Finite Group Databasebeta

← #30651327

Subgroups — #30651327

Order 16256.

386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550591C211yesmaximal, characteristic, complemented
2#22550531C211yesmaximal, characteristic, complemented
3#22550274C211yesmaximal, characteristic, complemented
4#22550199C212yesmaximal, complemented
5#22550193C212yesmaximal, complemented
6#22550166C211yesmaximal, characteristic, complemented
7#22550156C211yesmaximal, characteristic, complemented
8#22550126C212yesmaximal, complemented
9#22549955C211yesmaximal, characteristic, complemented
10#22549917C212yesmaximal, complemented
11C4×D8×C127C211yesmaximal, characteristic, complemented
12C2×#14523248—21no—
13C2×#14523231C2212yes—
14#16861209C2211yescharacteristic
15#16861209C2211yescharacteristic
16#16861195—22no—
17#16861191—22no—
18#16861188—21no—
19#16861183—21no—
20#16861180C2212yes—
21#16861180C2212yes—
22C4×D508C2211yescharacteristic
23#16861168—21no—
24#16861167C2212yes—
25C4⋊C4×D127C2212yes—
26#16861142C2212yes—
27#16861141C2212yes—
28#16861141C2212yes—
29#16861140C2212yes—
30C4⋊(C127⋊C8)C2211yescharacteristic
31#16861130C2212yes—
32#16861130C2212yes—
33(C2×D127)⋊C8C2212yes—
34C2×D8×C127C2211yescharacteristic
35C4×D4×C127C2211yescharacteristic
36C4×D4×C127C2211yescharacteristic
37#107×C127C2211yescharacteristic
38#102×C127C2211yescharacteristic
39#102×C127C2211yescharacteristic
40C4×C8×C127C2211yescharacteristic
41#14523248—44no—
42C2×Dic254—21no—
43C2×Dic254—21no—
44C2×D508D411yescharacteristic
45C2×C22⋊D127—22no—
46C2×C22⋊D127—22no—
47C2×C4×D127—22no—
48C2×C4×D127C2312yes—
49#14523231—24no—
50C2×C127⋊C8C2312yes—
51C4⋊Dic127—21no—
52C4⋊Dic127—21no—
53C4⋊Dic127D411yescharacteristic
54C4⋊Dic127C2312yes—
55C4⋊Dic127C2312yes—
56Dic127⋊C4—22no—
57C22⋊Dic127—22no—
58C22⋊Dic127—22no—
59(C2×D127)⋊C4—22no—
60(C2×D127)⋊C4D412yes—
61C4×Dic127—22no—
62C2×D4×C127C2311yescharacteristic
63C2×D4×C127C2311yescharacteristic
64C22×C4×C127—21no—
65C22×C4×C127—21no—
66D8×C127—22no—
67C2×C8×C127C2311yescharacteristic
68C2×C8×C127C2311yescharacteristic
69C4⋊C4×C127C2311yescharacteristic
70C4⋊C4×C127C2311yescharacteristic
71C22⋊C4×C127—21no—
72C22⋊C4×C127—21no—
73C42×C127C2311yescharacteristic
74C22×D127C2×D412yes—
75Dic254—42no—
76Dic254—81no—
77D508—22no—
78C22⋊D127—44no—
79C22⋊D127—82no—
80C2×Dic127—22no—
81C2×Dic127—22no—
82C2×Dic127C2×D412yes—
83C4×D127—22no—
84C4×D127—22no—
85C4×D127—44no—
86C127⋊C8—22no—
87C23×C127—21no—
88C23×C127—21no—
89D4×C127—21no—
90D4×C127—21no—
91D4×C127—21no—
92D4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127—21no—
95C2×C4×C127—42no—
96C2×C4×C127—42no—
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2411yescharacteristic
100C8×C127—21no—
101C8×C127#4112yescomplemented
102C22×C127—41no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127—41no—
106C22×C127—41no—
107C22×C127—41no—
108C22×C127C22×D411yescharacteristic
109C2×D127—22no—
110C2×D127—22no—
111C2×D127—22no—
112C4×C127—21no—
113C4×C127—41no—
114C4×C127—41no—
115C4×C127AGL(1,ℤ/8)12yes—
116C4×C127C2×#4111yescharacteristic
117C4×C1272−1+411yescharacteristic
118Dic127—22no—
119Dic127—42no—
120Dic127—42no—
121C2×C127—81no—
122C2×C127—81no—
123C2×C127#54711yescharacteristic
124C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
125C2×C127#57711yescharacteristic
126D127—42no—
127#3354—1271nomaximal, Sylow
128C127#335411yescharacteristic, complemented, Sylow
129#555—1271no—
130#547—1271no—
131#498—1271no—
132#482—1272no—
133#481—1272no—
134#476—1271no—
135#474—1271no—
136#469—1272no—
137C4×D8D12711yescharacteristic, complemented
138#436—1271no—
139#428—1272no—
140C2×#41—2541no—
141C2×D8C2×D12711yescharacteristic
142C2×M16—1272no—
143C4⋊Q8—1271no—
144C4⋊Q8—1271no—
145#124—2542no—
146#123—2542no—
147C22⋊Q8—2541no—
148#121—1272no—
149#121—1272no—
150#121—2541no—
151C4×D4—1271no—
152C4×D4C2×D12711yescharacteristic
153C4×D4C2×D12711yescharacteristic
154#117—1272no—
155#117—2541no—
156C2×C4⋊C4—1272no—
157#107—1272no—
158#107C2×D12711yescharacteristic
159#106—1272no—
160#106—1272no—
161#106—1272no—
162C4⋊C8—1271no—
163#102—1272no—
164#102—1272no—
165#102C2×D12711yescharacteristic
166#102C2×D12711yescharacteristic
167C22⋊C8—1272no—
168C4×C8C2×D12711yescharacteristic
169#41—5084no—
170C2×Q8—2541no—
171C2×Q8—2541no—
172C2×D4—1271no—
173C2×D4—2542no—
174C2×D4—2542no—
175C2×D4C22×D12711yescharacteristic
176C2×D4C22×D12711yescharacteristic
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—1272no—
180C22×C4—2542no—
181D8—22no—
182M16—2544no—
183C2×C8—1272no—
184C2×C8C22×D12711yescharacteristic
185C2×C8C22×D12711yescharacteristic
186C4⋊C4—1271no—
187C4⋊C4—1272no—
188C4⋊C4—1272no—
189C4⋊C4—2542no—
190C4⋊C4—2541no—
191C4⋊C4—2541no—
192C4⋊C4C22×D12711yescharacteristic
193C4⋊C4C22×D12711yescharacteristic
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—1272no—
197C22⋊C4—2542no—
198C22⋊C4—2542no—
199C22⋊C4—2542no—
200C42—2542no—
201C42C22×D12711yescharacteristic
202C23—21no—
203C23—21no—
204C23—1272no—
205Q8—10161no—
206Q8—5082no—
207D4—21no—
208D4—21no—
209D4—21no—
210D4—21no—
211D4—10162no—
212D4—5084no—
213D4—2542no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—1272no—
219C2×C4—5084no—
220C2×C4—2542no—
221C2×C4—2542no—
222C2×C4—2542no—
223C2×C4—2542no—
224C2×C4D4×D12711yescharacteristic
225C2×C4D4×D12711yescharacteristic
226C2×C4C23×D12711yescharacteristic
227C8—21no—
228C8—2542no—
229C8#1452324812yescomplemented
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—2542no—
237C22—2542no—
238C22—2542no—
239C22C2×D4×D12711yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—5082no—
244C4—5082no—
245C4—2542no—
246C4#1686123312yes—
247C4C2×#1452324811yescharacteristic
248C4#1686125611yescharacteristic
249C2—81no—
250C2—81no—
251C2—5082no—
252C2#2255053111yescharacteristic
253C2C2×#1686123311yescharacteristic
254C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×D8×C127C2
socleC22×C127C22×D4
hypercenterC4×D8D127
hypocenterC127#3354
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #22550591C2semidirect
2. #22550531C2semidirect
3. #22550274C2semidirect
4. #22550199C2semidirect
5. #22550193C2semidirect
6. #22550166C2semidirect
7. #22550156C2semidirect
8. #22550126C2semidirect
9. #22549955C2semidirect
10. #22549917C2semidirect
11. C4×D8×C127C2semidirect
101. C8×C127#41semidirect
128. C127#3354semidirect
137. C4×D8D127semidirect
229. C8#14523248semidirect

Maximal subgroups

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

subgroupindex
#225505912
#225505312
#225502742
#225501992
#225501932
#225501662
#225501562
#225501262
#225499552
#225499172
C4×D8×C1272
#3354127