Finite Group Databasebeta

← #27153567

Subgroups — #27153567

Order 12416.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261420C211yesmaximal, characteristic, complemented
2#20261379C211yesmaximal, characteristic
3#20261088C211yesmaximal, characteristic, complemented
4#20261065C211yesmaximal, characteristic, complemented
5#20261057C211yesmaximal, characteristic, complemented
6#20261033C211yesmaximal, characteristic, complemented
7#20260976C211yesmaximal, characteristic, complemented
8#20260963C211yesmaximal, characteristic, complemented
9#20260956C211yesmaximal, characteristic, complemented
10#20260898C211yesmaximal, characteristic, complemented
11#20260886C211yesmaximal, characteristic, complemented
12#20260826C211yesmaximal, characteristic, complemented
13C8×C22⋊D97C211yesmaximal, characteristic, complemented
14#20260728C211yesmaximal, characteristic, complemented
15C2×#102×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic
17C2×Dic388C2211yescharacteristic, complemented
18C2×#13095499C2211yescharacteristic
19C2×#13095499C2211yescharacteristic
20C2×C8×D97C2211yescharacteristic, complemented
21#15690805—21no—
22#15690801C2211yescharacteristic, complemented
23#15690800—21no—
24#15690797C2211yescharacteristic
25#15690793—21no—
26#15690791—21no—
27#15690790C2211yescharacteristic, complemented
28#15690789C2211yescharacteristic
29#15690789C2211yescharacteristic
30#15690782—21no—
31#15690781C2211yescharacteristic
32#15690779—21no—
33#15690775C2211yescharacteristic, complemented
34D4×Dic97C2211yescharacteristic
35C4×C22⋊D97C2211yescharacteristic
36#15690769—21no—
37#15690768C2211yescharacteristic, complemented
38C2×C22⋊Dic97—21no—
39#15690744C2211yescharacteristic, complemented
40#15690741C2211yescharacteristic, complemented
41#15690741C2211yescharacteristic, complemented
42#15690739C2211yescharacteristic, complemented
43#15690735C2211yescharacteristic, complemented
44#15690735C2211yescharacteristic, complemented
45#15690733C2211yescharacteristic, complemented
46#15690733C2211yescharacteristic, complemented
47#15690731C2211yescharacteristic
48#15690731C2211yescharacteristic
49C22⋊#11715895C2211yescharacteristic
50#15690722C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C22×D4×C97C2211yescharacteristic
53C22×C8×C97C2211yescharacteristic
54C2×C4⋊C4×C97C2211yescharacteristic
55#102×C97C2211yescharacteristic
56#102×C97C2211yescharacteristic
57#102×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic
59#13095510—22no—
60#13095510—41no—
61C2×Dic194C2311yescharacteristic
62C2×Dic194C2311yescharacteristic
63C2×C22⋊D97—21no—
64C2×C22⋊D97—21no—
65C2×C22⋊D97D411yescharacteristic
66C22×Dic97—21no—
67C22×Dic97—21no—
68C2×C4×D97C2311yescharacteristic
69Dic388—41no—
70#13095499—22no—
71#13095499—41no—
72C2×#11715895C2311yescharacteristic
73C2×#11715895C2311yescharacteristic
74C8×D97—41no—
75C4⋊Dic97—21no—
76C4⋊Dic97—21no—
77C4⋊Dic97C2311yescharacteristic
78C4⋊Dic97C2311yescharacteristic
79#13095490—21no—
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83#13095490D411yescharacteristic
84C22⋊Dic97—21no—
85C22⋊Dic97—21no—
86C22⋊Dic97—21no—
87C22⋊Dic97—21no—
88C22⋊Dic97—21no—
89C22⋊Dic97—21no—
90C22⋊Dic97D411yescharacteristic
91#13095488—21no—
92#13095488—21no—
93#13095488D411yescharacteristic
94C4×Dic97—21no—
95C4×Dic97—21no—
96C4×Dic97C2311yescharacteristic
97C24×C97—21no—
98C2×D4×C97—22no—
99C2×D4×C97—21no—
100C2×D4×C97—41no—
101C2×D4×C97C2311yescharacteristic
102C2×D4×C97C2311yescharacteristic
103C22×C4×C97—21no—
104C22×C4×C97C2311yescharacteristic
105C2×C8×C97—41no—
106C2×C8×C97C2311yescharacteristic
107C2×C8×C97C2311yescharacteristic
108C4⋊C4×C97—21no—
109C4⋊C4×C97C2311yescharacteristic
110C4⋊C4×C97C2311yescharacteristic
111C23×C97—21no—
112C23×C97—21no—
113C23×C97—42no—
114C23×C97—42no—
115C23×C97—81no—
116C23×C97C2×D411yescharacteristic
117C22×D97C2×D411yescharacteristic
118D4×C97—21no—
119D4×C97—21no—
120D4×C97—21no—
121D4×C97—42no—
122D4×C97#4112yes—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—42no—
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2411yescharacteristic
130C22⋊D97—22no—
131C22⋊D97—42no—
132C22⋊D97—81no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97—42no—
138C2×Dic97—81no—
139C2×Dic97C2×D411yescharacteristic
140C2×Dic97C2×D411yescharacteristic
141C2×Dic97C2×D411yescharacteristic
142C4×D97—21no—
143C4×D97—21no—
144Dic194—21no—
145Dic194—21no—
146Dic194—21no—
147Dic194—21no—
148C8×C97—21no—
149C8×C97—21no—
150#11715895—21no—
151#11715895—21no—
152C22×C97—21no—
153C22×C97—21no—
154C22×C97—22no—
155C22×C97—41no—
156C22×C97—41no—
157C22×C97—41no—
158C22×C97—41no—
159C22×C97—41no—
160C22×C97—84no—
161C22×C97#13512yes—
162C22×C97C22×D411yescharacteristic
163C2×D97—21no—
164C2×D97—21no—
165C2×D97—21no—
166C4×C97—21no—
167C4×C97—41no—
168C4×C97—41no—
169C4×C97C2×#4111yescharacteristic
170C4×C97H(5,𝔽2)11yescharacteristic
171Dic97—21no—
172Dic97—21no—
173Dic97—21no—
174Dic97—41no—
175Dic97—41no—
176Dic97—41no—
177Dic97—41no—
178C2×C97—22no—
179C2×C97—42no—
180C2×C97—81no—
181C2×C97#54611yescharacteristic
182C2×C97C2×#13511yescharacteristic
183C2×C97#57711yescharacteristic
184D97—41no—
185#3296—971nomaximal, Sylow
186C97#329611yescharacteristic, complemented, Sylow
187#552—971no—
188#546—971no—
189#488—971no—
190#483—971no—
191#482—971no—
192#478—971no—
193#467—971no—
194#465—971no—
195#464—971no—
196#452—971no—
197#450—971no—
198C4×SD16—971no—
199C8×D4—971no—
200#416—971no—
201C2×#102D9711yescharacteristic, complemented
202C2×#41—971no—
203C22×D4C2×D9711yescharacteristic
204C2×Q16—971no—
205C2×SD16—971no—
206C2×SD16—971no—
207C22×C8—971no—
208C22×C8C2×D9711yescharacteristic
209#126—1941no—
210#125—971no—
211#125—1941no—
212#124—971no—
213#123—1941no—
214#123—1941no—
215C22⋊Q8—971no—
216C22⋊Q8—971no—
217C22⋊Q8—971no—
218#121—971no—
219#121—1941no—
220#120—1941no—
221C4×Q8—971no—
222C4×D4—971no—
223C4×D4—971no—
224#117—971no—
225#117—1941no—
226C2×C4⋊C4C2×D9711yescharacteristic
227C2×C22⋊C4—1941no—
228#107—971no—
229#106—971no—
230#106—971no—
231C4⋊C8—971no—
232#103—971no—
233#103—971no—
234#103—971no—
235#103—971no—
236#102—971no—
237#102—971no—
238#102C2×D9711yescharacteristic
239#102C2×D9711yescharacteristic
240#102C2×D9711yescharacteristic
241#102C2×D9711yescharacteristic
242C22⋊C8—971no—
243C22⋊C8—971no—
244C4×C8—971no—
245C24—21no—
246#41—3881no—
247#41—1942no—
248C2×Q8—971no—
249C2×Q8—971no—
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—971no—
254C2×D4—1941no—
255C2×D4—1941no—
256C2×D4C22×D9711yescharacteristic
257C2×D4C22×D9711yescharacteristic
258C22×C4—21no—
259C22×C4—971no—
260C22×C4—1941no—
261C22×C4—1941no—
262C22×C4C22×D9711yescharacteristic
263Q16—3881no—
264SD16—3881no—
265SD16—1942no—
266C2×C8—41no—
267C2×C8—971no—
268C2×C8—971no—
269C2×C8—3881no—
270C2×C8C22×D9711yescharacteristic
271C2×C8C22×D9711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—971no—
274C4⋊C4—971no—
275C4⋊C4—971no—
276C4⋊C4—1941no—
277C4⋊C4—1941no—
278C4⋊C4—1941no—
279C4⋊C4—1941no—
280C4⋊C4—1941no—
281C4⋊C4—1941no—
282C4⋊C4C22×D9711yescharacteristic
283C4⋊C4C22×D9711yescharacteristic
284C22⋊C4—971no—
285C22⋊C4—971no—
286C22⋊C4—1941no—
287C22⋊C4—1941no—
288C22⋊C4—1941no—
289C22⋊C4—1941no—
290C22⋊C4—1941no—
291C22⋊C4—1941no—
292C22⋊C4—1941no—
293C22⋊C4—1941no—
294C42—971no—
295C42—1941no—
296C42—1941no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—971no—
303C23D4×D9711yescharacteristic
304Q8—1941no—
305Q8—1941no—
306Q8—1941no—
307Q8—1941no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—7761no—
313D4—3882no—
314D4—1942no—
315D4#1309551112yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—7761no—
322C2×C4—971no—
323C2×C4—971no—
324C2×C4—971no—
325C2×C4—3882no—
326C2×C4—1941no—
327C2×C4—1941no—
328C2×C4—1941no—
329C2×C4—1941no—
330C2×C4—1941no—
331C2×C4—1941no—
332C2×C4D4×D9711yescharacteristic
333C2×C4C23×D9711yescharacteristic
334C8—21no—
335C8—21no—
336C8—1941no—
337C8—1941no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22—1941no—
348C22—1941no—
349C22—1941no—
350C22#1569082912yes—
351C22C2×D4×D9711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—3881no—
356C4—3881no—
357C4—3881no—
358C4—3881no—
359C4—1941no—
360C4—1941no—
361C4—1941no—
362C4C2×#1309551111yescharacteristic
363C4#1569085511yescharacteristic
364C2—22no—
365C2—42no—
366C2—81no—
367C2—3881no—
368C2#2026137511yescharacteristic
369C2C2×#1569082911yescharacteristic
370C2#2026154111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#102×C97C2
socleC22×C97C22×D4
prime radical#20261379C2
hypercenterC2×#102D97
hypocenterC97#3296
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #20261420C2semidirect
3. #20261088C2semidirect
4. #20261065C2semidirect
5. #20261057C2semidirect
6. #20261033C2semidirect
7. #20260976C2semidirect
8. #20260963C2semidirect
9. #20260956C2semidirect
10. #20260898C2semidirect
11. #20260886C2semidirect
12. #20260826C2semidirect
13. C8×C22⋊D97C2semidirect
14. #20260728C2semidirect
15. C2×#102×C97C2semidirect
17. C2×Dic388C22semidirect
20. C2×C8×D97C22semidirect
22. #15690801C22semidirect
27. #15690790C22semidirect
33. #15690775C22semidirect
37. #15690768C22semidirect
39. #15690744C22semidirect
40. #15690741C22semidirect
41. #15690741C22semidirect
42. #15690739C22semidirect
43. #15690735C22semidirect
44. #15690735C22semidirect
45. #15690733C22semidirect
46. #15690733C22semidirect
50. #15690722C22semidirect
51. C8×Dic97C22semidirect
186. C97#3296semidirect
201. C2×#102D97semidirect

Maximal subgroups

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

subgroupindex
#202614202
#202613792
#202610882
#202610652
#202610572
#202610332
#202609762
#202609632
#202609562
#202608982
#202608862
#202608262
C8×C22⋊D972
#202607282
C2×#102×C972
#329697