Finite Group Databasebeta

← C2×#13049456 #72252784

Subgroups — C2×#13049456 #72252784

Order 2592.

1727 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 1727 conjugacy classes of subgroups — too many to draw. Only its 63 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×#66472C211yesmaximal, characteristic
2C2×#10620150C212yesmaximal
3#13049456C214yesmaximal, direct product factor
4C22×C3⋊(C3⋊(C3⋊D3))C2211yescharacteristic
5C2×#66472C414yescomplemented
6C2×#66472C2212yes—
7C2×C32×C3⋊Dic3—22no—
8#10620150—22no—
9#10620150C2214yes—
10C2×#6289—68no—
11C22×C34D411yescharacteristic
12C2×C3⋊(C3⋊(C3⋊D3))C2×C412yes—
13C2×C3⋊(C3⋊(C3⋊D3))D411yescharacteristic
14C2×C3⋊(C3⋊(C3⋊D3))Q812yes—
15C2×C3⋊(C3⋊(C3⋊D3))C2311yescharacteristic
16#66472C2×C414yescomplemented
17C32×C3⋊Dic3—24no—
18C2×#3773—94nomaximal
19C22×C3⋊(C3⋊D3)—68no—
20C22×C3⋊(C3⋊D3)—126no—
21C2×C3×C3⋊Dic3—28no—
22C2×C32×Dic3—68no—
23#6289—616no—
24#6289—616no—
25C2×C34C4⋊C412yescomplemented
26C2×C34C2×D411yescharacteristic
27C3⋊(C3⋊(C3⋊D3))C4⋊C412yescomplemented
28C3⋊(C3⋊(C3⋊D3))C22×C411yescharacteristic
29C3⋊(C3⋊(C3⋊D3))C2×Q811yescharacteristic
30C22×#157—94no—
31C22×#157—183no—
32C2×C4×C3⋊D3—182no—
33C2×#624—98no—
34C2×#624—1812no—
35#3773—916no—
36C22×C33—28no—
37C22×C33—46no—
38C2×C3⋊(C3⋊D3)—68no—
39C2×C3⋊(C3⋊D3)—68no—
40C2×C3⋊(C3⋊D3)—616no—
41C2×C3⋊(C3⋊D3)—616no—
42C2×C3⋊(C3⋊D3)—1212no—
43C2×C3⋊(C3⋊D3)—126no—
44C2×C3⋊(C3⋊D3)—1212no—
45C2×C3⋊(C3⋊D3)—126no—
46C3×C3⋊Dic3—216no—
47C32×Dic3—616no—
48C34C2×C4⋊C411yescharacteristic, complemented, Sylow
49C22×C3⋊D3—94no—
50C22×C3⋊D3—182no—
51C22×C3⋊D3—1812no—
52C22×C3⋊D3—183no—
53C22×C3⋊D3—3612no—
54C22×C3⋊D3—368no—
55C22×C3⋊D3—363no—
56C2×#157—916no—
57C2×#157—98no—
58C2×#157—1812no—
59C2×#157—186no—
60C2×C32×C4—182no—
61C2×C3⋊Dic3—22no—
62C4×C3⋊D3—184no—
63C4×C3⋊D3—184no—
64C2×C3×Dic3—624no—
65C2×C3×Dic3—68no—
66#624—916no—
67#624—188no—
68#624—1824no—
69#624—1824no—
70C2×C33—28no—
71C2×C33—216no—
72C2×C33—412no—
73C2×C33—46no—
74C3⋊(C3⋊D3)—616no—
75C3⋊(C3⋊D3)—68no—
76C3⋊(C3⋊D3)—68no—
77C3⋊(C3⋊D3)—1212no—
78C3⋊(C3⋊D3)—126no—
79C3⋊(C3⋊D3)—126no—
80C2×C4×D3—548no—
81C22×C32—22no—
82C22×C32—212no—
83C22×C32—23no—
84C22×C32—412no—
85C22×C32—48no—
86C22×C32—43no—
87C22×C32GO+(4,𝔽2)14yes—
88C2×C3⋊D3—94no—
89C2×C3⋊D3—94no—
90C2×C3⋊D3—98no—
91C2×C3⋊D3—98no—
92C2×C3⋊D3—182no—
93C2×C3⋊D3—182no—
94C2×C3⋊D3—184no—
95C2×C3⋊D3—184no—
96C2×C3⋊D3—1812no—
97C2×C3⋊D3—1812no—
98C2×C3⋊D3—1824no—
99C2×C3⋊D3—1824no—
100C2×C3⋊D3—186no—
101C2×C3⋊D3—183no—
102C2×C3⋊D3—183no—
103C2×C3⋊D3—186no—
104C2×C3⋊D3—3612no—
105C2×C3⋊D3—3612no—
106C2×C3⋊D3—3624no—
107C2×C3⋊D3—3624no—
108C2×C3⋊D3—368no—
109C2×C3⋊D3—368no—
110C2×C3⋊D3—3616no—
111C2×C3⋊D3—3616no—
112C2×C3⋊D3—366no—
113C2×C3⋊D3—363no—
114C2×C3⋊D3—366no—
115C2×C3⋊D3—363no—
116#157—916no—
117#157—1812no—
118C32×C4—184no—
119C3⋊Dic3—24no—
120C3×Dic3—648no—
121C3×Dic3—616no—
122C2×C4⋊C4—811noSylow
123C33—28no—
124C33—46no—
125C22×D3—548no—
126C22×D3—1086no—
127C2×C3×C4—188no—
128C2×Dic3—68no—
129C4×D3—5416no—
130C4×D3—5416no—
131C2×C32—22no—
132C2×C32—24no—
133C2×C32—212no—
134C2×C32—224no—
135C2×C32—23no—
136C2×C32—26no—
137C2×C32—412no—
138C2×C32—424no—
139C2×C32—48no—
140C2×C32—416no—
141C2×C32—43no—
142C2×C32—46no—
143C2×C32#377318yescomplemented
144C2×C32C2×GO+(4,𝔽2)14yes—
145C3⋊D3—94no—
146C3⋊D3—98no—
147C3⋊D3—94no—
148C3⋊D3—184no—
149C3⋊D3—182no—
150C3⋊D3—182no—
151C3⋊D3—1824no—
152C3⋊D3—1812no—
153C3⋊D3—1812no—
154C3⋊D3—183no—
155C3⋊D3—186no—
156C3⋊D3—183no—
157C3⋊D3—3624no—
158C3⋊D3—3612no—
159C3⋊D3—3612no—
160C3⋊D3—3616no—
161C3⋊D3—368no—
162C3⋊D3—368no—
163C3⋊D3—366no—
164C3⋊D3—363no—
165C3⋊D3—363no—
166C22×C4—812no—
167C22×C4—811no—
168C4⋊C4—814no—
169C22×C3—28no—
170C22×C3—46no—
171C2×D3—548no—
172C2×D3—548no—
173C2×D3—5416no—
174C2×D3—5416no—
175C2×D3—1086no—
176C2×D3—1086no—
177C2×D3—10812no—
178C2×D3—10812no—
179C3×C4—1816no—
180Dic3—616no—
181C32—22no—
182C32—212no—
183C32—23no—
184C32—412no—
185C32—48no—
186C32—43no—
187C32C2×#377314yescomplemented
188C23—811no—
189C2×C4—182no—
190C2×C4—814no—
191C2×C4—812no—
192C2×C4—814no—
193C2×C4—1622no—
194C2×C3—28no—
195C2×C3—216no—
196C2×C3—46no—
197C2×C3—412no—
198D3—548no—
199D3—548no—
200D3—5416no—
201D3—1086no—
202D3—1086no—
203D3—10812no—
204C22—811no—
205C22—811no—
206C22—812no—
207C22—812no—
208C22#1062020611yescharacteristic
209C4—184no—
210C4—814no—
211C3—28no—
212C3—46no—
213C2—811no—
214C2—811no—
215C2—812no—
216C2#1304945612yesdirect product factor
217C2C2×#1062020611yescharacteristic
subgroupG/H
center, hypercenterC22#10620206
commutatorC3⋊(C3⋊(C3⋊D3))C22×C4
FrattiniC2C2×#10620206
Fitting, socleC22×C34D4
prime radicalC22×C3⋊(C3⋊(C3⋊D3))C22
cosocleC2×C3⋊(C3⋊(C3⋊D3))C23
hypocenterC34C2×C4⋊C4
LCSG ▷ C3⋊(C3⋊(C3⋊D3)) ▷ C34
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
3. #13049456C2direct (N × H)
5. C2×#66472C4semidirect
16. #66472C2×C4semidirect
25. C2×C34C4⋊C4semidirect
27. C3⋊(C3⋊(C3⋊D3))C4⋊C4semidirect
48. C34C2×C4⋊C4semidirect
143. C2×C32#3773semidirect
187. C32C2×#3773semidirect
216. C2#13049456direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#664722
C2×#106201502
#130494562
C2×#37739