Finite Group Databasebeta

← #30637840

Subgroups — #30637840

Order 16256.

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

normal subgroups only

This group has 216 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#22550224C211yesmaximal, characteristic, complemented
2#22550076C211yesmaximal, characteristic, complemented
3#22550071C211yesmaximal, characteristic
4#22549622C211yesmaximal, characteristic, complemented
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
7#556×C127C211yesmaximal, characteristic, complemented
8C2×#14523235—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
17C4⋊Q8×C127C2211yescharacteristic
18#126×C127—22no—
19#124×C127—21no—
20C22⋊Q8×C127—21no—
21C4×Q8×C127—21no—
22C4×Q8×C127C2211yescharacteristic
23C4×D4×C127C2211yescharacteristic
24#117×C127—21no—
25C2×D508—21no—
26C2×D508—21no—
27C2×D508—21no—
28#14523235—42no—
29#14523233—42no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C2×Q8×C127—21no—
34C2×Q8×C127D411yescharacteristic
35C2×D4×C127D411yescharacteristic
36C22×C4×C127—21no—
37C4⋊C4×C127—22no—
38C4⋊C4×C127—21no—
39C4⋊C4×C127—21no—
40C4⋊C4×C127D411yescharacteristic
41C4⋊C4×C127D411yescharacteristic
42C4⋊C4×C127D412yes—
43C22⋊C4×C127—22no—
44C22⋊C4×C127—22no—
45C22⋊C4×C127—21no—
46C42×C127—22no—
47C42×C127—21no—
48C42×C127C2311yescharacteristic
49C22×D127—41no—
50D508—42no—
51D508—41no—
52D508—41no—
53D508—41no—
54D508—41no—
55C127⋊C8—41no—
56C127⋊C8—41no—
57C127⋊C8—41no—
58C23×C127—21no—
59Q8×C127—21no—
60Q8×C127—21no—
61Q8×C127—41no—
62Q8×C127—41no—
63D4×C127—21no—
64D4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—21no—
68C2×C4×C127—22no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C22×C127—41no—
75C22×C127—41no—
76C22×C127—41no—
77C22×C127#12011yescharacteristic
78C2×D127—81no—
79C2×D127—81no—
80C2×D127—81no—
81C4×C127—21no—
82C4×C127—41no—
83C4×C127—41no—
84C4×C127—42no—
85C4×C127—41no—
86C4×C127#13511yescharacteristic
87C4×C127#13511yescharacteristic
88C4×C127AGL(1,ℤ/8)11yescharacteristic
89C4×C127AGL(1,ℤ/8)11yescharacteristic
90C2×C127—81no—
91C2×C127#44911yescharacteristic
92C2×C127#44911yescharacteristic
93C2×C127#45411yescharacteristic
94D127—161no—
95#1628—1271nomaximal, Sylow
96C127#162811yescharacteristic, complemented, Sylow
97#556D12711yescharacteristic, complemented
98#486—1271no—
99#460—1271no—
100#459—1271no—
101#332—1271no—
102Q8⋊C8—1271no—
103D4⋊C8—1271no—
104C2×SD16—2541no—
105C2×D8—2541no—
106C4⋊Q8C2×D12711yescharacteristic
107#127—1271no—
108#126—22no—
109#124—21no—
110C22⋊Q8—21no—
111C4×Q8—21no—
112C4×Q8C2×D12711yescharacteristic
113C4×D4C2×D12711yescharacteristic
114#117—21no—
115C4⋊C8—1271no—
116C4⋊C8—1271no—
117#102—2541no—
118#102—2541no—
119#102—2542no—
120C4×C8—1271no—
121C2×Q8—21no—
122C2×Q8C22⋊D12711yescharacteristic
123C2×D4—2541no—
124C2×D4—2541no—
125C2×D4—2541no—
126C2×D4C22⋊D12711yescharacteristic
127C22×C4—21no—
128SD16—5082no—
129D8—5082no—
130C2×C8—2541no—
131C2×C8—2541no—
132C2×C8—2541no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4C22⋊D12711yescharacteristic
137C4⋊C4C22⋊D12711yescharacteristic
138C4⋊C4C22⋊D12712yes—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—21no—
142C42—22no—
143C42—21no—
144C42C22×D12711yescharacteristic
145C23—21no—
146C23—5081no—
147Q8—21no—
148Q8—21no—
149Q8—41no—
150Q8—41no—
151D4—21no—
152D4—21no—
153D4—5082no—
154D4—5081no—
155D4—5081no—
156D4—5081no—
157D4—5081no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4C2×C22⋊D12711yescharacteristic
165C2×C4C2×C22⋊D12711yescharacteristic
166C2×C4C2×C22⋊D12711yescharacteristic
167C8—5081no—
168C8—5081no—
169C8—5081no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—10161no—
174C22—10161no—
175C22—10161no—
176C22#1686117811yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—42no—
181C4—41no—
182C4#1686122911yescharacteristic
183C4#1686122911yescharacteristic
184C4#1686123411yescharacteristic
185C4#1686123411yescharacteristic
186C2—81no—
187C2—20321no—
188C2#2255002111yescharacteristic
189C2#2255002111yescharacteristic
190C2#2255004911yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#556×C127C2
socleC22×C127#120
prime radical#22550071C2
hypercenter#556D127
hypocenterC127#1628
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #22550224C2semidirect
2. #22550076C2semidirect
4. #22549622C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
6. D4⋊(C127⋊C8)C2semidirect
7. #556×C127C2semidirect
96. C127#1628semidirect
97. #556D127semidirect

Maximal subgroups

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

subgroupindex
#225502242
#225500762
#225500712
#225496222
Q8⋊(C127⋊C8)2
D4⋊(C127⋊C8)2
#556×C1272
#1628127