Finite Group Databasebeta

← #30637819

Subgroups — #30637819

Order 16256.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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#22550224C211yesmaximal, characteristic, complemented
2#22550071C211yesmaximal, characteristic, complemented
3#22550071C211yesmaximal, characteristic, complemented
4#22549622C211yesmaximal, characteristic, complemented
5D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
7#547×C127C211yesmaximal, characteristic, complemented
8C2×#14523233—21no—
9C2×#14523233—21no—
10#16861206C2211yescharacteristic
11C4⋊(C127⋊C8)C2211yescharacteristic
12C4⋊(C127⋊C8)C2211yescharacteristic
13#16861129—21no—
14#16861129—21no—
15#16861129—22no—
16C4×C127⋊C8C2211yescharacteristic
17C2×#41×C127—21no—
18C4⋊Q8×C127C2211yescharacteristic
19#123×C127—22no—
20C22⋊Q8×C127—21no—
21#121×C127—21no—
22C4×D4×C127C2211yescharacteristic
23C4×D4×C127C2211yescharacteristic
24C2×C4⋊C4×C127—21no—
25C2×D508—21no—
26C2×D508—21no—
27C2×D508—21no—
28#14523233—42no—
29#14523233—42no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33#41×C127—42no—
34#41×C127—42no—
35C2×Q8×C127—21no—
36C2×D4×C127—22no—
37C2×D4×C127D411yescharacteristic
38C2×D4×C127D411yescharacteristic
39C22×C4×C127—22no—
40C22×C4×C127—21no—
41C22×C4×C127—21no—
42C4⋊C4×C127—22no—
43C4⋊C4×C127—21no—
44C4⋊C4×C127D411yescharacteristic
45C4⋊C4×C127D411yescharacteristic
46C4⋊C4×C127D412yes—
47C22⋊C4×C127—22no—
48C22⋊C4×C127—21no—
49C22⋊C4×C127—21no—
50C42×C127C2311yescharacteristic
51C22×D127—41no—
52D508—42no—
53D508—41no—
54D508—41no—
55D508—41no—
56D508—41no—
57C127⋊C8—41no—
58C127⋊C8—41no—
59C127⋊C8—41no—
60C23×C127—21no—
61C23×C127—21no—
62Q8×C127—41no—
63Q8×C127—41no—
64D4×C127—21no—
65D4×C127—21no—
66D4×C127—21no—
67D4×C127—44no—
68D4×C127D812yescomplemented
69C2×C4×C127—21no—
70C2×C4×C127—21no—
71C2×C4×C127—22no—
72C2×C4×C127—44no—
73C2×C4×C127—41no—
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2×D411yescharacteristic
80C22×C127—22no—
81C22×C127—41no—
82C22×C127—41no—
83C22×C127—41no—
84C22×C127—41no—
85C22×C127—41no—
86C22×C127#12011yescharacteristic
87C2×D127—81no—
88C2×D127—81no—
89C2×D127—81no—
90C4×C127—21no—
91C4×C127—41no—
92C4×C127—42no—
93C4×C127—41no—
94C4×C127C2×D811yescharacteristic
95C4×C127#13511yescharacteristic
96C4×C127AGL(1,ℤ/8)11yescharacteristic
97C4×C127AGL(1,ℤ/8)11yescharacteristic
98C2×C127—42no—
99C2×C127—81no—
100C2×C127#44711yescharacteristic
101C2×C127#44911yescharacteristic
102C2×C127#45411yescharacteristic
103D127—161no—
104#1625—1271nomaximal, Sylow
105C127#162511yescharacteristic, complemented, Sylow
106#547D12711yescharacteristic, complemented
107#486—1271no—
108#459—1271no—
109#459—1271no—
110#332—1271no—
111D4⋊C8—1271no—
112D4⋊C8—1271no—
113C2×#41—21no—
114C2×D8—2541no—
115C2×D8—2541no—
116C4⋊Q8C2×D12711yescharacteristic
117#127—1271no—
118#123—22no—
119C22⋊Q8—21no—
120#121—21no—
121C4×D4C2×D12711yescharacteristic
122C4×D4C2×D12711yescharacteristic
123C2×C4⋊C4—21no—
124C4⋊C8—1271no—
125C4⋊C8—1271no—
126#102—2541no—
127#102—2541no—
128#102—2542no—
129C4×C8—1271no—
130#41—42no—
131#41—42no—
132C2×Q8—21no—
133C2×D4—22no—
134C2×D4—2541no—
135C2×D4—2541no—
136C2×D4—2541no—
137C2×D4C22⋊D12711yescharacteristic
138C2×D4C22⋊D12711yescharacteristic
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142D8—5082no—
143D8—5082no—
144C2×C8—2541no—
145C2×C8—2541no—
146C2×C8—2541no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4C22⋊D12711yescharacteristic
150C4⋊C4C22⋊D12711yescharacteristic
151C4⋊C4C22⋊D12712yes—
152C22⋊C4—22no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C42C22×D12711yescharacteristic
156C23—21no—
157C23—21no—
158C23—5081no—
159Q8—41no—
160Q8—41no—
161D4—21no—
162D4—21no—
163D4—21no—
164D4—44no—
165D4—5082no—
166D4—5081no—
167D4—5081no—
168D4—5081no—
169D4—5081no—
170D4#1452323312yescomplemented
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—44no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4C2×C22⋊D12711yescharacteristic
180C2×C4C2×C22⋊D12711yescharacteristic
181C2×C4C2×C22⋊D12711yescharacteristic
182C8—5081no—
183C8—5081no—
184C8—5081no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—10161no—
192C22—10161no—
193C22—10161no—
194C22#1686117811yescharacteristic
195C4—21no—
196C4—41no—
197C4—41no—
198C4—42no—
199C4C2×#1452323311yescharacteristic
200C4#1686122911yescharacteristic
201C4#1686123411yescharacteristic
202C4#1686123511yescharacteristic
203C2—42no—
204C2—81no—
205C2—20321no—
206C2#2255001011yescharacteristic
207C2#2255002111yescharacteristic
208C2#2255004911yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#547×C127C2
socleC22×C127#120
hypercenter#547D127
hypocenterC127#1625
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #22550224C2semidirect
2. #22550071C2semidirect
3. #22550071C2semidirect
4. #22549622C2semidirect
5. D4⋊(C127⋊C8)C2semidirect
6. D4⋊(C127⋊C8)C2semidirect
7. #547×C127C2semidirect
68. D4×C127D8semidirect
105. C127#1625semidirect
106. #547D127semidirect
170. D4#14523233semidirect

Maximal subgroups

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

subgroupindex
#225502242
#225500712
#225496222
D4⋊(C127⋊C8)2
#547×C1272
#1625127