Finite Group Databasebeta

← #30650195

Subgroups — #30650195

Order 16256.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550497C211yesmaximal, characteristic, complemented
2#22550166C211yesmaximal, characteristic, complemented
3#22550163C211yesmaximal, characteristic, complemented
4#22550161C211yesmaximal, characteristic, complemented
5#22550158C211yesmaximal, characteristic, complemented
6#22550132C211yesmaximal, characteristic, complemented
7#22550127C211yesmaximal, characteristic
8#22550107C211yesmaximal, characteristic, complemented
9#22550103C211yesmaximal, characteristic, complemented
10#22550098C211yesmaximal, characteristic, complemented
11#22550095C211yesmaximal, characteristic, complemented
12#22550091C211yesmaximal, characteristic, complemented
13#22550088C211yesmaximal, characteristic, complemented
14#22549903C211yesmaximal, characteristic, complemented
15#542×C127C211yesmaximal, characteristic, complemented
16C2×#14523238C2211yescharacteristic
17C2×#14523236C2211yescharacteristic
18C2×#14523235C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20C2×#14523232C2211yescharacteristic
21C22×C127⋊C8C2211yescharacteristic
22#16861209C2211yescharacteristic
23#16861204—21no—
24#16861197C2211yescharacteristic
25#16861195—21no—
26#16861191—21no—
27#16861188C2211yescharacteristic
28#16861187—21no—
29#16861183C2211yescharacteristic
30C4×Dic254C2211yescharacteristic
31C4×D508C2211yescharacteristic
32#16861140C2211yescharacteristic
33#16861140C2211yescharacteristic
34#16861140C2211yescharacteristic
35#16861140C2211yescharacteristic
36C4⋊(C127⋊C8)C2211yescharacteristic
37C4⋊(C127⋊C8)C2211yescharacteristic
38C4⋊(C127⋊C8)C2211yescharacteristic
39C4⋊(C127⋊C8)C2211yescharacteristic
40#16861133C2211yescharacteristic
41#16861133C2211yescharacteristic
42#16861132C2211yescharacteristic
43#16861132C2211yescharacteristic
44#16861130C2211yescharacteristic
45#16861130C2211yescharacteristic
46#16861129C2211yescharacteristic
47#16861129C2211yescharacteristic
48C4⋊Q8×C127C2211yescharacteristic
49#126×C127—21no—
50#124×C127—21no—
51#124×C127C2211yescharacteristic
52#123×C127—21no—
53C22⋊Q8×C127—21no—
54C22⋊Q8×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56C4×Q8×C127C2211yescharacteristic
57C4×D4×C127C2211yescharacteristic
58#117×C127C2211yescharacteristic
59C2×Dic254—21no—
60C2×Dic254C2311yescharacteristic
61C2×D508C2311yescharacteristic
62C2×C22⋊D127—21no—
63C2×C4×D127—21no—
64#14523238—41no—
65#14523236—41no—
66#14523235—41no—
67#14523233—41no—
68#14523232—41no—
69C2×C127⋊C8—41no—
70C2×C127⋊C8C2311yescharacteristic
71C2×C127⋊C8C2311yescharacteristic
72C2×C127⋊C8C2311yescharacteristic
73C2×C127⋊C8C2311yescharacteristic
74C4⋊Dic127—21no—
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77Dic127⋊C4—21no—
78Dic127⋊C4—21no—
79C22⋊Dic127—21no—
80(C2×D127)⋊C4—21no—
81(C2×D127)⋊C4—21no—
82(C2×D127)⋊C4—21no—
83(C2×D127)⋊C4—21no—
84C4×Dic127—21no—
85C2×Q8×C127—21no—
86C2×Q8×C127C2311yescharacteristic
87C2×D4×C127—21no—
88C2×D4×C127C2311yescharacteristic
89C22×C4×C127—21no—
90C22×C4×C127C2311yescharacteristic
91C4⋊C4×C127—21no—
92C4⋊C4×C127—21no—
93C4⋊C4×C127—21no—
94C4⋊C4×C127D411yescharacteristic
95C4⋊C4×C127D411yescharacteristic
96C4⋊C4×C127C2311yescharacteristic
97C4⋊C4×C127C2311yescharacteristic
98C22⋊C4×C127—21no—
99C22⋊C4×C127—21no—
100C22⋊C4×C127—21no—
101C22⋊C4×C127—21no—
102C22⋊C4×C127—21no—
103C22⋊C4×C127D411yescharacteristic
104C22⋊C4×C127D411yescharacteristic
105C42×C127—21no—
106C42×C127C2311yescharacteristic
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109Dic254—21no—
110Dic254—21no—
111Dic254—81no—
112D508—21no—
113D508—21no—
114C22⋊D127—81no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C4×D127—81no—
119C127⋊C8—21no—
120C127⋊C8—21no—
121C127⋊C8—21no—
122C127⋊C8—21no—
123C23×C127—21no—
124C23×C127C2×D411yescharacteristic
125Q8×C127—21no—
126Q8×C127—21no—
127Q8×C127—81no—
128D4×C127—21no—
129D4×C127—21no—
130D4×C127—81no—
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—81no—
137C2×C4×C127C2×D411yescharacteristic
138C2×C4×C127C2×D411yescharacteristic
139C2×C4×C127C2×D411yescharacteristic
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2411yescharacteristic
143C22×C127—21no—
144C22×C127—21no—
145C22×C127—21no—
146C22×C127—41no—
147C22×C127—41no—
148C22×C127—41no—
149C22×C127C22×D411yescharacteristic
150C2×D127—41no—
151C2×D127—41no—
152C2×D127—41no—
153C4×C127—21no—
154C4×C127—21no—
155C4×C127—21no—
156C4×C127—21no—
157C4×C127—21no—
158C4×C127—41no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127H(5,𝔽2)11yescharacteristic
162C4×C1272−1+411yescharacteristic
163Dic127—41no—
164Dic127—41no—
165Dic127—41no—
166C2×C127—41no—
167C2×C127—81no—
168C2×C127#53711yescharacteristic
169C2×C127#57711yescharacteristic
170C2×C127#57711yescharacteristic
171D127—81no—
172#3263—1271nomaximal, Sylow
173C127#326311yescharacteristic, complemented, Sylow
174#542—1271no—
175#542D12711yescharacteristic, complemented
176#476—1271no—
177#476—1271no—
178#475—1271no—
179#475—1271no—
180#470—1271no—
181#469—1271no—
182#465—1271no—
183#465—1271no—
184#464—1271no—
185#464—1271no—
186#463—1271no—
187#463—1271no—
188#424—1271no—
189C2×Q16—1271no—
190C2×SD16—1271no—
191C2×SD16—1271no—
192C2×D8—1271no—
193C2×M16—1271no—
194C22×C8—1271no—
195C4⋊Q8—1271no—
196C4⋊Q8C2×D12711yescharacteristic
197#126—21no—
198#126—2541no—
199#124—21no—
200#124—1271no—
201#124—2541no—
202#124C2×D12711yescharacteristic
203#123—21no—
204#123—2541no—
205C22⋊Q8—21no—
206C22⋊Q8—1271no—
207C22⋊Q8—2541no—
208C22⋊Q8C2×D12711yescharacteristic
209#121—1271no—
210#121C2×D12711yescharacteristic
211C4×Q8—1271no—
212C4×Q8C2×D12711yescharacteristic
213C4×D4—1271no—
214C4×D4C2×D12711yescharacteristic
215#117C2×D12711yescharacteristic
216#106—1271no—
217#106—1271no—
218#106—1271no—
219#106—1271no—
220C4⋊C8—1271no—
221C4⋊C8—1271no—
222C4⋊C8—1271no—
223C4⋊C8—1271no—
224#103—1271no—
225#103—1271no—
226#103—1271no—
227#103—1271no—
228#102—1271no—
229#102—1271no—
230#102—1271no—
231#102—1271no—
232C2×Q8—21no—
233C2×Q8—1271no—
234C2×Q8—2541no—
235C2×Q8C22×D12711yescharacteristic
236C2×D4—21no—
237C2×D4—1271no—
238C2×D4—2541no—
239C2×D4C22×D12711yescharacteristic
240C22×C4—21no—
241C22×C4—2541no—
242C22×C4C22×D12711yescharacteristic
243Q16—5081no—
244SD16—5081no—
245SD16—5081no—
246D8—5081no—
247M16—5081no—
248C2×C8—1271no—
249C2×C8—1271no—
250C2×C8—1271no—
251C2×C8—1271no—
252C2×C8—5081no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—1271no—
257C4⋊C4—1271no—
258C4⋊C4—2541no—
259C4⋊C4—2541no—
260C4⋊C4—2541no—
261C4⋊C4C22⋊D12711yescharacteristic
262C4⋊C4C22⋊D12711yescharacteristic
263C4⋊C4C22×D12711yescharacteristic
264C4⋊C4C22×D12711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—2541no—
271C22⋊C4—2541no—
272C22⋊C4—2541no—
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C22⋊C4C22⋊D12711yescharacteristic
276C22⋊C4C22⋊D12711yescharacteristic
277C42—21no—
278C42—2541no—
279C42C22×D12711yescharacteristic
280C42C22×D12711yescharacteristic
281C23—21no—
282C23—2541no—
283C23C2×C22⋊D12711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—81no—
287Q8—10161no—
288Q8—2541no—
289Q8—2541no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—10161no—
294D4—2541no—
295D4—2541no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—81no—
302C2×C4—10161no—
303C2×C4—2541no—
304C2×C4—2541no—
305C2×C4—2541no—
306C2×C4C2×C22⋊D12711yescharacteristic
307C2×C4C2×C22⋊D12711yescharacteristic
308C2×C4C2×C22⋊D12711yescharacteristic
309C2×C4C2×C22⋊D12711yescharacteristic
310C2×C4C2×C22⋊D12711yescharacteristic
311C2×C4C23×D12711yescharacteristic
312C8—2541no—
313C8—2541no—
314C8—2541no—
315C8—2541no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—5081no—
323C22—5081no—
324C22—5081no—
325C22C22×C22⋊D12711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—5081no—
335C4—5081no—
336C4—5081no—
337C4#1686125511yescharacteristic
338C4#1686125611yescharacteristic
339C2—41no—
340C2—81no—
341C2—10161no—
342C2#2255045411yescharacteristic
343C2#2255068411yescharacteristic
344C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#542×C127C2
socleC22×C127C22×D4
prime radical#22550127C2
hypercenter#542D127
hypocenterC127#3263
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #22550497C2semidirect
2. #22550166C2semidirect
3. #22550163C2semidirect
4. #22550161C2semidirect
5. #22550158C2semidirect
6. #22550132C2semidirect
8. #22550107C2semidirect
9. #22550103C2semidirect
10. #22550098C2semidirect
11. #22550095C2semidirect
12. #22550091C2semidirect
13. #22550088C2semidirect
14. #22549903C2semidirect
15. #542×C127C2semidirect
173. C127#3263semidirect
175. #542D127semidirect

Maximal subgroups

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

subgroupindex
#225504972
#225501662
#225501632
#225501612
#225501582
#225501322
#225501272
#225501072
#225501032
#225500982
#225500952
#225500912
#225500882
#225499032
#542×C1272
#3263127