Finite Group Databasebeta

← #30648651

Subgroups — #30648651

Order 16256.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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#22550383C211yesmaximal, characteristic
2C2×C4⋊C4×D127C211yesmaximal, characteristic, complemented
3#22550210C212yesmaximal, complemented
4#22550205C212yesmaximal, complemented
5#22550193C212yesmaximal, complemented
6#22550188C212yesmaximal, complemented
7#22549918C211yesmaximal, characteristic, complemented
8#22549916C211yesmaximal, characteristic, complemented
9#22549877C211yesmaximal, characteristic, complemented
10#22549839C211yesmaximal, characteristic, complemented
11#417×C127C211yesmaximal, characteristic, complemented
12C22×C4×D127—21no—
13C22×C4×D127C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15#16861190—21no—
16#16861185C2211yescharacteristic
17#16861185C2211yescharacteristic
18#16861180C2211yescharacteristic
19#16861180C2211yescharacteristic
20#16861177—21no—
21C4×C22⋊D127—21no—
22C4×C22⋊D127—21no—
23#16861168C2211yescharacteristic
24C2×C4⋊Dic127C2211yescharacteristic
25C2×Dic127⋊C4—21no—
26C4⋊C4×D127—21no—
27C4⋊C4×D127—21no—
28C4⋊C4×D127C2212yes—
29C4⋊C4×D127C2212yes—
30#16861143C2212yescomplemented
31#16861142C2212yescomplemented
32#16861141C2212yescomplemented
33#16861140C2212yescomplemented
34#16861133C2212yescomplemented
35#16861130C2212yescomplemented
36(C2×D127)⋊C8C2212yes—
37(C2×D127)⋊C8C2212yes—
38C2×#41×C127C2211yescharacteristic
39C2×M16×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41#103×C127C2212yescomplemented
42#102×C127C2212yescomplemented
43C23×D127D411yescharacteristic
44C2×C22⋊D127—21no—
45C2×C22⋊D127—21no—
46C22×Dic127—21no—
47C22×Dic127D411yescharacteristic
48C2×C4×D127—22no—
49C2×C4×D127—22no—
50C2×C4×D127—22no—
51C2×C4×D127—22no—
52C2×C4×D127—44no—
53C2×C4×D127D411yescharacteristic
54C2×C4×D127D411yescharacteristic
55C2×C4×D127C2311yescharacteristic
56C2×C4×D127C2311yescharacteristic
57#14523232—22no—
58C2×C127⋊C8C2312yes—
59C4⋊Dic127—21no—
60C4⋊Dic127C2311yescharacteristic
61C4⋊Dic127C2311yescharacteristic
62C4⋊Dic127C2312yes—
63Dic127⋊C4—21no—
64Dic127⋊C4—21no—
65Dic127⋊C4—22no—
66Dic127⋊C4—22no—
67C22⋊Dic127—21no—
68C22⋊Dic127—21no—
69C22⋊Dic127—21no—
70(C2×D127)⋊C4—21no—
71(C2×D127)⋊C4—21no—
72C4×Dic127—21no—
73#41×C127—22no—
74#41×C127—22no—
75C2×Q8×C127C2311yescharacteristic
76C2×D4×C127—21no—
77C2×D4×C127C2311yescharacteristic
78C22×C4×C127—21no—
79C22×C4×C127—21no—
80C22×C4×C127C2311yescharacteristic
81M16×C127—22no—
82C2×C8×C127C2312yes—
83C4⋊C4×C127—21no—
84C4⋊C4×C127C2312yes—
85C22×D127—22no—
86C22×D127—22no—
87C22×D127—22no—
88C22×D127C2×D411yescharacteristic
89C22×D127C2×D411yescharacteristic
90C22⋊D127—42no—
91C22⋊D127—42no—
92C2×Dic127—22no—
93C2×Dic127—21no—
94C2×Dic127—21no—
95C2×Dic127—22no—
96C2×Dic127—42no—
97C2×Dic127C2×D411yescharacteristic
98C2×Dic127C2×D411yescharacteristic
99C4×D127—22no—
100C4×D127—22no—
101C4×D127—21no—
102C4×D127—21no—
103C4×D127—21no—
104C4×D127—21no—
105C4×D127—44no—
106C4×D127—44no—
107C127⋊C8—22no—
108C23×C127—21no—
109C23×C127C2×D411yescharacteristic
110Q8×C127—21no—
111Q8×C127—21no—
112D4×C127—21no—
113D4×C127—21no—
114D4×C127—42no—
115C2×C4×C127—22no—
116C2×C4×C127—21no—
117C2×C4×C127—42no—
118C2×C4×C127—42no—
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127#4112yes—
121C2×C4×C127#4112yes—
122C2×C4×C127C2411yescharacteristic
123C8×C127—22no—
124C22×C127—21no—
125C22×C127—21no—
126C22×C127—41no—
127C22×C127—41no—
128C22×C127—41no—
129C22×C127#12312yes—
130C22×C127C22×D411yescharacteristic
131C2×D127—28no—
132C2×D127—21no—
133C2×D127—21no—
134C2×D127—21no—
135C2×D127—21no—
136C2×D127AGL(1,ℤ/8)12yes—
137C2×D127#13712yes—
138C4×C127—41no—
139C4×C127—42no—
140C4×C127#12312yes—
141C4×C127C2×#4111yescharacteristic
142C4×C127C2×#4111yescharacteristic
143Dic127—21no—
144Dic127—21no—
145Dic127—42no—
146Dic127—41no—
147Dic127—41no—
148C2×C127—22no—
149C2×C127—81no—
150C2×C127C2×#12311yescharacteristic
151C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C127C2×#13711yescharacteristic
153D127—22no—
154D127—22no—
155#3092—1271nomaximal, Sylow
156C127#309211yescharacteristic, complemented, Sylow
157#525—1271no—
158C22×C4⋊C4—1271no—
159#484—1272no—
160#483—1272no—
161#481—1272no—
162#480—1272no—
163#428—1271no—
164#428—1271no—
165#417—1271no—
166#417D12711yescharacteristic, complemented
167#407—1271no—
168C2×#41C2×D12711yescharacteristic
169C23×C4—1271no—
170C23×C4—2541no—
171C2×M16—1271no—
172C2×M16C2×D12711yescharacteristic
173#123—2541no—
174C22⋊Q8—1271no—
175C22⋊Q8—1271no—
176#121—1271no—
177#121—1271no—
178#120—2541no—
179C4×D4—2541no—
180C4×D4—2541no—
181#117—1271no—
182C2×C4⋊C4—1271no—
183C2×C4⋊C4—1272no—
184C2×C4⋊C4—1272no—
185C2×C4⋊C4—2541no—
186C2×C4⋊C4—2541no—
187C2×C4⋊C4—2541no—
188C2×C4⋊C4C2×D12711yescharacteristic
189#107—1272no—
190#107—1272no—
191#106—1272no—
192#106—1272no—
193#103—1272no—
194#103C2×D12712yescomplemented
195#102—1272no—
196#102C2×D12712yescomplemented
197C22⋊C8—1272no—
198C22⋊C8—1272no—
199C24—1271no—
200#41—22no—
201#41—22no—
202C2×Q8C22×D12711yescharacteristic
203C2×D4—21no—
204C2×D4—2541no—
205C2×D4—2541no—
206C2×D4C22×D12711yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—1271no—
210C22×C4—1271no—
211C22×C4—1271no—
212C22×C4—1271no—
213C22×C4—1271no—
214C22×C4—5084no—
215C22×C4—2542no—
216C22×C4—2542no—
217C22×C4—2541no—
218C22×C4—2542no—
219C22×C4—2542no—
220C22×C4C22×D12711yescharacteristic
221M16—22no—
222M16—2542no—
223C2×C8—1272no—
224C2×C8C22×D12712yes—
225C4⋊C4—21no—
226C4⋊C4—1272no—
227C4⋊C4—1271no—
228C4⋊C4—1271no—
229C4⋊C4—2542no—
230C4⋊C4—2542no—
231C4⋊C4—2541no—
232C4⋊C4—2541no—
233C4⋊C4—2541no—
234C4⋊C4C22×D12712yes—
235C22⋊C4—2541no—
236C22⋊C4—2541no—
237C22⋊C4—2541no—
238C22⋊C4—2541no—
239C22⋊C4—2541no—
240C42—2541no—
241C23—21no—
242C23—1271no—
243C23—1271no—
244C23—2542no—
245C23—2542no—
246C23—2542no—
247C23D4×D12711yescharacteristic
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—5082no—
254D4—5082no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—1271no—
260C2×C4—1271no—
261C2×C4—5084no—
262C2×C4—5084no—
263C2×C4—5082no—
264C2×C4—2541no—
265C2×C4—2541no—
266C2×C4—2542no—
267C2×C4—2542no—
268C2×C4—2542no—
269C2×C4—2542no—
270C2×C4—2541no—
271C2×C4—2541no—
272C2×C4—2541no—
273C2×C4—2541no—
274C2×C4D4×D12711yescharacteristic
275C2×C4#1452324712yes—
276C2×C4#1452324812yes—
277C2×C4C23×D12711yescharacteristic
278C8—22no—
279C8—2542no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—1272no—
286C22—1272no—
287C22—2548no—
288C22—2541no—
289C22—2541no—
290C22—2541no—
291C22—2541no—
292C22#1686119112yes—
293C22C2×D4×D12711yescharacteristic
294C4—41no—
295C4—42no—
296C4—5082no—
297C4—5081no—
298C4—5081no—
299C4—2541no—
300C4—2541no—
301C4#1686119112yes—
302C4C2×#1452324711yescharacteristic
303C4C2×#1452324811yescharacteristic
304C2—22no—
305C2—81no—
306C2—2542no—
307C2—2542no—
308C2C2×#1686119111yescharacteristic
309C2AGL(1,ℤ/8)×D12711yescharacteristic
310C2#137×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#417×C127C2
socleC22×C127C22×D4
prime radical#22550383C2
hypercenter#417D127
hypocenterC127#3092
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D127C2semidirect
3. #22550210C2semidirect
4. #22550205C2semidirect
5. #22550193C2semidirect
6. #22550188C2semidirect
7. #22549918C2semidirect
8. #22549916C2semidirect
9. #22549877C2semidirect
10. #22549839C2semidirect
11. #417×C127C2semidirect
30. #16861143C22semidirect
31. #16861142C22semidirect
32. #16861141C22semidirect
33. #16861140C22semidirect
34. #16861133C22semidirect
35. #16861130C22semidirect
41. #103×C127C22semidirect
42. #102×C127C22semidirect
156. C127#3092semidirect
166. #417D127semidirect
194. #103C2×D127semidirect
196. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225503832
C2×C4⋊C4×D1272
#225502102
#225502052
#225501932
#225501882
#225499182
#225499162
#225498772
#225498392
#417×C1272
#3092127