Finite Group Databasebeta

← #30637993

Subgroups — #30637993

Order 16256.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550272C211yesmaximal, characteristic, complemented
2#22550161C211yesmaximal, characteristic, complemented
3#22550081C211yesmaximal, characteristic
4#22549622C211yesmaximal, characteristic, complemented
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic
7D4×Q8×C127C211yesmaximal, characteristic
8C2×#14523236—21no—
9#16861209C2211yescharacteristic
10#16861140—22no—
11#16861140—21no—
12C4⋊(C127⋊C8)C2211yescharacteristic
13C4⋊(C127⋊C8)C2211yescharacteristic
14#16861133—21no—
15#16861132—21no—
16C4×C127⋊C8C2211yescharacteristic
17C22×Q8×C127—21no—
18C4⋊Q8×C127—21no—
19C4⋊Q8×C127C2211yescharacteristic
20C22⋊Q8×C127—21no—
21C22⋊Q8×C127—22no—
22C4×Q8×C127C2211yescharacteristic
23C4×D4×C127—21no—
24C4×D4×C127C2211yescharacteristic
25C2×Dic254—21no—
26#14523236—42no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32C2×Q8×C127—22no—
33C2×Q8×C127—21no—
34C2×Q8×C127—42no—
35C2×Q8×C127—42no—
36C2×Q8×C127D411yescharacteristic
37C2×D4×C127D411yescharacteristic
38C22×C4×C127—22no—
39C22×C4×C127—21no—
40C4⋊C4×C127—22no—
41C4⋊C4×C127—21no—
42C4⋊C4×C127—21no—
43C4⋊C4×C127D411yescharacteristic
44C4⋊C4×C127D411yescharacteristic
45C4⋊C4×C127D412yes—
46C22⋊C4×C127—22no—
47C22⋊C4×C127—21no—
48C42×C127—21no—
49C42×C127C2311yescharacteristic
50Dic254—41no—
51Dic254—41no—
52C2×Dic127—41no—
53C127⋊C8—22no—
54C127⋊C8—41no—
55C127⋊C8—41no—
56C23×C127—21no—
57Q8×C127—21no—
58Q8×C127—44no—
59Q8×C127—41no—
60Q8×C127—41no—
61Q8×C127SD1612yescomplemented
62D4×C127—21no—
63D4×C127SD1612yes—
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—22no—
67C2×C4×C127—21no—
68C2×C4×C127—44no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C22×C127—22no—
75C22×C127—41no—
76C22×C127—41no—
77C22×C127#12011yescharacteristic
78C4×C127—22no—
79C4×C127—21no—
80C4×C127—42no—
81C4×C127—41no—
82C4×C127—41no—
83C4×C127C2×SD1611yescharacteristic
84C4×C127C2×SD1611yescharacteristic
85C4×C127AGL(1,ℤ/8)11yescharacteristic
86C4×C127#13711yescharacteristic
87Dic127—81no—
88C2×C127—42no—
89C2×C127#44811yescharacteristic
90C2×C127#45011yescharacteristic
91C2×C127#45611yescharacteristic
92#1651—1271nomaximal, Sylow
93C127#165111yescharacteristic, complemented, Sylow
94D4×Q8D12711yescharacteristic
95#498—1271no—
96#475—1271no—
97#461—1271no—
98#332—1271no—
99Q8⋊C8—1271no—
100D4⋊C8—1271no—
101C22×Q8—21no—
102C2×SD16—2541no—
103C4⋊Q8—21no—
104C4⋊Q8—1271no—
105C4⋊Q8C2×D12711yescharacteristic
106C22⋊Q8—21no—
107C22⋊Q8—22no—
108C4×Q8C2×D12711yescharacteristic
109C4×D4—21no—
110C4×D4C2×D12711yescharacteristic
111#106—2541no—
112#106—2542no—
113C4⋊C8—1271no—
114C4⋊C8—1271no—
115#103—2541no—
116#103—2541no—
117C4×C8—1271no—
118C2×Q8—22no—
119C2×Q8—21no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—2541no—
123C2×Q8C22⋊D12711yescharacteristic
124C2×D4C22⋊D12711yescharacteristic
125C22×C4—22no—
126C22×C4—21no—
127SD16—5082no—
128C2×C8—2541no—
129C2×C8—2541no—
130C2×C8—2541no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—2541no—
135C4⋊C4—2541no—
136C4⋊C4C22⋊D12711yescharacteristic
137C4⋊C4C22⋊D12711yescharacteristic
138C4⋊C4C22⋊D12712yes—
139C22⋊C4—22no—
140C22⋊C4—21no—
141C42—21no—
142C42C22×D12711yescharacteristic
143C23—21no—
144Q8—21no—
145Q8—44no—
146Q8—41no—
147Q8—41no—
148Q8—5081no—
149Q8—5081no—
150Q8#1452323612yescomplemented
151D4—21no—
152D4#1452323512yes—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—44no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—5081no—
161C2×C4C2×C22⋊D12711yescharacteristic
162C2×C4C2×C22⋊D12711yescharacteristic
163C2×C4C2×C22⋊D12711yescharacteristic
164C8—5081no—
165C8—5081no—
166C8—2542no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1686117811yescharacteristic
171C4—22no—
172C4—21no—
173C4—42no—
174C4—41no—
175C4—41no—
176C4—10161no—
177C4C2×#1452323511yescharacteristic
178C4C2×#1452323611yescharacteristic
179C4#1686123511yescharacteristic
180C4#1686124011yescharacteristic
181C2—42no—
182C2#2255001511yescharacteristic
183C2#2255002711yescharacteristic
184C2#2255005911yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingD4×Q8×C127C2
socleC22×C127#120
prime radicalC2×D4×C127D4
hypercenterD4×Q8D127
hypocenterC127#1651
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #22550272C2semidirect
2. #22550161C2semidirect
4. #22549622C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
61. Q8×C127SD16semidirect
93. C127#1651semidirect
150. Q8#14523236semidirect

Maximal subgroups

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

subgroupindex
#225502722
#225501612
#225500812
#225496222
Q8⋊(C127⋊C8)2
D4⋊(C127⋊C8)2
D4×Q8×C1272
#1651127