Finite Group Databasebeta

← #30648709

Subgroups — #30648709

Order 16256.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550456C211yesmaximal, characteristic
2#22550357C211yesmaximal, characteristic, complemented
3#22550215C211yesmaximal, characteristic, complemented
4#22550214C211yesmaximal, characteristic, complemented
5#22550210C211yesmaximal, characteristic, complemented
6#22550203C211yesmaximal, characteristic, complemented
7#22550201C211yesmaximal, characteristic, complemented
8#22550191C211yesmaximal, characteristic, complemented
9#22550189C211yesmaximal, characteristic, complemented
10#22549917C211yesmaximal, characteristic, complemented
11#107×D127C211yesmaximal, characteristic, complemented
12#22549876C211yesmaximal, characteristic, complemented
13#22549871C211yesmaximal, characteristic, complemented
14#22549841C211yesmaximal, characteristic, complemented
15#484×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523248C2211yescharacteristic
18C2×#14523231C2211yescharacteristic
19C2×C8×D127C2211yescharacteristic
20#16861187—21no—
21#16861185C2211yescharacteristic
22#16861184—21no—
23#16861184C2211yescharacteristic
24#16861183C2211yescharacteristic
25#16861182—21no—
26#16861182—21no—
27#16861174—21no—
28#16861171—21no—
29C4×C22⋊D127—21no—
30C4×C22⋊D127—21no—
31#16861168C2211yescharacteristic
32#16861167C2211yescharacteristic
33#16861167C2211yescharacteristic
34C4⋊C4×D127C2211yescharacteristic
35C4⋊C4×D127C2211yescharacteristic
36C4⋊C4×D127C2211yescharacteristic
37#16861143C2211yescharacteristic
38#16861142C2211yescharacteristic
39#16861142C2211yescharacteristic
40#16861142C2211yescharacteristic
41#16861141C2211yescharacteristic
42#16861140C2211yescharacteristic
43#16861133C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic
52C22⋊Q8×C127C2211yescharacteristic
53C2×C4⋊C4×C127C2211yescharacteristic
54#107×C127C2211yescharacteristic
55#107×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic
59#14523249—22no—
60#14523249—22no—
61#14523248—22no—
62#14523248—22no—
63C2×Dic254D411yescharacteristic
64C2×D508—21no—
65C2×D508—21no—
66C2×D508D411yescharacteristic
67C2×D508C2311yescharacteristic
68C2×C22⋊D127—21no—
69C2×C22⋊D127D411yescharacteristic
70C2×C22⋊D127D411yescharacteristic
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127C2311yescharacteristic
76C2×C4×D127C2311yescharacteristic
77#14523231—22no—
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C8×D127—22no—
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88C22⋊Dic127—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94C4×Dic127—21no—
95C4×Dic127—21no—
96C4×Dic127—21no—
97C2×Q8×C127C2311yescharacteristic
98C22×C4×C127—21no—
99C22×C4×C127C2311yescharacteristic
100C2×C8×C127C2311yescharacteristic
101C2×C8×C127C2311yescharacteristic
102C4⋊C4×C127—21no—
103C4⋊C4×C127—21no—
104C4⋊C4×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C4⋊C4×C127C2311yescharacteristic
107C22⋊C4×C127—21no—
108C22×D127—21no—
109C22×D127C2×D411yescharacteristic
110C22×D127C2×D411yescharacteristic
111Dic254—22no—
112D508—22no—
113D508—21no—
114D508—21no—
115D508—42no—
116D508—81no—
117C22⋊D127—22no—
118C22⋊D127—22no—
119C22⋊D127—81no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127C2×D411yescharacteristic
124C2×Dic127C2×D411yescharacteristic
125C4×D127—21no—
126C4×D127—21no—
127C4×D127—42no—
128C4×D127—81no—
129C4×D127—81no—
130C4×D127—81no—
131C4×D127#4112yes—
132C4×D127#4112yes—
133C127⋊C8—21no—
134C127⋊C8—21no—
135C23×C127C2×D411yescharacteristic
136Q8×C127—21no—
137Q8×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—81no—
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2411yescharacteristic
147C8×C127—21no—
148C8×C127—21no—
149C22×C127—21no—
150C22×C127—21no—
151C22×C127—21no—
152C22×C127C22×D411yescharacteristic
153C2×D127—21no—
154C2×D127—21no—
155C2×D127—21no—
156C2×D127—21no—
157C2×D127—21no—
158C2×D127—41no—
159C2×D127—41no—
160C2×D127—41no—
161C2×D127#12312yes—
162C4×C127—21no—
163C4×C127—41no—
164C4×C127—41no—
165C4×C127—41no—
166C4×C127—41no—
167C4×C127C2×#4111yescharacteristic
168C4×C127C2×#4111yescharacteristic
169Dic127—21no—
170Dic127—41no—
171Dic127—41no—
172Dic127—41no—
173Dic127#12312yes—
174C2×C127—41no—
175C2×C127C2×#12311yescharacteristic
176C2×C127#57611yescharacteristic
177C2×C127#57811yescharacteristic
178D127—22no—
179D127—41no—
180D127—81no—
181#3098—1271nomaximal, Sylow
182C127#309811yescharacteristic, complemented, Sylow
183#537—1271no—
184#520—1271no—
185#485—1271no—
186#485—1271no—
187#484—1271no—
188#484D12711yescharacteristic, complemented
189#482—1271no—
190#482—1271no—
191#480—1271no—
192#480—1271no—
193#428—1271no—
194C2×#107—1271no—
195#417—1271no—
196#416—1271no—
197#408—1271no—
198C2×#41—1271no—
199C2×#41—1271no—
200C2×M16—1271no—
201C22×C8—1271no—
202C22⋊Q8—1271no—
203C22⋊Q8—2541no—
204C22⋊Q8C2×D12711yescharacteristic
205#121—1271no—
206#121—1271no—
207#121—2541no—
208#121—2541no—
209#121—2541no—
210C4×Q8—2541no—
211C4×D4—2541no—
212C4×D4—2541no—
213C4×D4—2541no—
214#117—1271no—
215#117—1271no—
216#117—1271no—
217C2×C4⋊C4—1271no—
218C2×C4⋊C4—1271no—
219C2×C4⋊C4—1271no—
220C2×C4⋊C4C2×D12711yescharacteristic
221#107—1271no—
222#107—1271no—
223#107—1271no—
224#107—1271no—
225#107C2×D12711yescharacteristic
226#107C2×D12711yescharacteristic
227#106—1271no—
228#106—1271no—
229#103—1271no—
230#103—1271no—
231#103C2×D12711yescharacteristic
232#103C2×D12711yescharacteristic
233#102—1271no—
234#102—1271no—
235#102—1271no—
236#102—1271no—
237C22⋊C8—1271no—
238C22⋊C8—1271no—
239C22⋊C8—1271no—
240C22⋊C8C2×D12711yescharacteristic
241#41—2542no—
242#41—2542no—
243#41—2542no—
244#41—2542no—
245C2×Q8—1271no—
246C2×Q8C22×D12711yescharacteristic
247C2×D4—1271no—
248C2×D4—1271no—
249C2×D4—1271no—
250C2×D4—1271no—
251C2×D4—2541no—
252C2×D4—2541no—
253C2×D4—2541no—
254C22×C4—21no—
255C22×C4—1271no—
256C22×C4—1271no—
257C22×C4—2541no—
258C22×C4—2541no—
259C22×C4—2541no—
260C22×C4—2541no—
261C22×C4C22×D12711yescharacteristic
262M16—2542no—
263C2×C8—1271no—
264C2×C8—1271no—
265C2×C8—2542no—
266C2×C8C22×D12711yescharacteristic
267C2×C8C22×D12711yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—1271no—
271C4⋊C4—1271no—
272C4⋊C4—1271no—
273C4⋊C4—2541no—
274C4⋊C4—2541no—
275C4⋊C4—2541no—
276C4⋊C4—2541no—
277C4⋊C4C22×D12711yescharacteristic
278C4⋊C4C22×D12711yescharacteristic
279C4⋊C4C22×D12711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C22⋊C4—2541no—
287C42—2541no—
288C42—2541no—
289C42—2541no—
290C23—1271no—
291C23—1271no—
292C23—2541no—
293C23D4×D12711yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—2542no—
297D4—10161no—
298D4—10161no—
299D4—5082no—
300D4—2542no—
301D4—2542no—
302D4—2542no—
303D4—2541no—
304D4—2541no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—1271no—
313C2×C4—1272no—
314C2×C4—1272no—
315C2×C4—1271no—
316C2×C4—10161no—
317C2×C4—10161no—
318C2×C4—10161no—
319C2×C4—5082no—
320C2×C4—2541no—
321C2×C4—2541no—
322C2×C4—2541no—
323C2×C4—2541no—
324C2×C4—2541no—
325C2×C4D4×D12711yescharacteristic
326C2×C4C23×D12711yescharacteristic
327C8—21no—
328C8—21no—
329C8—2541no—
330C8—2541no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—1272no—
335C22—5081no—
336C22—5081no—
337C22—5081no—
338C22—2541no—
339C22—2541no—
340C22—2541no—
341C22—2541no—
342C22—2541no—
343C22C2×D4×D12711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—1272no—
350C4—5081no—
351C4—5081no—
352C4—5081no—
353C4—2541no—
354C4#41×D12711yescharacteristic
355C4#41×D12711yescharacteristic
356C2—41no—
357C2—10161no—
358C2—5081no—
359C2—2542no—
360C2#123×D12711yescharacteristic
361C2#2255068011yescharacteristic
362C2#2255068911yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#484×C127C2
socleC22×C127C22×D4
prime radical#22550456C2
hypercenter#484D127
hypocenterC127#3098
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. #22550357C2semidirect
3. #22550215C2semidirect
4. #22550214C2semidirect
5. #22550210C2semidirect
6. #22550203C2semidirect
7. #22550201C2semidirect
8. #22550191C2semidirect
9. #22550189C2semidirect
10. #22549917C2semidirect
11. #107×D127C2semidirect
12. #22549876C2semidirect
13. #22549871C2semidirect
14. #22549841C2semidirect
15. #484×C127C2semidirect
182. C127#3098semidirect
188. #484D127semidirect

Maximal subgroups

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

subgroupindex
#225504562
#225503572
#225502152
#225502142
#225502102
#225502032
#225502012
#225501912
#225501892
#225499172
#107×D1272
#225498762
#225498712
#225498412
#484×C1272
#3098127