Finite Group Databasebeta

← #27158274

Subgroups — #27158274

Order 12416.

552 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261687C211yesmaximal, characteristic
2#20261685C212yesmaximal, complemented
3#20261685C212yesmaximal, complemented
4#20261684C212yesmaximal, complemented
5#20261683C211yesmaximal, characteristic
6C4⋊C4×#87604C212yesmaximal, complemented
7C22⋊C4×#87604C212yesmaximal, complemented
8C4×C22⋊#87604C212yesmaximal, complemented
9#123×D97C211yesmaximal, characteristic
10C23×#87604C2211yescharacteristic
11C2×C4⋊#87604C2212yes—
12C2×C4⋊#87604C2211yescharacteristic
13C2×C22⋊#87604C2212yes—
14C2×C22⋊#87604C2212yes—
15C2×C22⋊#87604C2212yes—
16C2×C22⋊#87604C2211yescharacteristic
17C2×C4×#87604C2212yes—
18C2×C4×#87604C2212yes—
19C2×C4×#87604C2212yes—
20C2×C4×#87604C2211yescharacteristic
21(C2×#87604)⋊C4C2212yes—
22(C2×#87604)⋊C4C2212yes—
23(C2×#87604)⋊C4C2212yes—
24(C2×#87604)⋊C4C2212yes—
25(C2×#87604)⋊C4C2212yes—
26C2×D4×D97C2211yescharacteristic
27C22×C4×D97C2211yescharacteristic
28#15690795C412yescomplemented
29#15690794C411yescharacteristic, complemented
30#15690793C411yescharacteristic, complemented
31#15690792C412yescomplemented
32#15690791C411yescharacteristic, complemented
33C4⋊C4×D97C2212yes—
34C22⋊C4×D97C2212yes—
35C22⋊C4×D97C2211yescharacteristic
36#123×C97C411yescharacteristic, complemented
37C22×#87604—24no—
38C22×#87604—41no—
39C22×#87604C2312yes—
40C22×#87604C2312yes—
41C22×#87604C2312yes—
42C22×#87604C2312yes—
43C4⋊#87604—24no—
44C4⋊#87604—41no—
45C22⋊#87604—24no—
46C22⋊#87604—24no—
47C22⋊#87604—24no—
48C22⋊#87604—41no—
49C4×#87604—24no—
50C4×#87604—42no—
51C4×#87604—42no—
52C4×#87604—41no—
53C23×D97C2311yescharacteristic
54C23×D97C2311yescharacteristic
55C2×D388C2×C411yescharacteristic
56C2×C22⋊D97C2×C411yescharacteristic
57C2×C22⋊D97C2×C411yescharacteristic
58D4×D97—41no—
59D4×D97—41no—
60C22×Dic97C2×C411yescharacteristic, complemented
61C2×C4×D97—41no—
62C2×C4×D97—41no—
63C2×C4×D97C2×C412yescomplemented
64C2×C4×D97C2312yes—
65C2×C4×D97C2312yes—
66C2×C4×D97C2311yescharacteristic
67C4⋊Dic97C2×C412yescomplemented
68#13095490C2×C412yescomplemented
69#13095490C2×C412yescomplemented
70C22⋊Dic97C2×C412yes—
71C22⋊Dic97C2×C411yescharacteristic, complemented
72#13095488C2×C412yes—
73#13095488C2×C412yes—
74#13095488C2×C412yescomplemented
75C2×D4×C97C2×C411yescharacteristic
76C22×C4×C97C2×C411yescharacteristic, complemented
77C4⋊C4×C97C2×C412yescomplemented
78C22⋊C4×C97C2×C412yes—
79C22⋊C4×C97C2×C411yescharacteristic, complemented
80C23×C97C22×C411yescharacteristic
81C23×C97C22×C411yescharacteristic
82C22×D97—22no—
83C22×D97—22no—
84C22×D97—22no—
85C22×D97—22no—
86C22×D97—21no—
87C22×D97—21no—
88C22×D97—21no—
89C22×D97—21no—
90C22×D97C22×C411yescharacteristic
91C22×D97C22×C411yescharacteristic
92C22×D97C2411yescharacteristic
93C2×#87604—22no—
94C2×#87604—22no—
95C2×#87604—22no—
96C2×#87604—22no—
97C2×#87604—22no—
98C2×#87604—22no—
99C2×#87604—22no—
100C2×#87604—22no—
101C2×#87604—22no—
102C2×#87604—22no—
103C2×#87604—44no—
104C2×#87604#4114yes—
105C2×#87604#4114yes—
106D4×C97—41no—
107C2×C4×C97—41no—
108C2×C4×C97C22×C412yes—
109C2×C4×C97C22×C412yes—
110C2×C4×C97C22×C411yescharacteristic
111C22⋊D97—41no—
112C22⋊D97—41no—
113C2×Dic97—41no—
114C2×Dic97C22×C412yes—
115C2×Dic97C22×C412yes—
116C2×Dic97C22×C411yescharacteristic
117D388—41no—
118C4×D97—22no—
119C4×D97—22no—
120C4×D97—22no—
121C4×D97—22no—
122C4×D97—21no—
123C4×D97—21no—
124C4×D97—42no—
125C22×C97—22no—
126C22×C97—22no—
127C22×C97—21no—
128C22×C97—21no—
129C22×C97C23×C411yescharacteristic
130C2×D97—22no—
131C2×D97—22no—
132C2×D97—21no—
133C2×D97—21no—
134C2×D97—41no—
135C2×D97—41no—
136C2×D97—42no—
137C2×D97—42no—
138C2×D97—41no—
139C2×D97—41no—
140C2×D97C2×#4112yes—
141C2×D97C2×#4112yes—
142C2×D97H(5,𝔽2)11yescharacteristic
143C2×D97H(5,𝔽2)11yescharacteristic
144#87604—24no—
145#87604—24no—
146#87604—42no—
147#87604—42no—
148C4×C97—22no—
149C4×C97—22no—
150C4×C97—21no—
151Dic97—22no—
152Dic97—22no—
153Dic97—21no—
154C2×C97—41no—
155C2×C97—41no—
156C2×C97C4×#4112yes—
157C2×C97#51811yescharacteristic
158D97—41no—
159D97—41no—
160D97#55111yescharacteristic
161D97#55312yes—
162D97#55911yescharacteristic
163#2374—971nomaximal, Sylow
164C97#237411yescharacteristic, complemented, Sylow
165C2×#123—971no—
166#390—971no—
167#388—972no—
168#388—972no—
169#387—972no—
170#386—971no—
171C4×C4⋊C4—972no—
172C4×C22⋊C4—972no—
173C4×C22⋊C4—972no—
174C22×D4—971no—
175C23×C4—971no—
176C23×C4—971no—
177#123—971no—
178#123—971no—
179#123—972no—
180#123—972no—
181#123—971no—
182#123#8760411yescharacteristic, complemented
183C2×C4⋊C4—972no—
184C2×C4⋊C4—971no—
185C2×C4⋊C4—972no—
186C2×C22⋊C4—972no—
187C2×C22⋊C4—971no—
188C2×C22⋊C4—971no—
189C2×C22⋊C4—972no—
190C2×C22⋊C4—972no—
191C2×C22⋊C4—972no—
192C2×C42—972no—
193C2×C42—972no—
194C2×C42—971no—
195C2×C42—972no—
196#95—972no—
197#95—972no—
198#95—972no—
199#95—972no—
200#95—972no—
201C24—971no—
202C24—971no—
203C2×D4—971no—
204C2×D4—971no—
205C2×D4—971no—
206C2×D4—3881no—
207C2×D4—3881no—
208C2×D4C2×#8760411yescharacteristic
209C22×C4—971no—
210C22×C4—972no—
211C22×C4—972no—
212C22×C4—972no—
213C22×C4—972no—
214C22×C4—972no—
215C22×C4—971no—
216C22×C4—972no—
217C22×C4—972no—
218C22×C4—3881no—
219C22×C4—3881no—
220C22×C4—3881no—
221C22×C4—1944no—
222C22×C4C2×#8760411yescharacteristic, complemented
223C4⋊C4—972no—
224C4⋊C4—972no—
225C4⋊C4—972no—
226C4⋊C4—3881no—
227C4⋊C4—1944no—
228C4⋊C4C2×#8760412yescomplemented
229C22⋊C4—972no—
230C22⋊C4—971no—
231C22⋊C4—972no—
232C22⋊C4—972no—
233C22⋊C4—972no—
234C22⋊C4—3881no—
235C22⋊C4—1944no—
236C22⋊C4—1944no—
237C22⋊C4—1944no—
238C22⋊C4C2×#8760412yes—
239C22⋊C4C2×#8760411yescharacteristic, complemented
240C42—3882no—
241C42—3882no—
242C42—3881no—
243C42—1944no—
244C23—971no—
245C23—971no—
246C23—971no—
247C23—1941no—
248C23—1942no—
249C23—1942no—
250C23—1942no—
251C23—1941no—
252C23—1941no—
253C23—1941no—
254C23—1942no—
255C23C22×#8760411yescharacteristic
256C23C22×#8760411yescharacteristic
257D4—41no—
258D4—3881no—
259D4—3881no—
260D4—3881no—
261C2×C4—41no—
262C2×C4—974no—
263C2×C4—974no—
264C2×C4—972no—
265C2×C4—972no—
266C2×C4—971no—
267C2×C4—3882no—
268C2×C4—3881no—
269C2×C4—3884no—
270C2×C4—1942no—
271C2×C4—1942no—
272C2×C4—1941no—
273C2×C4—1942no—
274C2×C4—1941no—
275C2×C4—1942no—
276C2×C4—1942no—
277C2×C4—1942no—
278C2×C4—1942no—
279C2×C4—1942no—
280C2×C4—1942no—
281C2×C4—1942no—
282C2×C4—1942no—
283C2×C4—1942no—
284C2×C4—1942no—
285C2×C4—1942no—
286C2×C4C22×#8760412yes—
287C2×C4C22×#8760412yes—
288C2×C4C22×#8760411yescharacteristic
289C22—22no—
290C22—22no—
291C22—21no—
292C22—21no—
293C22—972no—
294C22—972no—
295C22—971no—
296C22—971no—
297C22—3881no—
298C22—3881no—
299C22—3881no—
300C22—3882no—
301C22—3882no—
302C22—3881no—
303C22—1942no—
304C22—1942no—
305C22—1941no—
306C22—1941no—
307C22C23×#8760411yescharacteristic
308C4—22no—
309C4—22no—
310C4—21no—
311C4—3882no—
312C4—3882no—
313C4—1944no—
314C4—1944no—
315C4—1942no—
316C4—1942no—
317C4—1941no—
318C2—41no—
319C2—41no—
320C2—971no—
321C2—972no—
322C2—971no—
323C2—3881no—
324C2—3881no—
325C2#41×#8760412yes—
326C2#2026180211yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
Fitting#123×C97C4
prime radicalC2×D4×D97C22
cosocleC22×D97C24
hypercenter#123#87604
hypocenterC97#2374
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #20261685C2semidirect
3. #20261685C2semidirect
4. #20261684C2semidirect
6. C4⋊C4×#87604C2semidirect
7. C22⋊C4×#87604C2semidirect
8. C4×C22⋊#87604C2semidirect
28. #15690795C4semidirect
29. #15690794C4semidirect
30. #15690793C4semidirect
31. #15690792C4semidirect
32. #15690791C4semidirect
36. #123×C97C4semidirect
60. C22×Dic97C2×C4semidirect
63. C2×C4×D97C2×C4semidirect
67. C4⋊Dic97C2×C4semidirect
68. #13095490C2×C4semidirect
69. #13095490C2×C4semidirect
71. C22⋊Dic97C2×C4semidirect
74. #13095488C2×C4semidirect
76. C22×C4×C97C2×C4semidirect
77. C4⋊C4×C97C2×C4semidirect
79. C22⋊C4×C97C2×C4semidirect
164. C97#2374semidirect
182. #123#87604semidirect
222. C22×C4C2×#87604semidirect
228. C4⋊C4C2×#87604semidirect
239. C22⋊C4C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202616872
#202616852
#202616842
#202616832
C4⋊C4×#876042
C22⋊C4×#876042
C4×C22⋊#876042
#123×D972
#237497