Finite Group Databasebeta

← C2×#10592564 #11866960

Subgroups — C2×#10592564 #11866960

Order 1152.

1788 conjugacy classes of subgroups; only the 353 normal subgroups are shown.

normal subgroups only

This group has 1788 conjugacy classes of subgroups — too many to draw. Only its 353 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×#3745C211yesmaximal, characteristic, complemented
2C22×#3743C211yesmaximal, characteristic, complemented
3C2×#64104C212yesmaximal, complemented
4C2×C32×#106C211yesmaximal, characteristic, complemented
5C2×#63636C212yesmaximal, complemented
6#10592564C218yesmaximal, direct product factor
7C2×#4865—34nomaximal
8C22×C4⋊(C3⋊D3)—22no—
9C22×C4⋊(C3⋊D3)C2212yes—
10C22×C4×C3⋊D3C2211yescharacteristic
11C22×C32×C8C2211yescharacteristic
12C2×C32×C4⋊C4C2212yes—
13C2×#3752—22no—
14C22×C3⋊(C3⋊C8)C2211yescharacteristic
15C2×#3745—24no—
16C2×#3745C2214yescomplemented
17C2×#3743—24no—
18C2×#3743C2214yescomplemented
19#64104—24no—
20#64104C2218yescomplemented
21C32×#106C2214yescomplemented
22#63636C2218yescomplemented
23C22×D24—34no—
24C22×ΓL(1,𝔽25)—34no—
25C2×#895—38no—
26C2×C3×#106D314yescomplemented
27C2×#809—38no—
28#4865—332no—
29C23×C3⋊D3—22no—
30C23×C3⋊D3D411yescharacteristic
31C22×C32×C4—22no—
32C22×C32×C4C2311yescharacteristic
33C22×C3⋊Dic3D411yescharacteristic
34C2×C4⋊(C3⋊D3)—24no—
35C2×C4⋊(C3⋊D3)—24no—
36C2×C4⋊(C3⋊D3)—28no—
37C2×C4⋊(C3⋊D3)—22no—
38C2×C4⋊(C3⋊D3)—48no—
39C2×C4⋊(C3⋊D3)C2314yes—
40C2×C4×C3⋊D3—22no—
41C2×C4×C3⋊D3—22no—
42C2×C4×C3⋊D3D412yes—
43C2×C4×C3⋊D3C2312yes—
44C2×C32×C8D414yescomplemented
45C2×C32×C8C2312yes—
46C32×C4⋊C4—22no—
47C32×C4⋊C4C2314yes—
48#3752—28no—
49C2×C3⋊(C3⋊C8)—22no—
50C2×C3⋊(C3⋊C8)C2312yes—
51#3745—28no—
52#3743—28no—
53C2×#469—91noSylow
54C22×D12—38no—
55C22×D12—68no—
56C22×C4×D3—34no—
57C22×C3×C8C2×D314yes—
58C2×C3×C4⋊C4C2×D318yes—
59C2×(C2×D3)⋊C4—68no—
60C22×C3⋊C8—34no—
61C2×D24—316no—
62C2×D24—616no—
63C2×ΓL(1,𝔽25)—316no—
64C2×ΓL(1,𝔽25)—616no—
65#895—332no—
66#895—616no—
67C3×#106C2×D3116yescomplemented
68#809—332no—
69C23×C32C2×D411yescharacteristic
70C22×C3⋊D3—22no—
71C22×C3⋊D3—24no—
72C22×C3⋊D3—24no—
73C22×C3⋊D3—44no—
74C22×C3⋊D3—44no—
75C22×C3⋊D3—44no—
76C22×C3⋊D3C2×D412yes—
77C2×C32×C4—24no—
78C2×C32×C4—44no—
79C2×C32×C4C2×D412yes—
80C2×C32×C4C2×D411yescharacteristic
81C2×C32×C4#4112yes—
82C2×C32×C4C2411yescharacteristic
83C2×C3⋊Dic3—22no—
84C2×C3⋊Dic3C2×D412yes—
85C4⋊(C3⋊D3)—24no—
86C4⋊(C3⋊D3)—24no—
87C4⋊(C3⋊D3)—416no—
88C4⋊(C3⋊D3)—44no—
89C4×C3⋊D3—24no—
90C4×C3⋊D3—22no—
91C4×C3⋊D3—22no—
92C32×C8C2×D414yescomplemented
93C3⋊(C3⋊C8)—22no—
94C22×D8—91no—
95C22×M16—91no—
96C2×#121—92no—
97#469—98no—
98C2×#106C3⋊D311yescharacteristic, complemented
99C2×#102—92no—
100C23×D3—34no—
101C23×D3—68no—
102C22×C3×C4—28no—
103C22×C3×C4C22×D314yes—
104C22×Dic3—34no—
105C2×D12—316no—
106C2×D12—616no—
107C2×D12—616no—
108C2×D12—632no—
109C2×D12—68no—
110C2×D12—1232no—
111C2×C4×D3—38no—
112C2×C4×D3—38no—
113C2×C4×D3—68no—
114C2×C4×D3—68no—
115C2×C3×C8D12116yescomplemented
116C2×C3×C8C22×D318yes—
117C3×C4⋊C4—28no—
118C3×C4⋊C4C22×D3116yes—
119(C2×D3)⋊C4—632no—
120C2×C3⋊C8—38no—
121C2×C3⋊C8—68no—
122D24—632no—
123ΓL(1,𝔽25)—632no—
124C22×C32#12112yes—
125C22×C32AGL(1,ℤ/8)14yes—
126C22×C32C22×D411yescharacteristic
127C2×C3⋊D3—22no—
128C2×C3⋊D3—24no—
129C2×C3⋊D3—44no—
130C2×C3⋊D3—44no—
131C2×C3⋊D3—44no—
132C2×C3⋊D3—44no—
133C2×C3⋊D3—88no—
134C32×C4—44no—
135C32×C4#12112yes—
136C32×C4C22×D411yescharacteristic
137C32×C4C2×#4111yescharacteristic
138C3⋊Dic3—22no—
139C22×D4—92no—
140C22×D4—182no—
141C23×C4—91no—
142C2×D8—94no—
143C2×D8—184no—
144C2×M16—94no—
145C2×M16—184no—
146C22×C8—91no—
147C22×C8C2×C3⋊D311yescharacteristic
148#121—98no—
149#121—184no—
150C2×C4⋊C4C2×C3⋊D312yes—
151C2×C22⋊C4—182no—
152#106C2×C3⋊D314yescomplemented
153#102—98no—
154C23×C3D3×D414yes—
155C22×D3—38no—
156C22×D3—68no—
157C22×D3—616no—
158C22×D3—616no—
159C22×D3—1216no—
160C22×D3—1216no—
161C22×D3—1216no—
162C2×C3×C4—216no—
163C2×C3×C4—416no—
164C2×C3×C4C2×D1218yes—
165C2×C3×C4D3×D414yes—
166C2×C3×C4#23918yes—
167C2×C3×C4C23×D314yes—
168C2×Dic3—38no—
169C2×Dic3—68no—
170D12—616no—
171D12—616no—
172D12—1264no—
173D12—1216no—
174C4×D3—616no—
175C4×D3—68no—
176C4×D3—68no—
177C3×C8C2×D12116yescomplemented
178C3⋊C8—68no—
179C2×C32#46914yescomplemented
180C2×C32C2×#12111yescharacteristic
181C2×C32C2×AGL(1,ℤ/8)12yes—
182C3⋊D3—42no—
183C3⋊D3—84no—
184C24—91no—
185C24—182no—
186C2×D4—94no—
187C2×D4—188no—
188C2×D4—184no—
189C2×D4—182no—
190C2×D4—184no—
191C2×D4—368no—
192C22×C4—22no—
193C22×C4—91no—
194C22×C4—92no—
195C22×C4—92no—
196C22×C4—182no—
197C22×C4—182no—
198C22×C4C22×C3⋊D311yescharacteristic
199D8—188no—
200M16—188no—
201C2×C8—92no—
202C2×C8—182no—
203C2×C8C4⋊(C3⋊D3)14yescomplemented
204C2×C8C22×C3⋊D312yes—
205C4⋊C4—22no—
206C4⋊C4C22×C3⋊D314yes—
207C22⋊C4—188no—
208C22×C3#89518yes—
209C22×C3#914116yes—
210C22×C3C2×D3×D414yes—
211C2×D3—616no—
212C2×D3—68no—
213C2×D3—1216no—
214C2×D3—1216no—
215C2×D3—1216no—
216C2×D3—1216no—
217C2×D3—2432no—
218C3×C4—416no—
219C3×C4#89518yes—
220C3×C4C22×D1214yes—
221C3×C4C2×#23914yes—
222Dic3—68no—
223C32C2×#46911yescharacteristic, complemented, Sylow
224C23—92no—
225C23—182no—
226C23—184no—
227C23—184no—
228C23—364no—
229C23—364no—
230C23—364no—
231C23D4×C3⋊D311yescharacteristic
232D4—184no—
233D4—184no—
234D4—3616no—
235D4—364no—
236C2×C4—24no—
237C2×C4—44no—
238C2×C4—92no—
239C2×C4—182no—
240C2×C4—182no—
241C2×C4—184no—
242C2×C4—182no—
243C2×C4C2×C4⋊(C3⋊D3)12yes—
244C2×C4D4×C3⋊D311yescharacteristic
245C2×C4#383212yes—
246C2×C4C23×C3⋊D311yescharacteristic
247C8—182no—
248C8C2×C4⋊(C3⋊D3)14yescomplemented
249C2×C3#4865116yescomplemented
250C2×C3C2×#89514yes—
251C2×C3C2×#91418yes—
252D3—128no—
253D3—2416no—
254C22—184no—
255C22—182no—
256C22—364no—
257C22—364no—
258C22—364no—
259C22—364no—
260C22—728no—
261C22#6410412yes—
262C22#6412314yes—
263C22C2×D4×C3⋊D311yescharacteristic
264C4—44no—
265C4—182no—
266C4#6410412yes—
267C4C22×C4⋊(C3⋊D3)11yescharacteristic
268C4C2×#383211yescharacteristic
269C3C2×#486514yescomplemented
270C2—362no—
271C2—724no—
272C2#1059256414yesdirect product factor
273C2C2×#6410411yescharacteristic
274C2C2×#6412312yes—
subgroupG/H
centerC23D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC2×C32×#106C2
socleC23×C32C2×D4
hypercenterC2×#106C3⋊D3
hypocenterC32C2×#469
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
1. C22×#3745C2semidirect
2. C22×#3743C2semidirect
3. C2×#64104C2semidirect
4. C2×C32×#106C2semidirect
5. C2×#63636C2semidirect
6. #10592564C2direct (N × H)
16. C2×#3745C22semidirect
18. C2×#3743C22semidirect
20. #64104C22semidirect
21. C32×#106C22semidirect
22. #63636C22semidirect
26. C2×C3×#106D3semidirect
44. C2×C32×C8D4semidirect
67. C3×#106C2×D3semidirect
92. C32×C8C2×D4semidirect
98. C2×#106C3⋊D3semidirect
115. C2×C3×C8D12semidirect
152. #106C2×C3⋊D3semidirect
177. C3×C8C2×D12semidirect
179. C2×C32#469semidirect
203. C2×C8C4⋊(C3⋊D3)semidirect
223. C32C2×#469semidirect
248. C8C2×C4⋊(C3⋊D3)semidirect
249. C2×C3#4865semidirect
269. C3C2×#4865semidirect
272. C2#10592564direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#37452
C22×#37432
C2×#641042
C2×C32×#1062
C2×#636362
#105925642
C2×#48653