Finite Group Databasebeta

← #27151982

Subgroups — #27151982

Order 12416.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261345C211yesmaximal, characteristic, complemented
2#20261210C211yesmaximal, characteristic
3#20261077C211yesmaximal, characteristic, complemented
4#20261073C211yesmaximal, characteristic, complemented
5#20261037C211yesmaximal, characteristic, complemented
6#20261033C211yesmaximal, characteristic, complemented
7#20260938C211yesmaximal, characteristic, complemented
8#20260938C211yesmaximal, characteristic, complemented
9#20260886C211yesmaximal, characteristic
10#20260836C211yesmaximal, characteristic, complemented
11#20260832C211yesmaximal, characteristic, complemented
12#20260826C211yesmaximal, characteristic, complemented
13SD16×Dic97C211yesmaximal, characteristic, complemented
14#20260793C211yesmaximal, characteristic, complemented
15#450×C97C211yesmaximal, characteristic
16C2×#13095499C2211yescharacteristic
17C2×#13095499C2211yescharacteristic
18#15690808C2211yescharacteristic, complemented
19#15690803—21no—
20#15690803—21no—
21#15690801—21no—
22#15690801C2211yescharacteristic, complemented
23#15690800—21no—
24#15690789C2211yescharacteristic
25Q8×Dic97C2211yescharacteristic, complemented
26#15690775C2211yescharacteristic, complemented
27D4×Dic97—21no—
28D4×Dic97—21no—
29D4×Dic97C2211yescharacteristic
30D4×Dic97C2211yescharacteristic
31D4×Dic97C2211yescharacteristic
32D4×Dic97C2211yescharacteristic
33#15690769C2211yescharacteristic
34#15690769C2211yescharacteristic
35C2×C22⋊Dic97—21no—
36C2×C22⋊Dic97—21no—
37#15690744C2211yescharacteristic, complemented
38#15690743C2211yescharacteristic, complemented
39#15690742C2211yescharacteristic, complemented
40#15690741C2211yescharacteristic, complemented
41#15690739C2211yescharacteristic, complemented
42#15690739C2211yescharacteristic, complemented
43#15690735C2211yescharacteristic, complemented
44#15690735C2211yescharacteristic, complemented
45#15690734C2211yescharacteristic, complemented
46#15690734C2211yescharacteristic, complemented
47#15690731C2211yescharacteristic
48#15690731C2211yescharacteristic
49C22⋊#11715895C2211yescharacteristic
50#15690720C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C22×D4×C97C2211yescharacteristic
53C2×SD16×C97C2211yescharacteristic
54C2×SD16×C97C2211yescharacteristic
55C22⋊Q8×C97C2211yescharacteristic
56#102×C97C2211yescharacteristic
57#102×C97C2211yescharacteristic
58C22⋊C8×C97C2211yescharacteristic
59C2×Dic194C2311yescharacteristic
60C22×Dic97—21no—
61C22×Dic97—21no—
62C22×Dic97—21no—
63C22×Dic97—21no—
64#13095499—22no—
65#13095499—22no—
66C2×#11715895C2311yescharacteristic
67C2×#11715895C2311yescharacteristic
68C4⋊Dic97—21no—
69C4⋊Dic97—21no—
70C4⋊Dic97D411yescharacteristic
71C4⋊Dic97D411yescharacteristic
72C4⋊Dic97C2311yescharacteristic
73C4⋊Dic97C2311yescharacteristic
74C4⋊Dic97C2311yescharacteristic
75#13095490—21no—
76#13095490—21no—
77#13095490—21no—
78#13095490—21no—
79#13095490—21no—
80C22⋊Dic97—21no—
81C22⋊Dic97—21no—
82C22⋊Dic97—21no—
83C22⋊Dic97—21no—
84C22⋊Dic97—21no—
85C22⋊Dic97—21no—
86C22⋊Dic97D411yescharacteristic
87C22⋊Dic97D411yescharacteristic
88C4×Dic97—21no—
89C4×Dic97—21no—
90C4×Dic97—21no—
91C4×Dic97C2311yescharacteristic
92C4×Dic97C2311yescharacteristic
93C24×C97—21no—
94C2×Q8×C97C2311yescharacteristic
95C2×D4×C97—22no—
96C2×D4×C97—22no—
97C2×D4×C97—21no—
98C2×D4×C97C2311yescharacteristic
99C2×D4×C97C2311yescharacteristic
100C22×C4×C97C2311yescharacteristic
101SD16×C97—22no—
102SD16×C97—22no—
103C2×C8×C97C2311yescharacteristic
104C2×C8×C97C2311yescharacteristic
105C4⋊C4×C97—21no—
106C4⋊C4×C97C2311yescharacteristic
107C22⋊C4×C97—21no—
108C23×C97—24no—
109C23×C97—21no—
110C23×C97—21no—
111C23×C97—81no—
112C23×C97—81no—
113C23×C97C2×D411yescharacteristic
114Q8×C97—21no—
115Q8×C97—21no—
116D4×C97—24no—
117D4×C97—21no—
118D4×C97—21no—
119D4×C97#4112yes—
120D4×C97#4112yes—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2411yescharacteristic
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—81no—
132C2×Dic97—81no—
133C2×Dic97—81no—
134C2×Dic97—81no—
135C2×Dic97C2×D411yescharacteristic
136C2×Dic97C2×D411yescharacteristic
137C2×Dic97C2×D411yescharacteristic
138C2×Dic97C2×D411yescharacteristic
139Dic194—21no—
140Dic194—21no—
141C8×C97—21no—
142C8×C97—21no—
143#11715895—21no—
144#11715895—21no—
145C22×C97—21no—
146C22×C97—22no—
147C22×C97—22no—
148C22×C97—21no—
149C22×C97—21no—
150C22×C97—41no—
151C22×C97—41no—
152C22×C97—41no—
153C22×C97—41no—
154C22×C97—84no—
155C22×C97C22×D411yescharacteristic
156C4×C97—21no—
157C4×C97—41no—
158C4×C97—41no—
159C4×C97C2×#4111yescharacteristic
160C4×C97C2×#4111yescharacteristic
161Dic97—21no—
162Dic97—21no—
163Dic97—21no—
164Dic97—21no—
165Dic97—41no—
166Dic97—41no—
167Dic97—41no—
168Dic97—41no—
169C2×C97—41no—
170C2×C97—42no—
171C2×C97—42no—
172C2×C97#52511yescharacteristic
173C2×C97#57711yescharacteristic
174C2×C97#57711yescharacteristic
175#3123—971nomaximal, Sylow
176C97#312311yescharacteristic, complemented, Sylow
177#541—971no—
178#518—971no—
179#485—971no—
180#485—971no—
181#478—971no—
182#478—971no—
183#461—971no—
184#461—971no—
185#450—971no—
186#450D9711yescharacteristic
187#440—971no—
188#440—971no—
189C4×SD16—971no—
190C4×SD16—971no—
191#433—971no—
192C22×D4C2×D9711yescharacteristic
193C2×SD16—971no—
194C2×SD16—971no—
195C2×SD16C2×D9711yescharacteristic
196C2×SD16C2×D9711yescharacteristic
197C4⋊Q8—971no—
198#126—1941no—
199#126—1941no—
200#125—971no—
201#125—1941no—
202#125—1941no—
203C22⋊Q8—971no—
204C22⋊Q8C2×D9711yescharacteristic
205C4×Q8—971no—
206C4×Q8—971no—
207C4×D4—971no—
208C4×D4—971no—
209C4×D4—971no—
210C4×D4—971no—
211C4×D4—1941no—
212C4×D4—1941no—
213#117—971no—
214#117—971no—
215C2×C22⋊C4—1941no—
216C2×C22⋊C4—1941no—
217#107—971no—
218#107—971no—
219#106—971no—
220#106—971no—
221C4⋊C8—971no—
222C4⋊C8—971no—
223#103—971no—
224#103—971no—
225#103—971no—
226#103—971no—
227#102—971no—
228#102—971no—
229#102C2×D9711yescharacteristic
230#102C2×D9711yescharacteristic
231C22⋊C8—971no—
232C22⋊C8C2×D9711yescharacteristic
233#97—971no—
234C4×C8—971no—
235C24—21no—
236C2×Q8—971no—
237C2×Q8C22×D9711yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4C22×D9711yescharacteristic
242C2×D4C22×D9711yescharacteristic
243C22×C4—1941no—
244C22×C4—1941no—
245C22×C4—1941no—
246C22×C4—1941no—
247C22×C4C22×D9711yescharacteristic
248SD16—22no—
249SD16—22no—
250SD16—1942no—
251SD16—1942no—
252C2×C8—971no—
253C2×C8—971no—
254C2×C8C22×D9711yescharacteristic
255C2×C8C22×D9711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—971no—
258C4⋊C4—971no—
259C4⋊C4—971no—
260C4⋊C4—971no—
261C4⋊C4—971no—
262C4⋊C4—1941no—
263C4⋊C4—1941no—
264C4⋊C4—1941no—
265C4⋊C4—1941no—
266C4⋊C4—1941no—
267C4⋊C4—1941no—
268C4⋊C4—1941no—
269C4⋊C4C22×D9711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—971no—
272C22⋊C4—971no—
273C22⋊C4—1941no—
274C22⋊C4—1941no—
275C22⋊C4—1941no—
276C22⋊C4—1941no—
277C22⋊C4—1941no—
278C22⋊C4—1941no—
279C42—971no—
280C42—971no—
281C42—1941no—
282C42—1941no—
283C42—1941no—
284C23—24no—
285C23—21no—
286C23—21no—
287C23—81no—
288C23—81no—
289C23D4×D9711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—1941no—
293Q8—1941no—
294D4—24no—
295D4—21no—
296D4—21no—
297D4#1309551012yes—
298D4#1309551012yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—7761no—
304C2×C4—7761no—
305C2×C4—7761no—
306C2×C4—7761no—
307C2×C4—971no—
308C2×C4—971no—
309C2×C4—971no—
310C2×C4—971no—
311C2×C4—1941no—
312C2×C4—1941no—
313C2×C4—1941no—
314C2×C4—1941no—
315C2×C4D4×D9711yescharacteristic
316C2×C4C23×D9711yescharacteristic
317C8—21no—
318C8—21no—
319C8—1941no—
320C8—1941no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—84no—
331C22C2×D4×D9711yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—3881no—
336C4—3881no—
337C4—3881no—
338C4—3881no—
339C4—1941no—
340C4—1941no—
341C4—1941no—
342C4—1941no—
343C4C2×#1309551011yescharacteristic
344C4C2×#1309551011yescharacteristic
345C2—41no—
346C2—42no—
347C2—42no—
348C2#2026124111yescharacteristic
349C2#2026154111yescharacteristic
350C2#2026154511yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#450×C97C2
socleC22×C97C22×D4
prime radicalC22×D4×C97C22
hypercenter#450D97
hypocenterC97#3123
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20261345C2semidirect
3. #20261077C2semidirect
4. #20261073C2semidirect
5. #20261037C2semidirect
6. #20261033C2semidirect
7. #20260938C2semidirect
8. #20260938C2semidirect
10. #20260836C2semidirect
11. #20260832C2semidirect
12. #20260826C2semidirect
13. SD16×Dic97C2semidirect
14. #20260793C2semidirect
18. #15690808C22semidirect
22. #15690801C22semidirect
25. Q8×Dic97C22semidirect
26. #15690775C22semidirect
37. #15690744C22semidirect
38. #15690743C22semidirect
39. #15690742C22semidirect
40. #15690741C22semidirect
41. #15690739C22semidirect
42. #15690739C22semidirect
43. #15690735C22semidirect
44. #15690735C22semidirect
45. #15690734C22semidirect
46. #15690734C22semidirect
50. #15690720C22semidirect
51. C8×Dic97C22semidirect
176. C97#3123semidirect

Maximal subgroups

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

subgroupindex
#202613452
#202612102
#202610772
#202610732
#202610372
#202610332
#202609382
#202608862
#202608362
#202608322
#202608262
SD16×Dic972
#202607932
#450×C972
#312397