Finite Group Databasebeta

← #30639271

Subgroups — #30639271

Order 16256.

380 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550438C211yesmaximal, characteristic
2#22550349C211yesmaximal, characteristic, complemented
3C2×#16861123C211yesmaximal, characteristic, complemented
4C2×#16861123C211yesmaximal, characteristic, complemented
5C2×#16861123C211yesmaximal, characteristic, complemented
6#22549642C211yesmaximal, characteristic, complemented
7#342×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127C2211yescharacteristic
9C22×C22⋊D127C2211yescharacteristic
10#16861193—21no—
11#16861193C412yescomplemented
12#16861183—22no—
13#16861179—21no—
14#16861179C412yescomplemented
15C4×C22⋊D127—22no—
16(C22⋊D127)⋊C4—22no—
17#16861165—21no—
18C2×C22⋊Dic127C2211yescharacteristic
19C2×C22⋊Dic127C2211yescharacteristic
20C22⋊C4×D127—21no—
21#16861123—22no—
22#16861123—22no—
23#16861123—22no—
24C2×C22⋊C4×C127C2211yescharacteristic
25C2×C22⋊C4×C127C2211yescharacteristic
26C2×C22⋊C4×C127C2211yescharacteristic
27C23×D127C2×C411yescharacteristic, complemented
28C23×D127D411yescharacteristic
29C2×D508—22no—
30C2×C22⋊D127—21no—
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C2×C22⋊D127—21no—
35C2×C22⋊D127—21no—
36C2×C22⋊D127—21no—
37C2×C22⋊D127—21no—
38C2×C22⋊D127—21no—
39C2×C22⋊D127—21no—
40C2×C22⋊D127C2×C412yes—
41C2×C22⋊D127D412yes—
42C22×Dic127—21no—
43C22×Dic127—21no—
44C22×Dic127C2×C411yescharacteristic, complemented
45C22×Dic127D411yescharacteristic
46C2×C4×D127—22no—
47C4⋊Dic127—22no—
48Dic127⋊C4—22no—
49C22⋊Dic127—21no—
50C22⋊Dic127—21no—
51C22⋊Dic127—21no—
52C22⋊Dic127D412yescomplemented
53(C2×D127)⋊C4—22no—
54(C2×D127)⋊C4—22no—
55(C2×D127)⋊C4—22no—
56C4×Dic127—22no—
57C24×C127C2311yescharacteristic
58C22×C4×C127—21no—
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22⋊C4×C127—21no—
62C22⋊C4×C127—21no—
63C22⋊C4×C127—21no—
64C22⋊C4×C127—21no—
65C22⋊C4×C127C2×C412yescomplemented
66C22⋊C4×C127D412yescomplemented
67C22×D127—21no—
68C22×D127—21no—
69C22×D127—21no—
70C22×D127—21no—
71C22×D127—21no—
72C22×D127—81no—
73C22×D127—81no—
74C22×D127C22⋊C412yes—
75D508—82no—
76C22⋊D127—22no—
77C22⋊D127—22no—
78C22⋊D127—22no—
79C22⋊D127—22no—
80C22⋊D127—41no—
81C22⋊D127—41no—
82C22⋊D127—41no—
83C22⋊D127—41no—
84C2×Dic127—21no—
85C2×Dic127—21no—
86C2×Dic127—21no—
87C2×Dic127—21no—
88C2×Dic127—21no—
89C2×Dic127—22no—
90C2×Dic127—41no—
91C2×Dic127—81no—
92C2×Dic127—81no—
93C2×Dic127C22⋊C412yes—
94C4×D127—82no—
95C23×C127—81no—
96C23×C127C22×C411yescharacteristic
97C23×C127C2×D411yescharacteristic
98C23×C127C2×D411yescharacteristic
99C23×C127C2×D411yescharacteristic
100C23×C127C2×D411yescharacteristic
101C23×C127#4111yescharacteristic
102C23×C127#4111yescharacteristic
103C2×C4×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127—41no—
106C2×C4×C127—81no—
107C2×C4×C127—81no—
108C2×C4×C127—81no—
109C22×C127—41no—
110C22×C127—41no—
111C22×C127—41no—
112C22×C127—41no—
113C22×C127—41no—
114C22×C127—41no—
115C22×C127—41no—
116C22×C127C2×C22⋊C411yescharacteristic
117C22×C127C4×D411yescharacteristic
118C22×C127C4×D411yescharacteristic
119C22×C127#12011yescharacteristic
120C22×C127#12111yescharacteristic
121C22×C127#12111yescharacteristic
122C22×C127#12311yescharacteristic
123C2×D127—22no—
124C2×D127—42no—
125C2×D127—42no—
126C2×D127—42no—
127C2×D127—41no—
128C2×D127—81no—
129C2×D127—81no—
130C2×D127—81no—
131C4×C127—42no—
132C4×C127—42no—
133C4×C127—81no—
134Dic127—22no—
135Dic127—42no—
136Dic127—41no—
137Dic127—81no—
138C2×C127—21no—
139C2×C127—21no—
140C2×C127—21no—
141C2×C127—21no—
142C2×C127—21no—
143C2×C127—21no—
144C2×C127—21no—
145C2×C127#38611yescharacteristic
146D127—42no—
147D127—81no—
148#1913—1271nomaximal, Sylow
149C127#191311yescharacteristic, complemented, Sylow
150#534—1271no—
151#518—1271no—
152C2×#99—1271no—
153C2×#99—1271no—
154C2×#99—1271no—
155#342—1271no—
156#342D12711yescharacteristic, complemented
157C22×D4—1271no—
158C22×D4—1271no—
159#123—1272no—
160#123—2541no—
161#121—2542no—
162#120—1272no—
163#120—2541no—
164C4×D4—2542no—
165C4×D4—2542no—
166#117—2541no—
167C2×C22⋊C4—1271no—
168C2×C22⋊C4—1271no—
169C2×C22⋊C4—2541no—
170C2×C22⋊C4C2×D12711yescharacteristic
171C2×C22⋊C4C2×D12711yescharacteristic
172C2×C22⋊C4C2×D12711yescharacteristic
173#99—2542no—
174#99—2542no—
175#99—2542no—
176C24—1271no—
177C24—1271no—
178C24C22×D12711yescharacteristic
179C2×D4—1272no—
180C2×D4—1272no—
181C2×D4—2542no—
182C2×D4—2541no—
183C2×D4—2541no—
184C2×D4—2541no—
185C2×D4—2541no—
186C2×D4—2541no—
187C2×D4—2541no—
188C2×D4—2541no—
189C2×D4—2541no—
190C2×D4—2541no—
191C2×D4—2541no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—1271no—
196C22×C4—1271no—
197C22×C4—2542no—
198C22×C4—2541no—
199C22×C4—2541no—
200C4⋊C4—2542no—
201C4⋊C4—2542no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—1272no—
207C22⋊C4—2542no—
208C22⋊C4—2542no—
209C22⋊C4—2542no—
210C22⋊C4—2541no—
211C22⋊C4—2541no—
212C22⋊C4—2541no—
213C22⋊C4C4×D12712yescomplemented
214C22⋊C4D50812yescomplemented
215C42—2542no—
216C23—81no—
217C23—1272no—
218C23—10161no—
219C23—10161no—
220C23—2541no—
221C23—2541no—
222C23—2541no—
223C23—2541no—
224C23—2541no—
225C23C2×C4×D12711yescharacteristic
226C23D4×D12711yescharacteristic
227C23D4×D12711yescharacteristic
228C23D4×D12711yescharacteristic
229C23C2×D50811yescharacteristic
230C23#1452324811yescharacteristic
231C23#1452324911yescharacteristic
232D4—10162no—
233D4—5081no—
234D4—5081no—
235D4—5081no—
236D4—5081no—
237D4—2542no—
238D4—2542no—
239D4—2542no—
240D4—2542no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—1272no—
248C2×C4—10162no—
249C2×C4—10161no—
250C2×C4—10161no—
251C2×C4—5081no—
252C2×C4—2541no—
253C2×C4—2541no—
254C2×C4—2541no—
255C2×C4—2542no—
256C2×C4—2541no—
257C2×C4—2541no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—10161no—
266C22—10161no—
267C22—10161no—
268C22—5082no—
269C22—5082no—
270C22—5082no—
271C22—5081no—
272C22—2542no—
273C22C22⋊C4×D12711yescharacteristic
274C22#1686117111yescharacteristic
275C22C4×D50811yescharacteristic
276C22#1686117911yescharacteristic
277C22#1686118211yescharacteristic
278C22#1686118411yescharacteristic
279C22#1686119411yescharacteristic
280C4—42no—
281C4—42no—
282C4—81no—
283C4—10161no—
284C4—5082no—
285C4—5081no—
286C4—2542no—
287C2—21no—
288C2—21no—
289C2—21no—
290C2—21no—
291C2—21no—
292C2—21no—
293C2—21no—
294C2—10161no—
295C2—5082no—
296C2#2254975411yescharacteristic
subgroupG/H
centerC2#22549754
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#342×C127C2
socleC2×C127#386
prime radical#22550438C2
cosocleC24×C127C23
hypercenter#342D127
hypocenterC127#1913
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. #22550349C2semidirect
3. C2×#16861123C2semidirect
4. C2×#16861123C2semidirect
5. C2×#16861123C2semidirect
6. #22549642C2semidirect
7. #342×C127C2semidirect
11. #16861193C4semidirect
14. #16861179C4semidirect
27. C23×D127C2×C4semidirect
44. C22×Dic127C2×C4semidirect
52. C22⋊Dic127D4semidirect
65. C22⋊C4×C127C2×C4semidirect
66. C22⋊C4×C127D4semidirect
149. C127#1913semidirect
156. #342D127semidirect
213. C22⋊C4C4×D127semidirect
214. C22⋊C4D508semidirect

Maximal subgroups

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

subgroupindex
#225504382
#225503492
C2×#168611232
#225496422
#342×C1272
#1913127