Finite Group Databasebeta

← #27140203

Subgroups — #27140203

Order 12416.

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

normal subgroups only

This group has 246 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#20261282C211yesmaximal, characteristic
2#20260791C211yesmaximal, characteristic, complemented
3#20260786C211yesmaximal, characteristic, complemented
4#20260464C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×#13095511—21no—
9#15690809C411yescharacteristic
10#15690806C411yescharacteristic
11#15690796—21no—
12#15690785C411yescharacteristic
13#15690783—21no—
14#15690781C411yescharacteristic
15C4×D388C2211yescharacteristic
16C4×C22⋊D97—21no—
17C42×D97C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19#15690739—21no—
20#15690722—21no—
21#15690722—21no—
22#15690720—21no—
23C4×#11715895C2211yescharacteristic
24C4×D4×C97C2211yescharacteristic
25C4⋊C8×C97C2211yescharacteristic
26C4×C8×C97C2211yescharacteristic
27#13095511—42no—
28#13095511—42no—
29C2×Dic194—21no—
30C2×D388—21no—
31C2×D388C2×C411yescharacteristic
32C2×C22⋊D97—21no—
33C2×C22⋊D97—21no—
34C2×C22⋊D97—21no—
35C2×C4×D97—21no—
36C2×C4×D97C2×C411yescharacteristic
37C2×C4×D97D411yescharacteristic
38C2×C4×D97D411yescharacteristic
39C2×#11715895—21no—
40C2×#11715895—21no—
41C4⋊Dic97C2×C411yescharacteristic
42#13095490—21no—
43C22⋊Dic97—21no—
44#13095488—21no—
45#13095488—21no—
46C4×Dic97C2×C411yescharacteristic
47C4×Dic97D411yescharacteristic
48C4×Dic97D411yescharacteristic
49C2×D4×C97C2×C411yescharacteristic
50C22×C4×C97—21no—
51C2×C8×C97—21no—
52C2×C8×C97—21no—
53C4⋊C4×C97C2×C411yescharacteristic
54C22⋊C4×C97—21no—
55C42×C97C2311yescharacteristic
56C23×C97—21no—
57C22×D97—21no—
58C22×D97C22⋊C411yescharacteristic
59D4×C97—21no—
60D4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97—41no—
63C2×C4×C97—41no—
64C2×C4×C97C22×C411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C22⋊D97—42no—
68C22⋊D97—42no—
69C22⋊D97—42no—
70C2×Dic97—21no—
71C2×Dic97C22⋊C411yescharacteristic
72C2×Dic97C22⋊C411yescharacteristic
73C2×Dic97C22⋊C411yescharacteristic
74D388—21no—
75D388—21no—
76D388—42no—
77C4×D97—21no—
78C4×D97—21no—
79C4×D97—21no—
80C4×D97—21no—
81C4×D97—41no—
82C4×D97—41no—
83C4×D97—41no—
84Dic194—42no—
85C8×C97—41no—
86C8×C97—41no—
87#11715895—41no—
88#11715895—41no—
89C22×C97—41no—
90C22×C97—41no—
91C22×C97—41no—
92C22×C97C2×C22⋊C411yescharacteristic
93C2×D97—21no—
94C2×D97—21no—
95C2×D97—21no—
96C2×D97—41no—
97C2×D97—41no—
98C2×D97—41no—
99C4×C97—21no—
100C4×C97—41no—
101C4×C97#13111yescharacteristic
102C4×C97#13111yescharacteristic
103C4×C97AGL(1,ℤ/8)11yescharacteristic
104C4×C97AGL(1,ℤ/8)11yescharacteristic
105Dic97—21no—
106Dic97—21no—
107Dic97—21no—
108Dic97—41no—
109C2×C97—81no—
110C2×C97#40811yescharacteristic
111C2×C97#41811yescharacteristic
112C2×C97#42111yescharacteristic
113D97—41no—
114D97—81no—
115#1496—971nomaximal, Sylow
116C97#149611yescharacteristic, complemented, Sylow
117#532—971no—
118#432—971no—
119#431—971no—
120D4⋊C8—971no—
121D4⋊C8—971no—
122D4⋊C8—971no—
123D4⋊C8D9711yescharacteristic, complemented
124C2×#41—1941no—
125C4⋊Q8—971no—
126#127—971no—
127#124—1941no—
128#121—971no—
129#121—971no—
130#121—1941no—
131C4×D4—971no—
132C4×D4—1941no—
133C4×D4C2×D9711yescharacteristic
134C2×C42—971no—
135C4⋊C8—971no—
136C4⋊C8—1941no—
137C4⋊C8C2×D9711yescharacteristic
138C22⋊C8—1941no—
139C22⋊C8—1941no—
140#97—1941no—
141C4×C8—971no—
142C4×C8C2×D9711yescharacteristic
143#41—3882no—
144#41—3882no—
145C2×Q8—1941no—
146C2×D4—971no—
147C2×D4—1941no—
148C2×D4—1941no—
149C2×D4—1941no—
150C2×D4—1941no—
151C2×D4C4×D9711yescharacteristic
152C22×C4—21no—
153C22×C4—971no—
154C22×C4—971no—
155C22×C4—971no—
156C22×C4—1941no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—1941no—
160C2×C8—1941no—
161C4⋊C4—971no—
162C4⋊C4—1941no—
163C4⋊C4C4×D9711yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—1941no—
166C22⋊C4—1941no—
167C22⋊C4—1941no—
168C42—971no—
169C42—971no—
170C42—971no—
171C42C22×D9711yescharacteristic
172C23—21no—
173C23—971no—
174C23—1941no—
175Q8—3882no—
176D4—21no—
177D4—21no—
178D4—3882no—
179D4—3882no—
180D4—3882no—
181D4—3882no—
182D4—1941no—
183D4—1941no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—971no—
188C2×C4—971no—
189C2×C4—971no—
190C2×C4—3881no—
191C2×C4—3881no—
192C2×C4—3881no—
193C2×C4—1941no—
194C2×C4—1941no—
195C2×C4—1941no—
196C2×C4—1941no—
197C2×C4—1941no—
198C2×C4C2×C4×D9711yescharacteristic
199C2×C4D4×D9711yescharacteristic
200C2×C4D4×D9711yescharacteristic
201C8—41no—
202C8—41no—
203C8—3881no—
204C8—3881no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—3881no—
209C22—3881no—
210C22—3881no—
211C22—1941no—
212C22—1941no—
213C22—1941no—
214C22C22⋊C4×D9711yescharacteristic
215C4—21no—
216C4—41no—
217C4—3881no—
218C4—1941no—
219C4—1941no—
220C4—1941no—
221C4#1569081611yescharacteristic
222C4#1569081811yescharacteristic
223C4#1569083411yescharacteristic
224C4#1569083711yescharacteristic
225C2—81no—
226C2—7761no—
227C2—3881no—
228C2#2026070011yescharacteristic
229C2#2026073811yescharacteristic
230C2#2026075011yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261282C2
cosocleC42×C97C23
hypercenterD4⋊C8D97
hypocenterC97#1496
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. #20260791C2semidirect
3. #20260786C2semidirect
4. #20260464C2semidirect
5. D4⋊#11715895C2semidirect
6. #20260462C2semidirect
7. D4⋊C8×C97C2semidirect
116. C97#1496semidirect
123. D4⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202612822
#202607912
#202607862
#202604642
D4⋊#117158952
#202604622
D4⋊C8×C972
#149697