Finite Group Databasebeta

← #27142678

Subgroups — #27142678

Order 12416.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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#20261288C211yesmaximal, characteristic
2M16×Dic97C211yesmaximal, characteristic
3C4×#13095490C211yesmaximal, characteristic
4#20260496C211yesmaximal, characteristic
5#20260495C211yesmaximal, characteristic
6#20260494C211yesmaximal, characteristic
7#339×C97C211yesmaximal, characteristic
8C2×#13095495C2211yescharacteristic
9#15690809C411yescharacteristic, complemented
10#15690809C411yescharacteristic, complemented
11#15690801C411yescharacteristic, complemented
12#15690801C411yescharacteristic, complemented
13#15690787—21no—
14#15690787—21no—
15C4×Dic194—21no—
16C4×Dic194—21no—
17#15690769C2211yescharacteristic
18C2×#13095490—21no—
19C2×C4×Dic97C2211yescharacteristic
20C2×C4×Dic97C2211yescharacteristic
21#15690720—21no—
22C8×Dic97—21no—
23#15690717—21no—
24C2×M16×C97C2211yescharacteristic
25#117×C97C2211yescharacteristic
26C2×C42×C97C2211yescharacteristic
27C2×Dic194—21no—
28C2×Dic194—21no—
29C22×Dic97—21no—
30C22×Dic97D411yescharacteristic
31C22×Dic97D411yescharacteristic
32#13095495—21no—
33#13095495Q812yescomplemented
34C2×#11715895—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97—21no—
37C4⋊Dic97C2×C411yescharacteristic
38C4⋊Dic97C2×C411yescharacteristic
39#13095490—24no—
40#13095490—21no—
41#13095490—21no—
42#13095490—21no—
43#13095490—21no—
44C22⋊Dic97—21no—
45C4×Dic97—21no—
46C4×Dic97—21no—
47C4×Dic97—21no—
48C4×Dic97C2×C411yescharacteristic
49C4×Dic97C2×C411yescharacteristic
50C4×Dic97D411yescharacteristic
51C4×Dic97D411yescharacteristic
52C22×C4×C97—21no—
53C22×C4×C97C2311yescharacteristic
54M16×C97—21no—
55M16×C97Q812yescomplemented
56C2×C8×C97—21no—
57C4⋊C4×C97C2×C411yescharacteristic
58C4⋊C4×C97C2×C411yescharacteristic
59C22⋊C4×C97—21no—
60C42×C97—21no—
61C42×C97—21no—
62C42×C97—21no—
63C23×C97C2×D411yescharacteristic
64C2×C4×C97—22no—
65C2×C4×C97—21no—
66C2×C4×C97—21no—
67C2×C4×C97—41no—
68C2×C4×C97—41no—
69C2×C4×C97C22×C411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2×Q811yescharacteristic
72C2×C4×C97C2×Q811yescharacteristic
73C2×C4×C97#4111yescharacteristic
74C2×C4×C97#4111yescharacteristic
75C2×Dic97—22no—
76C2×Dic97—22no—
77C2×Dic97—21no—
78C2×Dic97—21no—
79C2×Dic97—41no—
80C2×Dic97—41no—
81C2×Dic97—41no—
82C2×Dic97C22⋊C411yescharacteristic
83C2×Dic97C22⋊C411yescharacteristic
84C2×Dic97C22⋊C411yescharacteristic
85C2×Dic97C22⋊C411yescharacteristic
86Dic194—42no—
87Dic194—42no—
88C8×C97—22no—
89#11715895—22no—
90C22×C97—21no—
91C22×C97—21no—
92C22×C97C2×C22⋊C411yescharacteristic
93C22×C97#12411yescharacteristic
94C22×C97C4⋊Q811yescharacteristic
95C4×C97—42no—
96C4×C97—41no—
97C4×C97—41no—
98C4×C97C4×Q811yescharacteristic
99C4×C97C4×Q811yescharacteristic
100C4×C97C22⋊Q811yescharacteristic
101C4×C97C22⋊Q811yescharacteristic
102Dic97—21no—
103Dic97—21no—
104Dic97—21no—
105Dic97—42no—
106Dic97—41no—
107Dic97—41no—
108Dic97#10412yes—
109C2×C97—21no—
110C2×C97—21no—
111C2×C97#39111yescharacteristic
112C2×C97C2×#10411yescharacteristic
113C2×C97#42111yescharacteristic
114#1986—971nomaximal, Sylow
115C97#198611yescharacteristic, complemented, Sylow
116#533—971no—
117C4×M16—971no—
118C4×C4⋊C4—971no—
119#339—971no—
120#339—971no—
121#339—971no—
122#339D9711yescharacteristic
123C2×M16—971no—
124C2×M16C2×D9711yescharacteristic
125C4⋊Q8—971no—
126C4⋊Q8—971no—
127#125—971no—
128#125—971no—
129C22⋊Q8—1941no—
130C22⋊Q8—1941no—
131C4×Q8—1941no—
132C4×Q8—1941no—
133#117—971no—
134#117C2×D9711yescharacteristic
135C2×C4⋊C4—1941no—
136C2×C42—971no—
137C2×C42—971no—
138C2×C42C2×D9711yescharacteristic
139#97—1941no—
140C4×C8—1941no—
141#95—1941no—
142C2×Q8—1941no—
143C2×Q8—1941no—
144C22×C4—21no—
145C22×C4—971no—
146C22×C4—971no—
147C22×C4—1941no—
148C22×C4C22×D9711yescharacteristic
149M16—21no—
150M16—972no—
151M16—1941no—
152M16Dic19412yescomplemented
153C2×C8—21no—
154C2×C8—1941no—
155C4⋊C4—971no—
156C4⋊C4—971no—
157C4⋊C4—1941no—
158C4⋊C4—1941no—
159C4⋊C4—1941no—
160C4⋊C4—1941no—
161C4⋊C4—1941no—
162C4⋊C4—1941no—
163C4⋊C4—1944no—
164C4⋊C4C4×D9711yescharacteristic
165C4⋊C4C4×D9711yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—1941no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—971no—
172C42—971no—
173C42—971no—
174C42—971no—
175C42—1941no—
176C42—1941no—
177C42—1941no—
178C23D4×D9711yescharacteristic
179Q8—3882no—
180Q8—3882no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—971no—
187C2×C4—971no—
188C2×C4—971no—
189C2×C4—971no—
190C2×C4—3881no—
191C2×C4—3881no—
192C2×C4—3881no—
193C2×C4—1942no—
194C2×C4—1942no—
195C2×C4—1941no—
196C2×C4—1941no—
197C2×C4C2×C4×D9711yescharacteristic
198C2×C4D4×D9711yescharacteristic
199C2×C4Q8×D9711yescharacteristic
200C2×C4C2×Dic19411yescharacteristic
201C2×C4#1309551011yescharacteristic
202C2×C4#1309551111yescharacteristic
203C8—22no—
204C8—1942no—
205C22—21no—
206C22—21no—
207C22C22⋊C4×D9711yescharacteristic
208C22#1569079611yescharacteristic
209C22#1569080911yescharacteristic
210C4—42no—
211C4—41no—
212C4—41no—
213C4—972no—
214C4—3882no—
215C4—3881no—
216C4—3881no—
217C4—1941no—
218C4—1941no—
219C4—1941no—
220C4#1569077511yescharacteristic
221C4C4×Dic19411yescharacteristic
222C4#1569078711yescharacteristic
223C4#1569078811yescharacteristic
224C2—21no—
225C2—21no—
226C2#2026063911yescharacteristic
227C2#104×D9711yescharacteristic
228C2#2026075011yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#339×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenter#339D97
hypocenterC97#1986
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
9. #15690809C4semidirect
10. #15690809C4semidirect
11. #15690801C4semidirect
12. #15690801C4semidirect
33. #13095495Q8semidirect
55. M16×C97Q8semidirect
115. C97#1986semidirect
152. M16Dic194semidirect

Maximal subgroups

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

subgroupindex
#202612882
M16×Dic972
C4×#130954902
#202604962
#202604952
#202604942
#339×C972
#198697