Finite Group Databasebeta

← #27157604

Subgroups — #27157604

Order 12416.

372 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 94 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#20261681C211yesmaximal, characteristic, complemented
4#20261608C212yesmaximal, complemented
5C4×D4×D97C211yesmaximal, characteristic
6C2×#13095519C2212yes—
7C2×C4⋊#87604—21no—
8C2×C4⋊#87604—21no—
9C2×C4⋊#87604C2212yes—
10#15690877—21no—
11C4⋊#11715897—21no—
12#15690872—24no—
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—
30C4⋊#87604—42no—
31C4⋊#87604—42no—
32C23×D97—21no—
33C2×D388C2×C411yescharacteristic
34C2×C22⋊D97—21no—
35D4×D97—21no—
36D4×D97—21no—
37D4×D97D412yes—
38D4×D97Q812yes—
39C22×Dic97—21no—
40C2×C4×D97—21no—
41C2×C4×D97—21no—
42C2×C4×D97—21no—
43C2×C4×D97—42no—
44C2×C4×D97—42no—
45C2×C4×D97D411yescharacteristic
46C2×C4×D97D411yescharacteristic
47C2×C4×D97C2311yescharacteristic
48C4⋊Dic97C2×C411yescharacteristic
49#13095490—21no—
50C22⋊Dic97—21no—
51#13095488—21no—
52#13095488—21no—
53C4×Dic97C2×C411yescharacteristic
54C4×Dic97D411yescharacteristic
55C4×Dic97D411yescharacteristic
56C2×D4×C97C2×C411yescharacteristic
57C22×C4×C97—21no—
58C4⋊C4×C97C2×C411yescharacteristic
59C22⋊C4×C97—21no—
60C42×C97C2×C411yescharacteristic
61C23×C97—21no—
62C22×D97—21no—
63C22×D97—22no—
64C22×D97—22no—
65C22×D97—41no—
66C22×D97—41no—
67C22×D97—41no—
68C22×D97—41no—
69C22×D97C2×D411yescharacteristic
70C2×#87604—42no—
71C2×#87604—42no—
72C2×#87604—42no—
73D4×C97—21no—
74D4×C97C4⋊C412yescomplemented
75C2×C4×C97—21no—
76C2×C4×C97—42no—
77C2×C4×C97C22⋊C411yescharacteristic
78C2×C4×C97C22⋊C411yescharacteristic
79C2×C4×C97C22×C411yescharacteristic
80C22⋊D97—22no—
81C22⋊D97—22no—
82C2×Dic97—21no—
83C2×Dic97—42no—
84C2×Dic97C22⋊C411yescharacteristic
85C2×Dic97C22⋊C411yescharacteristic
86C2×Dic97C2×D411yescharacteristic
87D388—21no—
88D388C4⋊C412yescomplemented
89C4×D97—21no—
90C4×D97—21no—
91C4×D97—41no—
92C4×D97—41no—
93C4×D97—42no—
94C4×D97—42no—
95C4×D97D812yes—
96C4×D97SD1612yes—
97C4×D97C2×D411yescharacteristic
98C4×D97C2×Q811yescharacteristic
99C4×D97#4111yescharacteristic
100C4×D97#4111yescharacteristic
101#11715897—42no—
102C22×C97—22no—
103C22×C97—41no—
104C22×C97—41no—
105C22×C97C2×C22⋊C411yescharacteristic
106C2×D97—22no—
107C2×D97—41no—
108C2×D97—41no—
109C2×D97—42no—
110C2×D97—42no—
111C2×D97—42no—
112C2×D97—42no—
113C2×D97#12111yescharacteristic
114C2×D97C22⋊Q811yescharacteristic
115C2×D97C2×D811yescharacteristic
116C2×D97C2×SD1611yescharacteristic
117C2×D97AGL(1,ℤ/8)11yescharacteristic
118C2×D97#13711yescharacteristic
119#87604—82no—
120C4×C97—21no—
121C4×C97—41no—
122C4×C97#10212yescomplemented
123C4×C97C2×C4⋊C411yescharacteristic
124C4×C97#11711yescharacteristic
125Dic97—21no—
126Dic97—41no—
127Dic97#10212yescomplemented
128Dic97#12111yescharacteristic
129Dic97C22⋊Q811yescharacteristic
130C2×C97—42no—
131C2×C97#38011yescharacteristic
132C2×C97C2×#10211yescharacteristic
133C2×C97#41711yescharacteristic
134D97—42no—
135D97#45911yescharacteristic
136D97#46111yescharacteristic
137D97#47411yescharacteristic
138D97#47611yescharacteristic
139#1802—971nomaximal, Sylow
140C97#180211yescharacteristic, complemented, Sylow
141C2×C4×D4—971no—
142C2×C4⋊C8—971no—
143C2×#102—972no—
144#384—971no—
145#340—972no—
146C22×D4—971no—
147C23×C4—1941no—
148C22×C8—972no—
149C4×D4—971no—
150C4×D4—971no—
151C4×D4—971no—
152C4×D4—1941no—
153C4×D4—1941no—
154C4×D4#8760411yescharacteristic, complemented
155C2×C4⋊C4—971no—
156C2×C4⋊C4—972no—
157C2×C4⋊C4—1941no—
158C2×C4⋊C4—1941no—
159C2×C22⋊C4—1941no—
160C2×C42—971no—
161C4⋊C8—1941no—
162C4⋊C8—1941no—
163#102—1944no—
164C24—1941no—
165C2×D4—972no—
166C2×D4—972no—
167C2×D4—971no—
168C2×D4—1941no—
169C2×D4—1941no—
170C2×D4—1941no—
171C2×D4C2×#8760411yescharacteristic
172C22×C4—21no—
173C22×C4—971no—
174C22×C4—971no—
175C22×C4—971no—
176C22×C4—3882no—
177C22×C4—3882no—
178C22×C4—1941no—
179C22×C4—1941no—
180C22×C4—1941no—
181C22×C4—1942no—
182C22×C4—1941no—
183C2×C8—1942no—
184C2×C8—1942no—
185C2×C8—1942no—
186C4⋊C4—971no—
187C4⋊C4—3882no—
188C4⋊C4—3882no—
189C4⋊C4—1941no—
190C4⋊C4—1942no—
191C4⋊C4—1942no—
192C4⋊C4C2×#8760411yescharacteristic
193C22⋊C4—21no—
194C22⋊C4—1941no—
195C22⋊C4—1941no—
196C22⋊C4—1941no—
197C42—971no—
198C42—971no—
199C42—971no—
200C42C2×#8760411yescharacteristic
201C23—21no—
202C23—971no—
203C23—3881no—
204C23—3881no—
205C23—3881no—
206C23—3881no—
207C23—1942no—
208C23—1942no—
209C23—1941no—
210D4—21no—
211D4—972no—
212D4—1942no—
213D4—1942no—
214D4—1941no—
215D4C4⋊#8760412yescomplemented
216C2×C4—21no—
217C2×C4—42no—
218C2×C4—971no—
219C2×C4—971no—
220C2×C4—972no—
221C2×C4—972no—
222C2×C4—971no—
223C2×C4—971no—
224C2×C4—971no—
225C2×C4—971no—
226C2×C4—971no—
227C2×C4—3882no—
228C2×C4—3881no—
229C2×C4—3881no—
230C2×C4—3882no—
231C2×C4—3882no—
232C2×C4—3882no—
233C2×C4—3882no—
234C2×C4—3882no—
235C2×C4—1941no—
236C2×C4—1941no—
237C2×C4—1941no—
238C2×C4C22⋊#8760411yescharacteristic
239C2×C4C22⋊#8760411yescharacteristic
240C2×C4C22×#8760411yescharacteristic
241C8—3882no—
242C22—22no—
243C22—41no—
244C22—41no—
245C22—971no—
246C22—971no—
247C22—971no—
248C22—971no—
249C22—971no—
250C22—971no—
251C22—3882no—
252C22—3882no—
253C22—3882no—
254C22—3882no—
255C22—3881no—
256C22—3881no—
257C22—1942no—
258C22C2×C22⋊#8760411yescharacteristic
259C4—21no—
260C4—41no—
261C4—7762no—
262C4—971no—
263C4—971no—
264C4—972no—
265C4—3881no—
266C4—1941no—
267C4#1569087212yescomplemented
268C4C2×C4⋊#8760411yescharacteristic
269C4#1569088911yescharacteristic
270C2—42no—
271C2—971no—
272C2—971no—
273C2—971no—
274C2—971no—
275C2—3882no—
276C2#2026167711yescharacteristic
277C2C2×#1569087211yescharacteristic
278C2#2026172611yescharacteristic
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#1802
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. #20261681C2semidirect
4. #20261608C2semidirect
15. C4×D388C4semidirect
16. D4×Dic97C4semidirect
17. #15690772C4semidirect
23. C4×D4×C97C4semidirect
74. D4×C97C4⋊C4semidirect
88. D388C4⋊C4semidirect
122. C4×C97#102semidirect
127. Dic97#102semidirect
140. C97#1802semidirect
154. C4×D4#87604semidirect
215. D4C4⋊#87604semidirect
267. C4#15690872semidirect

Maximal subgroups

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

subgroupindex
#202617362
C2×#156908722
#202616812
#202616082
C4×D4×D972
#180297