Finite Group Databasebeta

← #30648998

Subgroups — #30648998

Order 16256.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550498C211yesmaximal, characteristic, complemented
2#22550358C211yesmaximal, characteristic, complemented
3#22550210C211yesmaximal, characteristic, complemented
4#22550191C211yesmaximal, characteristic, complemented
5#22550169C211yesmaximal, characteristic, complemented
6#22550155C211yesmaximal, characteristic, complemented
7#22550100C211yesmaximal, characteristic, complemented
8#22550094C211yesmaximal, characteristic, complemented
9#22550037C211yesmaximal, characteristic, complemented
10#22550024C211yesmaximal, characteristic
11#22549974C211yesmaximal, characteristic, complemented
12#22549974C211yesmaximal, characteristic, complemented
13C4×D1016C211yesmaximal, characteristic, complemented
14#22549935C211yesmaximal, characteristic, complemented
15C4×Q16×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523235C2211yescharacteristic
18C2×#14523235C2211yescharacteristic
19C2×D1016C2211yescharacteristic
20#16861208C2211yescharacteristic
21#16861203—21no—
22#16861198C2211yescharacteristic
23#16861197—21no—
24#16861194—21no—
25#16861187—21no—
26#16861185C2211yescharacteristic
27#16861184C2211yescharacteristic
28Q8×Dic127C2211yescharacteristic
29C4×D508—21no—
30C4×D508C2211yescharacteristic
31C4×D508C2211yescharacteristic
32#16861171—21no—
33#16861171C2211yescharacteristic
34#16861166—21no—
35#16861166C2211yescharacteristic
36C4⋊C4×D127—21no—
37C4⋊C4×D127C2211yescharacteristic
38#16861143C2211yescharacteristic
39#16861142C2211yescharacteristic
40#16861142C2211yescharacteristic
41Dic127⋊C8C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861133C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51#16861120C2211yescharacteristic
52C2×Q16×C127C2211yescharacteristic
53C4×Q8×C127C2211yescharacteristic
54C4×Q8×C127C2211yescharacteristic
55#107×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58C4×C8×C127C2211yescharacteristic
59#14523249—22no—
60#14523249—22no—
61C2×Dic254—21no—
62C2×D508—21no—
63C2×D508—21no—
64C2×D508C2311yescharacteristic
65C2×D508C2311yescharacteristic
66C2×C4×D127—21no—
67C2×C4×D127—21no—
68C2×C4×D127—21no—
69C2×C4×D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127C2311yescharacteristic
72#14523235—22no—
73#14523235—22no—
74D1016—22no—
75C2×C127⋊C8C2311yescharacteristic
76C2×C127⋊C8C2311yescharacteristic
77C4⋊Dic127—21no—
78C4⋊Dic127C2311yescharacteristic
79C4⋊Dic127C2311yescharacteristic
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4D411yescharacteristic
84Dic127⋊C4D411yescharacteristic
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4D411yescharacteristic
93(C2×D127)⋊C4D411yescharacteristic
94C4×Dic127—21no—
95C4×Dic127C2311yescharacteristic
96C2×Q8×C127C2311yescharacteristic
97C2×Q8×C127C2311yescharacteristic
98Q16×C127—22no—
99C2×C8×C127C2311yescharacteristic
100C2×C8×C127C2311yescharacteristic
101C4⋊C4×C127—21no—
102C4⋊C4×C127—21no—
103C4⋊C4×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C42×C127—21no—
106C42×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127—21no—
110C22×D127C2×D411yescharacteristic
111Dic254—81no—
112D508—24no—
113D508—21no—
114D508—21no—
115D508—21no—
116D508—81no—
117D508#4112yes—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C2×Dic127C2×D411yescharacteristic
121C2×Dic127C2×D411yescharacteristic
122C2×Dic127C2×D411yescharacteristic
123C4×D127—24no—
124C4×D127—21no—
125C4×D127—21no—
126C4×D127—81no—
127C4×D127—81no—
128C4×D127—81no—
129C4×D127—81no—
130C127⋊C8—21no—
131C127⋊C8—21no—
132Q8×C127—21no—
133Q8×C127—21no—
134Q8×C127—21no—
135Q8×C127#4112yes—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2411yescharacteristic
143C8×C127—21no—
144C8×C127—21no—
145C22×C127C22×D411yescharacteristic
146C2×D127—21no—
147C2×D127—22no—
148C2×D127—21no—
149C2×D127—21no—
150C2×D127—41no—
151C2×D127—41no—
152C2×D127—41no—
153C2×D127—41no—
154C2×D127—41no—
155C4×C127—22no—
156C4×C127—21no—
157C4×C127—21no—
158C4×C127—41no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127C2×#4111yescharacteristic
162C4×C127C2×#4111yescharacteristic
163Dic127—21no—
164Dic127—21no—
165Dic127—21no—
166Dic127—41no—
167Dic127—41no—
168C2×C127#52511yescharacteristic
169C2×C127#57611yescharacteristic
170C2×C127#57811yescharacteristic
171D127—41no—
172D127—42no—
173D127—81no—
174#3124—1271nomaximal, Sylow
175C127#312411yescharacteristic, complemented, Sylow
176#542—1271no—
177#520—1271no—
178#484—1271no—
179#480—1271no—
180#477—1271no—
181#474—1271no—
182#464—1271no—
183#463—1271no—
184#452—1271no—
185#449—1271no—
186#440—1271no—
187#440—1271no—
188C4×Q16D12711yescharacteristic, complemented
189C4×D8—1271no—
190#433—1271no—
191C2×#41—1271no—
192C2×Q16C2×D12711yescharacteristic
193C2×SD16—1271no—
194C2×SD16—1271no—
195C2×D8—1271no—
196C4⋊Q8—1271no—
197#126—2541no—
198#124—1271no—
199#124—2541no—
200#123—2541no—
201C22⋊Q8—1271no—
202C22⋊Q8—2541no—
203#121—1271no—
204C4×Q8—1271no—
205C4×Q8C2×D12711yescharacteristic
206C4×Q8C2×D12711yescharacteristic
207C4×D4—1271no—
208C4×D4—1271no—
209C4×D4—1271no—
210C4×D4—2541no—
211C4×D4—2541no—
212#117—1271no—
213#117—2541no—
214C2×C4⋊C4—1271no—
215C2×C4⋊C4—2541no—
216#107—1271no—
217#107—1271no—
218#107—1271no—
219#107C2×D12711yescharacteristic
220C4⋊C8—1271no—
221C4⋊C8—1271no—
222#103—1271no—
223#103—1271no—
224#103C2×D12711yescharacteristic
225#103C2×D12711yescharacteristic
226#102—1271no—
227#102—1271no—
228#102—1271no—
229#102—1271no—
230C22⋊C8—1271no—
231C22⋊C8—1271no—
232#97—1271no—
233C4×C8C2×D12711yescharacteristic
234#41—2542no—
235#41—2542no—
236C2×Q8—2541no—
237C2×Q8C22×D12711yescharacteristic
238C2×Q8C22×D12711yescharacteristic
239C2×D4—1271no—
240C2×D4—1271no—
241C2×D4—2541no—
242C2×D4—2541no—
243C22×C4—1271no—
244C22×C4—2541no—
245C22×C4—2541no—
246C22×C4—2541no—
247C22×C4—2541no—
248C22×C4—2541no—
249Q16—22no—
250SD16—2542no—
251SD16—2542no—
252D8—2542no—
253C2×C8—1271no—
254C2×C8—1271no—
255C2×C8C22×D12711yescharacteristic
256C2×C8C22×D12711yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—1271no—
260C4⋊C4—1271no—
261C4⋊C4—1271no—
262C4⋊C4—1271no—
263C4⋊C4—2541no—
264C4⋊C4—2541no—
265C4⋊C4—2541no—
266C4⋊C4—2541no—
267C4⋊C4C22×D12711yescharacteristic
268C4⋊C4C22×D12711yescharacteristic
269C22⋊C4—1271no—
270C22⋊C4—1271no—
271C22⋊C4—2541no—
272C22⋊C4—2541no—
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C22⋊C4—2541no—
277C22⋊C4—2541no—
278C42—21no—
279C42—21no—
280C42—1271no—
281C42—2541no—
282C42C22×D12711yescharacteristic
283C23—1271no—
284C23—2541no—
285C23—2541no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—10161no—
290Q8#1452324812yes—
291D4—1272no—
292D4—10161no—
293D4—2544no—
294D4—2541no—
295D4—2541no—
296D4—2541no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—1271no—
302C2×C4—1271no—
303C2×C4—1271no—
304C2×C4—10161no—
305C2×C4—10161no—
306C2×C4—10161no—
307C2×C4—10161no—
308C2×C4—2544no—
309C2×C4—2541no—
310C2×C4—2541no—
311C2×C4—2541no—
312C2×C4—2541no—
313C2×C4D4×D12711yescharacteristic
314C2×C4D4×D12711yescharacteristic
315C2×C4C23×D12711yescharacteristic
316C8—21no—
317C8—21no—
318C8—2541no—
319C8—2541no—
320C22—5081no—
321C22—5081no—
322C22—5081no—
323C22—5081no—
324C22—5081no—
325C22—2542no—
326C22—2541no—
327C22—2541no—
328C22—2541no—
329C22C2×D4×D12711yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—5081no—
337C4—5081no—
338C4—2541no—
339C4—2541no—
340C4—2541no—
341C4#41×D12711yescharacteristic
342C4C2×#1452324811yescharacteristic
343C2—10161no—
344C2—5082no—
345C2—5081no—
346C2#2255038511yescharacteristic
347C2#2255067811yescharacteristic
348C2#2255068911yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×Q16×C127C2
socleC22×C127C22×D4
prime radical#22550024C2
hypercenterC4×Q16D127
hypocenterC127#3124
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. #22550498C2semidirect
2. #22550358C2semidirect
3. #22550210C2semidirect
4. #22550191C2semidirect
5. #22550169C2semidirect
6. #22550155C2semidirect
7. #22550100C2semidirect
8. #22550094C2semidirect
9. #22550037C2semidirect
11. #22549974C2semidirect
12. #22549974C2semidirect
13. C4×D1016C2semidirect
14. #22549935C2semidirect
15. C4×Q16×C127C2semidirect
175. C127#3124semidirect
188. C4×Q16D127semidirect

Maximal subgroups

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

subgroupindex
#225504982
#225503582
#225502102
#225501912
#225501692
#225501552
#225501002
#225500942
#225500372
#225500242
#225499742
C4×D10162
#225499352
C4×Q16×C1272
#3124127