Finite Group Databasebeta

← #30641921

Subgroups — #30641921

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×#16861204C211yesmaximal, characteristic
2#22549772C211yesmaximal, characteristic, complemented
3#22549771C211yesmaximal, characteristic, complemented
4#22549767C211yesmaximal, characteristic
5#22549766C211yesmaximal, characteristic
6#22549765C211yesmaximal, characteristic
7#22549762C211yesmaximal, characteristic, complemented
8#22549761C211yesmaximal, characteristic, complemented
9#22549760C211yesmaximal, characteristic, complemented
10#22549750C211yesmaximal, characteristic
11#22549747C211yesmaximal, characteristic
12#22549747C211yesmaximal, characteristic
13#22549745C211yesmaximal, characteristic, complemented
14#22549743C211yesmaximal, characteristic, complemented
15#389×C127C211yesmaximal, characteristic, complemented
16C22×C4×D127C2211yescharacteristic
17#16861204C418yescomplemented
18C2×C4⋊Dic127C2211yescharacteristic
19C2×C4⋊Dic127C2211yescharacteristic
20C2×Dic127⋊C4C2211yescharacteristic
21C2×Dic127⋊C4C2211yescharacteristic
22C2×Dic127⋊C4C2211yescharacteristic
23C2×Dic127⋊C4C2211yescharacteristic
24C2×Dic127⋊C4C2211yescharacteristic
25C2×Dic127⋊C4C2211yescharacteristic
26C2×(C2×D127)⋊C4C2211yescharacteristic
27C2×(C2×D127)⋊C4C2211yescharacteristic
28C2×(C2×D127)⋊C4C2211yescharacteristic
29C2×(C2×D127)⋊C4C2211yescharacteristic
30C2×(C2×D127)⋊C4C2211yescharacteristic
31C2×(C2×D127)⋊C4C2211yescharacteristic
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×C4⋊C4×C127C2211yescharacteristic
47C2×C4⋊C4×C127C2211yescharacteristic
48C2×C4⋊C4×C127C2211yescharacteristic
49C2×C42×C127C2211yescharacteristic
50#95×C127C2211yescharacteristic
51#95×C127C2211yescharacteristic
52C23×D127C2311yescharacteristic
53C22×Dic127C2311yescharacteristic
54C22×Dic127C2311yescharacteristic
55C22×Dic127C2311yescharacteristic
56C22×Dic127C2311yescharacteristic
57C22×Dic127C2311yescharacteristic
58C22×Dic127C2311yescharacteristic
59C22×Dic127C2311yescharacteristic
60C2×C4×D127—41no—
61C2×C4×D127—41no—
62C2×C4×D127—41no—
63C4⋊Dic127—41no—
64C4⋊Dic127C2×C414yescomplemented
65Dic127⋊C4—41no—
66Dic127⋊C4—41no—
67Dic127⋊C4—41no—
68Dic127⋊C4—41no—
69Dic127⋊C4—41no—
70Dic127⋊C4C2×C414yescomplemented
71(C2×D127)⋊C4—41no—
72(C2×D127)⋊C4—41no—
73(C2×D127)⋊C4—41no—
74(C2×D127)⋊C4C2×C414yes—
75(C2×D127)⋊C4C2×C414yes—
76(C2×D127)⋊C4C2×C414yes—
77C4×Dic127—41no—
78C4×Dic127—41no—
79C4×Dic127C2×C414yescomplemented
80C22×C4×C127C2311yescharacteristic
81C22×C4×C127C2311yescharacteristic
82C22×C4×C127C2311yescharacteristic
83C22×C4×C127C2311yescharacteristic
84C22×C4×C127C2311yescharacteristic
85C22×C4×C127C2311yescharacteristic
86C22×C4×C127C2311yescharacteristic
87C4⋊C4×C127—41no—
88C4⋊C4×C127—41no—
89C4⋊C4×C127—41no—
90C4⋊C4×C127C2×C414yescomplemented
91C42×C127—41no—
92C22×D127—21no—
93C22×D127—21no—
94C22×D127—21no—
95C22×D127—21no—
96C22×D127—21no—
97C22×D127—21no—
98C22×D127C22×C412yes—
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×Dic127—21no—
113C2×Dic127—21no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127C22×C412yes—
118C2×Dic127C22×C412yes—
119C2×Dic127C22×C412yes—
120C4×D127—44no—
121C23×C127C2411yescharacteristic
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—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127C22×C412yes—
141C2×C4×C127C22×C412yes—
142C2×C4×C127C22×C412yes—
143C22×C127C23×C411yescharacteristic
144C22×C127H(5,𝔽2)11yescharacteristic
145C22×C127H(5,𝔽2)11yescharacteristic
146C22×C127H(5,𝔽2)11yescharacteristic
147C22×C1272−1+411yescharacteristic
148C22×C1272−1+411yescharacteristic
149C22×C1272−1+411yescharacteristic
150C2×D127—22no—
151C2×D127—22no—
152C2×D127—22no—
153C2×D127—41no—
154C2×D127—41no—
155C2×D127—41no—
156C2×D127—41no—
157C4×C127—22no—
158C4×C127—22no—
159C4×C127—22no—
160C4×C127—41no—
161C4×C127—41no—
162C4×C127—41no—
163C4×C127—41no—
164Dic127—22no—
165Dic127—22no—
166Dic127—22no—
167Dic127—41no—
168Dic127—41no—
169Dic127—41no—
170Dic127—41no—
171C2×C127#52011yescharacteristic
172C2×C127#52011yescharacteristic
173C2×C127#52011yescharacteristic
174C2×C127#56011yescharacteristic
175C2×C127#56311yescharacteristic
176C2×C127#56311yescharacteristic
177C2×C127#56311yescharacteristic
178D127—42no—
179#2383—1271nomaximal, Sylow
180C127#238311yescharacteristic, complemented, Sylow
181C2×#126—1271no—
182#389—1271no—
183#389—1271no—
184#389D12711yescharacteristic, complemented
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#385—1271no—
194#384—1271no—
195#383—1271no—
196C23×C4—1271no—
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⋊C4C2×D12711yescharacteristic
207C2×C4⋊C4C2×D12711yescharacteristic
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⋊C4—1271no—
215C2×C22⋊C4—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
232C24—1271no—
233C22×C4—1271no—
234C22×C4—1271no—
235C22×C4—1271no—
236C22×C4—1271no—
237C22×C4—1271no—
238C22×C4—1271no—
239C22×C4—1271no—
240C22×C4—5081no—
241C22×C4—5081no—
242C22×C4—5081no—
243C22×C4C22×D12711yescharacteristic
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—41no—
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⋊C4C4×D12714yescomplemented
262C22⋊C4—1274no—
263C22⋊C4—1274no—
264C22⋊C4—1274no—
265C22⋊C4—5081no—
266C22⋊C4—5081no—
267C22⋊C4—5081no—
268C42—41no—
269C42—1274no—
270C42—5081no—
271C42—5081no—
272C23—1272no—
273C23—2541no—
274C23—2541no—
275C23—2541no—
276C23—2541no—
277C23—2541no—
278C23—2541no—
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—21no—
297C2×C4—21no—
298C2×C4—1272no—
299C2×C4—1272no—
300C2×C4—1272no—
301C2×C4—5084no—
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×C4C2×C4×D12712yes—
321C2×C4C2×C4×D12712yes—
322C2×C4C2×C4×D12712yes—
323C22—5081no—
324C22—5081no—
325C22—5081no—
326C22—5081no—
327C22—2542no—
328C22—2542no—
329C22—2542no—
330C22C22×C4×D12711yescharacteristic
331C22#1686125411yescharacteristic
332C22#1686125411yescharacteristic
333C22#1686125511yescharacteristic
334C22#1686125611yescharacteristic
335C22#1686125711yescharacteristic
336C22#1686125711yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—5081no—
345C4—5081no—
346C4—5081no—
347C4—5081no—
348C4—2542no—
349C4—2542no—
350C4—2542no—
351C2—5082no—
352C2#2255035511yescharacteristic
353C2#2255035711yescharacteristic
354C2#2255035811yescharacteristic
355C2#2255061411yescharacteristic
356C2#2255062511yescharacteristic
357C2#2255062711yescharacteristic
358C2#2255063011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#389×C127C2
socle, cosocleC23×C127C24
prime radicalC23×D127C23
hypercenter#389D127
hypocenterC127#2383
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. #22549772C2semidirect
3. #22549771C2semidirect
7. #22549762C2semidirect
8. #22549761C2semidirect
9. #22549760C2semidirect
13. #22549745C2semidirect
14. #22549743C2semidirect
15. #389×C127C2semidirect
17. #16861204C4semidirect
64. C4⋊Dic127C2×C4semidirect
70. Dic127⋊C4C2×C4semidirect
79. C4×Dic127C2×C4semidirect
90. C4⋊C4×C127C2×C4semidirect
180. C127#2383semidirect
184. #389D127semidirect
261. C4⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612042
#225497722
#225497712
#225497672
#225497662
#225497652
#225497622
#225497612
#225497602
#225497502
#225497472
#225497452
#225497432
#389×C1272
#2383127