Finite Group Databasebeta

← #30641585

Subgroups — #30641585

Order 16256.

532 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861167C211yesmaximal, characteristic
2#22549762C211yesmaximal, characteristic, complemented
3#22549761C211yesmaximal, characteristic, complemented
4#22549759C211yesmaximal, characteristic, complemented
5#22549750C211yesmaximal, characteristic
6#22549747C211yesmaximal, characteristic
7#22549743C211yesmaximal, characteristic, complemented
8#22549740C211yesmaximal, characteristic, complemented
9#22549738C211yesmaximal, characteristic
10C4⋊C4×Dic127C211yesmaximal, characteristic, complemented
11C4×(C2×D127)⋊C4C211yesmaximal, characteristic
12#22549723C211yesmaximal, characteristic
13#22549722C211yesmaximal, characteristic, complemented
14#95×D127C211yesmaximal, characteristic
15#387×C127C211yesmaximal, characteristic, complemented
16C22×C4×D127C2211yescharacteristic
17C22×C4×D127C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic
19#16861167C418yescomplemented
20C2×C4⋊Dic127C2211yescharacteristic
21C2×C4⋊Dic127C2211yescharacteristic
22C2×Dic127⋊C4C2211yescharacteristic
23C2×Dic127⋊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
33C127⋊#95C2211yescharacteristic
34C127⋊#95C2211yescharacteristic
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×C42×C127C2211yescharacteristic
48#95×C127C2211yescharacteristic
49#95×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—22no—
61C2×C4×D127—22no—
62C2×C4×D127—22no—
63C2×C4×D127—22no—
64C2×C4×D127—22no—
65C2×C4×D127—22no—
66C2×C4×D127—41no—
67C2×C4×D127—41no—
68C2×C4×D127C2×C414yes—
69C4⋊Dic127—22no—
70C4⋊Dic127C2×C414yescomplemented
71Dic127⋊C4—41no—
72Dic127⋊C4—41no—
73(C2×D127)⋊C4—22no—
74(C2×D127)⋊C4—22no—
75(C2×D127)⋊C4C2×C414yes—
76(C2×D127)⋊C4C2×C414yes—
77C4×Dic127—22no—
78C4×Dic127—41no—
79C4×Dic127—41no—
80C4×Dic127C2×C414yescomplemented
81C4×Dic127C2×C414yescomplemented
82C22×C4×C127C2311yescharacteristic
83C22×C4×C127C2311yescharacteristic
84C22×C4×C127C2311yescharacteristic
85C22×C4×C127C2311yescharacteristic
86C22×C4×C127C2311yescharacteristic
87C22×C4×C127C2311yescharacteristic
88C22×C4×C127C2311yescharacteristic
89C4⋊C4×C127—22no—
90C4⋊C4×C127C2×C414yescomplemented
91C42×C127—22no—
92C22×D127—21no—
93C22×D127—21no—
94C22×D127—21no—
95C22×D127—21no—
96C22×D127C22×C412yes—
97C22×D127#4112yes—
98C22×D127#4112yes—
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×Dic127C22×C412yes—
117C2×Dic127C22×C412yes—
118C2×Dic127C22×C412yes—
119C2×Dic127#4112yes—
120C4×D127—28no—
121C4×D127—44no—
122C4×D127—44no—
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—21no—
141C2×C4×C127C22×C412yes—
142C2×C4×C127C22×C412yes—
143C2×C4×C127C22×C412yes—
144C2×C4×C127#4112yes—
145C22×C127C23×C411yescharacteristic
146C22×C127C2×#4111yescharacteristic
147C22×C127C2×#4111yescharacteristic
148C22×C127C2×#4111yescharacteristic
149C22×C127C2×#4111yescharacteristic
150C22×C127H(5,𝔽2)11yescharacteristic
151C22×C1272−1+411yescharacteristic
152C2×D127—22no—
153C2×D127—22no—
154C2×D127—22no—
155C2×D127—22no—
156C2×D127—22no—
157C2×D127—22no—
158C2×D127#11714yes—
159C4×C127—22no—
160C4×C127—22no—
161C4×C127—22no—
162C4×C127—22no—
163C4×C127—41no—
164C4×C127—41no—
165C4×C127—41no—
166Dic127—22no—
167Dic127—22no—
168Dic127—22no—
169Dic127—22no—
170Dic127—41no—
171Dic127—41no—
172Dic127—41no—
173C2×C127C2×#11711yescharacteristic
174C2×C127C4×#4111yescharacteristic
175C2×C127#52011yescharacteristic
176C2×C127#55111yescharacteristic
177C2×C127#55111yescharacteristic
178C2×C127#55211yescharacteristic
179C2×C127#55211yescharacteristic
180D127—24no—
181#2343—1271nomaximal, Sylow
182C127#234311yescharacteristic, complemented, Sylow
183C2×#117—1271no—
184#387—1271no—
185#387—1271no—
186#387—1271no—
187#387D12711yescharacteristic, complemented
188#385—1271no—
189#385—1271no—
190#383—1271no—
191#382—1271no—
192#381—1271no—
193C4×C4⋊C4—1271no—
194C4×C22⋊C4—1271no—
195C4×C22⋊C4—1271no—
196#376—1271no—
197C2×#95—1271no—
198C23×C4—1271no—
199C23×C4—1271no—
200C23×C4—1271no—
201#117—1278no—
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×C22⋊C4—1271no—
209C2×C22⋊C4—1271no—
210C2×C22⋊C4—1271no—
211C2×C22⋊C4—1271no—
212C2×C42—1271no—
213C2×C42—1271no—
214C2×C42—1271no—
215C2×C42—1271no—
216C2×C42—1271no—
217C2×C42C2×D12711yescharacteristic
218#95—1271no—
219#95—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
232#95C2×D12711yescharacteristic
233#95C2×D12711yescharacteristic
234C24—1271no—
235C22×C4—1271no—
236C22×C4—1271no—
237C22×C4—1271no—
238C22×C4—1271no—
239C22×C4—1274no—
240C22×C4—1271no—
241C22×C4—1271no—
242C22×C4—1271no—
243C22×C4—5081no—
244C22×C4—5081no—
245C22×C4—2542no—
246C22×C4—2542no—
247C22×C4—2542no—
248C22×C4—2542no—
249C22×C4—2542no—
250C22×C4—2542no—
251C22×C4C22×D12711yescharacteristic
252C22×C4C22×D12711yescharacteristic
253C22×C4C22×D12711yescharacteristic
254C22×C4C22×D12711yescharacteristic
255C22×C4C22×D12711yescharacteristic
256C22×C4C22×D12711yescharacteristic
257C22×C4C22×D12711yescharacteristic
258C4⋊C4—22no—
259C4⋊C4—1274no—
260C4⋊C4—5081no—
261C4⋊C4—5081no—
262C4⋊C4—2542no—
263C4⋊C4C4×D12714yescomplemented
264C22⋊C4—1274no—
265C22⋊C4—1274no—
266C22⋊C4—2542no—
267C22⋊C4—2542no—
268C42—22no—
269C42—1274no—
270C42—1274no—
271C42—5081no—
272C42—5081no—
273C42—2542no—
274C23—1272no—
275C23—1272no—
276C23—1272no—
277C23—2541no—
278C23—2541no—
279C23—2541no—
280C23—2541no—
281C23C23×D12711yescharacteristic
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—21no—
299C2×C4—1272no—
300C2×C4—1272no—
301C2×C4—1272no—
302C2×C4—1272no—
303C2×C4—5084no—
304C2×C4—5084no—
305C2×C4—2548no—
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×C4—2541no—
322C2×C4—2541no—
323C2×C4C2×C4×D12712yes—
324C2×C4C2×C4×D12712yes—
325C2×C4C2×C4×D12712yes—
326C2×C4#1452324712yes—
327C22—1274no—
328C22—2542no—
329C22—2542no—
330C22—2542no—
331C22—2542no—
332C22—2542no—
333C22—2542no—
334C22C22×C4×D12711yescharacteristic
335C22#41×D12711yescharacteristic
336C22#41×D12711yescharacteristic
337C22#41×D12711yescharacteristic
338C22C2×#1452324711yescharacteristic
339C22#1686125511yescharacteristic
340C22#1686125711yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—5081no—
349C4—5081no—
350C4—5081no—
351C4—2542no—
352C4—2542no—
353C4—2542no—
354C4—2542no—
355C2—2544no—
356C2#117×D12711yescharacteristic
357C2C4×#1452324711yescharacteristic
358C2#2255035811yescharacteristic
359C2#2255054911yescharacteristic
360C2#2255055111yescharacteristic
361C2#2255055811yescharacteristic
362C2#2255056011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#387×C127C2
socle, cosocleC23×C127C24
prime radicalC23×D127C23
hypercenter#387D127
hypocenterC127#2343
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #22549762C2semidirect
3. #22549761C2semidirect
4. #22549759C2semidirect
7. #22549743C2semidirect
8. #22549740C2semidirect
10. C4⋊C4×Dic127C2semidirect
13. #22549722C2semidirect
15. #387×C127C2semidirect
19. #16861167C4semidirect
70. C4⋊Dic127C2×C4semidirect
80. C4×Dic127C2×C4semidirect
81. C4×Dic127C2×C4semidirect
90. C4⋊C4×C127C2×C4semidirect
182. C127#2343semidirect
187. #387D127semidirect
263. C4⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611672
#225497622
#225497612
#225497592
#225497502
#225497472
#225497432
#225497402
#225497382
C4⋊C4×Dic1272
C4×(C2×D127)⋊C42
#225497232
#225497222
#95×D1272
#387×C1272
#2343127