Finite Group Databasebeta

← #30650703

Subgroups — #30650703

Order 16256.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550579C211yesmaximal, characteristic, complemented
2#22550530C211yesmaximal, characteristic, complemented
3#22550246C211yesmaximal, characteristic, complemented
4#22550219C211yesmaximal, characteristic, complemented
5#22550208C211yesmaximal, characteristic, complemented
6#22550123C211yesmaximal, characteristic, complemented
7#22550106C211yesmaximal, characteristic, complemented
8#22550082C211yesmaximal, characteristic, complemented
9#22550022C211yesmaximal, characteristic, complemented
10#22550022C211yesmaximal, characteristic, complemented
11#22549986C211yesmaximal, characteristic, complemented
12#22549946C211yesmaximal, characteristic, complemented
13#22549874C211yesmaximal, characteristic, complemented
14#22549869C211yesmaximal, characteristic, complemented
15#478×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C2×#14523237C2211yescharacteristic
19C2×#14523237C2211yescharacteristic
20C2×#14523233C2211yescharacteristic
21C2×#14523231C2211yescharacteristic
22C2×C8×D127C2211yescharacteristic
23#16861209C2211yescharacteristic
24#16861204—21no—
25#16861194—21no—
26#16861193—21no—
27#16861193—21no—
28#16861189C2211yescharacteristic
29#16861189C2211yescharacteristic
30#16861184—21no—
31#16861184C2211yescharacteristic
32#16861184C2211yescharacteristic
33#16861183—21no—
34C4×D508C2211yescharacteristic
35C4×D508C2211yescharacteristic
36#16861171—21no—
37#16861167C2211yescharacteristic
38C2×C4⋊Dic127—21no—
39C4⋊C4×D127C2211yescharacteristic
40#16861140C2211yescharacteristic
41#16861134C2211yescharacteristic
42#16861134C2211yescharacteristic
43#16861132C2211yescharacteristic
44#16861132C2211yescharacteristic
45#16861132C2211yescharacteristic
46#16861132C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861129C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51#16861120C2211yescharacteristic
52#125×C127C2211yescharacteristic
53C4×D4×C127C2211yescharacteristic
54#107×C127C2211yescharacteristic
55#106×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#14523247—22no—
60#14523247—22no—
61#14523247—41no—
62#14523247—41no—
63C2×Dic254C2311yescharacteristic
64C2×Dic254C2311yescharacteristic
65C2×D508—21no—
66C2×D508—21no—
67C2×D508D411yescharacteristic
68C2×D508C2311yescharacteristic
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C22×Dic127—21no—
72C22×Dic127—21no—
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127C2311yescharacteristic
76C2×C4×D127C2311yescharacteristic
77#14523237—41no—
78#14523237—41no—
79#14523233—22no—
80#14523231—41no—
81C2×C127⋊C8C2311yescharacteristic
82C2×C127⋊C8C2311yescharacteristic
83C8×D127—41no—
84C4⋊Dic127—21no—
85C4⋊Dic127—21no—
86C4⋊Dic127—21no—
87C4⋊Dic127—21no—
88C4⋊Dic127D411yescharacteristic
89C4⋊Dic127C2311yescharacteristic
90Dic127⋊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—
97(C2×D127)⋊C4D411yescharacteristic
98(C2×D127)⋊C4D411yescharacteristic
99C4×Dic127—21no—
100C2×D4×C127C2311yescharacteristic
101C22×C4×C127—21no—
102C2×C8×C127C2311yescharacteristic
103C2×C8×C127C2311yescharacteristic
104C4⋊C4×C127—21no—
105C4⋊C4×C127—21no—
106C4⋊C4×C127C2311yescharacteristic
107C4⋊C4×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C22⋊C4×C127—21no—
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112C22×D127C2×D411yescharacteristic
113C22×D127C2×D411yescharacteristic
114Dic254—21no—
115Dic254—21no—
116Dic254—21no—
117Dic254—21no—
118D508—21no—
119D508—21no—
120D508—41no—
121D508—42no—
122D508—81no—
123C22⋊D127—42no—
124C22⋊D127—42no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—42no—
129C2×Dic127—42no—
130C2×Dic127C2×D411yescharacteristic
131C2×Dic127C2×D411yescharacteristic
132C4×D127—21no—
133C4×D127—21no—
134C4×D127—21no—
135C4×D127—21no—
136C4×D127—81no—
137C4×D127—81no—
138C127⋊C8—21no—
139C127⋊C8—21no—
140C23×C127—21no—
141D4×C127—21no—
142D4×C127#4112yes—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—42no—
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2411yescharacteristic
150C8×C127—21no—
151C8×C127—21no—
152C22×C127—22no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127C22×D411yescharacteristic
156C2×D127—21no—
157C2×D127—21no—
158C2×D127—21no—
159C2×D127—21no—
160C2×D127—21no—
161C2×D127—21no—
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—41no—
165C4×C127—21no—
166C4×C127—21no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127—41no—
170C4×C127C2×#4111yescharacteristic
171C4×C127H(5,𝔽2)11yescharacteristic
172Dic127—21no—
173Dic127—21no—
174Dic127—41no—
175Dic127—41no—
176Dic127—41no—
177C2×C127—42no—
178C2×C127#54611yescharacteristic
179C2×C127#57511yescharacteristic
180C2×C127#57711yescharacteristic
181D127—41no—
182D127—41no—
183D127—81no—
184#3307—1271nomaximal, Sylow
185C127#330711yescharacteristic, complemented, Sylow
186#553—1271no—
187#547—1271no—
188#491—1271no—
189#485—1271no—
190#483—1271no—
191#478D12711yescharacteristic, complemented
192#468—1271no—
193#465—1271no—
194#461—1271no—
195#449—1271no—
196#449—1271no—
197#442—1271no—
198#435—1271no—
199#417—1271no—
200#416—1271no—
201C2×#41—1271no—
202C2×#41—1271no—
203C2×SD16—1271no—
204C2×SD16—1271no—
205C2×D8—1271no—
206C2×M16—1271no—
207C22×C8—1271no—
208C4⋊Q8—1271no—
209#126—2541no—
210#125C2×D12711yescharacteristic
211#123—2541no—
212#123—2541no—
213#123—2541no—
214C22⋊Q8—1271no—
215C22⋊Q8—1271no—
216#121—1271no—
217#121—1271no—
218#121—2541no—
219#121—2541no—
220C4×D4—1271no—
221C4×D4—1271no—
222C4×D4—2541no—
223C4×D4C2×D12711yescharacteristic
224#117—1271no—
225C2×C4⋊C4—1271no—
226C2×C4⋊C4—2541no—
227#107C2×D12711yescharacteristic
228#106—1271no—
229#106C2×D12711yescharacteristic
230C4⋊C8C2×D12711yescharacteristic
231#103—1271no—
232#103—1271no—
233#103—1271no—
234#103—1271no—
235#103—1271no—
236#103—1271no—
237#102—1271no—
238#102—1271no—
239#102C2×D12711yescharacteristic
240#102C2×D12711yescharacteristic
241C22⋊C8—1271no—
242C22⋊C8—1271no—
243#97—1271no—
244#41—5081no—
245#41—5081no—
246#41—2542no—
247#41—2542no—
248C2×Q8—1271no—
249C2×Q8—1271no—
250C2×D4—1271no—
251C2×D4—1271no—
252C2×D4—2541no—
253C2×D4—2541no—
254C2×D4—2541no—
255C2×D4—2541no—
256C2×D4C22×D12711yescharacteristic
257C22×C4—21no—
258C22×C4—1271no—
259C22×C4—1271no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4—2541no—
263C22×C4—2541no—
264SD16—5081no—
265SD16—5081no—
266D8—2542no—
267M16—5081no—
268C2×C8—1271no—
269C2×C8—1271no—
270C2×C8—5081no—
271C2×C8C22×D12711yescharacteristic
272C2×C8C22×D12711yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—1271no—
276C4⋊C4—1271no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4—2541no—
282C4⋊C4C22×D12711yescharacteristic
283C4⋊C4C22×D12711yescharacteristic
284C4⋊C4C22×D12711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—1271no—
287C22⋊C4—1271no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C22⋊C4—2541no—
294C42—2541no—
295C42C22×D12711yescharacteristic
296C23—21no—
297C23—1271no—
298C23—1271no—
299C23—2541no—
300Q8—2541no—
301Q8—2541no—
302Q8—2541no—
303Q8—2541no—
304D4—21no—
305D4—10161no—
306D4—5081no—
307D4—5082no—
308D4—5082no—
309D4—5082no—
310D4—2541no—
311D4—2541no—
312D4#1452324912yes—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—42no—
317C2×C4—1271no—
318C2×C4—1271no—
319C2×C4—10161no—
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×C4—2541no—
330C2×C4D4×D12711yescharacteristic
331C2×C4D4×D12711yescharacteristic
332C2×C4C23×D12711yescharacteristic
333C8—21no—
334C8—21no—
335C8—2541no—
336C8—2541no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—5081no—
341C22—5081no—
342C22—5081no—
343C22—2541no—
344C22—2541no—
345C22—2541no—
346C22—2541no—
347C22—2541no—
348C22—2541no—
349C22C2×D4×D12711yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—5081no—
356C4—5081no—
357C4—5081no—
358C4—2541no—
359C4—2541no—
360C4C2×#1452324911yescharacteristic
361C4#1686125511yescharacteristic
362C2—42no—
363C2—10161no—
364C2—5081no—
365C2—5081no—
366C2#2255051711yescharacteristic
367C2#2255067211yescharacteristic
368C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#478×C127C2
socleC22×C127C22×D4
hypercenter#478D127
hypocenterC127#3307
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #22550579C2semidirect
2. #22550530C2semidirect
3. #22550246C2semidirect
4. #22550219C2semidirect
5. #22550208C2semidirect
6. #22550123C2semidirect
7. #22550106C2semidirect
8. #22550082C2semidirect
9. #22550022C2semidirect
10. #22550022C2semidirect
11. #22549986C2semidirect
12. #22549946C2semidirect
13. #22549874C2semidirect
14. #22549869C2semidirect
15. #478×C127C2semidirect
185. C127#3307semidirect
191. #478D127semidirect

Maximal subgroups

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

subgroupindex
#225505792
#225505302
#225502462
#225502192
#225502082
#225501232
#225501062
#225500822
#225500222
#225499862
#225499462
#225498742
#225498692
#478×C1272
#3307127