Finite Group Databasebeta

← #30638000

Subgroups — #30638000

Order 16256.

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

normal subgroups only

This group has 200 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#22550272C211yesmaximal, characteristic, complemented
2#22550156C211yesmaximal, characteristic
3#22550086C211yesmaximal, characteristic, complemented
4#22549622C211yesmaximal, characteristic, complemented
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic
7#556×C127C211yesmaximal, characteristic
8C2×#14523238—21no—
9#16861209C2211yescharacteristic
10#16861142—21no—
11#16861140—22no—
12C4⋊(C127⋊C8)C2211yescharacteristic
13C4⋊(C127⋊C8)C2211yescharacteristic
14#16861132—21no—
15#16861130—21no—
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×Dic254—21no—
26#14523238—42no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32C2×Q8×C127—21no—
33C2×Q8×C127D411yescharacteristic
34C2×D4×C127D411yescharacteristic
35C22×C4×C127—21no—
36C4⋊C4×C127—22no—
37C4⋊C4×C127—21no—
38C4⋊C4×C127—21no—
39C4⋊C4×C127D411yescharacteristic
40C4⋊C4×C127D411yescharacteristic
41C4⋊C4×C127D412yes—
42C22⋊C4×C127—22no—
43C22⋊C4×C127—22no—
44C22⋊C4×C127—21no—
45C42×C127—22no—
46C42×C127—21no—
47C42×C127C2311yescharacteristic
48Dic254—41no—
49Dic254—41no—
50C2×Dic127—41no—
51C127⋊C8—22no—
52C127⋊C8—41no—
53C127⋊C8—41no—
54C23×C127—21no—
55Q8×C127—21no—
56Q8×C127—21no—
57Q8×C127—41no—
58Q8×C127—41no—
59D4×C127—21no—
60D4×C127—21no—
61C2×C4×C127—21no—
62C2×C4×C127—21no—
63C2×C4×C127—21no—
64C2×C4×C127—22no—
65C2×C4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C22×C127—41no—
71C22×C127—41no—
72C22×C127—41no—
73C22×C127#12011yescharacteristic
74C4×C127—21no—
75C4×C127—41no—
76C4×C127—41no—
77C4×C127—42no—
78C4×C127—41no—
79C4×C127#13511yescharacteristic
80C4×C127#13511yescharacteristic
81C4×C127AGL(1,ℤ/8)11yescharacteristic
82C4×C127#13711yescharacteristic
83Dic127—81no—
84C2×C127—81no—
85C2×C127#44911yescharacteristic
86C2×C127#45211yescharacteristic
87C2×C127#45611yescharacteristic
88#1652—1271nomaximal, Sylow
89C127#165211yescharacteristic, complemented, Sylow
90#556D12711yescharacteristic
91#498—1271no—
92#474—1271no—
93#462—1271no—
94#332—1271no—
95Q8⋊C8—1271no—
96D4⋊C8—1271no—
97C2×Q16—2541no—
98C4⋊Q8—1271no—
99C4⋊Q8C2×D12711yescharacteristic
100#126—22no—
101#124—21no—
102C22⋊Q8—21no—
103C4×Q8—21no—
104C4×Q8C2×D12711yescharacteristic
105C4×D4C2×D12711yescharacteristic
106#117—21no—
107#107—2541no—
108#106—2542no—
109C4⋊C8—1271no—
110C4⋊C8—1271no—
111#103—2541no—
112#102—2541no—
113C4×C8—1271no—
114C2×Q8—21no—
115C2×Q8—2541no—
116C2×Q8C22⋊D12711yescharacteristic
117C2×D4C22⋊D12711yescharacteristic
118C22×C4—21no—
119Q16—5082no—
120C2×C8—2541no—
121C2×C8—2541no—
122C2×C8—2541no—
123C4⋊C4—22no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—2541no—
127C4⋊C4—2541no—
128C4⋊C4C22⋊D12711yescharacteristic
129C4⋊C4C22⋊D12712yes—
130C4⋊C4C22⋊D12711yescharacteristic
131C22⋊C4—22no—
132C22⋊C4—22no—
133C22⋊C4—21no—
134C42—22no—
135C42—21no—
136C42C22×D12711yescharacteristic
137C23—21no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—5081no—
143Q8—5081no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—5081no—
153C2×C4C2×C22⋊D12711yescharacteristic
154C2×C4C2×C22⋊D12711yescharacteristic
155C2×C4C2×C22⋊D12711yescharacteristic
156C8—5081no—
157C8—5081no—
158C8—2542no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#1686117811yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—42no—
167C4—41no—
168C4—10161no—
169C4#1686122911yescharacteristic
170C4#1686122911yescharacteristic
171C4#1686123411yescharacteristic
172C4#1686123911yescharacteristic
173C2—81no—
174C2#2255002111yescharacteristic
175C2#2255003811yescharacteristic
176C2#2255005911yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#556×C127C2
socleC22×C127#120
prime radicalC2×D4×C127D4
hypercenter#556D127
hypocenterC127#1652
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. #22550272C2semidirect
3. #22550086C2semidirect
4. #22549622C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
89. C127#1652semidirect

Maximal subgroups

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

subgroupindex
#225502722
#225501562
#225500862
#225496222
Q8⋊(C127⋊C8)2
D4⋊(C127⋊C8)2
#556×C1272
#1652127