Finite Group Databasebeta

← #27157903

Subgroups — #27157903

Order 12416.

428 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 88 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
1#20261736C211yesmaximal, characteristic, complemented
2C2×#15690872C214yesmaximal, complemented
3C4×C4⋊#87604C211yesmaximal, characteristic, complemented
4#127×D97C211yesmaximal, characteristic
5C2×#13095519C2212yes—
6C2×C4⋊#87604C2212yes—
7C2×C4×#87604—21no—
8#15690877—21no—
9C4⋊#11715897—21no—
10#15690872—28no—
11(C2×#87604)⋊C4—21no—
12C2×D4×D97—21no—
13C2×D4×D97—21no—
14C2×D4×D97C2212yes—
15#15690807C411yescharacteristic, complemented
16#15690806—21no—
17#15690806—21no—
18#15690806C412yescomplemented
19C42×D97C2211yescharacteristic
20#127×C97C411yescharacteristic, complemented
21C22×#87604—22no—
22#13095519—24no—
23C2×#11715897—22no—
24C4⋊#87604—22no—
25C4⋊#87604D414yescomplemented
26C4×#87604—42no—
27C23×D97—22no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388C2×C412yes—
31C2×C22⋊D97—22no—
32C2×C22⋊D97—22no—
33C2×C22⋊D97—22no—
34D4×D97—24no—
35D4×D97—24no—
36D4×D97—44no—
37D4×D97—42no—
38D4×D97—42no—
39C2×C4×D97D411yescharacteristic
40C2×C4×D97D411yescharacteristic
41C2×C4×D97C2311yescharacteristic
42C4×Dic97C2×C411yescharacteristic
43C4×Dic97D411yescharacteristic
44C4×Dic97D411yescharacteristic
45C2×D4×C97—21no—
46C2×D4×C97—21no—
47C2×D4×C97C2×C412yes—
48C42×C97C2×C411yescharacteristic
49C23×C97—22no—
50C22×D97—22no—
51C22×D97—44no—
52C22×D97—42no—
53C22×D97—42no—
54C22×D97—44no—
55C22×D97C2×D411yescharacteristic
56C2×#87604—24no—
57C2×#87604—42no—
58C2×#87604—42no—
59D4×C97—24no—
60D4×C97—42no—
61D4×C97—42no—
62C2×C4×C97C22⋊C411yescharacteristic
63C2×C4×C97C22⋊C411yescharacteristic
64C2×C4×C97C22×C411yescharacteristic
65C22⋊D97—44no—
66C22⋊D97—44no—
67C22⋊D97—42no—
68C22⋊D97—42no—
69C2×Dic97C22⋊C411yescharacteristic
70C2×Dic97C22⋊C411yescharacteristic
71C2×Dic97C2×D411yescharacteristic
72D388—24no—
73D388—42no—
74D388—42no—
75C4×D97—22no—
76C4×D97D812yes—
77C4×D97SD1612yes—
78C4×D97C2×D412yes—
79C4×D97#4112yes—
80#11715897—42no—
81C22×C97—42no—
82C22×C97—42no—
83C22×C97—42no—
84C22×C97C2×C22⋊C411yescharacteristic
85C2×D97—42no—
86C2×D97—42no—
87C2×D97—42no—
88C2×D97—84no—
89C2×D97—84no—
90C2×D97#12112yes—
91C2×D97C2×D811yescharacteristic
92C2×D97C2×SD1611yescharacteristic
93C2×D97AGL(1,ℤ/8)12yes—
94#87604—44no—
95C4×C97—21no—
96C4×C97#10212yescomplemented
97C4×C97C4×D412yes—
98Dic97—21no—
99Dic97#10212yescomplemented
100Dic97#12411yescharacteristic
101Dic97#12711yescharacteristic
102C2×C97—82no—
103C2×C97#39011yescharacteristic
104C2×C97C2×#10211yescharacteristic
105C2×C97#41811yescharacteristic
106D97—82no—
107D97#45912yes—
108D97#46012yes—
109#1982—971nomaximal, Sylow
110C97#198211yescharacteristic, complemented, Sylow
111C2×#127—971no—
112C2×C4⋊C8—971no—
113C2×#102—974no—
114C4×C4⋊C4—971no—
115C22×D4—972no—
116C22×D4—1941no—
117C22×D4—1941no—
118C22×C8—972no—
119#127—972no—
120#127—971no—
121#127—1941no—
122#127—1941no—
123#127#8760411yescharacteristic, complemented
124C2×C4⋊C4—972no—
125C2×C42—971no—
126C2×C42—1941no—
127C4⋊C8—1941no—
128C4⋊C8—1941no—
129#102—1948no—
130#95—1941no—
131C24—1942no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—972no—
135C2×D4—3882no—
136C2×D4—3882no—
137C2×D4—3884no—
138C2×D4—1944no—
139C2×D4—1944no—
140C2×D4—1942no—
141C2×D4—1942no—
142C2×D4—1942no—
143C2×D4—1941no—
144C2×D4—1941no—
145C2×D4C2×#8760412yes—
146C22×C4—971no—
147C22×C4—971no—
148C22×C4—971no—
149C22×C4—1942no—
150C2×C8—1942no—
151C2×C8—1944no—
152C4⋊C4—974no—
153C4⋊C4—1942no—
154C42—971no—
155C42—971no—
156C42—971no—
157C42—3882no—
158C42C2×#8760411yescharacteristic
159C23—22no—
160C23—971no—
161C23—3882no—
162C23—3882no—
163C23—3884no—
164C23—3884no—
165C23—1942no—
166D4—24no—
167D4—42no—
168D4—42no—
169D4—3882no—
170D4—3884no—
171D4—3882no—
172D4—3882no—
173D4—3884no—
174D4—3882no—
175D4—1944no—
176C2×C4—971no—
177C2×C4—971no—
178C2×C4—972no—
179C2×C4—972no—
180C2×C4—972no—
181C2×C4—972no—
182C2×C4—971no—
183C2×C4—3882no—
184C2×C4—3882no—
185C2×C4—1942no—
186C2×C4—1944no—
187C2×C4C22⋊#8760411yescharacteristic
188C2×C4C22⋊#8760411yescharacteristic
189C2×C4C22×#8760411yescharacteristic
190C8—3882no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—7764no—
195C22—7764no—
196C22—972no—
197C22—971no—
198C22—971no—
199C22—972no—
200C22—3882no—
201C22—3882no—
202C22—3882no—
203C22C2×C22⋊#8760411yescharacteristic
204C4—21no—
205C4—971no—
206C4—971no—
207C4—972no—
208C4—3884no—
209C4—1941no—
210C4#1569087212yescomplemented
211C4D4×#8760412yes—
212C2—82no—
213C2—7762no—
214C2—972no—
215C2—972no—
216C2#2026168711yescharacteristic
217C2C2×#1569087211yescharacteristic
218C2#2026172711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#127×C97C4
socleC22×C97C2×C22⋊C4
prime radical#127×D97C2
cosocleC2×C4×D97C23
hypercenter#127#87604
hypocenterC97#1982
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #20261736C2semidirect
2. C2×#15690872C2semidirect
3. C4×C4⋊#87604C2semidirect
15. #15690807C4semidirect
18. #15690806C4semidirect
20. #127×C97C4semidirect
25. C4⋊#87604D4semidirect
96. C4×C97#102semidirect
99. Dic97#102semidirect
110. C97#1982semidirect
123. #127#87604semidirect
210. C4#15690872semidirect

Maximal subgroups

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

subgroupindex
#202617362
C2×#156908722
C4×C4⋊#876042
#127×D972
#198297