Finite Group Databasebeta

← #30647123

Subgroups — #30647123

Order 16256.

418 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22549768C211yesmaximal, characteristic
2#22549768C211yesmaximal, characteristic
3#22549767C211yesmaximal, characteristic
4#22549766C211yesmaximal, characteristic
5#22549766C211yesmaximal, characteristic
6#22549765C211yesmaximal, characteristic
7#22549765C211yesmaximal, characteristic
8#22549763C211yesmaximal, characteristic, complemented
9#22549762C211yesmaximal, characteristic, complemented
10#22549761C211yesmaximal, characteristic, complemented
11#22549759C211yesmaximal, characteristic, complemented
12#22549758C211yesmaximal, characteristic, complemented
13#22549757C211yesmaximal, characteristic, complemented
14C42×Dic127C211yesmaximal, characteristic, complemented
15#387×C127C211yesmaximal, characteristic, complemented
16C2×C4⋊Dic127C2211yescharacteristic
17C2×C4⋊Dic127C2211yescharacteristic
18C2×Dic127⋊C4C2211yescharacteristic
19C2×Dic127⋊C4C2211yescharacteristic
20C2×Dic127⋊C4C2211yescharacteristic
21C2×(C2×D127)⋊C4C2211yescharacteristic
22C2×(C2×D127)⋊C4C2211yescharacteristic
23C2×(C2×D127)⋊C4C2211yescharacteristic
24C2×(C2×D127)⋊C4C2211yescharacteristic
25C2×(C2×D127)⋊C4C2211yescharacteristic
26C2×(C2×D127)⋊C4C2211yescharacteristic
27C2×(C2×D127)⋊C4C2211yescharacteristic
28C2×C4×Dic127C2211yescharacteristic
29C2×C4×Dic127C2211yescharacteristic
30C2×C4×Dic127C2211yescharacteristic
31C2×C4×Dic127C2211yescharacteristic
32C2×C4×Dic127C2211yescharacteristic
33C2×C4×Dic127C2211yescharacteristic
34C127⋊#95C2211yescharacteristic
35C127⋊#95C2211yescharacteristic
36C127⋊#95C2211yescharacteristic
37C127⋊#95C2211yescharacteristic
38C127⋊#95C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C2×C4⋊C4×C127C2211yescharacteristic
45C2×C4⋊C4×C127C2211yescharacteristic
46C2×C42×C127C2211yescharacteristic
47#95×C127C2211yescharacteristic
48#95×C127C2211yescharacteristic
49#95×C127C2211yescharacteristic
50#95×C127C2211yescharacteristic
51C23×D127C2311yescharacteristic
52C22×Dic127C2311yescharacteristic
53C22×Dic127C2311yescharacteristic
54C22×Dic127C2311yescharacteristic
55C22×Dic127C2311yescharacteristic
56C22×Dic127C2311yescharacteristic
57C22×Dic127C2311yescharacteristic
58C22×Dic127C2311yescharacteristic
59C4⋊Dic127—22no—
60C4⋊Dic127—22no—
61Dic127⋊C4—22no—
62Dic127⋊C4—22no—
63Dic127⋊C4—22no—
64(C2×D127)⋊C4—22no—
65(C2×D127)⋊C4—22no—
66(C2×D127)⋊C4—22no—
67(C2×D127)⋊C4—22no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4—22no—
71C4×Dic127—22no—
72C4×Dic127—22no—
73C4×Dic127—22no—
74C4×Dic127—22no—
75C4×Dic127—22no—
76C4×Dic127—22no—
77C22×C4×C127C2311yescharacteristic
78C22×C4×C127C2311yescharacteristic
79C22×C4×C127C2311yescharacteristic
80C22×C4×C127C2311yescharacteristic
81C22×C4×C127C2311yescharacteristic
82C22×C4×C127C2311yescharacteristic
83C22×C4×C127C2311yescharacteristic
84C4⋊C4×C127—22no—
85C4⋊C4×C127—22no—
86C42×C127—22no—
87C22×D127—21no—
88C22×D127—21no—
89C22×D127—21no—
90C22×D127—21no—
91C22×D127—21no—
92C22×D127—21no—
93C22×D127—21no—
94C2×Dic127—21no—
95C2×Dic127—21no—
96C2×Dic127—21no—
97C2×Dic127—21no—
98C2×Dic127—21no—
99C2×Dic127—21no—
100C2×Dic127—21no—
101C2×Dic127—21no—
102C2×Dic127—21no—
103C2×Dic127—21no—
104C2×Dic127—21no—
105C2×Dic127—21no—
106C2×Dic127—21no—
107C2×Dic127—21no—
108C2×Dic127—21no—
109C2×Dic127—21no—
110C2×Dic127—21no—
111C2×Dic127#4112yes—
112C2×Dic127#4112yes—
113C2×Dic127#4112yes—
114C2×Dic127#4112yes—
115C23×C127C2411yescharacteristic
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—21no—
120C2×C4×C127—21no—
121C2×C4×C127—21no—
122C2×C4×C127—21no—
123C2×C4×C127—21no—
124C2×C4×C127—21no—
125C2×C4×C127—21no—
126C2×C4×C127—21no—
127C2×C4×C127—21no—
128C2×C4×C127—21no—
129C2×C4×C127—21no—
130C2×C4×C127—21no—
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127#4112yes—
135C2×C4×C127#4112yes—
136C2×C4×C127#4112yes—
137C22×C127C2×#4111yescharacteristic
138C22×C127C2×#4111yescharacteristic
139C22×C127C2×#4111yescharacteristic
140C22×C127C2×#4111yescharacteristic
141C22×C127C2×#4111yescharacteristic
142C22×C127C2×#4111yescharacteristic
143C22×C127C2×#4111yescharacteristic
144C2×D127—41no—
145C2×D127—41no—
146C2×D127—41no—
147C2×D127—41no—
148C2×D127—41no—
149C2×D127—41no—
150C2×D127—41no—
151C4×C127—22no—
152C4×C127—22no—
153C4×C127—22no—
154C4×C127—41no—
155C4×C127—41no—
156C4×C127—41no—
157C4×C127—41no—
158Dic127—22no—
159Dic127—22no—
160Dic127—22no—
161Dic127—22no—
162Dic127—41no—
163Dic127—41no—
164Dic127—41no—
165C2×C127#52911yescharacteristic
166C2×C127#52911yescharacteristic
167C2×C127#52911yescharacteristic
168C2×C127#52911yescharacteristic
169C2×C127#52911yescharacteristic
170C2×C127#52911yescharacteristic
171C2×C127#52911yescharacteristic
172D127—81no—
173#2866—1271nomaximal, Sylow
174C127#286611yescharacteristic, complemented, Sylow
175#388—1271no—
176#388—1271no—
177#388—1271no—
178#388—1271no—
179#388—1271no—
180#388—1271no—
181#388—1271no—
182#387—1271no—
183#387—1271no—
184#387—1271no—
185#387—1271no—
186#387—1271no—
187#387—1271no—
188#387D12711yescharacteristic, complemented
189C43—1271no—
190C2×C4⋊C4—1271no—
191C2×C4⋊C4—1271no—
192C2×C4⋊C4—1271no—
193C2×C4⋊C4—1271no—
194C2×C4⋊C4—1271no—
195C2×C4⋊C4C2×D12711yescharacteristic
196C2×C4⋊C4C2×D12711yescharacteristic
197C2×C22⋊C4—1271no—
198C2×C22⋊C4—1271no—
199C2×C22⋊C4—1271no—
200C2×C22⋊C4—1271no—
201C2×C22⋊C4—1271no—
202C2×C22⋊C4—1271no—
203C2×C22⋊C4—1271no—
204C2×C42—1271no—
205C2×C42—1271no—
206C2×C42—1271no—
207C2×C42—1271no—
208C2×C42—1271no—
209C2×C42—1271no—
210C2×C42C2×D12711yescharacteristic
211#95—1271no—
212#95—1271no—
213#95—1271no—
214#95—1271no—
215#95—1271no—
216#95—1271no—
217#95—1271no—
218#95—1271no—
219#95—1271no—
220#95—1271no—
221#95C2×D12711yescharacteristic
222#95C2×D12711yescharacteristic
223#95C2×D12711yescharacteristic
224#95C2×D12711yescharacteristic
225C24—1271no—
226C22×C4—1271no—
227C22×C4—1271no—
228C22×C4—1271no—
229C22×C4—1271no—
230C22×C4—1271no—
231C22×C4—1271no—
232C22×C4—1271no—
233C22×C4C22×D12711yescharacteristic
234C22×C4C22×D12711yescharacteristic
235C22×C4C22×D12711yescharacteristic
236C22×C4C22×D12711yescharacteristic
237C22×C4C22×D12711yescharacteristic
238C22×C4C22×D12711yescharacteristic
239C22×C4C22×D12711yescharacteristic
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—2542no—
243C4⋊C4—2542no—
244C4⋊C4—2542no—
245C4⋊C4—2542no—
246C4⋊C4—2542no—
247C22⋊C4—2542no—
248C22⋊C4—2542no—
249C22⋊C4—2542no—
250C22⋊C4—2542no—
251C22⋊C4—2542no—
252C22⋊C4—2542no—
253C22⋊C4—2542no—
254C42—22no—
255C42—2542no—
256C42—2542no—
257C42—2542no—
258C42—2542no—
259C42—2542no—
260C42—2542no—
261C23—2541no—
262C23—2541no—
263C23—2541no—
264C23—2541no—
265C23—2541no—
266C23—2541no—
267C23—2541no—
268C23C23×D12711yescharacteristic
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—1272no—
288C2×C4—1272no—
289C2×C4—1272no—
290C2×C4—1272no—
291C2×C4—2541no—
292C2×C4—2541no—
293C2×C4—2541no—
294C2×C4—2541no—
295C2×C4—2541no—
296C2×C4—2541no—
297C2×C4—2541no—
298C2×C4—2541no—
299C2×C4—2541no—
300C2×C4—2541no—
301C2×C4—2541no—
302C2×C4—2541no—
303C2×C4—2541no—
304C2×C4—2541no—
305C2×C4—2541no—
306C2×C4—2541no—
307C2×C4—2541no—
308C2×C4#1452324712yes—
309C2×C4#1452324812yes—
310C2×C4#1452324912yes—
311C22—5081no—
312C22—5081no—
313C22—5081no—
314C22—5081no—
315C22—5081no—
316C22—5081no—
317C22—5081no—
318C22#41×D12711yescharacteristic
319C22#41×D12711yescharacteristic
320C22#41×D12711yescharacteristic
321C22#41×D12711yescharacteristic
322C22C2×#1452324711yescharacteristic
323C22C2×#1452324811yescharacteristic
324C22C2×#1452324911yescharacteristic
325C4—22no—
326C4—22no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—5081no—
333C4—5081no—
334C4—5081no—
335C4—2542no—
336C4—2542no—
337C4—2542no—
338C4—2542no—
339C2—10161no—
340C2#2255040311yescharacteristic
341C2#2255040411yescharacteristic
342C2#2255040611yescharacteristic
343C2#2255040911yescharacteristic
344C2#2255041111yescharacteristic
345C2#2255041211yescharacteristic
346C2#2255041311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#387×C127C2
prime radicalC23×D127C23
hypercenter#387D127
hypocenterC127#2866
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
8. #22549763C2semidirect
9. #22549762C2semidirect
10. #22549761C2semidirect
11. #22549759C2semidirect
12. #22549758C2semidirect
13. #22549757C2semidirect
14. C42×Dic127C2semidirect
15. #387×C127C2semidirect
174. C127#2866semidirect
188. #387D127semidirect

Maximal subgroups

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

subgroupindex
#225497682
#225497672
#225497662
#225497652
#225497632
#225497622
#225497612
#225497592
#225497582
#225497572
C42×Dic1272
#387×C1272
#2866127