Finite Group Databasebeta

← #27140141

Subgroups — #27140141

Order 12416.

212 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 65 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#20261267C211yesmaximal, characteristic
2#20260787C211yesmaximal, characteristic, complemented
3#20260786C211yesmaximal, characteristic, complemented
4#20260467C211yesmaximal, characteristic, complemented
5#20260465C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic, complemented
8#15690803—21no—
9#15690801C411yescharacteristic
10#15690797C411yescharacteristic
11#15690792—21no—
12#15690790C411yescharacteristic
13#15690781C411yescharacteristic
14C4×Dic194C2211yescharacteristic
15C4×C22⋊D97—21no—
16#15690769—21no—
17C42×D97C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19#15690739—21no—
20#15690722—21no—
21#15690722—21no—
22C4×#11715895C2211yescharacteristic
23C8×Dic97—21no—
24C4×D4×C97C2211yescharacteristic
25C4⋊C8×C97C2211yescharacteristic
26C4×C8×C97C2211yescharacteristic
27C2×Dic194C2×C411yescharacteristic
28C2×C22⋊D97—21no—
29C2×C4×D97C2×C411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C2×C4×D97D411yescharacteristic
32C2×#11715895—21no—
33C2×#11715895—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97C2×C411yescharacteristic
37#13095490—21no—
38#13095490—21no—
39C22⋊Dic97—21no—
40C22⋊Dic97—21no—
41C22⋊Dic97—21no—
42#13095488—21no—
43C4×Dic97—21no—
44C4×Dic97C2×C411yescharacteristic
45C4×Dic97D411yescharacteristic
46C4×Dic97D411yescharacteristic
47C2×D4×C97C2×C411yescharacteristic
48C22×C4×C97—21no—
49C2×C8×C97—21no—
50C2×C8×C97—21no—
51C4⋊C4×C97C2×C411yescharacteristic
52C22⋊C4×C97—21no—
53C42×C97C2311yescharacteristic
54C23×C97—21no—
55C22×D97C22⋊C411yescharacteristic
56D4×C97—21no—
57D4×C97—21no—
58C2×C4×C97—21no—
59C2×C4×C97—41no—
60C2×C4×C97—41no—
61C2×C4×C97C22×C411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97C2×D411yescharacteristic
64C22⋊D97—42no—
65C2×Dic97—21no—
66C2×Dic97—21no—
67C2×Dic97C22⋊C411yescharacteristic
68C2×Dic97C22⋊C411yescharacteristic
69C2×Dic97C22⋊C411yescharacteristic
70C4×D97—21no—
71C4×D97—21no—
72C4×D97—21no—
73C4×D97—21no—
74C4×D97—41no—
75Dic194—21no—
76Dic194—21no—
77C8×C97—41no—
78C8×C97—41no—
79#11715895—41no—
80#11715895—41no—
81C22×C97—41no—
82C22×C97—41no—
83C22×C97—41no—
84C22×C97C2×C22⋊C411yescharacteristic
85C2×D97—21no—
86C2×D97—21no—
87C2×D97—21no—
88C4×C97—21no—
89C4×C97—41no—
90C4×C97#13111yescharacteristic
91C4×C97#13111yescharacteristic
92C4×C97#13511yescharacteristic
93C4×C97#13511yescharacteristic
94Dic97—21no—
95Dic97—21no—
96Dic97—21no—
97Dic97—41no—
98Dic97—41no—
99C2×C97—81no—
100C2×C97#40811yescharacteristic
101C2×C97#41611yescharacteristic
102C2×C97#42111yescharacteristic
103D97—41no—
104#1482—971nomaximal, Sylow
105C97#148211yescharacteristic, complemented, Sylow
106#529—971no—
107#431—971no—
108#431—971no—
109Q8⋊C8—971no—
110Q8⋊C8—971no—
111D4⋊C8—971no—
112D4⋊C8D9711yescharacteristic, complemented
113#126—1941no—
114#125—971no—
115#124—971no—
116#123—1941no—
117C22⋊Q8—971no—
118#121—971no—
119C4×Q8—971no—
120C4×D4—1941no—
121C4×D4C2×D9711yescharacteristic
122#117—1941no—
123C2×C42—971no—
124C4⋊C8—971no—
125C4⋊C8—1941no—
126C4⋊C8C2×D9711yescharacteristic
127C22⋊C8—1941no—
128C22⋊C8—1941no—
129C4×C8—971no—
130C4×C8—1941no—
131C4×C8C2×D9711yescharacteristic
132C2×Q8—971no—
133C2×D4—1941no—
134C2×D4C4×D9711yescharacteristic
135C22×C4—21no—
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—971no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—1941no—
142C2×C8—1941no—
143C4⋊C4—971no—
144C4⋊C4—1941no—
145C4⋊C4—1941no—
146C4⋊C4—1941no—
147C4⋊C4—1941no—
148C4⋊C4C4×D9711yescharacteristic
149C22⋊C4—21no—
150C22⋊C4—1941no—
151C22⋊C4—1941no—
152C22⋊C4—1941no—
153C22⋊C4—1941no—
154C42—971no—
155C42—971no—
156C42—971no—
157C42—1941no—
158C42C22×D9711yescharacteristic
159C23—21no—
160C23—971no—
161Q8—1941no—
162Q8—1941no—
163D4—21no—
164D4—21no—
165D4—3882no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—971no—
170C2×C4—971no—
171C2×C4—971no—
172C2×C4—3881no—
173C2×C4—1941no—
174C2×C4—1941no—
175C2×C4—1941no—
176C2×C4—1941no—
177C2×C4—1941no—
178C2×C4—1941no—
179C2×C4C2×C4×D9711yescharacteristic
180C2×C4D4×D9711yescharacteristic
181C2×C4D4×D9711yescharacteristic
182C8—41no—
183C8—41no—
184C8—3881no—
185C8—3881no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—1941no—
190C22—1941no—
191C22—1941no—
192C22C22⋊C4×D9711yescharacteristic
193C4—21no—
194C4—41no—
195C4—3881no—
196C4—3881no—
197C4—1941no—
198C4—1941no—
199C4—1941no—
200C4#1569081611yescharacteristic
201C4#1569081811yescharacteristic
202C4#1569082911yescharacteristic
203C4#1569083111yescharacteristic
204C2—81no—
205C2—3881no—
206C2#2026070011yescharacteristic
207C2#2026072811yescharacteristic
208C2#2026075011yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97C2×C22⋊C4
prime radical#15690781C4
cosocleC42×C97C23
hypercenterD4⋊C8D97
hypocenterC97#1482
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20260787C2semidirect
3. #20260786C2semidirect
4. #20260467C2semidirect
5. #20260465C2semidirect
6. D4⋊#11715895C2semidirect
7. D4⋊C8×C97C2semidirect
105. C97#1482semidirect
112. D4⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202612672
#202607872
#202607862
#202604672
#202604652
D4⋊#117158952
D4⋊C8×C972
#148297