Finite Group Databasebeta

← #27153590

Subgroups — #27153590

Order 12416.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261437C211yesmaximal, characteristic, complemented
2#20261379C211yesmaximal, characteristic, complemented
3#20261088C211yesmaximal, characteristic, complemented
4#20261071C211yesmaximal, characteristic, complemented
5#20261057C211yesmaximal, characteristic, complemented
6#20261033C211yesmaximal, characteristic, complemented
7#20260968C211yesmaximal, characteristic, complemented
8#20260962C211yesmaximal, characteristic, complemented
9#20260948C211yesmaximal, characteristic, complemented
10#20260877C211yesmaximal, characteristic, complemented
11D8×Dic97C211yesmaximal, characteristic, complemented
12#20260798C211yesmaximal, characteristic, complemented
13#20260728C211yesmaximal, characteristic, complemented
14C2×#15690731C211yesmaximal, characteristic, complemented
15#447×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic
17C2×#13095499C2211yescharacteristic, complemented
18C22×#11715895C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic, complemented
20#15690803—21no—
21#15690801C2211yescharacteristic, complemented
22#15690797C2211yescharacteristic
23#15690794—21no—
24#15690791—21no—
25#15690789C2211yescharacteristic
26#15690782—21no—
27#15690782—21no—
28#15690781C2211yescharacteristic
29#15690781C2211yescharacteristic, complemented
30#15690779—21no—
31D4×Dic97—21no—
32D4×Dic97C2211yescharacteristic
33D4×Dic97C2211yescharacteristic, complemented
34C4×C22⋊D97C2211yescharacteristic
35#15690768C2211yescharacteristic, complemented
36C2×C4⋊Dic97C2211yescharacteristic
37C2×C22⋊Dic97—21no—
38#15690744C2211yescharacteristic, complemented
39#15690744C2211yescharacteristic, complemented
40#15690741C2211yescharacteristic, complemented
41#15690739C2211yescharacteristic, complemented
42#15690735C2211yescharacteristic, complemented
43#15690735C2211yescharacteristic, complemented
44#15690733C2211yescharacteristic, complemented
45#15690731C2211yescharacteristic
46#15690731C2211yescharacteristic
47#15690731C2211yescharacteristic
48#15690731C2211yescharacteristic
49#15690731C2211yescharacteristic, complemented
50#15690722C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C22×D4×C97C2211yescharacteristic
53C2×D8×C97C2211yescharacteristic
54C2×D8×C97C2211yescharacteristic
55#121×C97C2211yescharacteristic
56#102×C97C2211yescharacteristic
57#102×C97C2211yescharacteristic
58C22⋊C8×C97C2211yescharacteristic
59#13095510—22no—
60#13095510—41no—
61C2×Dic194C2311yescharacteristic
62C2×C22⋊D97—21no—
63C2×C22⋊D97—21no—
64C2×C22⋊D97—21no—
65C2×C22⋊D97D411yescharacteristic
66C22×Dic97—21no—
67C22×Dic97—21no—
68C22×Dic97—21no—
69C22×Dic97—21no—
70C2×C4×D97C2311yescharacteristic
71#13095499—41no—
72C2×#11715895—41no—
73C2×#11715895C2311yescharacteristic
74C2×#11715895C2311yescharacteristic
75C8×D97—41no—
76C4⋊Dic97—21no—
77C4⋊Dic97—21no—
78C4⋊Dic97C2311yescharacteristic
79C4⋊Dic97C2311yescharacteristic
80C4⋊Dic97C2311yescharacteristic
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⋊Dic97—21no—
91C22⋊Dic97D411yescharacteristic
92#13095488—21no—
93#13095488D411yescharacteristic
94C4×Dic97—21no—
95C4×Dic97C2311yescharacteristic
96C24×C97—21no—
97C2×D4×C97—22no—
98C2×D4×C97—21no—
99C2×D4×C97—21no—
100C2×D4×C97—41no—
101C2×D4×C97C2311yescharacteristic
102C2×D4×C97C2311yescharacteristic
103C2×D4×C97C2311yescharacteristic
104C22×C4×C97C2311yescharacteristic
105D8×C97—22no—
106D8×C97—41no—
107C2×C8×C97C2311yescharacteristic
108C2×C8×C97C2311yescharacteristic
109C4⋊C4×C97C2311yescharacteristic
110C22⋊C4×C97—21no—
111C23×C97—21no—
112C23×C97—21no—
113C23×C97—21no—
114C23×C97—42no—
115C23×C97—42no—
116C23×C97—81no—
117C23×C97C2×D411yescharacteristic
118C22×D97C2×D411yescharacteristic
119D4×C97—21no—
120D4×C97—21no—
121D4×C97—21no—
122D4×C97—21no—
123D4×C97—21no—
124D4×C97—42no—
125D4×C97—42no—
126D4×C97#4112yes—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97—21no—
130C2×C4×C97C2×D411yescharacteristic
131C2×C4×C97C2411yescharacteristic
132C22⋊D97—22no—
133C22⋊D97—42no—
134C22⋊D97—81no—
135C22⋊D97—81no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—21no—
139C2×Dic97—21no—
140C2×Dic97—42no—
141C2×Dic97—81no—
142C2×Dic97—81no—
143C2×Dic97—81no—
144C2×Dic97C2×D411yescharacteristic
145C2×Dic97C2×D411yescharacteristic
146C2×Dic97C2×D411yescharacteristic
147C4×D97—21no—
148C4×D97—21no—
149Dic194—21no—
150Dic194—21no—
151C8×C97—21no—
152C8×C97—21no—
153#11715895—21no—
154#11715895—21no—
155C22×C97—21no—
156C22×C97—21no—
157C22×C97—22no—
158C22×C97—41no—
159C22×C97—41no—
160C22×C97—41no—
161C22×C97—41no—
162C22×C97—41no—
163C22×C97—41no—
164C22×C97—41no—
165C22×C97—41no—
166C22×C97—84no—
167C22×C97#13512yes—
168C22×C97C22×D411yescharacteristic
169C2×D97—21no—
170C2×D97—21no—
171C2×D97—21no—
172C4×C97—21no—
173C4×C97—41no—
174C4×C97C2×#4111yescharacteristic
175C4×C97H(5,𝔽2)11yescharacteristic
176Dic97—21no—
177Dic97—21no—
178Dic97—21no—
179Dic97—41no—
180Dic97—41no—
181Dic97—41no—
182Dic97—41no—
183C2×C97—22no—
184C2×C97—42no—
185C2×C97—81no—
186C2×C97—81no—
187C2×C97#54611yescharacteristic
188C2×C97C2×#13511yescharacteristic
189C2×C97#57611yescharacteristic
190D97—41no—
191#3297—971nomaximal, Sylow
192C97#329711yescharacteristic, complemented, Sylow
193#553—971no—
194#546—971no—
195#488—971no—
196#484—971no—
197#482—971no—
198#478—971no—
199#466—971no—
200#465—971no—
201#463—971no—
202#449—971no—
203#447D9711yescharacteristic, complemented
204C4×D8—971no—
205C8×D4—971no—
206#416—971no—
207C2×#102—971no—
208C2×#41—971no—
209C22×D4C2×D9711yescharacteristic
210C2×SD16—971no—
211C2×D8C2×D9711yescharacteristic
212C2×D8C2×D9711yescharacteristic
213C22×C8—971no—
214C22×C8—971no—
215#126—1941no—
216#125—971no—
217#124—971no—
218#123—1941no—
219#123—1941no—
220C22⋊Q8—971no—
221#121—971no—
222#121—971no—
223#121—1941no—
224#121—1941no—
225#121C2×D9711yescharacteristic
226#120—1941no—
227C4×D4—971no—
228C4×D4—971no—
229C4×D4—971no—
230C4×D4—1941no—
231#117—971no—
232C2×C4⋊C4—971no—
233C2×C22⋊C4—1941no—
234#107—971no—
235#107—971no—
236#106—971no—
237C4⋊C8—971no—
238#103—971no—
239#103—971no—
240#103—971no—
241#102—971no—
242#102—971no—
243#102—971no—
244#102—971no—
245#102—971no—
246#102C2×D9711yescharacteristic
247#102C2×D9711yescharacteristic
248C22⋊C8—971no—
249C22⋊C8C2×D9711yescharacteristic
250C4×C8—971no—
251C24—21no—
252#41—3881no—
253#41—1942no—
254C2×Q8—971no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C2×D4—971no—
260C2×D4—1941no—
261C2×D4—1941no—
262C2×D4—1941no—
263C2×D4C22×D9711yescharacteristic
264C2×D4C22×D9711yescharacteristic
265C2×D4C22×D9711yescharacteristic
266C22×C4—971no—
267C22×C4—1941no—
268C22×C4—1941no—
269C22×C4—1941no—
270C22×C4—1941no—
271C22×C4C22×D9711yescharacteristic
272SD16—3881no—
273D8—22no—
274D8—41no—
275C2×C8—971no—
276C2×C8—971no—
277C2×C8—3881no—
278C2×C8—3881no—
279C2×C8C22×D9711yescharacteristic
280C2×C8C22×D9711yescharacteristic
281C4⋊C4—971no—
282C4⋊C4—971no—
283C4⋊C4—971no—
284C4⋊C4—971no—
285C4⋊C4—1941no—
286C4⋊C4—1941no—
287C4⋊C4—1941no—
288C4⋊C4—1941no—
289C4⋊C4C22×D9711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—971no—
292C22⋊C4—971no—
293C22⋊C4—1941no—
294C22⋊C4—1941no—
295C22⋊C4—1941no—
296C22⋊C4—1941no—
297C22⋊C4—1941no—
298C22⋊C4—1941no—
299C22⋊C4—1941no—
300C22⋊C4—1941no—
301C42—971no—
302C42—1941no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—42no—
307C23—42no—
308C23—81no—
309C23—971no—
310C23D4×D9711yescharacteristic
311Q8—1941no—
312Q8—1941no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—42no—
319D4—42no—
320D4—7761no—
321D4—7761no—
322D4—3882no—
323D4—1942no—
324D4#1309551012yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—7761no—
329C2×C4—7761no—
330C2×C4—7761no—
331C2×C4—971no—
332C2×C4—971no—
333C2×C4—971no—
334C2×C4—3882no—
335C2×C4—1941no—
336C2×C4—1941no—
337C2×C4—1941no—
338C2×C4—1941no—
339C2×C4—1941no—
340C2×C4—1941no—
341C2×C4D4×D9711yescharacteristic
342C2×C4C23×D9711yescharacteristic
343C8—21no—
344C8—21no—
345C8—1941no—
346C8—1941no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—1941no—
360C22—1941no—
361C22—1941no—
362C22#1569082912yes—
363C22C2×D4×D9711yescharacteristic
364C4—21no—
365C4—41no—
366C4—3881no—
367C4—3881no—
368C4—3881no—
369C4—3881no—
370C4—1941no—
371C4—1941no—
372C4—1941no—
373C4C2×#1309551011yescharacteristic
374C4#1569085511yescharacteristic
375C2—22no—
376C2—42no—
377C2—81no—
378C2—81no—
379C2—3881no—
380C2#2026137211yescharacteristic
381C2C2×#1569082911yescharacteristic
382C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#447×C97C2
socleC22×C97C22×D4
hypercenter#447D97
hypocenterC97#3297
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #20261437C2semidirect
2. #20261379C2semidirect
3. #20261088C2semidirect
4. #20261071C2semidirect
5. #20261057C2semidirect
6. #20261033C2semidirect
7. #20260968C2semidirect
8. #20260962C2semidirect
9. #20260948C2semidirect
10. #20260877C2semidirect
11. D8×Dic97C2semidirect
12. #20260798C2semidirect
13. #20260728C2semidirect
14. C2×#15690731C2semidirect
15. #447×C97C2semidirect
17. C2×#13095499C22semidirect
19. C2×C8×D97C22semidirect
21. #15690801C22semidirect
29. #15690781C22semidirect
33. D4×Dic97C22semidirect
35. #15690768C22semidirect
38. #15690744C22semidirect
39. #15690744C22semidirect
40. #15690741C22semidirect
41. #15690739C22semidirect
42. #15690735C22semidirect
43. #15690735C22semidirect
44. #15690733C22semidirect
49. #15690731C22semidirect
50. #15690722C22semidirect
51. C8×Dic97C22semidirect
192. C97#3297semidirect
203. #447D97semidirect

Maximal subgroups

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

subgroupindex
#202614372
#202613792
#202610882
#202610712
#202610572
#202610332
#202609682
#202609622
#202609482
#202608772
D8×Dic972
#202607982
#202607282
C2×#156907312
#447×C972
#329797