Finite Group Databasebeta

← #30649020

Subgroups — #30649020

Order 16256.

382 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#22550477C211yesmaximal, characteristic
2#22550352C211yesmaximal, characteristic, complemented
3#22550203C211yesmaximal, characteristic, complemented
4#22550198C211yesmaximal, characteristic, complemented
5#22550185C211yesmaximal, characteristic, complemented
6#22550182C211yesmaximal, characteristic, complemented
7#22550075C211yesmaximal, characteristic, complemented
8#22550074C211yesmaximal, characteristic, complemented
9#22550016C211yesmaximal, characteristic, complemented
10#22550013C211yesmaximal, characteristic, complemented
11#22549978C211yesmaximal, characteristic, complemented
12C4×#14523235C211yesmaximal, characteristic, complemented
13C4×#14523236C211yesmaximal, characteristic, complemented
14#22549938C211yesmaximal, characteristic, complemented
15#440×C127C211yesmaximal, characteristic, complemented
16C2×Q8×D127C2211yescharacteristic
17C2×#14523237C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523235C2211yescharacteristic
20#16861205C2211yescharacteristic
21#16861200C2211yescharacteristic
22#16861192—21no—
23#16861191—21no—
24#16861184C2211yescharacteristic
25#16861183—21no—
26#16861182—21no—
27#16861180C2211yescharacteristic
28Q8×Dic127C2211yescharacteristic
29C4×Dic254—21no—
30C4×Dic254C2211yescharacteristic
31#16861174—21no—
32#16861174C2211yescharacteristic
33C4×D508C2211yescharacteristic
34#16861167—21no—
35#16861167C2211yescharacteristic
36#16861166—21no—
37#16861166C2211yescharacteristic
38#16861143C2211yescharacteristic
39#16861140C2211yescharacteristic
40#16861140C2211yescharacteristic
41Dic127⋊C8C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861132C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861129C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C4×C127⋊C8C2211yescharacteristic
52C2×SD16×C127C2211yescharacteristic
53C4×Q8×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic
55#107×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#97×C127C2211yescharacteristic
59C2×Dic254—21no—
60C2×Dic254C2311yescharacteristic
61Q8×D127—22no—
62Q8×D127—22no—
63C2×D508—21no—
64C2×D508C2311yescharacteristic
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C4×D127—21no—
69C2×C4×D127—21no—
70C2×C4×D127C2311yescharacteristic
71#14523237—22no—
72#14523236—22no—
73#14523235—22no—
74C2×C127⋊C8C2311yescharacteristic
75C2×C127⋊C8C2311yescharacteristic
76C4⋊Dic127—21no—
77C4⋊Dic127—21no—
78C4⋊Dic127C2311yescharacteristic
79C4⋊Dic127C2311yescharacteristic
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4D411yescharacteristic
84Dic127⋊C4D411yescharacteristic
85C22⋊Dic127—21no—
86(C2×D127)⋊C4—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4D411yescharacteristic
90(C2×D127)⋊C4D411yescharacteristic
91C4×Dic127—21no—
92C4×Dic127—21no—
93C4×Dic127—21no—
94C4×Dic127—21no—
95C4×Dic127C2311yescharacteristic
96C2×Q8×C127C2311yescharacteristic
97C2×D4×C127C2311yescharacteristic
98C22×C4×C127—21no—
99SD16×C127—22no—
100C2×C8×C127C2311yescharacteristic
101C2×C8×C127C2311yescharacteristic
102C4⋊C4×C127—21no—
103C4⋊C4×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C22⋊C4×C127—21no—
106C42×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127C2×D411yescharacteristic
110Dic254—24no—
111Dic254—21no—
112Dic254#4112yes—
113D508—21no—
114D508—21no—
115D508—81no—
116C22⋊D127—81no—
117C22⋊D127—81no—
118C22⋊D127—81no—
119C2×Dic127—21no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127C2×D411yescharacteristic
123C2×Dic127C2×D411yescharacteristic
124C2×Dic127C2×D411yescharacteristic
125C4×D127—24no—
126C4×D127—21no—
127C4×D127—21no—
128C4×D127—81no—
129C127⋊C8—21no—
130C127⋊C8—21no—
131C23×C127—21no—
132Q8×C127—21no—
133Q8×C127#4112yes—
134D4×C127—21no—
135D4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—81no—
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2411yescharacteristic
143C8×C127—21no—
144C8×C127—21no—
145C22×C127—41no—
146C22×C127—41no—
147C22×C127—41no—
148C22×C127C22×D411yescharacteristic
149C2×D127—21no—
150C2×D127—21no—
151C2×D127—21no—
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×C127C2×#4111yescharacteristic
161C4×C127C2×#4111yescharacteristic
162Dic127—22no—
163Dic127—21no—
164Dic127—21no—
165Dic127—21no—
166Dic127—41no—
167Dic127—41no—
168C2×C127—81no—
169C2×C127#52511yescharacteristic
170C2×C127#57711yescharacteristic
171C2×C127#57711yescharacteristic
172D127—41no—
173D127—81no—
174#3126—1271nomaximal, Sylow
175C127#312611yescharacteristic, complemented, Sylow
176#540—1271no—
177#519—1271no—
178#482—1271no—
179#482—1271no—
180#479—1271no—
181#479—1271no—
182#460—1271no—
183#460—1271no—
184#448—1271no—
185#448—1271no—
186#440—1271no—
187#440D12711yescharacteristic, complemented
188C4×SD16—1271no—
189C4×SD16—1271no—
190#433—1271no—
191C22×Q8—1271no—
192C2×SD16—1271no—
193C2×SD16—1271no—
194C2×SD16—1271no—
195C2×SD16C2×D12711yescharacteristic
196#127—1271no—
197#125—1271no—
198#123—2541no—
199#123—2541no—
200#121—1271no—
201#121—1271no—
202#121—2541no—
203#121—2541no—
204C4×Q8—1271no—
205C4×Q8—1271no—
206C4×Q8—1271no—
207C4×Q8—2541no—
208C4×Q8—2541no—
209C4×Q8C2×D12711yescharacteristic
210C4×D4—1271no—
211C4×D4C2×D12711yescharacteristic
212#117—1271no—
213#117—1271no—
214#117—2541no—
215#117—2541no—
216#107—1271no—
217#107C2×D12711yescharacteristic
218#106—1271no—
219#106—1271no—
220C4⋊C8—1271no—
221C4⋊C8—1271no—
222#103—1271no—
223#103—1271no—
224#103—1271no—
225#103C2×D12711yescharacteristic
226#102—1271no—
227#102—1271no—
228#102—1271no—
229#102C2×D12711yescharacteristic
230C22⋊C8—1271no—
231C22⋊C8—1271no—
232#97C2×D12711yescharacteristic
233C4×C8—1271no—
234C2×Q8—1271no—
235C2×Q8—2541no—
236C2×Q8—2542no—
237C2×Q8—2542no—
238C2×Q8C22×D12711yescharacteristic
239C2×D4—1271no—
240C2×D4—2541no—
241C2×D4—2541no—
242C2×D4—2541no—
243C2×D4—2541no—
244C2×D4C22×D12711yescharacteristic
245C22×C4—21no—
246C22×C4—1271no—
247C22×C4—2541no—
248C22×C4—2541no—
249SD16—22no—
250SD16—2542no—
251SD16—2542no—
252SD16—2542no—
253C2×C8—1271no—
254C2×C8—1271no—
255C2×C8C22×D12711yescharacteristic
256C2×C8C22×D12711yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—1271no—
259C4⋊C4—1271no—
260C4⋊C4—1271no—
261C4⋊C4—1271no—
262C4⋊C4—2541no—
263C4⋊C4—2541no—
264C4⋊C4—2541no—
265C4⋊C4—2541no—
266C4⋊C4—2541no—
267C4⋊C4C22×D12711yescharacteristic
268C4⋊C4C22×D12711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—1271no—
271C22⋊C4—1271no—
272C22⋊C4—2541no—
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C42—21no—
277C42—1271no—
278C42—2541no—
279C42—2541no—
280C42—2541no—
281C42—2541no—
282C42C22×D12711yescharacteristic
283C23—21no—
284C23—1271no—
285C23—2541no—
286Q8—21no—
287Q8—1272no—
288Q8—2544no—
289Q8—2541no—
290Q8#1452324812yes—
291D4—21no—
292D4—21no—
293D4—10161no—
294D4—10161no—
295D4—10161no—
296D4—10161no—
297D4—2541no—
298D4—2541no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—1271no—
304C2×C4—1271no—
305C2×C4—1271no—
306C2×C4—10161no—
307C2×C4—2544no—
308C2×C4—2541no—
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—41no—
321C22—41no—
322C22—41no—
323C22—5081no—
324C22—5081no—
325C22—5081no—
326C22—2541no—
327C22—2541no—
328C22—2541no—
329C22C2×D4×D12711yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—5081no—
336C4—5081no—
337C4—2541no—
338C4—2542no—
339C4—2541no—
340C4—2541no—
341C4#41×D12711yescharacteristic
342C4C2×#1452324811yescharacteristic
343C2—81no—
344C2—10161no—
345C2—5081no—
346C2#2255038511yescharacteristic
347C2#2255068511yescharacteristic
348C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#440×C127C2
socleC22×C127C22×D4
prime radical#22550477C2
hypercenter#440D127
hypocenterC127#3126
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #22550352C2semidirect
3. #22550203C2semidirect
4. #22550198C2semidirect
5. #22550185C2semidirect
6. #22550182C2semidirect
7. #22550075C2semidirect
8. #22550074C2semidirect
9. #22550016C2semidirect
10. #22550013C2semidirect
11. #22549978C2semidirect
12. C4×#14523235C2semidirect
13. C4×#14523236C2semidirect
14. #22549938C2semidirect
15. #440×C127C2semidirect
175. C127#3126semidirect
187. #440D127semidirect

Maximal subgroups

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

subgroupindex
#225504772
#225503522
#225502032
#225501982
#225501852
#225501822
#225500752
#225500742
#225500162
#225500132
#225499782
C4×#145232352
C4×#145232362
#225499382
#440×C1272
#3126127