Finite Group Databasebeta

← #30637817

Subgroups — #30637817

Order 16256.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550530C211yesmaximal, characteristic, complemented
2#22550222C211yesmaximal, characteristic, complemented
3#22550068C211yesmaximal, characteristic, complemented
4#22549623C211yesmaximal, characteristic, complemented
5D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#459×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9C2×#14523233—21no—
10#16861209C2211yescharacteristic
11#16861193—22no—
12#16861189—21no—
13#16861183—21no—
14C4×D508C2211yescharacteristic
15C2×C4⋊Dic127—21no—
16C4⋊(C127⋊C8)C2211yescharacteristic
17#16861130—21no—
18#16861130—22no—
19C4×C127⋊C8C2211yescharacteristic
20C2×D8×C127—21no—
21#127×C127C2211yescharacteristic
22C4×D4×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24#102×C127—21no—
25#14523247—42no—
26#14523247—42no—
27C2×Dic254—21no—
28C2×D508D411yescharacteristic
29C2×C22⋊D127—22no—
30C22×Dic127—22no—
31C2×C4×D127—21no—
32#14523233—42no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—22no—
36C4⋊Dic127—21no—
37C4⋊Dic127D412yes—
38C4⋊Dic127D411yescharacteristic
39(C2×D127)⋊C4—22no—
40(C2×D127)⋊C4—21no—
41C2×D4×C127—21no—
42C2×D4×C127—21no—
43C2×D4×C127—21no—
44C2×D4×C127D411yescharacteristic
45C22×C4×C127—21no—
46D8×C127—42no—
47C2×C8×C127—21no—
48C4⋊C4×C127D411yescharacteristic
49C22⋊C4×C127—21no—
50C42×C127C2311yescharacteristic
51C22×D127—21no—
52Dic254—41no—
53Dic254—41no—
54D508—21no—
55D508—21no—
56C22⋊D127—44no—
57C2×Dic127—21no—
58C2×Dic127—22no—
59C2×Dic127—44no—
60C4×D127—41no—
61C4×D127—41no—
62C127⋊C8—41no—
63C127⋊C8—41no—
64C23×C127—21no—
65C23×C127—41no—
66D4×C127—21no—
67D4×C127—42no—
68D4×C127—41no—
69D4×C127—41no—
70D4×C127—41no—
71D4×C127—41no—
72D4×C127D812yescomplemented
73C2×C4×C127—21no—
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2×D411yescharacteristic
79C8×C127—41no—
80C22×C127—22no—
81C22×C127—41no—
82C22×C127—41no—
83C22×C127—81no—
84C22×C127—81no—
85C22×C127—81no—
86C22×C127#12011yescharacteristic
87C2×D127—41no—
88C2×D127—41no—
89C2×D127—41no—
90C4×C127—21no—
91C4×C127—41no—
92C4×C127C2×D811yescharacteristic
93C4×C127#13511yescharacteristic
94C4×C127AGL(1,ℤ/8)11yescharacteristic
95C4×C127AGL(1,ℤ/8)11yescharacteristic
96Dic127—41no—
97Dic127—42no—
98C2×C127—42no—
99C2×C127—161no—
100C2×C127#44711yescharacteristic
101C2×C127#44911yescharacteristic
102C2×C127#45411yescharacteristic
103D127—81no—
104#1625—1271nomaximal, Sylow
105C127#162511yescharacteristic, complemented, Sylow
106#547—1271no—
107#486—1271no—
108#459—1271no—
109#459D12711yescharacteristic, complemented
110#332—1271no—
111D4⋊C8—1271no—
112D4⋊C8—1271no—
113C2×#41—2541no—
114C2×D8—21no—
115C2×D8—2541no—
116C4⋊Q8—1271no—
117#127C2×D12711yescharacteristic
118#123—2542no—
119C22⋊Q8—2541no—
120#121—2541no—
121C4×D4—1271no—
122C4×D4C2×D12711yescharacteristic
123C2×C4⋊C4—2541no—
124C4⋊C8—1271no—
125C4⋊C8C2×D12711yescharacteristic
126#102—21no—
127#102—2541no—
128#102—2542no—
129C4×C8—1271no—
130#41—5082no—
131#41—5082no—
132C2×Q8—2541no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—1271no—
137C2×D4—2542no—
138C2×D4C22⋊D12711yescharacteristic
139C22×C4—21no—
140C22×C4—2542no—
141C22×C4—2541no—
142D8—42no—
143D8—5082no—
144C2×C8—21no—
145C2×C8—2541no—
146C2×C8—2541no—
147C4⋊C4—1271no—
148C4⋊C4—1272no—
149C4⋊C4—2542no—
150C4⋊C4—2541no—
151C4⋊C4C22⋊D12711yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—2542no—
154C22⋊C4—2541no—
155C42C22×D12711yescharacteristic
156C23—21no—
157C23—41no—
158C23—2541no—
159Q8—5081no—
160Q8—5081no—
161D4—21no—
162D4—42no—
163D4—41no—
164D4—41no—
165D4—41no—
166D4—41no—
167D4—5084no—
168D4—2541no—
169D4—2541no—
170D4#1452323312yescomplemented
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—5084no—
175C2×C4—5081no—
176C2×C4—5081no—
177C2×C4—2541no—
178C2×C4—2542no—
179C2×C4D4×D12711yescharacteristic
180C2×C4D4×D12711yescharacteristic
181C2×C4C2×C22⋊D12711yescharacteristic
182C8—41no—
183C8—5081no—
184C8—5081no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—5081no—
192C22—5081no—
193C22—5081no—
194C22#1686117711yescharacteristic
195C4—21no—
196C4—41no—
197C4—5081no—
198C4—5082no—
199C4C2×#1452323311yescharacteristic
200C4#1686122811yescharacteristic
201C4#1686123311yescharacteristic
202C4#1686123411yescharacteristic
203C2—42no—
204C2—161no—
205C2—10161no—
206C2#2255000911yescharacteristic
207C2#2255001811yescharacteristic
208C2#2255004711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#459×C127C2
socleC22×C127#120
hypercenter#459D127
hypocenterC127#1625
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #22550530C2semidirect
2. #22550222C2semidirect
3. #22550068C2semidirect
4. #22549623C2semidirect
5. D4⋊(C127⋊C8)C2semidirect
6. #22549603C2semidirect
7. #459×C127C2semidirect
72. D4×C127D8semidirect
105. C127#1625semidirect
109. #459D127semidirect
170. D4#14523233semidirect

Maximal subgroups

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

subgroupindex
#225505302
#225502222
#225500682
#225496232
D4⋊(C127⋊C8)2
#225496032
#459×C1272
#1625127