Finite Group Databasebeta

← #11800893

Subgroups — #11800893

Order 1152.

1601 conjugacy classes of subgroups; only the 428 normal subgroups are shown.

normal subgroups only

This group has 1601 conjugacy classes of subgroups — too many to draw. Only its 428 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×#3750C211yesmaximal, characteristic
2C2×#63957C212yesmaximal
3C3×C4×#217C212yesmaximal
4#10591705C212yesmaximal
5#10591243C218yesmaximal, complemented
6#73435—32nomaximal
7C23×C32×C4C2211yescharacteristic
8C23×C3⋊Dic3C2211yescharacteristic
9C23×C3⋊Dic3C2211yescharacteristic
10C2×#3750C418yescomplemented
11C2×#3750C2214yes—
12C2×C3×#217C2214yes—
13C2×C3×C4×Dic3C2214yes—
14C2×#3724C2214yes—
15#63957C4116yescomplemented
16C2×Dic32C2214yes—
17C2×Dic32C2214yes—
18C3×C3⋊#95C2214yes—
19#63579C2214yes—
20C22×Dic3⋊C4—32no—
21C22×Dic3⋊C4—61no—
22C2×#877—34no—
23C3×C4×C22⋊C4—32no—
24C4×#217D312yes—
25Dic3×C22⋊C4—34no—
26C4×Dic3⋊C4—38no—
27#4649—38no—
28C24×C32C2311yescharacteristic
29C22×C32×C4—42no—
30C22×C32×C4C2×C414yes—
31C22×C32×C4C2312yes—
32C22×C3⋊Dic3—42no—
33C22×C3⋊Dic3C2×C414yes—
34C22×C3⋊Dic3C2×C414yes—
35C22×C3⋊Dic3C2×C418yes—
36C22×C3⋊Dic3C2312yes—
37C22×C3⋊Dic3C2312yes—
38C22×C3×Dic3C2318yes—
39#3750C2×C4116yes—
40C3×#217C2×C4116yes—
41C3×C4×Dic3—44no—
42#3724C2×C4116yescomplemented
43Dic32—44no—
44Dic32C2×C4116yescomplemented
45#2274—91noSylow
46C23×C3×C4—21no—
47C23×C3×C4C2×D312yes—
48C23×Dic3—32no—
49C23×Dic3—32no—
50C23×Dic3—61no—
51C23×Dic3—61no—
52C2×C3×C22⋊C4—34no—
53C2×C3×C42—34no—
54C2×#217C2×D314yes—
55C2×Dic3⋊C4—316no—
56C2×Dic3⋊C4—34no—
57C2×Dic3⋊C4—34no—
58C2×Dic3⋊C4—38no—
59C2×Dic3⋊C4—68no—
60C2×Dic3⋊C4—64no—
61C2×C4×Dic3—38no—
62C2×C4×Dic3—38no—
63C2×C4×Dic3C2×D314yes—
64#877—332no—
65C3×#95—34no—
66C3⋊#95—38no—
67C3⋊#95C2×D314yes—
68C23×C32—22no—
69C23×C32—22no—
70C23×C32C22×C414yes—
71C23×C32C22×C412yes—
72C23×C32C2411yescharacteristic
73C2×C32×C4—24no—
74C2×C32×C4—44no—
75C2×C32×C4C22×C414yes—
76C2×C3⋊Dic3—24no—
77C2×C3⋊Dic3—44no—
78C2×C3⋊Dic3C42116yescomplemented
79C2×C3⋊Dic3C22×C414yes—
80C2×C3⋊Dic3C22×C414yes—
81C2×C3⋊Dic3C22×C414yes—
82C2×C3⋊Dic3C22×C414yes—
83C2×C3×Dic3—28no—
84C2×C3×Dic3—28no—
85C2×C3×Dic3C22×C4116yes—
86C2×#117—92no—
87C22×C4⋊C4—91no—
88C4×C4⋊C4—98no—
89C4×C22⋊C4—32no—
90C4×C22⋊C4—92no—
91C24×C3—21no—
92C24×C3C22×D312yes—
93C22×C3×C4—24no—
94C22×C3×C4—22no—
95C22×C3×C4—38no—
96C22×C3×C4—44no—
97C22×C3×C4—44no—
98C22×C3×C4C4×D318yes—
99C22×C3×C4C22×D312yes—
100C22×C3×C4C22×D312yes—
101C22×Dic3—316no—
102C22×Dic3—38no—
103C22×Dic3—34no—
104C22×Dic3—34no—
105C22×Dic3—38no—
106C22×Dic3—64no—
107C22×Dic3—62no—
108C22×Dic3—68no—
109C22×Dic3—64no—
110C22×Dic3—62no—
111C22×Dic3—124no—
112C22×Dic3—124no—
113C22×Dic3C22×D318yes—
114C3×C22⋊C4—316no—
115C3×C42—124no—
116#217C4×D3116yes—
117Dic3⋊C4—316no—
118Dic3⋊C4—316no—
119Dic3⋊C4—332no—
120Dic3⋊C4—616no—
121C4×Dic3—332no—
122C4×Dic3—44no—
123C4×Dic3—128no—
124C22×C32—22no—
125C22×C32—22no—
126C22×C32—28no—
127C22×C32C2×C4214yes—
128C22×C32C23×C411yescharacteristic
129C22×C32C23×C412yes—
130C22×C32H(5,𝔽2)12yes—
131C22×C322−1+412yes—
132C32×C4—24no—
133C3⋊Dic3—24no—
134C3⋊Dic3C2×C4218yes—
135C3×Dic3—216no—
136C23×C4—91no—
137C23×C4—91no—
138C23×C4D3211yescharacteristic
139#117—916no—
140C2×C4⋊C4—94no—
141C2×C4⋊C4—98no—
142C2×C4⋊C4—94no—
143C2×C22⋊C4—34no—
144C2×C42—34no—
145C2×C42—94no—
146C2×C42—94no—
147#95—34no—
148#95—94no—
149C23×C3—24no—
150C23×C3—24no—
151C23×C3—24no—
152C23×C3—24no—
153C23×C3—21no—
154C23×C3—24no—
155C23×C3—22no—
156C23×C3C2×C4×D318yes—
157C23×C3C2×C4×D314yes—
158C23×C3C23×D312yes—
159C2×C3×C4—24no—
160C2×C3×C4—24no—
161C2×C3×C4—24no—
162C2×C3×C4—316no—
163C2×C3×C4—44no—
164C2×C3×C4—48no—
165C2×C3×C4—48no—
166C2×C3×C4—68no—
167C2×C3×C4—68no—
168C2×C3×C4C2×C4×D314yes—
169C2×C3×C4C2×C4×D314yes—
170C2×Dic3—28no—
171C2×Dic3—28no—
172C2×Dic3—332no—
173C2×Dic3—38no—
174C2×Dic3—38no—
175C2×Dic3—38no—
176C2×Dic3—38no—
177C2×Dic3—616no—
178C2×Dic3—64no—
179C2×Dic3—64no—
180C2×Dic3—64no—
181C2×Dic3—64no—
182C2×Dic3—68no—
183C2×Dic3—124no—
184C2×Dic3—128no—
185C2×Dic3—128no—
186C2×Dic3C2×C4×D3116yes—
187C2×C32—24no—
188C2×C32C22×C4211yescharacteristic
189C2×C32#51811yescharacteristic
190C2×C32#51911yescharacteristic
191C2×C32#52014yes—
192C24C2×D3211yescharacteristic
193C22×C4—38no—
194C22×C4—42no—
195C22×C4—92no—
196C22×C4—98no—
197C22×C4—94no—
198C22×C4—92no—
199C22×C4—94no—
200C22×C4—362no—
201C22×C4#62414yes—
202C22×C4C2×D3212yes—
203C4⋊C4—916no—
204C4⋊C4—916no—
205C22⋊C4—316no—
206C42—916no—
207C42—124no—
208C42—364no—
209C22×C3—24no—
210C22×C3—24no—
211C22×C3—216no—
212C22×C3—24no—
213C22×C3—24no—
214C22×C3—216no—
215C22×C3—22no—
216C22×C3—22no—
217C22×C3—21no—
218C22×C3—22no—
219C22×C3—24no—
220C22×C3C42×D318yes—
221C22×C3C22×C4×D314yes—
222C22×C3C22×C4×D312yes—
223C22×C3#100414yes—
224C22×C3#101014yes—
225C3×C4—24no—
226C3×C4—24no—
227C3×C4—44no—
228C3×C4—616no—
229Dic3—216no—
230Dic3—316no—
231Dic3—68no—
232Dic3—68no—
233Dic3—124no—
234C32#227411yescharacteristic, complemented, Sylow
235C23—22no—
236C23—22no—
237C23C4×D3214yes—
238C23C2×#62412yes—
239C23C22×D3211yescharacteristic
240C2×C4—24no—
241C2×C4—316no—
242C2×C4—44no—
243C2×C4—68no—
244C2×C4—68no—
245C2×C4—94no—
246C2×C4—94no—
247C2×C4—94no—
248C2×C4—916no—
249C2×C4—94no—
250C2×C4—184no—
251C2×C4—364no—
252C2×C4C2×#62414yes—
253C2×C3—28no—
254C2×C3—28no—
255C2×C3—21no—
256C2×C3—24no—
257C2×C3—21no—
258C2×C3—21no—
259C2×C3C2×C42×D312yes—
260C2×C3#548612yes—
261C2×C3#551612yes—
262C2×C3#554618yes—
263C22—22no—
264C22—28no—
265C22—22no—
266C22C4×#62414yes—
267C22#6429812yes—
268C22C2×C4×D3212yes—
269C22C22×#62411yescharacteristic
270C22#6433012yes—
271C4—24no—
272C4—616no—
273C4—98no—
274C4—184no—
275C3—21no—
276C3#7343512yescomplemented
277C2—24no—
278C2#1059482811yescharacteristic
279C2#1059487614yes—
280C2C2×C4×#62411yescharacteristic
281C2#1059551011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC2×C32C22×C42
FittingC23×C32×C4C22
socle, cosocleC23×C32C24
prime radicalC24×C32C23
hypercenterC23×C4D32
hypocenterC32#2274
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
5. #10591243C2semidirect
10. C2×#3750C4semidirect
15. #63957C4semidirect
42. #3724C2×C4semidirect
44. Dic32C2×C4semidirect
78. C2×C3⋊Dic3C42semidirect
234. C32#2274semidirect
276. C3#73435semidirect

Maximal subgroups

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

subgroupindex
C22×#37502
C2×#639572
C3×C4×#2172
#105917052
#105912432
#734353