Finite Group Databasebeta

← #30648650

Subgroups — #30648650

Order 16256.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×C4⋊Dic127C211yesmaximal, characteristic, complemented
2#22550209C212yesmaximal, complemented
3#22550204C212yesmaximal, complemented
4#22550192C212yesmaximal, complemented
5#22550187C212yesmaximal, complemented
6#22549919C212yesmaximal, complemented
7#22549877C212yesmaximal, complemented
8#22549840C211yesmaximal, characteristic, complemented
9#525×C127C211yesmaximal, characteristic
10C23×Dic127—21no—
11C2×#14523232C2212yes—
12C2×C4⋊Dic127—21no—
13C2×C4⋊Dic127—21no—
14C2×C4⋊Dic127—21no—
15C2×C4⋊Dic127C2212yes—
16C2×C4⋊Dic127C2212yes—
17C2×C4⋊Dic127C2212yes—
18#16861142C2214yescomplemented
19#16861140C2214yescomplemented
20#16861133C2214yescomplemented
21#16861130C2214yescomplemented
22C22⋊(C127⋊C8)C2212yes—
23C22⋊(C127⋊C8)C2212yes—
24C2×#41×C127C2211yescharacteristic
25C23×C4×C127C2211yescharacteristic
26#123×C127—21no—
27C22⋊Q8×C127C2211yescharacteristic
28C22⋊Q8×C127C2211yescharacteristic
29#121×C127C2211yescharacteristic
30#121×C127C2211yescharacteristic
31#120×C127—21no—
32C4×D4×C127—21no—
33C4×D4×C127—21no—
34#117×C127C2211yescharacteristic
35C22×Dic127—22no—
36C22×Dic127—22no—
37C22×Dic127—22no—
38C22×Dic127—44no—
39#14523232—24no—
40C2×C127⋊C8C2314yes—
41C4⋊Dic127—22no—
42C4⋊Dic127—22no—
43C4⋊Dic127—22no—
44C4⋊Dic127C2314yes—
45C24×C127D411yescharacteristic
46#41×C127—24no—
47C2×Q8×C127C2311yescharacteristic
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127C2311yescharacteristic
52C22×C4×C127—24no—
53C22×C4×C127—21no—
54C22×C4×C127D411yescharacteristic
55C22×C4×C127D411yescharacteristic
56C22×C4×C127D411yescharacteristic
57C22×C4×C127C2311yescharacteristic
58C22×C4×C127C2311yescharacteristic
59C22×C4×C127C2311yescharacteristic
60C4⋊C4×C127—21no—
61C4⋊C4×C127—21no—
62C4⋊C4×C127C2311yescharacteristic
63C4⋊C4×C127C2311yescharacteristic
64C22⋊C4×C127—21no—
65C22⋊C4×C127—21no—
66C22⋊C4×C127—21no—
67C22⋊C4×C127—21no—
68C22⋊C4×C127—21no—
69C42×C127—21no—
70C2×Dic127—24no—
71C2×Dic127—44no—
72C2×Dic127—44no—
73C2×Dic127—44no—
74C127⋊C8—24no—
75C23×C127—22no—
76C23×C127—22no—
77C23×C127—22no—
78C23×C127—21no—
79C23×C127C2×D411yescharacteristic
80C23×C127C2×D411yescharacteristic
81C23×C127C2×D411yescharacteristic
82Q8×C127—22no—
83D4×C127—22no—
84D4×C127—42no—
85D4×C127—42no—
86D4×C127—42no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127—21no—
95C2×C4×C127—42no—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127#4114yes—
100C2×C4×C127C2411yescharacteristic
101C22×C127—28no—
102C22×C127—21no—
103C22×C127—21no—
104C22×C127—21no—
105C22×C127—21no—
106C22×C127—21no—
107C22×C127—21no—
108C22×C127—41no—
109C22×C127—41no—
110C22×C127—41no—
111C22×C127#12312yes—
112C22×C127AGL(1,ℤ/8)12yes—
113C22×C127#13712yes—
114C22×C127C22×D411yescharacteristic
115C4×C127—21no—
116C4×C127—21no—
117C4×C127—41no—
118C4×C127—41no—
119C4×C127—41no—
120C4×C127#12312yes—
121C4×C127C2×#4112yes—
122Dic127—44no—
123C2×C127—22no—
124C2×C127—22no—
125C2×C127—22no—
126C2×C127—81no—
127C2×C127C2×#12311yescharacteristic
128C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
129C2×C127C2×#13711yescharacteristic
130#3092—1271nomaximal, Sylow
131C127#309211yescharacteristic, complemented, Sylow
132#525D12711yescharacteristic
133C22×C4⋊C4—1271no—
134#484—1272no—
135#483—1272no—
136#481—1272no—
137#480—1272no—
138#428—1272no—
139#417—1272no—
140#407—1271no—
141C2×#41C2×D12711yescharacteristic
142C23×C4—2541no—
143C23×C4C2×D12711yescharacteristic
144C2×M16—1272no—
145#123—21no—
146C22⋊Q8C2×D12711yescharacteristic
147C22⋊Q8C2×D12711yescharacteristic
148#121C2×D12711yescharacteristic
149#121C2×D12711yescharacteristic
150#120—21no—
151C4×D4—21no—
152C4×D4—21no—
153#117C2×D12711yescharacteristic
154C2×C4⋊C4—1272no—
155C2×C4⋊C4—1272no—
156C2×C4⋊C4—1272no—
157C2×C4⋊C4—2541no—
158C2×C4⋊C4—2541no—
159C2×C4⋊C4—2541no—
160#107—1274no—
161#106—1274no—
162#103—1274no—
163#102—1274no—
164C22⋊C8—1272no—
165C22⋊C8—1272no—
166C24C22⋊D12711yescharacteristic
167#41—24no—
168C2×Q8C22×D12711yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4C22×D12711yescharacteristic
173C22×C4—24no—
174C22×C4—21no—
175C22×C4—5084no—
176C22×C4—2542no—
177C22×C4—2542no—
178C22×C4—2542no—
179C22×C4C22⋊D12711yescharacteristic
180C22×C4C22⋊D12711yescharacteristic
181C22×C4C22⋊D12711yescharacteristic
182C22×C4C22×D12711yescharacteristic
183C22×C4C22×D12711yescharacteristic
184C22×C4C22×D12711yescharacteristic
185M16—2544no—
186C2×C8—1274no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—1274no—
190C4⋊C4—2542no—
191C4⋊C4—2542no—
192C4⋊C4—2542no—
193C4⋊C4C22×D12711yescharacteristic
194C4⋊C4C22×D12711yescharacteristic
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C42—21no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—21no—
205C23C2×C22⋊D12711yescharacteristic
206C23C2×C22⋊D12711yescharacteristic
207C23C2×C22⋊D12711yescharacteristic
208Q8—22no—
209D4—22no—
210D4—42no—
211D4—42no—
212D4—42no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—5084no—
223C2×C4—5084no—
224C2×C4—5084no—
225C2×C4—2544no—
226C2×C4C2×C22⋊D12711yescharacteristic
227C2×C4C2×C22⋊D12711yescharacteristic
228C2×C4C2×C22⋊D12711yescharacteristic
229C2×C4#1452324714yes—
230C2×C4C23×D12711yescharacteristic
231C8—2544no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22#1686119012yes—
243C22#1686123412yes—
244C22#1686123912yes—
245C22C22×C22⋊D12711yescharacteristic
246C4—21no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—41no—
251C4—5084no—
252C4#1686119012yes—
253C4C2×#1452324712yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—81no—
258C2C2×#1686119011yescharacteristic
259C2C2×#1686123411yescharacteristic
260C2C2×#1686123911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#525×C127C2
socleC22×C127C22×D4
hypercenter#525D127
hypocenterC127#3092
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊Dic127C2semidirect
2. #22550209C2semidirect
3. #22550204C2semidirect
4. #22550192C2semidirect
5. #22550187C2semidirect
6. #22549919C2semidirect
7. #22549877C2semidirect
8. #22549840C2semidirect
18. #16861142C22semidirect
19. #16861140C22semidirect
20. #16861133C22semidirect
21. #16861130C22semidirect
131. C127#3092semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic1272
#225502092
#225502042
#225501922
#225501872
#225499192
#225498772
#225498402
#525×C1272
#3092127