Finite Group Databasebeta

← #30650582

Subgroups — #30650582

Order 16256.

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#22550572C211yesmaximal, characteristic, complemented
2#22550522C211yesmaximal, characteristic, complemented
3#22550231C211yesmaximal, characteristic, complemented
4#22550212C211yesmaximal, characteristic, complemented
5#22550201C211yesmaximal, characteristic, complemented
6#22550177C211yesmaximal, characteristic, complemented
7#22550112C211yesmaximal, characteristic, complemented
8#22550105C211yesmaximal, characteristic, complemented
9#22550092C211yesmaximal, characteristic, complemented
10#22550023C211yesmaximal, characteristic, complemented
11#22550012C211yesmaximal, characteristic, complemented
12C4×D1016C211yesmaximal, characteristic, complemented
13#22549873C211yesmaximal, characteristic, complemented
14C2×#16861131C211yesmaximal, characteristic, complemented
15C8×D4×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C22×D508C2211yescharacteristic
18C2×#14523237C2211yescharacteristic, complemented
19C2×D1016C2211yescharacteristic
20C2×D1016C2211yescharacteristic
21#16861204—21no—
22#16861201C2211yescharacteristic, complemented
23#16861197C2211yescharacteristic
24#16861194—21no—
25#16861191—21no—
26#16861188C2211yescharacteristic
27#16861183C2211yescharacteristic
28#16861183C2211yescharacteristic
29#16861183C2211yescharacteristic, complemented
30#16861182—21no—
31#16861181—21no—
32#16861177—21no—
33C4×D508C2211yescharacteristic
34C4×D508C2211yescharacteristic, complemented
35(C22⋊D127)⋊C4—21no—
36#16861168C2211yescharacteristic, complemented
37C2×C4⋊Dic127C2211yescharacteristic
38C22⋊C4×D127—21no—
39#16861143C2211yescharacteristic, complemented
40#16861143C2211yescharacteristic, complemented
41#16861141C2211yescharacteristic, complemented
42#16861134C2211yescharacteristic, complemented
43#16861134C2211yescharacteristic, complemented
44#16861134C2211yescharacteristic, complemented
45#16861131C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861131C2211yescharacteristic
48#16861131C2211yescharacteristic
49#16861131C2211yescharacteristic
50#16861131C2211yescharacteristic
51#16861131C2211yescharacteristic, complemented
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×D508—22no—
64C2×D508—21no—
65C2×D508—41no—
66C2×D508C2311yescharacteristic
67C2×D508C2311yescharacteristic
68C2×D508C2311yescharacteristic
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C22×Dic127—21no—
74C2×C4×D127—21no—
75C2×C4×D127—21no—
76C2×C4×D127—21no—
77#14523237—41no—
78D1016—22no—
79D1016—41no—
80C4⋊Dic127—21no—
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84C4⋊Dic127C2311yescharacteristic
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—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—21no—
114C22×D127—42no—
115C22×D127—42no—
116C22×D127—81no—
117Dic254—21no—
118Dic254—21no—
119D508—21no—
120D508—21no—
121D508—21no—
122D508—21no—
123D508—21no—
124D508—42no—
125D508#4112yes—
126C22⋊D127—42no—
127C22⋊D127—42no—
128C22⋊D127—81no—
129C22⋊D127—81no—
130C2×Dic127—21no—
131C2×Dic127—21no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—21no—
135C2×Dic127—81no—
136C4×D127—42no—
137C4×D127—81no—
138C4×D127—81no—
139C23×C127C2×D411yescharacteristic
140C23×C127C2×D411yescharacteristic
141D4×C127—22no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2×D411yescharacteristic
150C2×C4×C127C2411yescharacteristic
151C8×C127—21no—
152C8×C127—21no—
153C8×C127—21no—
154C8×C127—21no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—21no—
158C22×C127—21no—
159C22×C127—21no—
160C22×C127#13512yes—
161C22×C127C22×D411yescharacteristic
162C2×D127—22no—
163C2×D127—41no—
164C2×D127—41no—
165C2×D127—41no—
166C2×D127—41no—
167C2×D127—41no—
168C2×D127—41no—
169C2×D127—41no—
170C2×D127—41no—
171C2×D127—84no—
172C4×C127—21no—
173C4×C127—21no—
174C4×C127—21no—
175C4×C127—21no—
176C4×C127C2×#4111yescharacteristic
177C4×C127H(5,𝔽2)11yescharacteristic
178Dic127—41no—
179Dic127—41no—
180Dic127—41no—
181Dic127—41no—
182Dic127—41no—
183C2×C127—22no—
184C2×C127—41no—
185C2×C127#54611yescharacteristic
186C2×C127C2×#13511yescharacteristic
187C2×C127#57611yescharacteristic
188D127—42no—
189D127—81no—
190D127—81no—
191#3297—1271nomaximal, Sylow
192C127#329711yescharacteristic, complemented, Sylow
193#553—1271no—
194#546—1271no—
195#488—1271no—
196#484—1271no—
197#482—1271no—
198#478—1271no—
199#466—1271no—
200#465—1271no—
201#463—1271no—
202#449—1271no—
203#447—1271no—
204C4×D8—1271no—
205C8×D4D12711yescharacteristic, complemented
206#416—1271no—
207C2×#102—1271no—
208C2×#41—1271no—
209C22×D4—1271no—
210C2×SD16—1271no—
211C2×D8—1271no—
212C2×D8—1271no—
213C22×C8C2×D12711yescharacteristic
214C22×C8C2×D12711yescharacteristic, complemented
215#126—2541no—
216#125—1271no—
217#124—1271no—
218#123—2541no—
219#123—2541no—
220C22⋊Q8—1271no—
221#121—1271no—
222#121—1271no—
223#121—1271no—
224#121—2541no—
225#121—2541no—
226#120—2541no—
227C4×D4—1271no—
228C4×D4—1271no—
229C4×D4—2541no—
230C4×D4C2×D12711yescharacteristic
231#117—1271no—
232C2×C4⋊C4—1271no—
233C2×C22⋊C4—2541no—
234#107—1271no—
235#107—1271no—
236#106—1271no—
237C4⋊C8C2×D12711yescharacteristic, complemented
238#103—1271no—
239#103—1271no—
240#103—1271no—
241#102—1271no—
242#102—1271no—
243#102—1271no—
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248C22⋊C8C2×D12711yescharacteristic
249C22⋊C8C2×D12711yescharacteristic, complemented
250C4×C8C2×D12711yescharacteristic, complemented
251C24—2541no—
252#41—5081no—
253#41—2542no—
254C2×Q8—1271no—
255C2×D4—1271no—
256C2×D4—1271no—
257C2×D4—1271no—
258C2×D4—5081no—
259C2×D4—2542no—
260C2×D4—2541no—
261C2×D4—2541no—
262C2×D4—2541no—
263C2×D4—2541no—
264C2×D4—2541no—
265C2×D4D50811yescharacteristic
266C22×C4—2541no—
267C22×C4—2541no—
268C22×C4—2541no—
269C22×C4—2541no—
270C22×C4C22×D12711yescharacteristic
271C22×C4C22×D12711yescharacteristic
272SD16—5081no—
273D8—5081no—
274D8—2542no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C22×D12711yescharacteristic
278C2×C8C22×D12711yescharacteristic
279C2×C8C22×D12711yescharacteristic
280C2×C8C22×D12711yescharacteristic
281C4⋊C4—1271no—
282C4⋊C4—1271no—
283C4⋊C4—1271no—
284C4⋊C4—1271no—
285C4⋊C4—2541no—
286C4⋊C4—2541no—
287C4⋊C4—2541no—
288C4⋊C4—2541no—
289C4⋊C4D50811yescharacteristic
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C22⋊C4—2541no—
294C22⋊C4—2541no—
295C22⋊C4—2541no—
296C22⋊C4—2541no—
297C22⋊C4—2541no—
298C22⋊C4—2541no—
299C22⋊C4D50811yescharacteristic
300C22⋊C4D50811yescharacteristic
301C42—2541no—
302C42C22×D12711yescharacteristic
303C23—10161no—
304C23—5082no—
305C23—5082no—
306C23—2541no—
307C23—2541no—
308C23—2541no—
309C23C2×D50811yescharacteristic
310C23C2×D50811yescharacteristic
311Q8—2541no—
312Q8—2541no—
313D4—22no—
314D4—1272no—
315D4—10161no—
316D4—10161no—
317D4—5082no—
318D4—5082no—
319D4—5082no—
320D4—2541no—
321D4—2541no—
322D4—2541no—
323D4—2541no—
324D4—2541no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—10161no—
330C2×C4—10161no—
331C2×C4—10161no—
332C2×C4—5082no—
333C2×C4—2541no—
334C2×C4—2541no—
335C2×C4—2541no—
336C2×C4—2541no—
337C2×C4—2541no—
338C2×C4C2×D50811yescharacteristic
339C2×C4C2×D50811yescharacteristic
340C2×C4C2×D50811yescharacteristic
341C2×C4C2×D50811yescharacteristic
342C2×C4C23×D12711yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—10164no—
353C22—5081no—
354C22—5081no—
355C22—5081no—
356C22—5081no—
357C22—5081no—
358C22—5081no—
359C22—5081no—
360C22—5081no—
361C22—2542no—
362C22#1686123112yes—
363C22C22×D50811yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—5081no—
369C4—5081no—
370C4—5081no—
371C4—5081no—
372C4—5081no—
373C4#41×D12711yescharacteristic
374C4#1686125411yescharacteristic
375C2—22no—
376C2—41no—
377C2—10161no—
378C2—10161no—
379C2—5082no—
380C2#2255052111yescharacteristic
381C2C2×#1686123111yescharacteristic
382C2#2255068111yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC8×D4×C127C2
socleC22×C127C22×D4
hypercenterC8×D4D127
hypocenterC127#3297
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. #22550572C2semidirect
2. #22550522C2semidirect
3. #22550231C2semidirect
4. #22550212C2semidirect
5. #22550201C2semidirect
6. #22550177C2semidirect
7. #22550112C2semidirect
8. #22550105C2semidirect
9. #22550092C2semidirect
10. #22550023C2semidirect
11. #22550012C2semidirect
12. C4×D1016C2semidirect
13. #22549873C2semidirect
14. C2×#16861131C2semidirect
15. C8×D4×C127C2semidirect
18. C2×#14523237C22semidirect
22. #16861201C22semidirect
29. #16861183C22semidirect
34. C4×D508C22semidirect
36. #16861168C22semidirect
39. #16861143C22semidirect
40. #16861143C22semidirect
41. #16861141C22semidirect
42. #16861134C22semidirect
43. #16861134C22semidirect
44. #16861134C22semidirect
51. #16861131C22semidirect
53. C22×C8×C127C22semidirect
55. C4⋊C8×C127C22semidirect
57. C22⋊C8×C127C22semidirect
58. C4×C8×C127C22semidirect
192. C127#3297semidirect
205. C8×D4D127semidirect
214. C22×C8C2×D127semidirect
237. C4⋊C8C2×D127semidirect
249. C22⋊C8C2×D127semidirect
250. C4×C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505722
#225505222
#225502312
#225502122
#225502012
#225501772
#225501122
#225501052
#225500922
#225500232
#225500122
C4×D10162
#225498732
C2×#168611312
C8×D4×C1272
#3297127