Finite Group Databasebeta

← #30641224

Subgroups — #30641224

Order 16256.

524 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861165C211yesmaximal, characteristic
2#22549770C211yesmaximal, characteristic
3#22549769C211yesmaximal, characteristic
4#22549764C211yesmaximal, characteristic
5#22549763C211yesmaximal, characteristic, complemented
6#22549760C211yesmaximal, characteristic, complemented
7#22549758C211yesmaximal, characteristic, complemented
8#22549757C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic127C211yesmaximal, characteristic, complemented
10#22549727C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic127C211yesmaximal, characteristic
12C22⋊C4×Dic127C211yesmaximal, characteristic
13#22549721C211yesmaximal, characteristic, complemented
14C42×Dic127C211yesmaximal, characteristic, complemented
15#388×C127C211yesmaximal, characteristic
16C23×Dic127C2211yescharacteristic
17#16861165C418yescomplemented
18C2×C4⋊Dic127C2211yescharacteristic
19C2×Dic127⋊C4C2211yescharacteristic
20C2×Dic127⋊C4C2211yescharacteristic
21C2×Dic127⋊C4C2211yescharacteristic
22C2×Dic127⋊C4C2211yescharacteristic
23C2×C22⋊Dic127C2211yescharacteristic
24C2×C22⋊Dic127C2211yescharacteristic
25C2×C22⋊Dic127C2211yescharacteristic
26C2×C4×Dic127C2211yescharacteristic
27C2×C4×Dic127C2211yescharacteristic
28C2×C4×Dic127C2211yescharacteristic
29C2×C4×Dic127C2211yescharacteristic
30C2×C4×Dic127C2211yescharacteristic
31C2×C4×Dic127C2211yescharacteristic
32C2×C4×Dic127C2211yescharacteristic
33C2×C4×Dic127C2211yescharacteristic
34C2×C4×Dic127C2211yescharacteristic
35C127⋊#95C2211yescharacteristic
36C127⋊#95C2211yescharacteristic
37C127⋊#95C2211yescharacteristic
38C127⋊#95C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C2×C4⋊C4×C127C2211yescharacteristic
46C2×C22⋊C4×C127C2211yescharacteristic
47C2×C22⋊C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C42×C127C2211yescharacteristic
50#95×C127C2211yescharacteristic
51#95×C127C2211yescharacteristic
52C22×Dic127—22no—
53C22×Dic127—22no—
54C22×Dic127C2×C414yes—
55C22×Dic127C2311yescharacteristic
56C22×Dic127C2311yescharacteristic
57C22×Dic127C2311yescharacteristic
58C22×Dic127C2311yescharacteristic
59C22×Dic127C2311yescharacteristic
60C22×Dic127C2311yescharacteristic
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C4⋊Dic127—22no—
64Dic127⋊C4—22no—
65Dic127⋊C4—22no—
66Dic127⋊C4C2×C414yes—
67Dic127⋊C4C2×C414yes—
68C22⋊Dic127—22no—
69C22⋊Dic127—22no—
70C22⋊Dic127C2×C414yes—
71C4×Dic127—22no—
72C4×Dic127—22no—
73C4×Dic127—22no—
74C4×Dic127—22no—
75C4×Dic127—22no—
76C4×Dic127—22no—
77C4×Dic127—22no—
78C4×Dic127C2×C414yes—
79C4×Dic127C2×C414yes—
80C24×C127C2311yescharacteristic
81C22×C4×C127C2311yescharacteristic
82C22×C4×C127C2311yescharacteristic
83C22×C4×C127C2311yescharacteristic
84C22×C4×C127C2311yescharacteristic
85C22×C4×C127C2311yescharacteristic
86C22×C4×C127C2311yescharacteristic
87C4⋊C4×C127—22no—
88C22⋊C4×C127—22no—
89C22⋊C4×C127—22no—
90C22⋊C4×C127C2×C414yes—
91C42×C127—22no—
92C2×Dic127—21no—
93C2×Dic127—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—28no—
103C2×Dic127—21no—
104C2×Dic127—21no—
105C2×Dic127—21no—
106C2×Dic127—21no—
107C2×Dic127—21no—
108C2×Dic127—21no—
109C2×Dic127C22×C412yes—
110C2×Dic127C22×C412yes—
111C2×Dic127C22×C412yes—
112C2×Dic127C22×C412yes—
113C2×Dic127#4112yes—
114C2×Dic127#4112yes—
115C2×Dic127#4112yes—
116C2×Dic127#4112yes—
117C23×C127—21no—
118C23×C127—21no—
119C23×C127—21no—
120C23×C127—21no—
121C23×C127—21no—
122C23×C127—21no—
123C23×C127C22×C412yes—
124C23×C127C2411yescharacteristic
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—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127C22×C412yes—
140C2×C4×C127C22×C412yes—
141C2×C4×C127#4112yes—
142C2×C4×C127#4112yes—
143C22×C127—22no—
144C22×C127—22no—
145C22×C127—22no—
146C22×C127—41no—
147C22×C127—41no—
148C22×C127—41no—
149C22×C127—41no—
150C22×C127C23×C411yescharacteristic
151C22×C127C2×#4111yescharacteristic
152C22×C127C2×#4111yescharacteristic
153C22×C127C2×#4111yescharacteristic
154C22×C127C2×#4111yescharacteristic
155C22×C127C2×#4111yescharacteristic
156C22×C127C2×#4111yescharacteristic
157C4×C127—22no—
158C4×C127—22no—
159C4×C127—22no—
160C4×C127—22no—
161C4×C127—41no—
162C4×C127—41no—
163Dic127—22no—
164Dic127—22no—
165Dic127—22no—
166Dic127—22no—
167Dic127—22no—
168Dic127—41no—
169Dic127—41no—
170Dic127#11714yes—
171C2×C127—42no—
172C2×C127C2×#11711yescharacteristic
173C2×C127C4×#4111yescharacteristic
174C2×C127C4×#4111yescharacteristic
175C2×C127#52911yescharacteristic
176C2×C127#52911yescharacteristic
177C2×C127#52911yescharacteristic
178C2×C127#52911yescharacteristic
179#2298—1271nomaximal, Sylow
180C127#229811yescharacteristic, complemented, Sylow
181C2×#117—1271no—
182#388—1271no—
183#388—1271no—
184#388—1271no—
185#388D12711yescharacteristic
186#387—1271no—
187#387—1271no—
188#387—1271no—
189#387—1271no—
190C4×C4⋊C4—1271no—
191C4×C4⋊C4—1271no—
192C4×C22⋊C4—1271no—
193C4×C22⋊C4—1271no—
194#376—1271no—
195C43—1271no—
196C23×C4—1271no—
197#117—1278no—
198C2×C4⋊C4—1271no—
199C2×C4⋊C4—1271no—
200C2×C4⋊C4—1271no—
201C2×C4⋊C4—1271no—
202C2×C4⋊C4—1271no—
203C2×C4⋊C4C2×D12711yescharacteristic
204C2×C22⋊C4—1271no—
205C2×C22⋊C4—1271no—
206C2×C22⋊C4—1271no—
207C2×C22⋊C4C2×D12711yescharacteristic
208C2×C22⋊C4C2×D12711yescharacteristic
209C2×C22⋊C4C2×D12711yescharacteristic
210C2×C42—1271no—
211C2×C42—1271no—
212C2×C42—1271no—
213C2×C42—1271no—
214C2×C42—1271no—
215C2×C42—1271no—
216C2×C42—1271no—
217C2×C42—1271no—
218C2×C42—1271no—
219C2×C42C2×D12711yescharacteristic
220#95—1271no—
221#95—1271no—
222#95—1271no—
223#95—1271no—
224#95—1271no—
225#95—1271no—
226#95—1271no—
227#95—1271no—
228#95—1271no—
229#95—1271no—
230#95C2×D12711yescharacteristic
231#95C2×D12711yescharacteristic
232C24C22×D12711yescharacteristic
233C22×C4—1271no—
234C22×C4—1274no—
235C22×C4—1271no—
236C22×C4—1271no—
237C22×C4—1271no—
238C22×C4—1271no—
239C22×C4—1271no—
240C22×C4—1271no—
241C22×C4—1271no—
242C22×C4—2542no—
243C22×C4—2542no—
244C22×C4C22×D12711yescharacteristic
245C22×C4C22×D12711yescharacteristic
246C22×C4C22×D12711yescharacteristic
247C22×C4C22×D12711yescharacteristic
248C22×C4C22×D12711yescharacteristic
249C22×C4C22×D12711yescharacteristic
250C4⋊C4—22no—
251C4⋊C4—1274no—
252C4⋊C4—1274no—
253C4⋊C4—2542no—
254C4⋊C4—2542no—
255C4⋊C4—2542no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—1274no—
259C22⋊C4—2542no—
260C22⋊C4—2542no—
261C22⋊C4C4×D12714yes—
262C42—22no—
263C42—1274no—
264C42—1274no—
265C42—2542no—
266C42—2542no—
267C42—2542no—
268C42—2542no—
269C42—2542no—
270C42—2542no—
271C42—2542no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D12712yes—
279C23C23×D12711yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—1272no—
295C2×C4—1272no—
296C2×C4—1272no—
297C2×C4—1272no—
298C2×C4—1272no—
299C2×C4—1272no—
300C2×C4—1272no—
301C2×C4—1272no—
302C2×C4—2548no—
303C2×C4—2541no—
304C2×C4—2541no—
305C2×C4—2541no—
306C2×C4—2541no—
307C2×C4—2541no—
308C2×C4—2541no—
309C2×C4—2541no—
310C2×C4—2541no—
311C2×C4—2541no—
312C2×C4—2541no—
313C2×C4—2541no—
314C2×C4—2541no—
315C2×C4—2541no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4C2×C4×D12712yes—
320C2×C4C2×C4×D12712yes—
321C2×C4#1452324712yes—
322C2×C4#1452324712yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C22×C4×D12711yescharacteristic
331C22#41×D12711yescharacteristic
332C22#41×D12711yescharacteristic
333C22#41×D12711yescharacteristic
334C22#41×D12711yescharacteristic
335C22C2×#1452324711yescharacteristic
336C22C2×#1452324711yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—1274no—
344C4—5081no—
345C4—5081no—
346C4—2542no—
347C4—2542no—
348C4—2542no—
349C4—2542no—
350C4—2542no—
351C2—42no—
352C2#117×D12711yescharacteristic
353C2C4×#1452324711yescharacteristic
354C2C4×#1452324711yescharacteristic
355C2#2255040311yescharacteristic
356C2#2255040511yescharacteristic
357C2#2255040711yescharacteristic
358C2#2255040911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#388×C127C2
socle, cosocleC23×C127C24
prime radicalC24×C127C23
hypercenter#388D127
hypocenterC127#2298
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
5. #22549763C2semidirect
6. #22549760C2semidirect
7. #22549758C2semidirect
8. #22549757C2semidirect
9. C4⋊C4×Dic127C2semidirect
10. #22549727C2semidirect
13. #22549721C2semidirect
14. C42×Dic127C2semidirect
17. #16861165C4semidirect
180. C127#2298semidirect

Maximal subgroups

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

subgroupindex
C2×#168611652
#225497702
#225497692
#225497642
#225497632
#225497602
#225497582
#225497572
C4⋊C4×Dic1272
#225497272
C22⋊C4×Dic1272
#225497212
C42×Dic1272
#388×C1272
#2298127