Finite Group Databasebeta

← #30641918

Subgroups — #30641918

Order 16256.

456 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×#16861202C211yesmaximal, characteristic
2#22549773C211yesmaximal, characteristic, complemented
3#22549772C211yesmaximal, characteristic, complemented
4#22549771C211yesmaximal, characteristic, complemented
5#22549770C211yesmaximal, characteristic
6#22549769C211yesmaximal, characteristic
7#22549764C211yesmaximal, characteristic
8#22549759C211yesmaximal, characteristic, complemented
9#22549758C211yesmaximal, characteristic, complemented
10#22549757C211yesmaximal, characteristic, complemented
11#22549749C211yesmaximal, characteristic
12#22549749C211yesmaximal, characteristic
13#22549745C211yesmaximal, characteristic, complemented
14#22549743C211yesmaximal, characteristic, complemented
15#385×C127C211yesmaximal, characteristic
16#16861202C418yescomplemented
17C2×C4⋊Dic127C2211yescharacteristic
18C2×C4⋊Dic127C2211yescharacteristic
19C2×Dic127⋊C4C2211yescharacteristic
20C2×Dic127⋊C4C2211yescharacteristic
21C2×Dic127⋊C4C2211yescharacteristic
22C2×Dic127⋊C4C2211yescharacteristic
23C2×Dic127⋊C4C2211yescharacteristic
24C2×Dic127⋊C4C2211yescharacteristic
25C2×Dic127⋊C4C2211yescharacteristic
26C2×Dic127⋊C4C2211yescharacteristic
27C2×C22⋊Dic127C2211yescharacteristic
28C2×C22⋊Dic127C2211yescharacteristic
29C2×C22⋊Dic127C2211yescharacteristic
30C2×C22⋊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
45C23×C4×C127C2211yescharacteristic
46C2×C4⋊C4×C127C2211yescharacteristic
47C2×C4⋊C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C22⋊C4×C127C2211yescharacteristic
50#95×C127C2211yescharacteristic
51#95×C127C2211yescharacteristic
52C22×Dic127C2311yescharacteristic
53C22×Dic127C2311yescharacteristic
54C22×Dic127C2311yescharacteristic
55C22×Dic127C2311yescharacteristic
56C22×Dic127C2311yescharacteristic
57C22×Dic127C2311yescharacteristic
58C22×Dic127C2311yescharacteristic
59C22×Dic127C2311yescharacteristic
60C4⋊Dic127—41no—
61C4⋊Dic127C2×C414yescomplemented
62Dic127⋊C4—41no—
63Dic127⋊C4—41no—
64Dic127⋊C4—41no—
65Dic127⋊C4—41no—
66Dic127⋊C4—41no—
67Dic127⋊C4—41no—
68Dic127⋊C4C2×C414yescomplemented
69Dic127⋊C4C2×C414yescomplemented
70C22⋊Dic127—41no—
71C22⋊Dic127—41no—
72C22⋊Dic127C2×C414yes—
73C22⋊Dic127C2×C414yes—
74C4×Dic127—41no—
75C4×Dic127—41no—
76C4×Dic127—41no—
77C4×Dic127C2×C414yescomplemented
78C24×C127C2311yescharacteristic
79C22×C4×C127—41no—
80C22×C4×C127—41no—
81C22×C4×C127—41no—
82C22×C4×C127C2311yescharacteristic
83C22×C4×C127C2311yescharacteristic
84C22×C4×C127C2311yescharacteristic
85C22×C4×C127C2311yescharacteristic
86C22×C4×C127C2311yescharacteristic
87C22×C4×C127C2311yescharacteristic
88C4⋊C4×C127—41no—
89C4⋊C4×C127—41no—
90C22⋊C4×C127—41no—
91C22⋊C4×C127C2×C414yes—
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—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—21no—
112C2×Dic127C22×C412yes—
113C2×Dic127C22×C412yes—
114C2×Dic127C22×C412yes—
115C2×Dic127C22×C412yes—
116C23×C127—21no—
117C23×C127—21no—
118C23×C127—21no—
119C23×C127—21no—
120C23×C127—21no—
121C23×C127—21no—
122C23×C127C22×C412yes—
123C23×C127C2411yescharacteristic
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—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—44no—
141C2×C4×C127C22×C412yes—
142C2×C4×C127C22×C412yes—
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×C127H(5,𝔽2)11yescharacteristic
152C22×C127H(5,𝔽2)11yescharacteristic
153C22×C127H(5,𝔽2)11yescharacteristic
154C22×C1272−1+411yescharacteristic
155C22×C1272−1+411yescharacteristic
156C22×C1272−1+411yescharacteristic
157C4×C127—22no—
158C4×C127—22no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127—41no—
162C4×C127—41no—
163Dic127—22no—
164Dic127—22no—
165Dic127—22no—
166Dic127—22no—
167Dic127—41no—
168Dic127—41no—
169Dic127—41no—
170Dic127—41no—
171C2×C127—42no—
172C2×C127#52011yescharacteristic
173C2×C127#52011yescharacteristic
174C2×C127#52011yescharacteristic
175C2×C127#56011yescharacteristic
176C2×C127#56311yescharacteristic
177C2×C127#56311yescharacteristic
178C2×C127#56311yescharacteristic
179#2383—1271nomaximal, Sylow
180C127#238311yescharacteristic, complemented, Sylow
181C2×#126—1271no—
182#389—1271no—
183#389—1271no—
184#389—1271no—
185#388—1271no—
186#388—1271no—
187#388—1271no—
188#387—1271no—
189#387—1271no—
190#387—1271no—
191#385—1271no—
192#385—1271no—
193#385D12711yescharacteristic
194#384—1271no—
195#383—1271no—
196C23×C4C2×D12711yescharacteristic
197#126—1278no—
198C2×C4⋊C4—1271no—
199C2×C4⋊C4—1271no—
200C2×C4⋊C4—1271no—
201C2×C4⋊C4—1271no—
202C2×C4⋊C4—1271no—
203C2×C4⋊C4—1271no—
204C2×C4⋊C4—1271no—
205C2×C4⋊C4—1271no—
206C2×C4⋊C4—1271no—
207C2×C4⋊C4—1271no—
208C2×C4⋊C4C2×D12711yescharacteristic
209C2×C4⋊C4C2×D12711yescharacteristic
210C2×C22⋊C4—1271no—
211C2×C22⋊C4—1271no—
212C2×C22⋊C4—1271no—
213C2×C22⋊C4—1271no—
214C2×C22⋊C4C2×D12711yescharacteristic
215C2×C22⋊C4C2×D12711yescharacteristic
216C2×C42—1271no—
217C2×C42—1271no—
218C2×C42—1271no—
219C2×C42—1271no—
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—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—1271no—
237C22×C4—1271no—
238C22×C4—1271no—
239C22×C4—1271no—
240C22×C4—1271no—
241C22×C4—1271no—
242C22×C4—1271no—
243C22×C4—1271no—
244C22×C4C22×D12711yescharacteristic
245C22×C4C22×D12711yescharacteristic
246C22×C4C22×D12711yescharacteristic
247C22×C4C22×D12711yescharacteristic
248C22×C4C22×D12711yescharacteristic
249C22×C4C22×D12711yescharacteristic
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—1274no—
253C4⋊C4—1274no—
254C4⋊C4—1274no—
255C4⋊C4—5081no—
256C4⋊C4—5081no—
257C4⋊C4—5081no—
258C4⋊C4—5081no—
259C4⋊C4—5081no—
260C4⋊C4—5081no—
261C4⋊C4—5081no—
262C22⋊C4—41no—
263C22⋊C4—1274no—
264C22⋊C4—1274no—
265C22⋊C4—5081no—
266C22⋊C4—5081no—
267C22⋊C4C4×D12714yes—
268C42—1274no—
269C42—5081no—
270C42—5081no—
271C42—5081no—
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—21no—
295C2×C4—21no—
296C2×C4—44no—
297C2×C4—1272no—
298C2×C4—1272no—
299C2×C4—1272no—
300C2×C4—1272no—
301C2×C4—2541no—
302C2×C4—2541no—
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×C4—2541no—
320C2×C4—2541no—
321C2×C4C2×C4×D12712yes—
322C2×C4C2×C4×D12712yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C22×C4×D12711yescharacteristic
331C22#1686125411yescharacteristic
332C22#1686125411yescharacteristic
333C22#1686125411yescharacteristic
334C22#1686125611yescharacteristic
335C22#1686125611yescharacteristic
336C22#1686125711yescharacteristic
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—5081no—
344C4—5081no—
345C4—5081no—
346C4—5081no—
347C4—2542no—
348C4—2542no—
349C4—2542no—
350C4—2542no—
351C2—42no—
352C2#2255035511yescharacteristic
353C2#2255035511yescharacteristic
354C2#2255035711yescharacteristic
355C2#2255061511yescharacteristic
356C2#2255062711yescharacteristic
357C2#2255062711yescharacteristic
358C2#2255062911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#385×C127C2
socle, cosocleC23×C127C24
prime radicalC24×C127C23
hypercenter#385D127
hypocenterC127#2383
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #22549773C2semidirect
3. #22549772C2semidirect
4. #22549771C2semidirect
8. #22549759C2semidirect
9. #22549758C2semidirect
10. #22549757C2semidirect
13. #22549745C2semidirect
14. #22549743C2semidirect
16. #16861202C4semidirect
61. C4⋊Dic127C2×C4semidirect
68. Dic127⋊C4C2×C4semidirect
69. Dic127⋊C4C2×C4semidirect
77. C4×Dic127C2×C4semidirect
180. C127#2383semidirect

Maximal subgroups

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

subgroupindex
C2×#168612022
#225497732
#225497722
#225497712
#225497702
#225497692
#225497642
#225497592
#225497582
#225497572
#225497492
#225497452
#225497432
#385×C1272
#2383127