Finite Group Databasebeta

← #27154336

Subgroups — #27154336

Order 12416.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261431C211yesmaximal, characteristic, complemented
2#20261134C211yesmaximal, characteristic, complemented
3#20261059C211yesmaximal, characteristic, complemented
4#20261056C211yesmaximal, characteristic, complemented
5#20261048C211yesmaximal, characteristic, complemented
6#20261046C211yesmaximal, characteristic, complemented
7#20261035C211yesmaximal, characteristic, complemented
8#20261032C211yesmaximal, characteristic, complemented
9#20260986C211yesmaximal, characteristic, complemented
10#20260986C211yesmaximal, characteristic, complemented
11C4×#13095496C211yesmaximal, characteristic, complemented
12#20260813C211yesmaximal, characteristic, complemented
13#20260778C211yesmaximal, characteristic, complemented
14#20260775C211yesmaximal, characteristic, complemented
15#553×C97C211yesmaximal, characteristic, complemented
16C2×#13095496C2211yescharacteristic
17C2×#13095495C2211yescharacteristic
18C2×#13095495C2211yescharacteristic
19#15690805—21no—
20#15690802C2211yescharacteristic
21#15690792—21no—
22#15690784C2211yescharacteristic
23#15690784C2211yescharacteristic
24#15690782—21no—
25C4×D388C2211yescharacteristic
26#15690770—21no—
27#15690769C2211yescharacteristic
28C2×C4⋊Dic97C2211yescharacteristic
29#15690743C2211yescharacteristic
30#15690743C2211yescharacteristic
31#15690743C2211yescharacteristic
32#15690743C2211yescharacteristic
33#15690743C2211yescharacteristic
34#15690741C2211yescharacteristic
35#15690741C2211yescharacteristic
36#15690741C2211yescharacteristic
37#15690741C2211yescharacteristic
38C4⋊#11715895C2211yescharacteristic
39#15690731C2211yescharacteristic
40#15690731C2211yescharacteristic
41#15690731C2211yescharacteristic
42#15690730C2211yescharacteristic
43#15690730C2211yescharacteristic
44#15690730C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46C22⋊#11715895C2211yescharacteristic
47C4×#11715895C2211yescharacteristic
48#126×C97—21no—
49#125×C97C2211yescharacteristic
50#123×C97—21no—
51#121×C97—21no—
52#121×C97C2211yescharacteristic
53#121×C97C2211yescharacteristic
54C4×D4×C97—21no—
55C4×D4×C97C2211yescharacteristic
56C4×D4×C97C2211yescharacteristic
57#117×C97C2211yescharacteristic
58C2×C4⋊C4×C97C2211yescharacteristic
59C2×D388C2311yescharacteristic
60C2×C22⋊D97—21no—
61C2×C22⋊D97—21no—
62C22×Dic97—21no—
63C2×C4×D97—21no—
64#13095496—22no—
65#13095495—22no—
66#13095495—22no—
67C2×#11715895C2311yescharacteristic
68C2×#11715895C2311yescharacteristic
69C2×#11715895C2311yescharacteristic
70C2×#11715895C2311yescharacteristic
71C4⋊Dic97—21no—
72C4⋊Dic97C2311yescharacteristic
73C4⋊Dic97C2311yescharacteristic
74C4⋊Dic97C2311yescharacteristic
75#13095490—21no—
76#13095490—21no—
77C22⋊Dic97—21no—
78#13095488—21no—
79#13095488—21no—
80#13095488—21no—
81C4×Dic97—21no—
82C2×D4×C97—21no—
83C2×D4×C97—21no—
84C2×D4×C97D411yescharacteristic
85C2×D4×C97C2311yescharacteristic
86C22×C4×C97—21no—
87C22×C4×C97—21no—
88C22×C4×C97C2311yescharacteristic
89C22×C4×C97C2311yescharacteristic
90C4⋊C4×C97—21no—
91C4⋊C4×C97—21no—
92C4⋊C4×C97—21no—
93C4⋊C4×C97D411yescharacteristic
94C4⋊C4×C97C2311yescharacteristic
95C4⋊C4×C97C2311yescharacteristic
96C4⋊C4×C97C2311yescharacteristic
97C22⋊C4×C97—21no—
98C22⋊C4×C97—21no—
99C22⋊C4×C97—21no—
100C22⋊C4×C97—21no—
101C22⋊C4×C97D411yescharacteristic
102C22⋊C4×C97D411yescharacteristic
103C42×C97—21no—
104C42×C97C2311yescharacteristic
105C23×C97—21no—
106C23×C97C2×D411yescharacteristic
107C23×C97C2×D411yescharacteristic
108C22×D97—21no—
109D4×C97—21no—
110D4×C97—21no—
111D4×C97—41no—
112D4×C97—42no—
113D4×C97—81no—
114C2×C4×C97—21no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—81no—
122C2×C4×C97—81no—
123C2×C4×C97C2×D411yescharacteristic
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2411yescharacteristic
128C22⋊D97—42no—
129C22⋊D97—81no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—81no—
134D388—21no—
135D388—21no—
136C4×D97—81no—
137#11715895—21no—
138#11715895—21no—
139#11715895—21no—
140#11715895#4112yes—
141C22×C97—21no—
142C22×C97—21no—
143C22×C97—21no—
144C22×C97—21no—
145C22×C97—21no—
146C22×C97—21no—
147C22×C97—41no—
148C22×C97—41no—
149C22×C97—41no—
150C22×C97C22×D411yescharacteristic
151C2×D97—41no—
152C2×D97—41no—
153C2×D97—41no—
154C4×C97—21no—
155C4×C97—21no—
156C4×C97—21no—
157C4×C97—21no—
158C4×C97—41no—
159C4×C97—41no—
160C4×C97—41no—
161C4×C97C2×#4111yescharacteristic
162C4×C972−1+411yescharacteristic
163Dic97—41no—
164Dic97—41no—
165Dic97—41no—
166C2×C97—41no—
167C2×C97—41no—
168C2×C97—81no—
169C2×C97#54711yescharacteristic
170C2×C97#57511yescharacteristic
171C2×C97#57611yescharacteristic
172D97—81no—
173#3355—971nomaximal, Sylow
174C97#335511yescharacteristic, complemented, Sylow
175#553—971no—
176#553D9711yescharacteristic, complemented
177#499—971no—
178#482—971no—
179#482—971no—
180#480—971no—
181#480—971no—
182#478—971no—
183#478—971no—
184#469—971no—
185#469—971no—
186C4×D8—971no—
187#436—971no—
188#428—971no—
189#428—971no—
190C2×D8—971no—
191C2×M16—971no—
192C2×M16—971no—
193#126—21no—
194#126—1941no—
195#125—971no—
196#125C2×D9711yescharacteristic
197#123—21no—
198#123—1941no—
199#121—21no—
200#121—971no—
201#121—971no—
202#121—1941no—
203#121C2×D9711yescharacteristic
204#121C2×D9711yescharacteristic
205C4×D4—21no—
206C4×D4—971no—
207C4×D4—1941no—
208C4×D4C2×D9711yescharacteristic
209C4×D4C2×D9711yescharacteristic
210#117—971no—
211#117C2×D9711yescharacteristic
212C2×C4⋊C4—971no—
213C2×C4⋊C4C2×D9711yescharacteristic
214#107—971no—
215#107—971no—
216#107—971no—
217#107—971no—
218#107—971no—
219#106—971no—
220#106—971no—
221#106—971no—
222#106—971no—
223C4⋊C8—971no—
224#102—971no—
225#102—971no—
226#102—971no—
227#102—971no—
228#102—971no—
229#102—971no—
230C22⋊C8—971no—
231C22⋊C8—971no—
232C4×C8—971no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—971no—
236C2×D4—1941no—
237C2×D4—1941no—
238C2×D4C22⋊D9711yescharacteristic
239C2×D4C22×D9711yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—1941no—
243C22×C4—1941no—
244C22×C4C22×D9711yescharacteristic
245C22×C4C22×D9711yescharacteristic
246D8—1942no—
247M16—1942no—
248M16—1942no—
249C2×C8—971no—
250C2×C8—971no—
251C2×C8—971no—
252C2×C8—971no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—971no—
257C4⋊C4—971no—
258C4⋊C4—971no—
259C4⋊C4—1941no—
260C4⋊C4—1941no—
261C4⋊C4—1941no—
262C4⋊C4C22⋊D9711yescharacteristic
263C4⋊C4C22×D9711yescharacteristic
264C4⋊C4C22×D9711yescharacteristic
265C4⋊C4C22×D9711yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—1941no—
271C22⋊C4—1941no—
272C22⋊C4—1941no—
273C22⋊C4—1941no—
274C22⋊C4C22⋊D9711yescharacteristic
275C22⋊C4C22⋊D9711yescharacteristic
276C42—21no—
277C42—1941no—
278C42C22×D9711yescharacteristic
279C23—21no—
280C23—1941no—
281C23C2×C22⋊D9711yescharacteristic
282C23C2×C22⋊D9711yescharacteristic
283D4—21no—
284D4—21no—
285D4—41no—
286D4—42no—
287D4—81no—
288D4—7761no—
289D4—3882no—
290D4—1941no—
291D4—1941no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—7761no—
302C2×C4—7761no—
303C2×C4—1941no—
304C2×C4—1941no—
305C2×C4—1941no—
306C2×C4C2×C22⋊D9711yescharacteristic
307C2×C4C2×C22⋊D9711yescharacteristic
308C2×C4C2×C22⋊D9711yescharacteristic
309C2×C4C2×C22⋊D9711yescharacteristic
310C2×C4C23×D9711yescharacteristic
311C8—972no—
312C8—1941no—
313C8—1941no—
314C8—1941no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—3881no—
325C22—3881no—
326C22—3881no—
327C22C22×C22⋊D9711yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—3881no—
336C4—3881no—
337C4—3881no—
338C4#41×D9711yescharacteristic
339C4#1569085711yescharacteristic
340C2—41no—
341C2—41no—
342C2—81no—
343C2—7761no—
344C2#2026138411yescharacteristic
345C2#2026153311yescharacteristic
346C2#2026153811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#553×C97C2
socleC22×C97C22×D4
hypercenter#553D97
hypocenterC97#3355
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #20261431C2semidirect
2. #20261134C2semidirect
3. #20261059C2semidirect
4. #20261056C2semidirect
5. #20261048C2semidirect
6. #20261046C2semidirect
7. #20261035C2semidirect
8. #20261032C2semidirect
9. #20260986C2semidirect
10. #20260986C2semidirect
11. C4×#13095496C2semidirect
12. #20260813C2semidirect
13. #20260778C2semidirect
14. #20260775C2semidirect
15. #553×C97C2semidirect
174. C97#3355semidirect
176. #553D97semidirect

Maximal subgroups

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

subgroupindex
#202614312
#202611342
#202610592
#202610562
#202610482
#202610462
#202610352
#202610322
#202609862
C4×#130954962
#202608132
#202607782
#202607752
#553×C972
#335597