Finite Group Databasebeta

← #27140151

Subgroups — #27140151

Order 12416.

262 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 67 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#20260787C211yesmaximal, characteristic, complemented
3C4×#13095494C211yesmaximal, characteristic, complemented
4#20260464C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×#13095511—21no—
9C2×#13095494—21no—
10#15690809C411yescharacteristic
11#15690806C411yescharacteristic
12#15690796—21no—
13#15690785C411yescharacteristic
14#15690783—21no—
15#15690781C411yescharacteristic
16C4×D388C2211yescharacteristic
17C4×C22⋊D97—21no—
18C42×D97C2211yescharacteristic
19C4⋊#11715895C2211yescharacteristic
20#15690722—21no—
21#15690720—21no—
22C4×#11715895C2211yescharacteristic
23C8×Dic97—21no—
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
39#13095494—24no—
40C2×#11715895—21no—
41C2×#11715895—21no—
42C4⋊Dic97C2×C411yescharacteristic
43#13095490—21no—
44C22⋊Dic97—21no—
45#13095488—21no—
46#13095488—21no—
47C4×Dic97C2×C411yescharacteristic
48C4×Dic97D411yescharacteristic
49C4×Dic97D411yescharacteristic
50C2×D4×C97C2×C411yescharacteristic
51C22×C4×C97—21no—
52C2×C8×C97—21no—
53C2×C8×C97—21no—
54C4⋊C4×C97C2×C411yescharacteristic
55C22⋊C4×C97—21no—
56C42×C97C2311yescharacteristic
57C23×C97—21no—
58C22×D97—21no—
59C22×D97C22⋊C411yescharacteristic
60D4×C97—21no—
61D4×C97—21no—
62C2×C4×C97—21no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97C22×C411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68C22⋊D97—42no—
69C22⋊D97—42no—
70C22⋊D97—42no—
71C2×Dic97—21no—
72C2×Dic97C22⋊C411yescharacteristic
73C2×Dic97C22⋊C411yescharacteristic
74C2×Dic97C22⋊C411yescharacteristic
75D388—21no—
76D388—21no—
77D388—42no—
78C4×D97—21no—
79C4×D97—21no—
80C4×D97—21no—
81C4×D97—21no—
82C4×D97—22no—
83C4×D97—41no—
84C4×D97—41no—
85Dic194—42no—
86C8×C97—22no—
87C8×C97—41no—
88#11715895—22no—
89#11715895—41no—
90C22×C97—41no—
91C22×C97—41no—
92C22×C97—41no—
93C22×C97C2×C22⋊C411yescharacteristic
94C2×D97—21no—
95C2×D97—21no—
96C2×D97—21no—
97C2×D97—41no—
98C2×D97—41no—
99C2×D97—41no—
100C4×C97—21no—
101C4×C97—41no—
102C4×C97#13111yescharacteristic
103C4×C97#13111yescharacteristic
104C4×C97AGL(1,ℤ/8)11yescharacteristic
105C4×C97AGL(1,ℤ/8)11yescharacteristic
106Dic97—21no—
107Dic97—21no—
108Dic97—41no—
109Dic97#10412yes—
110C2×C97—81no—
111C2×C97#40811yescharacteristic
112C2×C97#41811yescharacteristic
113C2×C97C2×#10411yescharacteristic
114D97—41no—
115D97—81no—
116#1484—971nomaximal, Sylow
117C97#148411yescharacteristic, complemented, Sylow
118#532—971no—
119#431—971no—
120C4×M16—971no—
121D4⋊C8—971no—
122D4⋊C8—971no—
123D4⋊C8—971no—
124D4⋊C8D9711yescharacteristic, complemented
125C2×#41—1941no—
126C2×M16—1941no—
127C4⋊Q8—971no—
128#127—971no—
129#124—1941no—
130#121—971no—
131#121—971no—
132#121—1941no—
133C4×D4—971no—
134C4×D4—1941no—
135C4×D4C2×D9711yescharacteristic
136C2×C42—971no—
137C4⋊C8—971no—
138C4⋊C8C2×D9711yescharacteristic
139C22⋊C8—1941no—
140#97—1941no—
141C4×C8—971no—
142C4×C8—1941no—
143C4×C8C2×D9711yescharacteristic
144#41—3882no—
145#41—3882no—
146C2×Q8—1941no—
147C2×D4—971no—
148C2×D4—1941no—
149C2×D4—1941no—
150C2×D4—1941no—
151C2×D4—1941no—
152C2×D4C4×D9711yescharacteristic
153C22×C4—21no—
154C22×C4—971no—
155C22×C4—971no—
156C22×C4—971no—
157C22×C4—1941no—
158M16—1944no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—1941no—
162C2×C8—1941no—
163C4⋊C4—971no—
164C4⋊C4—1941no—
165C4⋊C4C4×D9711yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—1941no—
168C22⋊C4—1941no—
169C22⋊C4—1941no—
170C42—971no—
171C42—971no—
172C42—971no—
173C42C22×D9711yescharacteristic
174C23—21no—
175C23—971no—
176C23—1941no—
177Q8—3882no—
178D4—21no—
179D4—21no—
180D4—3882no—
181D4—3882no—
182D4—3882no—
183D4—3882no—
184D4—1941no—
185D4—1941no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—971no—
190C2×C4—971no—
191C2×C4—971no—
192C2×C4—3881no—
193C2×C4—3881no—
194C2×C4—1942no—
195C2×C4—1941no—
196C2×C4—1941no—
197C2×C4—1941no—
198C2×C4—1941no—
199C2×C4—1941no—
200C2×C4C2×C4×D9711yescharacteristic
201C2×C4D4×D9711yescharacteristic
202C2×C4D4×D9711yescharacteristic
203C8—22no—
204C8—41no—
205C8—3881no—
206C8—1942no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—3881no—
211C22—3881no—
212C22—3881no—
213C22—1941no—
214C22—1941no—
215C22—1941no—
216C22C22⋊C4×D9711yescharacteristic
217C4—21no—
218C4—41no—
219C4—972no—
220C4—3881no—
221C4—1941no—
222C4—1941no—
223C4#1569081611yescharacteristic
224C4#1569081811yescharacteristic
225C4#1569083411yescharacteristic
226C4#1569083711yescharacteristic
227C2—81no—
228C2—7761no—
229C2—3881no—
230C2#2026070011yescharacteristic
231C2#2026073811yescharacteristic
232C2#104×D9711yescharacteristic
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#1484
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. C4×#13095494C2semidirect
4. #20260464C2semidirect
5. D4⋊#11715895C2semidirect
6. #20260462C2semidirect
7. D4⋊C8×C97C2semidirect
117. C97#1484semidirect
124. D4⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202612822
#202607872
C4×#130954942
#202604642
D4⋊#117158952
#202604622
D4⋊C8×C972
#148497