Finite Group Databasebeta

← #27157606

Subgroups — #27157606

Order 12416.

356 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 84 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#20261736C211yesmaximal, characteristic, complemented
2C2×#15690872C212yesmaximal
3#20261679C211yesmaximal, characteristic, complemented
4#20261608C212yesmaximal, complemented
5C4×D4×D97C211yesmaximal, characteristic
6C2×#13095519C2212yes—
7C2×C4⋊#87604C2212yes—
8#15690877—21no—
9C4⋊#11715897—21no—
10#15690872—24no—
11(C2×#87604)⋊C4—21no—
12(C2×#87604)⋊C4—21no—
13C2×D4×D97C2211yescharacteristic
14C22×C4×D97—21no—
15C4×D388C411yescharacteristic, complemented
16D4×Dic97C411yescharacteristic, complemented
17#15690772C411yescharacteristic, complemented
18C4×C22⋊D97—21no—
19#15690770—21no—
20C4⋊C4×D97C2211yescharacteristic
21C22⋊C4×D97—21no—
22C42×D97C2211yescharacteristic
23C4×D4×C97C411yescharacteristic, complemented
24C22×#87604—22no—
25#13095519—22no—
26#13095519—22no—
27C2×#11715897—22no—
28C4⋊#87604—22no—
29C4⋊#87604—22no—
30C23×D97—21no—
31C2×D388C2×C411yescharacteristic
32C2×C22⋊D97—21no—
33D4×D97—21no—
34D4×D97—21no—
35D4×D97D412yes—
36D4×D97Q812yes—
37C22×Dic97—21no—
38C2×C4×D97—21no—
39C2×C4×D97—21no—
40C2×C4×D97—21no—
41C2×C4×D97—44no—
42C2×C4×D97D411yescharacteristic
43C2×C4×D97D411yescharacteristic
44C2×C4×D97C2311yescharacteristic
45C4⋊Dic97C2×C411yescharacteristic
46#13095490—21no—
47C22⋊Dic97—21no—
48#13095488—21no—
49#13095488—21no—
50C4×Dic97C2×C411yescharacteristic
51C4×Dic97D411yescharacteristic
52C4×Dic97D411yescharacteristic
53C2×D4×C97C2×C411yescharacteristic
54C22×C4×C97—21no—
55C4⋊C4×C97C2×C411yescharacteristic
56C22⋊C4×C97—21no—
57C42×C97C2×C411yescharacteristic
58C23×C97—21no—
59C22×D97—21no—
60C22×D97—22no—
61C22×D97—22no—
62C22×D97—42no—
63C22×D97—42no—
64C22×D97C2×D411yescharacteristic
65C2×#87604—42no—
66C2×#87604—42no—
67C2×#87604—42no—
68D4×C97—21no—
69D4×C97C4⋊C412yes—
70C2×C4×C97—21no—
71C2×C4×C97—42no—
72C2×C4×C97C22⋊C411yescharacteristic
73C2×C4×C97C22⋊C411yescharacteristic
74C2×C4×C97C22×C411yescharacteristic
75C22⋊D97—22no—
76C22⋊D97—22no—
77C2×Dic97—21no—
78C2×Dic97—42no—
79C2×Dic97C22⋊C411yescharacteristic
80C2×Dic97C22⋊C411yescharacteristic
81C2×Dic97C2×D411yescharacteristic
82D388—21no—
83D388C4⋊C412yes—
84C4×D97—22no—
85C4×D97—22no—
86C4×D97—42no—
87C4×D97—42no—
88C4×D97—42no—
89C4×D97C2×D411yescharacteristic
90C4×D97C2×Q811yescharacteristic
91C4×D97#4111yescharacteristic
92C4×D97#4111yescharacteristic
93#11715897—42no—
94C22×C97—22no—
95C22×C97—41no—
96C22×C97—41no—
97C22×C97C2×C22⋊C411yescharacteristic
98C2×D97—22no—
99C2×D97—41no—
100C2×D97—41no—
101C2×D97—44no—
102C2×D97—44no—
103C2×D97#12111yescharacteristic
104C2×D97C22⋊Q811yescharacteristic
105C2×D97#13512yes—
106C2×D97AGL(1,ℤ/8)12yes—
107#87604—82no—
108C4×C97—21no—
109C4×C97—21no—
110C4×C97—41no—
111C4×C97C2×C4⋊C411yescharacteristic
112C4×C97#11711yescharacteristic
113Dic97—21no—
114Dic97—21no—
115Dic97—41no—
116Dic97#12111yescharacteristic
117Dic97C22⋊Q811yescharacteristic
118C2×C97—42no—
119C2×C97#38011yescharacteristic
120C2×C97#41611yescharacteristic
121C2×C97#41811yescharacteristic
122D97—42no—
123D97#46312yes—
124D97#47812yes—
125#1804—971nomaximal, Sylow
126C97#180411yescharacteristic, complemented, Sylow
127C2×C4×D4—971no—
128C2×C4⋊C8—971no—
129C2×#102—972no—
130#382—971no—
131#340—972no—
132C22×D4—971no—
133C23×C4—1941no—
134C22×C8—972no—
135C4×D4—971no—
136C4×D4—971no—
137C4×D4—971no—
138C4×D4—1941no—
139C4×D4—1941no—
140C4×D4#8760411yescharacteristic, complemented
141C2×C4⋊C4—971no—
142C2×C4⋊C4—972no—
143C2×C22⋊C4—1941no—
144C2×C42—971no—
145C4⋊C8—1941no—
146C4⋊C8—1941no—
147#102—1944no—
148#95—1941no—
149#95—1941no—
150C24—1941no—
151C2×D4—972no—
152C2×D4—972no—
153C2×D4—971no—
154C2×D4—1941no—
155C2×D4—1941no—
156C2×D4—1941no—
157C2×D4C2×#8760411yescharacteristic
158C22×C4—21no—
159C22×C4—971no—
160C22×C4—971no—
161C22×C4—971no—
162C22×C4—3884no—
163C22×C4—1941no—
164C22×C4—1941no—
165C22×C4—1941no—
166C22×C4—1942no—
167C22×C4—1941no—
168C2×C8—1942no—
169C2×C8—1942no—
170C2×C8—1942no—
171C4⋊C4—971no—
172C4⋊C4—1941no—
173C4⋊C4—1942no—
174C4⋊C4—1942no—
175C4⋊C4C2×#8760411yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—1941no—
178C22⋊C4—1941no—
179C22⋊C4—1941no—
180C42—971no—
181C42—971no—
182C42—971no—
183C42C2×#8760411yescharacteristic
184C23—21no—
185C23—971no—
186C23—3882no—
187C23—3882no—
188C23—1942no—
189C23—1942no—
190C23—1941no—
191D4—21no—
192D4—972no—
193D4—1942no—
194D4—1942no—
195D4—1941no—
196D4C4⋊#8760412yes—
197C2×C4—21no—
198C2×C4—42no—
199C2×C4—971no—
200C2×C4—971no—
201C2×C4—971no—
202C2×C4—971no—
203C2×C4—971no—
204C2×C4—971no—
205C2×C4—971no—
206C2×C4—3882no—
207C2×C4—3882no—
208C2×C4—3882no—
209C2×C4—3882no—
210C2×C4—3882no—
211C2×C4—3882no—
212C2×C4—3882no—
213C2×C4—1942no—
214C2×C4—1942no—
215C2×C4—1941no—
216C2×C4C22⋊#8760411yescharacteristic
217C2×C4C22⋊#8760411yescharacteristic
218C2×C4C22×#8760411yescharacteristic
219C8—3882no—
220C22—22no—
221C22—41no—
222C22—41no—
223C22—971no—
224C22—971no—
225C22—972no—
226C22—972no—
227C22—3884no—
228C22—3884no—
229C22—3881no—
230C22—3881no—
231C22—1942no—
232C22C2×C22⋊#8760411yescharacteristic
233C4—21no—
234C4—21no—
235C4—41no—
236C4—7762no—
237C4—971no—
238C4—971no—
239C4—3881no—
240C4—1941no—
241C4—1941no—
242C4C2×C4⋊#8760411yescharacteristic
243C4#1569088911yescharacteristic
244C2—42no—
245C2—972no—
246C2—972no—
247C2—3882no—
248C2#2026167711yescharacteristic
249C2#2026172511yescharacteristic
250C2#2026172711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4×D4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×D4×D97C22
cosocleC2×C4×D97C23
hypercenterC4×D4#87604
hypocenterC97#1804
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #20261736C2semidirect
3. #20261679C2semidirect
4. #20261608C2semidirect
15. C4×D388C4semidirect
16. D4×Dic97C4semidirect
17. #15690772C4semidirect
23. C4×D4×C97C4semidirect
126. C97#1804semidirect
140. C4×D4#87604semidirect

Maximal subgroups

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

subgroupindex
#202617362
C2×#156908722
#202616792
#202616082
C4×D4×D972
#180497