Finite Group Databasebeta

← C2×#20261286 #27155020

Subgroups — C2×#20261286 #27155020

Order 12416.

1728 conjugacy classes of subgroups; only the 557 normal subgroups are shown.

normal subgroups only

This group has 1728 conjugacy classes of subgroups — too many to draw. Only its 557 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×#13095512C212yesmaximal, complemented
2#20261286C2116yesmaximal, direct product factor
3C2×#15690807C211yesmaximal, characteristic, complemented
4C2×#15690799C212yesmaximal, complemented
5C2×#15690785C214yesmaximal, complemented
6C2×#15690772C214yesmaximal, complemented
7C2×C42×D97C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C97C211yesmaximal, characteristic, complemented
9C2×#13095512—216no—
10C2×#13095512C22116yescomplemented
11C22×D388C2214yes—
12C22×D388C2214yes—
13C22×D388C2212yes—
14C22×C4×D97C2212yes—
15C22×C4×D97C2214yes—
16C22×C4×D97C2211yescharacteristic
17#15690807C2218yescomplemented
18#15690799C22116yescomplemented
19#15690785C22132yescomplemented
20#15690772C22132yescomplemented
21C2×#13095488C2218yes—
22C2×C4×Dic97C2212yes—
23C2×C4×Dic97C2211yescharacteristic
24C42×D97C2218yescomplemented
25C22×Q8×C97C2212yes—
26C4⋊Q8×C97C2218yescomplemented
27C2×C4⋊C4×C97C2214yes—
28C2×C42×C97C2211yescharacteristic
29C23×D97C2314yes—
30C23×D97C2311yescharacteristic
31#13095512—264no—
32C2×D388—216no—
33C2×D388—28no—
34C2×D388—216no—
35C2×D388C23116yes—
36C2×D388C23116yes—
37C2×D388C2318yes—
38C22×Dic97C2312yes—
39C22×Dic97C2311yescharacteristic
40C2×C4×D97—28no—
41C2×C4×D97—216no—
42C2×C4×D97D418yescomplemented
43C2×C4×D97C2318yes—
44C2×C4×D97C23116yes—
45C2×C4×D97C2314yes—
46#13095488C23132yes—
47C4×Dic97C2318yes—
48C4×Dic97C2314yes—
49C2×Q8×C97—28no—
50C2×Q8×C97C2318yes—
51C22×C4×C97C2314yes—
52C22×C4×C97C2311yescharacteristic
53C22×C4×C97C2312yes—
54C4⋊C4×C97C23116yes—
55C42×C97C2314yes—
56C23×C97C2411yescharacteristic
57C22×D97—24no—
58C22×D97—28no—
59C22×D97—216no—
60C22×D97C2×D414yes—
61C22×D97C2418yes—
62C22×D97C2412yes—
63Q8×C97—216no—
64C2×C4×C97—28no—
65C2×C4×C97C2×D414yes—
66C2×C4×C97#4118yescomplemented
67C2×C4×C97C2418yes—
68C2×C4×C97C2412yes—
69C2×C4×C97C2414yes—
70C2×Dic97—24no—
71C2×Dic97C2×D414yes—
72C2×Dic97C2414yes—
73C2×Dic97C2412yes—
74D388—232no—
75D388—216no—
76D388—232no—
77C4×D97—216no—
78C4×D97—232no—
79C4×D97C2×D4116yescomplemented
80C22×C97C22×D412yes—
81C22×C97C2×#4114yes—
82C22×C97C2511yescharacteristic
83C2×D97—28no—
84C2×D97—24no—
85C2×D97—28no—
86C2×D97—216no—
87C2×D97—416no—
88C2×D97C22×D414yes—
89C4×C97—28no—
90C4×C97C22×D414yes—
91C4×C97C2×#4118yescomplemented
92Dic97—24no—
93Dic97C22×D414yes—
94C2×C97#53214yescomplemented
95C2×C97C23×D411yescharacteristic
96C2×C97C22×#4112yes—
97D97—24no—
98D97—48no—
99C2×#532—971nomaximal, Sylow
100C97C2×#53211yescharacteristic, complemented, Sylow
101C22×#41—972no—
102#532—9716no—
103C2×C4⋊Q8D9711yescharacteristic, complemented
104C2×#127—971no—
105C2×#124—972no—
106C2×#121—974no—
107C2×C4×D4—974no—
108C22×C42—971no—
109C2×#41—9716no—
110C2×#41—19416no—
111C22×Q8C2×D9712yes—
112C22×D4—974no—
113C22×D4—974no—
114C22×D4—972no—
115C23×C4—971no—
116C23×C4—974no—
117C23×C4—972no—
118C4⋊Q8C2×D9718yescomplemented
119#127—978no—
120#124—9716no—
121#121—9732no—
122C4×D4—9732no—
123C2×C4⋊C4C2×D9714yes—
124C2×C22⋊C4—978no—
125C2×C42—972no—
126C2×C42—971no—
127C2×C42—978no—
128C2×C42C2×D9711yescharacteristic
129C24—971no—
130C24—974no—
131#41—19464no—
132C2×Q8—28no—
133C2×Q8C22×D9718yes—
134C2×D4—9716no—
135C2×D4—9716no—
136C2×D4—978no—
137C2×D4—19416no—
138C2×D4—1948no—
139C2×D4—19416no—
140C22×C4—971no—
141C22×C4—978no—
142C22×C4—974no—
143C22×C4—972no—
144C22×C4—978no—
145C22×C4—9716no—
146C22×C4—19416no—
147C22×C4—1948no—
148C22×C4C22×D9714yes—
149C22×C4C22×D9711yescharacteristic
150C22×C4C22×D9712yes—
151C4⋊C4C22×D97116yes—
152C22⋊C4—9732no—
153C42—978no—
154C42—974no—
155C42C22×D9714yes—
156C23—974no—
157C23—972no—
158C23—978no—
159C23—1948no—
160C23—19416no—
161C23—1944no—
162C23C23×D9711yescharacteristic
163Q8—216no—
164D4—19432no—
165D4—19416no—
166D4—19432no—
167C2×C4—28no—
168C2×C4—974no—
169C2×C4—972no—
170C2×C4—9716no—
171C2×C4—974no—
172C2×C4—19432no—
173C2×C4—1944no—
174C2×C4—19416no—
175C2×C4D4×D9714yes—
176C2×C4#1309551218yescomplemented
177C2×C4C23×D9718yes—
178C2×C4C23×D9712yes—
179C2×C4C23×D9714yes—
180C22—974no—
181C22—38816no—
182C22—1948no—
183C22—19416no—
184C22—1948no—
185C22—1944no—
186C22C2×D4×D9712yes—
187C22C2×#1309551214yes—
188C22C24×D9711yescharacteristic
189C4—28no—
190C4—974no—
191C4—1944no—
192C4C2×D4×D9714yes—
193C4C2×#1309551218yescomplemented
194C2—3888no—
195C2—1944no—
196C2#2026128614yesdirect product factor
197C2C22×D4×D9711yescharacteristic
198C2C22×#1309551212yes—
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×C4⋊Q8×C97C2
socleC23×C97C24
hypercenterC2×C4⋊Q8D97
hypocenterC97C2×#532
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13095512C2semidirect
2. #20261286C2direct (N × H)
3. C2×#15690807C2semidirect
4. C2×#15690799C2semidirect
5. C2×#15690785C2semidirect
6. C2×#15690772C2semidirect
7. C2×C42×D97C2semidirect
8. C2×C4⋊Q8×C97C2semidirect
10. C2×#13095512C22semidirect
17. #15690807C22semidirect
18. #15690799C22semidirect
19. #15690785C22semidirect
20. #15690772C22semidirect
24. C42×D97C22semidirect
26. C4⋊Q8×C97C22semidirect
42. C2×C4×D97D4semidirect
66. C2×C4×C97#41semidirect
79. C4×D97C2×D4semidirect
91. C4×C97C2×#41semidirect
94. C2×C97#532semidirect
100. C97C2×#532semidirect
103. C2×C4⋊Q8D97semidirect
118. C4⋊Q8C2×D97semidirect
176. C2×C4#13095512semidirect
193. C4C2×#13095512semidirect
196. C2#20261286direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130955122
#202612862
C2×#156908072
C2×#156907992
C2×#156907852
C2×#156907722
C2×C42×D972
C2×C4⋊Q8×C972
C2×#53297