Finite Group Databasebeta

← C2×#26012532 #36370316

Subgroups — C2×#26012532 #36370316

Order 22144.

608 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 170 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×D4×#10621924C212yesmaximal, complemented
2#26012532C218yesmaximal, direct product factor
3C2×#19149747C211yesmaximal, characteristic, complemented
4C2×#19149741C212yesmaximal, complemented
5C2×#19149733C211yesmaximal, characteristic, complemented
6C2×D8×D173C211yesmaximal, characteristic
7C23×#10621924—22no—
8C2×D173⋊C8C2211yescharacteristic
9D4×#10621924—24no—
10D4×#10621924C2218yescomplemented
11C2×C4⋊#10621924C2212yes—
12C2×C22⋊#10621924—22no—
13C2×C4×#10621924C2211yescharacteristic
14#19149747C2214yescomplemented
15#19149741C2218yescomplemented
16#19149733C2214yescomplemented
17C2×D4×D173C2212yes—
18C2×D1384C411yescharacteristic, complemented
19C2×#14583811C412yescomplemented
20D8×D173C414yescomplemented
21D8×D173C2214yes—
22C2×C8×D173C2211yescharacteristic
23C2×D8×C173C411yescharacteristic, complemented
24C22×#10621924—28no—
25C22×#10621924—22no—
26C22×#10621924—48no—
27C22×#10621924D412yes—
28D173⋊C8—21no—
29D173⋊C8C2312yes—
30C2×#13063443—21no—
31C4⋊#10621924—22no—
32C4⋊#10621924C2314yes—
33C22⋊#10621924—28no—
34C4×#10621924D412yes—
35C4×#10621924C2312yes—
36C23×D173—22no—
37C2×D692C2×C412yescomplemented
38C2×C22⋊D173—22no—
39D4×D173—22no—
40D4×D173—22no—
41D4×D173C2×C414yescomplemented
42D4×D173C2314yes—
43C2×C4×D173C2311yescharacteristic
44D1384C2×C414yescomplemented
45#14583811C2×C418yescomplemented
46C2×#13063436C2×C411yescharacteristic, complemented
47C8×D173C2×C412yescomplemented
48C8×D173C2312yes—
49C2×D4×C173C2×C412yescomplemented
50D8×C173C2×C414yescomplemented
51C2×C8×C173C2×C411yescharacteristic, complemented
52C2×#10621924—24no—
53C2×#10621924—24no—
54C2×#10621924—416no—
55C2×#10621924—44no—
56C2×#10621924C2×D414yes—
57#13063443—22no—
58C22×D173—22no—
59C22×D173—24no—
60C22×D173—24no—
61C22×D173—44no—
62C22×D173—44no—
63C22×D173C2×D411yescharacteristic
64D692—22no—
65D692C22×C414yescomplemented
66C22⋊D173—24no—
67C22⋊D173—24no—
68C2×Dic173#4111yescharacteristic
69C4×D173C22×C411yescharacteristic
70C4×D173C2×D411yescharacteristic
71C4×D173#4111yescharacteristic
72C4×D173C2411yescharacteristic
73#13063436C22×C412yescomplemented
74C23×C173—22no—
75D4×C173—22no—
76D4×C173C22×C414yescomplemented
77C2×C4×C173C22×C411yescharacteristic
78C8×C173C22×C412yescomplemented
79C22×C173—24no—
80C22×C173—44no—
81C22×C173C4×D411yescharacteristic
82C2×D173—24no—
83C2×D173—44no—
84C2×D173—44no—
85C2×D173—44no—
86C2×D173—48no—
87C2×D173C4×D411yescharacteristic
88C2×D173AGL(1,ℤ/8)14yes—
89C2×D173C22×D411yescharacteristic
90#10621924—24no—
91#10621924—44no—
92C4×C173C4×D411yescharacteristic
93C4×C173C23×C411yescharacteristic
94Dic173C4×D411yescharacteristic
95Dic173C2×#4111yescharacteristic
96C2×C173—44no—
97C2×C173#44212yescomplemented
98C2×C173C2×C4×D411yescharacteristic
99D173—44no—
100D173#44212yescomplemented
101D173C2×AGL(1,ℤ/8)12yes—
102C173C2×#44211yescharacteristic, complemented, Sylow
103C2×#442—1731nomaximal, Sylow
104C22×D8—1731no—
105C2×C4×D4—1732no—
106#442—1738no—
107C2×#106—1731no—
108C2×#102—1732no—
109C2×#97—1731no—
110C22×D4—1732no—
111C23×C4—3462no—
112C2×D8—1732no—
113C2×D8—1734no—
114C2×D8—1734no—
115C2×D8—1731no—
116C2×D8#1062192411yescharacteristic, complemented
117C22×C8—1731no—
118C22×C8—1731no—
119C4×D4—3464no—
120C4×D4—1738no—
121C2×C4⋊C4—1732no—
122C2×C22⋊C4—3462no—
123C2×C42—1731no—
124#106—1734no—
125#102—1738no—
126#97—1734no—
127C24—3462no—
128C2×D4—3462no—
129C2×D4—3462no—
130C2×D4—3462no—
131C2×D4—1734no—
132C2×D4—1734no—
133C2×D4—1732no—
134C2×D4C2×#1062192412yescomplemented
135C22×C4—3462no—
136C22×C4—3468no—
137C22×C4—1732no—
138C22×C4—1731no—
139C22×C4—6928no—
140D8—1738no—
141D8—1734no—
142D8C2×#1062192414yescomplemented
143C2×C8—3461no—
144C2×C8—3461no—
145C2×C8—1732no—
146C2×C8—1731no—
147C2×C8—1732no—
148C2×C8—1732no—
149C2×C8C2×#1062192411yescharacteristic, complemented
150C4⋊C4—3462no—
151C4⋊C4—1734no—
152C22⋊C4—3468no—
153C42—1732no—
154C42—1732no—
155C23—22no—
156C23—3464no—
157C23—3464no—
158C23—3462no—
159C23—1731no—
160C23—6924no—
161C23—6924no—
162D4—22no—
163D4—3464no—
164D4—3464no—
165D4—3462no—
166D4—1734no—
167D4C22×#1062192414yescomplemented
168C2×C4—3464no—
169C2×C4—3464no—
170C2×C4—1734no—
171C2×C4—1731no—
172C2×C4—1731no—
173C2×C4—1731no—
174C2×C4—1731no—
175C2×C4—1731no—
176C2×C4—69216no—
177C2×C4—6924no—
178C2×C4C22×#1062192411yescharacteristic
179C8—3462no—
180C8—1732no—
181C8C22×#1062192412yescomplemented
182C22—24no—
183C22—44no—
184C22—3464no—
185C22—1734no—
186C22—1731no—
187C22—1731no—
188C22—6924no—
189C22—6924no—
190C22—6928no—
191C22—6924no—
192C22D4×#1062192411yescharacteristic
193C4—3464no—
194C4—1731no—
195C4—1731no—
196C4—6924no—
197C4D4×#1062192411yescharacteristic
198C4C23×#1062192411yescharacteristic
199C2—44no—
200C2—1732no—
201C2—1732no—
202C2—6924no—
203C2#2601253212yesdirect product factor
204C2C2×D4×#1062192411yescharacteristic
subgroupG/H
centerC22D4×#10621924
commutatorC4×C173C23×C4
FrattiniC4C23×#10621924
FittingC2×D8×C173C4
socleC22×C173C4×D4
prime radicalC2×D8×D173C2
cosocleC4×D173C24
hypercenterC2×D8#10621924
hypocenterC173C2×#442
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C2×D4×#10621924C2semidirect
2. #26012532C2direct (N × H)
3. C2×#19149747C2semidirect
4. C2×#19149741C2semidirect
5. C2×#19149733C2semidirect
10. D4×#10621924C22semidirect
14. #19149747C22semidirect
15. #19149741C22semidirect
16. #19149733C22semidirect
18. C2×D1384C4semidirect
19. C2×#14583811C4semidirect
20. D8×D173C4semidirect
23. C2×D8×C173C4semidirect
37. C2×D692C2×C4semidirect
41. D4×D173C2×C4semidirect
44. D1384C2×C4semidirect
45. #14583811C2×C4semidirect
46. C2×#13063436C2×C4semidirect
47. C8×D173C2×C4semidirect
49. C2×D4×C173C2×C4semidirect
50. D8×C173C2×C4semidirect
51. C2×C8×C173C2×C4semidirect
65. D692C22×C4semidirect
73. #13063436C22×C4semidirect
76. D4×C173C22×C4semidirect
78. C8×C173C22×C4semidirect
97. C2×C173#442semidirect
100. D173#442semidirect
102. C173C2×#442semidirect
116. C2×D8#10621924semidirect
134. C2×D4C2×#10621924semidirect
142. D8C2×#10621924semidirect
149. C2×C8C2×#10621924semidirect
167. D4C22×#10621924semidirect
181. C8C22×#10621924semidirect
203. C2#26012532direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×#106219242
#260125322
C2×#191497472
C2×#191497412
C2×#191497332
C2×D8×D1732
C2×#442173