Finite Group Databasebeta

← #30650567

Subgroups — #30650567

Order 16256.

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#22550566C211yesmaximal, characteristic, complemented
2#22550522C211yesmaximal, characteristic
3#22550231C211yesmaximal, characteristic, complemented
4#22550207C211yesmaximal, characteristic, complemented
5#22550201C211yesmaximal, characteristic, complemented
6#22550177C211yesmaximal, characteristic, complemented
7#22550117C211yesmaximal, characteristic, complemented
8#22550105C211yesmaximal, characteristic, complemented
9#22550099C211yesmaximal, characteristic, complemented
10#22550040C211yesmaximal, characteristic, complemented
11#22550029C211yesmaximal, characteristic, complemented
12C4×#14523237C211yesmaximal, characteristic, complemented
13#22549873C211yesmaximal, characteristic, complemented
14C2×#16861131C211yesmaximal, characteristic, complemented
15C8×D4×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C22×D508C2211yescharacteristic
18C2×Dic508C2211yescharacteristic, complemented
19C2×#14523237C2211yescharacteristic
20C2×#14523237C2211yescharacteristic
21#16861202—21no—
22#16861201C2211yescharacteristic, complemented
23#16861199—21no—
24#16861197C2211yescharacteristic
25#16861194—21no—
26#16861190—21no—
27#16861188C2211yescharacteristic
28#16861188C2211yescharacteristic
29#16861188C2211yescharacteristic, complemented
30#16861183C2211yescharacteristic
31#16861182—21no—
32#16861177—21no—
33C4×Dic254C2211yescharacteristic, complemented
34C4×D508C2211yescharacteristic
35#16861168C2211yescharacteristic, complemented
36#16861165—21no—
37C2×C4⋊Dic127C2211yescharacteristic
38C22⋊C4×D127—21no—
39#16861143C2211yescharacteristic, complemented
40#16861141C2211yescharacteristic, complemented
41#16861141C2211yescharacteristic, complemented
42#16861134C2211yescharacteristic, complemented
43#16861134C2211yescharacteristic, complemented
44#16861134C2211yescharacteristic, complemented
45#16861134C2211yescharacteristic, complemented
46#16861131C2211yescharacteristic
47#16861131C2211yescharacteristic
48#16861131C2211yescharacteristic
49#16861131C2211yescharacteristic
50#16861131C2211yescharacteristic
51#16861131C2211yescharacteristic
52C22×C8×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic, complemented
54C4×D4×C127C2211yescharacteristic
55C4⋊C8×C127C2211yescharacteristic, complemented
56C22⋊C8×C127C2211yescharacteristic
57C22⋊C8×C127C2211yescharacteristic, complemented
58C4×C8×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60#14523248—22no—
61#14523248—41no—
62C2×Dic254C2311yescharacteristic
63C2×Dic254C2311yescharacteristic
64C2×D508—22no—
65C2×D508—21no—
66C2×D508—41no—
67C2×D508C2311yescharacteristic
68C2×D508C2311yescharacteristic
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C22×Dic127—21no—
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74Dic508—41no—
75#14523237—22no—
76#14523237—41no—
77C4⋊Dic127—21no—
78C4⋊Dic127C2311yescharacteristic
79C4⋊Dic127C2311yescharacteristic
80C4⋊Dic127C2311yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82Dic127⋊C4—21no—
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96C4×Dic127—21no—
97C4×Dic127—21no—
98C2×D4×C127D411yescharacteristic
99C22×C4×C127C2311yescharacteristic
100C22×C4×C127C2311yescharacteristic
101C2×C8×C127—41no—
102C2×C8×C127—41no—
103C2×C8×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C4⋊C4×C127D411yescharacteristic
108C22⋊C4×C127D411yescharacteristic
109C22⋊C4×C127D411yescharacteristic
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112C22×D127—21no—
113C22×D127—42no—
114C22×D127—42no—
115C22×D127—81no—
116Dic254—21no—
117Dic254—21no—
118Dic254—21no—
119Dic254—21no—
120D508—21no—
121D508—21no—
122D508—21no—
123D508—42no—
124D508#4112yes—
125C22⋊D127—42no—
126C22⋊D127—81no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127—21no—
130C2×Dic127—21no—
131C2×Dic127—21no—
132C2×Dic127—21no—
133C2×Dic127—42no—
134C4×D127—42no—
135C4×D127—81no—
136C23×C127C2×D411yescharacteristic
137C23×C127C2×D411yescharacteristic
138D4×C127—22no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2411yescharacteristic
148C8×C127—21no—
149C8×C127—21no—
150C8×C127—21no—
151C8×C127—21no—
152C22×C127—21no—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127#13512yes—
158C22×C127C22×D411yescharacteristic
159C2×D127—22no—
160C2×D127—41no—
161C2×D127—41no—
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—41no—
165C2×D127—84no—
166C4×C127—21no—
167C4×C127—21no—
168C4×C127—21no—
169C4×C127—21no—
170C4×C127C2×#4111yescharacteristic
171C4×C127H(5,𝔽2)11yescharacteristic
172Dic127—41no—
173Dic127—41no—
174Dic127—41no—
175Dic127—41no—
176Dic127—41no—
177Dic127—41no—
178C2×C127—22no—
179C2×C127—41no—
180C2×C127#54611yescharacteristic
181C2×C127C2×#13511yescharacteristic
182C2×C127#57711yescharacteristic
183D127—42no—
184D127—81no—
185#3296—1271nomaximal, Sylow
186C127#329611yescharacteristic, complemented, Sylow
187#552—1271no—
188#546—1271no—
189#488—1271no—
190#483—1271no—
191#482—1271no—
192#478—1271no—
193#467—1271no—
194#465—1271no—
195#464—1271no—
196#452—1271no—
197#450—1271no—
198C4×SD16—1271no—
199C8×D4D12711yescharacteristic, complemented
200#416—1271no—
201C2×#102—1271no—
202C2×#41—1271no—
203C22×D4—1271no—
204C2×Q16—1271no—
205C2×SD16—1271no—
206C2×SD16—1271no—
207C22×C8C2×D12711yescharacteristic
208C22×C8C2×D12711yescharacteristic, complemented
209#126—2541no—
210#125—1271no—
211#125—2541no—
212#124—1271no—
213#123—2541no—
214#123—2541no—
215C22⋊Q8—1271no—
216C22⋊Q8—1271no—
217C22⋊Q8—1271no—
218#121—1271no—
219#121—2541no—
220#120—2541no—
221C4×Q8—1271no—
222C4×D4—1271no—
223C4×D4C2×D12711yescharacteristic
224#117—1271no—
225#117—2541no—
226C2×C4⋊C4—1271no—
227C2×C22⋊C4—2541no—
228#107—1271no—
229#106—1271no—
230#106—1271no—
231C4⋊C8C2×D12711yescharacteristic, complemented
232#103—1271no—
233#103—1271no—
234#103—1271no—
235#103—1271no—
236#102—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102—1271no—
241#102—1271no—
242C22⋊C8C2×D12711yescharacteristic
243C22⋊C8C2×D12711yescharacteristic, complemented
244C4×C8C2×D12711yescharacteristic, complemented
245C24—2541no—
246#41—5081no—
247#41—2542no—
248C2×Q8—1271no—
249C2×Q8—1271no—
250C2×D4—1271no—
251C2×D4—1271no—
252C2×D4—5081no—
253C2×D4—2542no—
254C2×D4—2541no—
255C2×D4—2541no—
256C2×D4—2541no—
257C2×D4D50811yescharacteristic
258C22×C4—2541no—
259C22×C4—2541no—
260C22×C4—2541no—
261C22×C4C22×D12711yescharacteristic
262C22×C4C22×D12711yescharacteristic
263Q16—5081no—
264SD16—5081no—
265SD16—2542no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D12711yescharacteristic
269C2×C8C22×D12711yescharacteristic
270C2×C8C22×D12711yescharacteristic
271C2×C8C22×D12711yescharacteristic
272C4⋊C4—1271no—
273C4⋊C4—1271no—
274C4⋊C4—1271no—
275C4⋊C4—1271no—
276C4⋊C4—2541no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4—2541no—
282C4⋊C4—2541no—
283C4⋊C4D50811yescharacteristic
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4D50811yescharacteristic
293C22⋊C4D50811yescharacteristic
294C42—2541no—
295C42—2541no—
296C42C22×D12711yescharacteristic
297C23—10161no—
298C23—5082no—
299C23—5082no—
300C23—2541no—
301C23—2541no—
302C23C2×D50811yescharacteristic
303C23C2×D50811yescharacteristic
304Q8—2541no—
305Q8—2541no—
306Q8—2541no—
307Q8—2541no—
308D4—22no—
309D4—1272no—
310D4—10161no—
311D4—5082no—
312D4—5082no—
313D4—2541no—
314D4—2541no—
315D4—2541no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—10161no—
321C2×C4—5082no—
322C2×C4—5082no—
323C2×C4—2541no—
324C2×C4—2541no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4C2×D50811yescharacteristic
330C2×C4C2×D50811yescharacteristic
331C2×C4C2×D50811yescharacteristic
332C2×C4C2×D50811yescharacteristic
333C2×C4C23×D12711yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—10164no—
344C22—5081no—
345C22—5081no—
346C22—5081no—
347C22—5081no—
348C22—5081no—
349C22—2542no—
350C22#1686123112yes—
351C22C22×D50811yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—5081no—
357C4—5081no—
358C4—5081no—
359C4—5081no—
360C4—5081no—
361C4—5081no—
362C4#41×D12711yescharacteristic
363C4#1686125411yescharacteristic
364C2—22no—
365C2—41no—
366C2—10161no—
367C2—5082no—
368C2#2255052111yescharacteristic
369C2C2×#1686123111yescharacteristic
370C2#2255068811yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC8×D4×C127C2
socleC22×C127C22×D4
prime radical#22550522C2
hypercenterC8×D4D127
hypocenterC127#3296
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #22550566C2semidirect
3. #22550231C2semidirect
4. #22550207C2semidirect
5. #22550201C2semidirect
6. #22550177C2semidirect
7. #22550117C2semidirect
8. #22550105C2semidirect
9. #22550099C2semidirect
10. #22550040C2semidirect
11. #22550029C2semidirect
12. C4×#14523237C2semidirect
13. #22549873C2semidirect
14. C2×#16861131C2semidirect
15. C8×D4×C127C2semidirect
18. C2×Dic508C22semidirect
22. #16861201C22semidirect
29. #16861188C22semidirect
33. C4×Dic254C22semidirect
35. #16861168C22semidirect
39. #16861143C22semidirect
40. #16861141C22semidirect
41. #16861141C22semidirect
42. #16861134C22semidirect
43. #16861134C22semidirect
44. #16861134C22semidirect
45. #16861134C22semidirect
53. C22×C8×C127C22semidirect
55. C4⋊C8×C127C22semidirect
57. C22⋊C8×C127C22semidirect
58. C4×C8×C127C22semidirect
186. C127#3296semidirect
199. C8×D4D127semidirect
208. C22×C8C2×D127semidirect
231. C4⋊C8C2×D127semidirect
243. C22⋊C8C2×D127semidirect
244. C4×C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505662
#225505222
#225502312
#225502072
#225502012
#225501772
#225501172
#225501052
#225500992
#225500402
#225500292
C4×#145232372
#225498732
C2×#168611312
C8×D4×C1272
#3296127