Finite Group Databasebeta

← C2×#13049399 #72252772

Subgroups — C2×#13049399 #72252772

Order 2592.

1484 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 1484 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C2×#10620150C211yesmaximal, characteristic
2C2×C3⋊(C3⋊#622)C211yesmaximal, characteristic, complemented
3C2×C32×#622C211yesmaximal, characteristic, complemented
4#13049399C214yesmaximal, direct product factor
5C2×#88924—34nomaximal
6C22×C3⋊(C3⋊(C3⋊D3))C411yescharacteristic
7C2×C32×C3⋊Dic3C411yescharacteristic
8C2×C32×C3⋊Dic3C2211yescharacteristic
9#10620150C412yes—
10#10620150C2212yes—
11C3⋊(C3⋊#622)C2212yescomplemented
12C32×#622C2212yescomplemented
13C2×#6289—34no—
14C2×#6289—62no—
15C2×C3⋊#622—34no—
16C2×C3×#622D314yescomplemented
17#88924—316no—
18C22×C34C2×C411yescharacteristic
19C2×C3⋊(C3⋊(C3⋊D3))C814yescomplemented
20C2×C3⋊(C3⋊(C3⋊D3))C2×C412yes—
21C32×C3⋊Dic3C2×C412yes—
22C32×C3⋊Dic3C2×C411yescharacteristic
23C32×C3⋊Dic3C2311yescharacteristic
24C2×#3787—91no—
25C2×C8×C3⋊D3—91nomaximal
26C22×C3⋊(C3⋊D3)—34no—
27C22×C3⋊(C3⋊D3)—62no—
28C22×C3⋊(C3⋊D3)—128no—
29C2×C3×C3⋊Dic3—22no—
30C2×C3×C3⋊Dic3C2×D314yes—
31C2×C32×Dic3—34no—
32C2×C32×Dic3—62no—
33#6289—38no—
34#6289—38no—
35#6289—64no—
36#6289—64no—
37C3⋊#622—38no—
38C3×#622C2×D318yescomplemented
39C2×C34C2×C812yescomplemented
40C2×C34C22×C411yescharacteristic
41C3⋊(C3⋊(C3⋊D3))C2×C814yescomplemented
42C2×C4×C3⋊D3—91no—
43C2×C4×C3⋊D3—91no—
44C2×#624—188no—
45C2×#622—91no—
46C2×#622C3⋊D311yescharacteristic, complemented
47#3787—94no—
48C2×C32×C8—91no—
49C2×C3⋊(C3⋊C8)—91no—
50C8×C3⋊D3—94no—
51C22×C33—22no—
52C22×C33—48no—
53C22×C33C4×D314yes—
54C2×C3⋊(C3⋊D3)—316no—
55C2×C3⋊(C3⋊D3)—38no—
56C2×C3⋊(C3⋊D3)—64no—
57C2×C3⋊(C3⋊D3)—68no—
58C2×C3⋊(C3⋊D3)—1232no—
59C2×C3⋊(C3⋊D3)—1216no—
60C3×C3⋊Dic3—24no—
61C3×C3⋊Dic3C4×D314yes—
62C3×C3⋊Dic3C22×D314yes—
63C32×Dic3—38no—
64C32×Dic3—62no—
65C32×Dic3—62no—
66C2×D3×C8—274no—
67C34C22×C811yescharacteristic, complemented, Sylow
68C22×C3⋊D3—91no—
69C22×C3⋊D3—91no—
70C22×C3⋊D3—188no—
71C22×C3⋊D3—368no—
72C22×C3⋊D3—368no—
73C22×C3⋊D3—3612no—
74C2×C32×C4—91no—
75C2×C32×C4—91no—
76C2×C3⋊Dic3#15711yescharacteristic
77C2×C3⋊Dic3C2×C3⋊D311yescharacteristic
78C4×C3⋊D3—92no—
79C4×C3⋊D3—92no—
80C4×C3⋊D3—92no—
81C4×C3⋊D3—92no—
82C2×C3×Dic3—68no—
83C2×C3×Dic3—68no—
84#624—1816no—
85#624—1816no—
86#622—92no—
87#622C2×C3⋊D312yescomplemented
88C32×C8—92no—
89C3⋊(C3⋊C8)—92no—
90C2×C33—22no—
91C2×C33—24no—
92C2×C33—416no—
93C2×C33—48no—
94C2×C33D3×C818yescomplemented
95C2×C33C2×C4×D314yes—
96C3⋊(C3⋊D3)—316no—
97C3⋊(C3⋊D3)—68no—
98C3⋊(C3⋊D3)—1232no—
99C2×C4×D3—274no—
100C2×C4×D3—542no—
101C2×C3×C8—94no—
102C2×C3⋊C8—274no—
103D3×C8—2716no—
104C22×C32—28no—
105C22×C32—48no—
106C22×C32—412no—
107C22×C32—48no—
108C22×C32C4×C3⋊D311yescharacteristic
109C22×C32C2×#15711yescharacteristic
110C2×C3⋊D3—92no—
111C2×C3⋊D3—94no—
112C2×C3⋊D3—94no—
113C2×C3⋊D3—92no—
114C2×C3⋊D3—1816no—
115C2×C3⋊D3—1832no—
116C2×C3⋊D3—3616no—
117C2×C3⋊D3—3632no—
118C2×C3⋊D3—3616no—
119C2×C3⋊D3—3632no—
120C2×C3⋊D3—3648no—
121C2×C3⋊D3—3624no—
122C32×C4—91no—
123C32×C4—91no—
124C32×C4—92no—
125C3⋊Dic3C4×C3⋊D311yescharacteristic
126C3⋊Dic3C2×#15712yes—
127C3⋊Dic3C22×C3⋊D311yescharacteristic
128C3×Dic3—616no—
129C3×Dic3—68no—
130C3×Dic3—68no—
131C22×C8—811noSylow
132C33—22no—
133C33—48no—
134C33C2×D3×C814yescomplemented
135C22×D3—274no—
136C22×D3—542no—
137C22×D3—1088no—
138C2×C3×C4—94no—
139C2×C3×C4—182no—
140C2×Dic3—34no—
141C2×Dic3—62no—
142C4×D3—278no—
143C4×D3—278no—
144C4×D3—544no—
145C4×D3—544no—
146C3×C8—98no—
147C3⋊C8—278no—
148C2×C32—28no—
149C2×C32—216no—
150C2×C32—416no—
151C2×C32—48no—
152C2×C32—424no—
153C2×C32—412no—
154C2×C32—48no—
155C2×C32—416no—
156C2×C32C8×C3⋊D312yescomplemented
157C2×C32#378712yescomplemented
158C2×C32C2×C4×C3⋊D311yescharacteristic
159C2×C32C22×#15711yescharacteristic
160C3⋊D3—94no—
161C3⋊D3—94no—
162C3⋊D3—1832no—
163C3⋊D3—3632no—
164C3⋊D3—3632no—
165C3⋊D3—3648no—
166C22×C4—811no—
167C2×C8—91no—
168C2×C8—814no—
169C2×C8—811no—
170C22×C3—22no—
171C22×C3—48no—
172C22×C3D3×#15714yes—
173C2×D3—278no—
174C2×D3—2716no—
175C2×D3—544no—
176C2×D3—548no—
177C2×D3—10832no—
178C2×D3—10816no—
179C3×C4—94no—
180C3×C4—94no—
181C3×C4—184no—
182Dic3—38no—
183Dic3—62no—
184Dic3—62no—
185C32—28no—
186C32—48no—
187C32—412no—
188C32—48no—
189C32C2×C8×C3⋊D311yescharacteristic, complemented
190C32C2×#378711yescharacteristic, complemented
191C23—811no—
192C2×C4—91no—
193C2×C4—91no—
194C2×C4—812no—
195C2×C4—812no—
196C8—92no—
197C8—812no—
198C2×C3—24no—
199C2×C3—22no—
200C2×C3—416no—
201C2×C3—48no—
202C2×C3#8892418yescomplemented
203C2×C3C2×D3×#15714yes—
204D3—2716no—
205D3—548no—
206D3—10832no—
207C22—814no—
208C22—812no—
209C22C3⋊D3×#15711yescharacteristic
210C4—91no—
211C4—92no—
212C4—91no—
213C3—22no—
214C3—48no—
215C3C2×#8892414yescomplemented
216C2—814no—
217C2#1304939912yesdirect product factor
218C2C2×C3⋊D3×#15711yescharacteristic
subgroupG/H
center, hypercenterC22C3⋊D3×#157
commutator, hypocenterC34C22×C8
FrattiniC2C2×C3⋊D3×#157
Fitting, socleC22×C34C2×C4
prime radicalC22×C3⋊(C3⋊(C3⋊D3))C4
cosocleC32×C3⋊Dic3C23
LCSG ▷ C34
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊(C3⋊#622)C2semidirect
3. C2×C32×#622C2semidirect
4. #13049399C2direct (N × H)
11. C3⋊(C3⋊#622)C22semidirect
12. C32×#622C22semidirect
16. C2×C3×#622D3semidirect
19. C2×C3⋊(C3⋊(C3⋊D3))C8semidirect
38. C3×#622C2×D3semidirect
39. C2×C34C2×C8semidirect
41. C3⋊(C3⋊(C3⋊D3))C2×C8semidirect
46. C2×#622C3⋊D3semidirect
67. C34C22×C8semidirect
87. #622C2×C3⋊D3semidirect
94. C2×C33D3×C8semidirect
134. C33C2×D3×C8semidirect
156. C2×C32C8×C3⋊D3semidirect
157. C2×C32#3787semidirect
189. C32C2×C8×C3⋊D3semidirect
190. C32C2×#3787semidirect
202. C2×C3#88924semidirect
215. C3C2×#88924semidirect
217. C2#13049399direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#106201502
C2×C3⋊(C3⋊#622)2
C2×C32×#6222
#130493992
C2×#889243
C2×C8×C3⋊D39