Finite Group Databasebeta

← #27140823

Subgroups — #27140823

Order 12416.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261389C211yesmaximal, characteristic, complemented
2#20261081C211yesmaximal, characteristic, complemented
3#20260927C211yesmaximal, characteristic, complemented
4#20260482C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7#459×C97C211yesmaximal, characteristic, complemented
8C2×#13095510—21no—
9C2×#13095496—21no—
10#15690810C2211yescharacteristic
11#15690794—22no—
12#15690790—21no—
13#15690784—21no—
14C4×D388C2211yescharacteristic
15C2×C4⋊Dic97—21no—
16C4⋊#11715895C2211yescharacteristic
17#15690731—21no—
18#15690731—22no—
19C4×#11715895C2211yescharacteristic
20C2×D8×C97—21no—
21#127×C97C2211yescharacteristic
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24#102×C97—21no—
25#13095510—42no—
26#13095510—42no—
27C2×Dic194—21no—
28C2×D388D411yescharacteristic
29C2×C22⋊D97—22no—
30C22×Dic97—22no—
31C2×C4×D97—21no—
32#13095496—42no—
33C2×#11715895—21no—
34C2×#11715895—21no—
35C4⋊Dic97—22no—
36C4⋊Dic97—21no—
37C4⋊Dic97D411yescharacteristic
38C4⋊Dic97D412yes—
39#13095488—22no—
40#13095488—21no—
41C2×D4×C97—21no—
42C2×D4×C97—21no—
43C2×D4×C97—21no—
44C2×D4×C97D411yescharacteristic
45C22×C4×C97—21no—
46D8×C97—42no—
47C2×C8×C97—21no—
48C4⋊C4×C97D411yescharacteristic
49C22⋊C4×C97—21no—
50C42×C97C2311yescharacteristic
51C23×C97—21no—
52C23×C97—41no—
53C22×D97—21no—
54D4×C97—21no—
55D4×C97—42no—
56D4×C97—41no—
57D4×C97—41no—
58D4×C97—41no—
59D4×C97—41no—
60D4×C97D812yescomplemented
61C2×C4×C97—21no—
62C2×C4×C97—41no—
63C2×C4×C97—41no—
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C22⋊D97—44no—
68C2×Dic97—21no—
69C2×Dic97—22no—
70C2×Dic97—44no—
71D388—21no—
72D388—21no—
73C4×D97—41no—
74C4×D97—41no—
75Dic194—41no—
76Dic194—41no—
77C8×C97—41no—
78#11715895—41no—
79#11715895—41no—
80C22×C97—22no—
81C22×C97—41no—
82C22×C97—41no—
83C22×C97—81no—
84C22×C97—81no—
85C22×C97—81no—
86C22×C97#12011yescharacteristic
87C2×D97—41no—
88C2×D97—41no—
89C2×D97—41no—
90C4×C97—21no—
91C4×C97—41no—
92C4×C97C2×D811yescharacteristic
93C4×C97#13511yescharacteristic
94C4×C97AGL(1,ℤ/8)11yescharacteristic
95C4×C97AGL(1,ℤ/8)11yescharacteristic
96Dic97—42no—
97Dic97—41no—
98C2×C97—42no—
99C2×C97—161no—
100C2×C97#44711yescharacteristic
101C2×C97#44911yescharacteristic
102C2×C97#45411yescharacteristic
103D97—81no—
104#1625—971nomaximal, Sylow
105C97#162511yescharacteristic, complemented, Sylow
106#547—971no—
107#486—971no—
108#459—971no—
109#459D9711yescharacteristic, complemented
110#332—971no—
111D4⋊C8—971no—
112D4⋊C8—971no—
113C2×#41—1941no—
114C2×D8—21no—
115C2×D8—1941no—
116C4⋊Q8—971no—
117#127C2×D9711yescharacteristic
118#123—1942no—
119C22⋊Q8—1941no—
120#121—1941no—
121C4×D4—971no—
122C4×D4C2×D9711yescharacteristic
123C2×C4⋊C4—1941no—
124C4⋊C8—971no—
125C4⋊C8C2×D9711yescharacteristic
126#102—21no—
127#102—1941no—
128#102—1942no—
129C4×C8—971no—
130#41—3882no—
131#41—3882no—
132C2×Q8—1941no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—971no—
137C2×D4—1942no—
138C2×D4C22⋊D9711yescharacteristic
139C22×C4—21no—
140C22×C4—1942no—
141C22×C4—1941no—
142D8—42no—
143D8—3882no—
144C2×C8—21no—
145C2×C8—1941no—
146C2×C8—1941no—
147C4⋊C4—971no—
148C4⋊C4—972no—
149C4⋊C4—1942no—
150C4⋊C4—1941no—
151C4⋊C4C22⋊D9711yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—1942no—
154C22⋊C4—1941no—
155C42C22×D9711yescharacteristic
156C23—21no—
157C23—41no—
158C23—1941no—
159Q8—3881no—
160Q8—3881no—
161D4—21no—
162D4—42no—
163D4—41no—
164D4—41no—
165D4—41no—
166D4—41no—
167D4—3884no—
168D4—1941no—
169D4—1941no—
170D4#1309549612yescomplemented
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—3884no—
175C2×C4—3881no—
176C2×C4—3881no—
177C2×C4—1941no—
178C2×C4—1942no—
179C2×C4D4×D9711yescharacteristic
180C2×C4D4×D9711yescharacteristic
181C2×C4C2×C22⋊D9711yescharacteristic
182C8—41no—
183C8—3881no—
184C8—3881no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—3881no—
192C22—3881no—
193C22—3881no—
194C22#1569077811yescharacteristic
195C4—21no—
196C4—41no—
197C4—3881no—
198C4—3882no—
199C4C2×#1309549611yescharacteristic
200C4#1569082911yescharacteristic
201C4#1569083411yescharacteristic
202C4#1569083511yescharacteristic
203C2—42no—
204C2—7761no—
205C2—161no—
206C2#2026086811yescharacteristic
207C2#2026087711yescharacteristic
208C2#2026090611yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#459×C97C2
socleC22×C97#120
hypercenter#459D97
hypocenterC97#1625
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #20261389C2semidirect
2. #20261081C2semidirect
3. #20260927C2semidirect
4. #20260482C2semidirect
5. D4⋊#11715895C2semidirect
6. #20260462C2semidirect
7. #459×C97C2semidirect
60. D4×C97D8semidirect
105. C97#1625semidirect
109. #459D97semidirect
170. D4#13095496semidirect

Maximal subgroups

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

subgroupindex
#202613892
#202610812
#202609272
#202604822
D4⋊#117158952
#202604622
#459×C972
#162597