋊D3—212no—
59C2×C3×C3⋊D3C22×C414yes—
60C33×C4C22×C413yes—
61C33×C4C2411yescharacteristic
62C3×C3⋊Dic3C22×C414yes—
63C23×C3×C4—91no—
64C22×C4×D3—32no—
65C22×C4×D3—61no—
66C22×C3×C8—92no—
67C2×#208—912no—
68C22×C3⋊C8—32no—
69C22×C3⋊C8—32no—
70C22×C3⋊C8—62no—
71C2×ΓL(1,𝔽25)—924no—
72C23×C32—22no—
73C23×C32—21no—
74C23×C32—41no—
75C23×C32C4×D312yes—
76C23×C32C2×Dic311yescharacteristic
77C22×C3⋊D3—24no—
78C22×C3⋊D3C2×Dic316yes—
79C22×C3×D3—312no—
80C22×C3×D3—66no—
81C22×C3×D3—68no—
82C22×C3×D3—68no—
83C2×C32×C4—26no—
84C2×C32×C4—23no—
85C2×C32×C4—26no—
86C2×C32×C4—23no—
87C2×C32×C4—43no—
88C2×C32×C4—43no—
89C2×C32×C4C4×D316yes—
90C2×C32×C4C2×Dic313yes—
91C2×C32×C4C22×D316yes—
92C2×C32×C4C22×D313yes—
93C2×C3⋊Dic3C2×Dic316yes—
94C4×C3⋊D3C2×Dic3112yes—
95C4×C3⋊D3C22×D314yes—
96C2×C3×Dic3—312no—
97C2×C3×Dic3—66no—
98C3×C4×D3—324no—
99C3×C4×D3—38no—
100C3×C4×D3—612no—
101C3×C4×D3—64no—
102C3⋊(C3⋊C8)C22×D318yescomplemented
103C3×C3⋊C8—38no—
104C3×C3⋊C8—38no—
105C3×C3⋊C8—68no—
106C22×M16—271noSylow
107C2×C33C2×M1616yescomplemented
108C2×C33C23×C411yescharacteristic
109C3×C3⋊D3—24no—
110C24×C3—91no—
111C23×D3—32no—
112C23×D3—61no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4—41no—
116C22×C3×C4—91no—
117C22×C3×C4—96no—
118C22×C3×C4—96no—
119C22×C3×C4D3212yes—
120C22×C3×C4D3211yescharacteristic
121C22×Dic3—32no—
122C22×Dic3—61no—
123C2×C4×D3—312no—
124C2×C4×D3—312no—
125C2×C4×D3—66no—
126C2×C4×D3—66no—
127C2×C3×C8—912no—
128#208—916no—
129C2×C3⋊C8—312no—
130C2×C3⋊C8—312no—
131C2×C3⋊C8—612no—
132ΓL(1,𝔽25)—932no—
133C22×C32—28no—
134C22×C32—24no—
135C22×C32—26no—
136C22×C32—23no—
137C22×C32—44no—
138C22×C32—43no—
139C22×C32ΓL(1,𝔽25)18yescomplemented
140C22×C32#20814yescomplemented
141C22×C32C2×C4×D316yes—
142C22×C32C22×Dic313yes—
143C2×C3⋊D3—212no—
144C2×C3⋊D3C22×Dic314yes—
145C2×C3×D3—38no—
146C2×C3×D3—64no—
147C2×C3×D3—624no—
148C2×C3×D3—624no—
149C32×C4—22no—
150C32×C4—21no—
151C32×C4—26no—
152C32×C4—23no—
153C32×C4—43no—
154C32×C4—41no—
155C32×C4C2×C4×D316yes—
156C32×C4C22×Dic313yes—
157C32×C4C23×D312yes—
158C32×C4C23×D311yescharacteristic
159C3⋊Dic3C22×Dic314yes—
160C3×Dic3—38no—
161C3×Dic3—64no—
162C23×C4—91no—
163C2×M16—2712no—
164C22×C8—92no—
165C33C22×M1611yescharacteristic, complemented, Sylow
166C23×C3—22no—
167C23×C3—21no—
168C23×C3—41no—
169C23×C3—96no—
170C23×C3—184no—
171C23×C3D3×Dic312yes—
172C23×C3#62411yescharacteristic
173C22×D3—312no—
174C22×D3—66no—
175C22×D3—68no—
176C22×D3—68no—
177C2×C3×C4—26no—
178C2×C3×C4—23no—
179C2×C3×C4—26no—
180C2×C3×C4—23no—
181C2×C3×C4—43no—
182C2×C3×C4—43no—
183C2×C3×C4—96no—
184C2×C3×C4—94no—
185C2×C3×C4—912no—
186C2×C3×C4D3×Dic316yes—
187C2×C3×C4#62413yes—
188C2×C3×C4C2×D3216yes—
189C2×C3×C4C2×D3213yes—
190C2×Dic3—312no—
191C2×Dic3—66no—
192C4×D3—38no—
193C4×D3—324no—
194C4×D3—612no—
195C4×D3—64no—
196C3×C8—98no—
197C3⋊C8—38no—
198C3⋊C8—38no—
199C3⋊C8—68no—
200C2×C32—22no—
201C2×C32—21no—
202C2×C32—212no—
203C2×C32—26no—
204C2×C32—46no—
205C2×C32—41no—
206C2×C32C2×ΓL(1,𝔽25)112yescomplemented
207C2×C32C2×#20816yescomplemented
208C2×C32C22×C4×D312yes—
209C2×C32C23×Dic311yescharacteristic
210C3⋊D3—24no—
211C3×D3—68no—
212C3×D3—68no—
213C24—91no—
214C22×C4—91no—
215C22×C4—96no—
216C22×C4—96no—
217C22×C4#111311yescharacteristic
218M16—2716no—
219C2×C8—912no—
220C22×C3—23no—
221C22×C3—26no—
222C22×C3—24no—
223C22×C3—28no—
224C22×C3—43no—
225C22×C3—44no—
226C22×C3—94no—
227C22×C3—1812no—
228C22×C3#371118yescomplemented
229C22×C3#371214yescomplemented
230C22×C3C2×D3×Dic316yes—
231C22×C3C2×#62413yes—
232C2×D3—38no—
233C2×D3—64no—
234C2×D3—624no—
235C2×D3—624no—
236C3×C4—22no—
237C3×C4—21no—
238C3×C4—23no—
239C3×C4—26no—
240C3×C4—41no—
241C3×C4—43no—
242C3×C4—94no—
243C3×C4C2×D3×Dic316yes—
244C3×C4C2×#62413yes—
245C3×C4C22×D3212yes—
246C3×C4C22×D3211yescharacteristic
247Dic3—38no—
248Dic3—64no—
249C32—22no—
250C32—21no—
251C32—41no—
252C32C22×ΓL(1,𝔽25)12yescomplemented
253C32C22×#20811yescharacteristic, complemented
254C23—96no—
255C23—184no—
256C23#629411yescharacteristic
257C2×C4—96no—
258C2×C4—94no—
259C2×C4—912no—
260C2×C4#629413yes—
261C2×C4C2×#111313yes—
262C8—98no—
263C2×C3—21no—
264C2×C3—22no—
265C2×C3—26no—
266C2×C3—212no—
267C2×C3—41no—
268C2×C3—46no—
269C2×C3—184no—
270C2×C3C2×#3711112yescomplemented
271C2×C3C2×#371216yescomplemented
272C2×C3C22×D3×Dic312yes—
273C2×C3C22×#62411yescharacteristic
274D3—68no—
275D3—68no—
276C22—94no—
277C22—1812no—
278C22#8880914yesdirect product factor
279C22C2×#629413yes—
280C4—94no—
281C4C2×#629413yes—
282C4C22×#111311yescharacteristic
283C3—21no—
284C3—22no—
285C3—41no—
286C3C22×#371112yescomplemented
287C3C22×#371211yescharacteristic, complemented
288C2—184no—
289C2C2×#8880916yesdirect product factor
290C2C22×#629411yescharacteristic
subgroupG/H
center, hypercenterC22×C4#1113
commutatorC2×C33C23×C4
FrattiniC4C22×#1113
FittingC22×C33×C4C22
socleC23×C33C2×C4
prime radicalC23×C3×C3⋊D3C4
cosocleC33×C4C24
hypocenterC33C22×M16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×C3⋊(C3⋊C8)C2semidirect
3. C2×#88809C2direct (N × H)
11. C2×C3×C3⋊(C3⋊C8)C22semidirect
12. #88809C22direct (N × H)
16. C22×C3⋊(C3⋊C8)D3semidirect
30. C3×C3⋊(C3⋊C8)C23semidirect
50. C2×C3⋊(C3⋊C8)C2×D3semidirect
56. C22×C33M16semidirect
102. C3⋊(C3⋊C8)C22×D3semidirect
107. C2×C33C2×M16semidirect
139. C22×C32ΓL(1,𝔽25)semidirect
140. C22×C32#208semidirect
165. C33C22×M16semidirect
206. C2×C32C2×ΓL(1,𝔽25)semidirect
207. C2×C32C2×#208semidirect
228. C22×C3#3711semidirect
229. C22×C3#3712semidirect
252. C32C22×ΓL(1,𝔽25)semidirect
253. C32C22×#208semidirect
270. C2×C3C2×#3711semidirect
271. C2×C3C2×#3712semidirect
278. C22#88809direct (N × H)
286. C3C22×#3711semidirect
287. C3C22×#3712semidirect
289. C2C2×#88809direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C22×C3×C3⋊(C3⋊C8)2
C2×#888092
C22×#37123
C22×#37113