Finite Group Databasebeta

← #11800865

Subgroups — #11800865

Order 1152.

2161 conjugacy classes of subgroups.

The subgroup lattice has 277 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3757C211yesmaximal, characteristic
2C22×C3×#217C212yesmaximal
3#10591705C214yesmaximal, complemented
4Dic3×#217C218yesmaximal, complemented
5#82083—32nomaximal
6C25×C32C2211yescharacteristic
7C23×C3⋊Dic3C2212yes—
8C23×C3×Dic3C2214yes—
9C2×#3757C418yescomplemented
10C2×#3757C2214yes—
11C2×C3×#217C4116yescomplemented
12C2×C3×#217C2214yes—
13C2×C3×#217C2214yes—
14C2×Dic32C2218yescomplemented
15#63579C2218yescomplemented
16C22×C3×C22⋊C4—32no—
17C22×#217—32no—
18C22×#217—61no—
19C22×#217D312yes—
20C4×#217—38no—
21Dic3×C22⋊C4—38no—
22Dic3×C22⋊C4—38no—
23C24×C32—44no—
24C24×C32C2×C414yes—
25C24×C32C2×C418yes—
26C24×C32C2311yescharacteristic
27C24×C32C2312yes—
28C22×C3⋊Dic3—44no—
29C22×C3⋊Dic3C2×C418yescomplemented
30C22×C3⋊Dic3C2314yes—
31C22×C3×Dic3—48no—
32C22×C3×Dic3C2×C4116yescomplemented
33C22×C3×Dic3C2318yes—
34#3757C2×C4116yescomplemented
35C3×#217C2×C4116yescomplemented
36C3×#217C2×C4116yescomplemented
37Dic32—48no—
38#2273—91noSylow
39C25×C3—21no—
40C25×C3C2×D312yes—
41C23×C3×C4—34no—
42C23×Dic3—34no—
43C23×Dic3—62no—
44C23×Dic3C2×D314yes—
45C2×C3×C22⋊C4—316no—
46C2×C3×C22⋊C4—34no—
47C2×C3×C22⋊C4—34no—
48C2×#217—316no—
49C2×#217—34no—
50C2×#217—34no—
51C2×#217—68no—
52C2×#217—64no—
53C2×#217Dic3116yescomplemented
54C2×#217C2×D314yes—
55C2×#217C2×D314yes—
56C2×C4×Dic3—316no—
57C3⋊#95—316no—
58C23×C32—24no—
59C23×C32—28no—
60C23×C32—48no—
61C23×C32—416no—
62C23×C32C42116yescomplemented
63C23×C32C22×C414yes—
64C23×C32C22×C412yes—
65C23×C32C22×C414yes—
66C23×C32C22×C414yes—
67C23×C32C22×C414yes—
68C23×C32C2411yescharacteristic
69C2×C3⋊Dic3—28no—
70C2×C3⋊Dic3—48no—
71C2×C3⋊Dic3C22×C418yes—
72C2×C3×Dic3—216no—
73C2×C3×Dic3—416no—
74C2×C3×Dic3C22×C4116yes—
75C22×C22⋊C4—32no—
76C22×C22⋊C4—91no—
77C4×C22⋊C4—94no—
78C4×C22⋊C4—98no—
79C24×C3—28no—
80C24×C3—24no—
81C24×C3—21no—
82C24×C3—22no—
83C24×C3—48no—
84C24×C3—48no—
85C24×C3C4×D318yes—
86C24×C3C4×D318yes—
87C24×C3C2×Dic318yes—
88C24×C3C22×D312yes—
89C24×C3C22×D312yes—
90C24×C3C22×D312yes—
91C22×C3×C4—38no—
92C22×C3×C4—316no—
93C22×C3×C4—128no—
94C22×Dic3—38no—
95C22×Dic3—316no—
96C22×Dic3—48no—
97C22×Dic3—68no—
98C22×Dic3—64no—
99C22×Dic3—128no—
100C22×Dic3—128no—
101C22×Dic3C2×Dic3116yescomplemented
102C22×Dic3C22×D318yes—
103C3×C22⋊C4—316no—
104C3×C22⋊C4—316no—
105#217—316no—
106#217—316no—
107#217—616no—
108#217C2×Dic3116yescomplemented
109#217C2×Dic3116yescomplemented
110C4×Dic3—1216no—
111C22×C32—24no—
112C22×C32—28no—
113C22×C32—28no—
114C22×C32—28no—
115C22×C32—28no—
116C22×C32—416no—
117C22×C32C2×C4214yes—
118C22×C32C2×C4218yes—
119C22×C32C23×C411yescharacteristic
120C22×C32C23×C412yes—
121C22×C32H(5,𝔽2)14yes—
122C3⋊Dic3—28no—
123C3×Dic3—216no—
124C25D3211yescharacteristic
125C23×C4—34no—
126C23×C4—92no—
127C2×C22⋊C4—316no—
128C2×C22⋊C4—34no—
129C2×C22⋊C4—34no—
130C2×C22⋊C4—98no—
131C2×C22⋊C4—94no—
132C2×C42—98no—
133#95—98no—
134C23×C3—28no—
135C23×C3—28no—
136C23×C3—28no—
137C23×C3—28no—
138C23×C3—21no—
139C23×C3—216no—
140C23×C3—24no—
141C23×C3—24no—
142C23×C3—24no—
143C23×C3—22no—
144C23×C3—24no—
145C23×C3—48no—
146C23×C3—416no—
147C23×C3—416no—
148C23×C3—416no—
149C23×C3—432no—
150C23×C3—416no—
151C23×C3C4×Dic3132yescomplemented
152C23×C3C2×C4×D314yes—
153C23×C3C2×C4×D314yes—
154C23×C3C2×C4×D314yes—
155C23×C3C2×C4×D314yes—
156C23×C3C2×C4×D314yes—
157C23×C3C2×C4×D314yes—
158C23×C3C22×Dic314yes—
159C23×C3C22×Dic314yes—
160C23×C3C22×Dic314yes—
161C23×C3C23×D312yes—
162C2×C3×C4—316no—
163C2×C3×C4—616no—
164C2×C3×C4—1216no—
165C2×Dic3—216no—
166C2×Dic3—316no—
167C2×Dic3—416no—
168C2×Dic3—68no—
169C2×Dic3—616no—
170C2×Dic3—128no—
171C2×Dic3—1216no—
172C2×Dic3—1216no—
173C2×Dic3C22×Dic3116yes—
174C2×C32—24no—
175C2×C32—28no—
176C2×C32C22×C4211yescharacteristic
177C2×C32#51812yes—
178C2×C32#51814yes—
179C24—44no—
180C24D3×Dic318yes—
181C24#62414yes—
182C24C2×D3211yescharacteristic
183C24C2×D3212yes—
184C22×C4—38no—
185C22×C4—316no—
186C22×C4—94no—
187C22×C4—98no—
188C22×C4—128no—
189C22×C4—364no—
190C22⋊C4—316no—
191C22⋊C4—316no—
192C22⋊C4—916no—
193C42—368no—
194C22×C3—28no—
195C22×C3—216no—
196C22×C3—28no—
197C22×C3—28no—
198C22×C3—28no—
199C22×C3—28no—
200C22×C3—28no—
201C22×C3—28no—
202C22×C3—28no—
203C22×C3—28no—
204C22×C3—28no—
205C22×C3—22no—
206C22×C3—21no—
207C22×C3—24no—
208C22×C3—24no—
209C22×C3—28no—
210C22×C3—48no—
211C22×C3—48no—
212C22×C3—48no—
213C22×C3—432no—
214C22×C3—432no—
215C22×C3C2×C4×Dic318yes—
216C22×C3C2×C4×Dic318yes—
217C22×C3C2×C4×Dic318yes—
218C22×C3C22×C4×D312yes—
219C22×C3C22×C4×D312yes—
220C22×C3#100418yes—
221C22×C3C23×Dic312yes—
222C3×C4—616no—
223Dic3—216no—
224Dic3—616no—
225Dic3—128no—
226C32#227311yescharacteristic, complemented, Sylow
227C23—28no—
228C23—24no—
229C23—416no—
230C23—48no—
231C23Dic32116yescomplemented
232C23C2×D3×Dic314yes—
233C23C2×D3×Dic314yes—
234C23C2×D3×Dic314yes—
235C23C2×#62414yes—
236C23C2×#62412yes—
237C23C22×D3211yescharacteristic
238C2×C4—316no—
239C2×C4—616no—
240C2×C4—98no—
241C2×C4—1216no—
242C2×C4—188no—
243C2×C4—368no—
244C2×C3—28no—
245C2×C3—28no—
246C2×C3—28no—
247C2×C3—28no—
248C2×C3—21no—
249C2×C3—24no—
250C2×C3—22no—
251C2×C3—48no—
252C2×C3#548614yes—
253C2×C3#548614yes—
254C2×C3C22×C4×Dic312yes—
255C2×C3#579814yes—
256C22—28no—
257C22—28no—
258C22—28no—
259C22—24no—
260C22—28no—
261C22—416no—
262C22C2×Dic3214yes—
263C22C2×Dic3218yes—
264C22C22×D3×Dic312yes—
265C22C22×#62411yescharacteristic
266C22#6433014yes—
267C4—616no—
268C4—188no—
269C3—21no—
270C3#8208312yescomplemented
271C2—24no—
272C2—28no—
273C2C22×Dic3211yescharacteristic
274C2#1059550614yes—
275C2#1059551012yes—
subgroupG/H
center, FrattiniC23C22×D32
commutatorC2×C32C22×C42
Fitting, prime radicalC25×C32C22
socle, cosocleC23×C32C24
hypercenterC25D32
hypocenterC32#2273
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C25

Complements

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

NH ≅ G/Ntype
3. #10591705C2semidirect
4. Dic3×#217C2semidirect
9. C2×#3757C4semidirect
11. C2×C3×#217C4semidirect
14. C2×Dic32C22semidirect
15. #63579C22semidirect
29. C22×C3⋊Dic3C2×C4semidirect
32. C22×C3×Dic3C2×C4semidirect
34. #3757C2×C4semidirect
35. C3×#217C2×C4semidirect
36. C3×#217C2×C4semidirect
53. C2×#217Dic3semidirect
62. C23×C32C42semidirect
101. C22×Dic3C2×Dic3semidirect
108. #217C2×Dic3semidirect
109. #217C2×Dic3semidirect
151. C23×C3C4×Dic3semidirect
226. C32#2273semidirect
231. C23Dic32semidirect
270. C3#82083semidirect

Maximal subgroups

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

subgroupindex
C22×#37572
C22×C3×#2172
#105917052
Dic3×#2172
#820833