Finite Group Databasebeta

← #27158243

Subgroups — #27158243

Order 12416.

532 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261684C212yesmaximal, complemented
2#20261684C212yesmaximal, complemented
3#20261682C212yesmaximal
4#20261680C211yesmaximal, characteristic, complemented
5#20261679C211yesmaximal, characteristic, complemented
6#20261678C211yesmaximal, characteristic
7C4×C4⋊#87604C211yesmaximal, characteristic, complemented
8C22⋊C4×#87604C212yesmaximal
9#20261671C211yesmaximal, characteristic, complemented
10C2×(C2×#87604)⋊C4C211yesmaximal, characteristic
11#117×D97C211yesmaximal, characteristic
12C23×#87604C2212yes—
13C2×C4⋊#87604C2212yes—
14C2×C4⋊#87604C2212yes—
15C2×C22⋊#87604C2212yes—
16C2×C4×#87604C2212yes—
17C2×C4×#87604C2212yes—
18(C2×#87604)⋊C4C2211yescharacteristic
19(C2×#87604)⋊C4C2211yescharacteristic
20(C2×#87604)⋊C4C2212yes—
21(C2×#87604)⋊C4C2211yescharacteristic
22(C2×#87604)⋊C4C2212yes—
23(C2×#87604)⋊C4C2212yes—
24(C2×#87604)⋊C4C2212yes—
25(C2×#87604)⋊C4C2211yescharacteristic
26(C2×#87604)⋊C4C2212yes—
27(C2×#87604)⋊C4C2212yes—
28C22×C4×D97C2211yescharacteristic
29#15690769C411yescharacteristic, complemented
30#15690768C411yescharacteristic, complemented
31#15690768C411yescharacteristic, complemented
32#15690767C411yescharacteristic, complemented
33#15690767C411yescharacteristic, complemented
34#15690766C411yescharacteristic, complemented
35#15690766C411yescharacteristic, complemented
36C4⋊C4×D97C2211yescharacteristic
37C4⋊C4×D97C2211yescharacteristic
38C22⋊C4×D97C2211yescharacteristic
39C22⋊C4×D97C2211yescharacteristic
40C42×D97C2211yescharacteristic
41C42×D97C2211yescharacteristic
42#117×C97C411yescharacteristic, complemented
43C22×#87604—24no—
44C22×#87604—24no—
45C22×#87604—42no—
46C22×#87604C2312yes—
47C22×#87604C2312yes—
48C22×#87604C2312yes—
49C22×#87604C2312yes—
50C4⋊#87604—24no—
51C4⋊#87604—42no—
52C22⋊#87604—24no—
53C4×#87604—24no—
54C4×#87604—42no—
55C23×D97C2311yescharacteristic
56C22×Dic97C2×C411yescharacteristic
57C2×C4×D97—24no—
58C2×C4×D97C2×C411yescharacteristic
59C2×C4×D97C2×C411yescharacteristic
60C2×C4×D97C2311yescharacteristic
61C2×C4×D97C2311yescharacteristic
62C2×C4×D97C2311yescharacteristic
63C2×C4×D97C2311yescharacteristic
64C2×C4×D97C2311yescharacteristic
65C2×C4×D97C2311yescharacteristic
66C4⋊Dic97C2×C411yescharacteristic, complemented
67C4⋊Dic97C2×C411yescharacteristic, complemented
68#13095490C2×C411yescharacteristic, complemented
69#13095490C2×C411yescharacteristic, complemented
70#13095490C2×C411yescharacteristic, complemented
71#13095490C2×C411yescharacteristic, complemented
72C22⋊Dic97C2×C411yescharacteristic
73C22⋊Dic97C2×C411yescharacteristic
74#13095488C2×C411yescharacteristic
75#13095488C2×C411yescharacteristic
76#13095488C2×C411yescharacteristic
77#13095488C2×C411yescharacteristic
78C4×Dic97C2×C411yescharacteristic, complemented
79C4×Dic97C2×C411yescharacteristic, complemented
80C4×Dic97C2×C411yescharacteristic, complemented
81C4×Dic97C2×C411yescharacteristic, complemented
82C4×Dic97C2×C411yescharacteristic, complemented
83C4×Dic97C2×C411yescharacteristic, complemented
84C22×C4×C97C2×C411yescharacteristic
85C4⋊C4×C97C2×C411yescharacteristic, complemented
86C4⋊C4×C97C2×C411yescharacteristic, complemented
87C22⋊C4×C97C2×C411yescharacteristic
88C22⋊C4×C97C2×C411yescharacteristic
89C42×C97C2×C411yescharacteristic, complemented
90C42×C97C2×C411yescharacteristic, complemented
91C23×C97C22×C411yescharacteristic
92C22×D97—22no—
93C22×D97—21no—
94C22×D97—21no—
95C22×D97C22×C411yescharacteristic
96C22×D97#4114yes—
97C22×D97C2411yescharacteristic
98C2×#87604—22no—
99C2×#87604—22no—
100C2×#87604—22no—
101C2×#87604—22no—
102C2×#87604—22no—
103C2×#87604—22no—
104C2×#87604—22no—
105C2×#87604—22no—
106C2×#87604—22no—
107C2×#87604—22no—
108C2×#87604—22no—
109C2×#87604—48no—
110C2×#87604#4114yes—
111C2×C4×C97—22no—
112C2×C4×C97C22×C411yescharacteristic
113C2×C4×C97C22×C411yescharacteristic
114C2×C4×C97C22×C411yescharacteristic
115C2×C4×C97C22×C411yescharacteristic
116C2×C4×C97C22×C411yescharacteristic
117C2×C4×C97C22×C411yescharacteristic
118C2×Dic97—22no—
119C2×Dic97C22×C411yescharacteristic
120C2×Dic97C22×C411yescharacteristic
121C2×Dic97C22×C411yescharacteristic
122C2×Dic97C22×C411yescharacteristic
123C2×Dic97C22×C411yescharacteristic
124C2×Dic97C22×C411yescharacteristic
125C4×D97—22no—
126C4×D97—22no—
127C4×D97—22no—
128C4×D97—22no—
129C4×D97—22no—
130C4×D97—22no—
131C4×D97—21no—
132C4×D97—21no—
133C4×D97—21no—
134C4×D97—21no—
135C22×C97—21no—
136C22×C97—21no—
137C22×C97#11712yes—
138C22×C97C23×C411yescharacteristic
139C2×D97—21no—
140C2×D97—21no—
141C2×D97—24no—
142C2×D97—24no—
143C2×D97#11712yes—
144C2×D97C2×#4112yes—
145C2×D97C2×#4112yes—
146C2×D97H(5,𝔽2)11yescharacteristic
147C2×D972−1+411yescharacteristic
148#87604—24no—
149#87604—42no—
150#87604—42no—
151#87604—42no—
152C4×C97—21no—
153C4×C97—21no—
154C4×C97—21no—
155C4×C97—21no—
156C4×C97—21no—
157C4×C97—21no—
158Dic97—21no—
159Dic97—21no—
160Dic97—21no—
161Dic97—21no—
162Dic97—21no—
163Dic97—21no—
164C2×C97—22no—
165C2×C97C2×#11711yescharacteristic
166C2×C97C4×#4111yescharacteristic
167C2×C97#52011yescharacteristic
168D97—22no—
169D97#55112yes—
170D97#55212yes—
171#2343—971nomaximal, Sylow
172C97#234311yescharacteristic, complemented, Sylow
173C2×#117—971no—
174#387—972no—
175#387—972no—
176#385—972no—
177#383—971no—
178#382—971no—
179#381—971no—
180C4×C4⋊C4—971no—
181C4×C22⋊C4—972no—
182#376—971no—
183C2×#95—971no—
184C23×C4—971no—
185C23×C4—972no—
186#117—971no—
187#117—971no—
188#117—971no—
189#117—971no—
190#117—971no—
191#117—971no—
192#117—971no—
193#117#8760411yescharacteristic, complemented
194C2×C4⋊C4—971no—
195C2×C4⋊C4—971no—
196C2×C4⋊C4—972no—
197C2×C4⋊C4—972no—
198C2×C22⋊C4—971no—
199C2×C22⋊C4—971no—
200C2×C22⋊C4—972no—
201C2×C42—971no—
202C2×C42—971no—
203C2×C42—972no—
204C2×C42—972no—
205#95—971no—
206#95—971no—
207#95—972no—
208#95—971no—
209#95—972no—
210#95—972no—
211#95—972no—
212#95—971no—
213#95—972no—
214#95—972no—
215C24—971no—
216C22×C4—972no—
217C22×C4—971no—
218C22×C4—971no—
219C22×C4—971no—
220C22×C4—971no—
221C22×C4—971no—
222C22×C4—971no—
223C22×C4—971no—
224C22×C4—971no—
225C22×C4—971no—
226C22×C4—972no—
227C22×C4—972no—
228C22×C4—972no—
229C22×C4—3882no—
230C22×C4—1944no—
231C22×C4—1944no—
232C22×C4—1944no—
233C22×C4C2×#8760411yescharacteristic
234C4⋊C4—971no—
235C4⋊C4—971no—
236C4⋊C4—971no—
237C4⋊C4—971no—
238C4⋊C4—971no—
239C4⋊C4—971no—
240C4⋊C4—3882no—
241C4⋊C4—1944no—
242C4⋊C4C2×#8760411yescharacteristic, complemented
243C4⋊C4C2×#8760411yescharacteristic, complemented
244C22⋊C4—971no—
245C22⋊C4—971no—
246C22⋊C4—971no—
247C22⋊C4—971no—
248C22⋊C4—971no—
249C22⋊C4—971no—
250C22⋊C4—1944no—
251C22⋊C4C2×#8760411yescharacteristic
252C22⋊C4C2×#8760411yescharacteristic
253C42—971no—
254C42—971no—
255C42—971no—
256C42—971no—
257C42—971no—
258C42—971no—
259C42—3882no—
260C42—1944no—
261C42C2×#8760411yescharacteristic, complemented
262C42C2×#8760411yescharacteristic, complemented
263C23—971no—
264C23—974no—
265C23—971no—
266C23—1941no—
267C23—1942no—
268C23—1941no—
269C23C22×#8760411yescharacteristic
270C2×C4—22no—
271C2×C4—974no—
272C2×C4—971no—
273C2×C4—971no—
274C2×C4—971no—
275C2×C4—971no—
276C2×C4—971no—
277C2×C4—971no—
278C2×C4—3888no—
279C2×C4—1941no—
280C2×C4—1942no—
281C2×C4—1941no—
282C2×C4—1942no—
283C2×C4—1942no—
284C2×C4—1942no—
285C2×C4—1942no—
286C2×C4—1942no—
287C2×C4—1941no—
288C2×C4—1941no—
289C2×C4—1942no—
290C2×C4—1942no—
291C2×C4—1942no—
292C2×C4—1942no—
293C2×C4—1942no—
294C2×C4—1942no—
295C2×C4—1942no—
296C2×C4—1942no—
297C2×C4—1942no—
298C2×C4—1942no—
299C2×C4—1942no—
300C2×C4—1942no—
301C2×C4C22×#8760411yescharacteristic
302C2×C4C22×#8760411yescharacteristic
303C2×C4C22×#8760411yescharacteristic
304C2×C4C22×#8760411yescharacteristic
305C2×C4C22×#8760411yescharacteristic
306C2×C4C22×#8760411yescharacteristic
307C22—21no—
308C22—21no—
309C22—972no—
310C22—972no—
311C22—971no—
312C22—971no—
313C22—972no—
314C22—1944no—
315C22—1944no—
316C22—1941no—
317C22—1941no—
318C22#1569088912yes—
319C22C23×#8760411yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—3882no—
327C4—3882no—
328C4—3882no—
329C4—1944no—
330C4—1941no—
331C4—1941no—
332C4—1941no—
333C4—1941no—
334C4—1941no—
335C4—1941no—
336C2—22no—
337C2—972no—
338C2—972no—
339C2—1942no—
340C2C2×#1569088911yescharacteristic
341C2#41×#8760411yescharacteristic
342C2#2026180411yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
Fitting#117×C97C4
prime radicalC23×D97C23
cosocleC22×D97C24
hypercenter#117#87604
hypocenterC97#2343
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #20261684C2semidirect
2. #20261684C2semidirect
4. #20261680C2semidirect
5. #20261679C2semidirect
7. C4×C4⋊#87604C2semidirect
9. #20261671C2semidirect
29. #15690769C4semidirect
30. #15690768C4semidirect
31. #15690768C4semidirect
32. #15690767C4semidirect
33. #15690767C4semidirect
34. #15690766C4semidirect
35. #15690766C4semidirect
42. #117×C97C4semidirect
66. C4⋊Dic97C2×C4semidirect
67. C4⋊Dic97C2×C4semidirect
68. #13095490C2×C4semidirect
69. #13095490C2×C4semidirect
70. #13095490C2×C4semidirect
71. #13095490C2×C4semidirect
78. C4×Dic97C2×C4semidirect
79. C4×Dic97C2×C4semidirect
80. C4×Dic97C2×C4semidirect
81. C4×Dic97C2×C4semidirect
82. C4×Dic97C2×C4semidirect
83. C4×Dic97C2×C4semidirect
85. C4⋊C4×C97C2×C4semidirect
86. C4⋊C4×C97C2×C4semidirect
89. C42×C97C2×C4semidirect
90. C42×C97C2×C4semidirect
172. C97#2343semidirect
193. #117#87604semidirect
242. C4⋊C4C2×#87604semidirect
243. C4⋊C4C2×#87604semidirect
261. C42C2×#87604semidirect
262. C42C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202616842
#202616822
#202616802
#202616792
#202616782
C4×C4⋊#876042
C22⋊C4×#876042
#202616712
C2×(C2×#87604)⋊C42
#117×D972
#234397