Finite Group Databasebeta

← #30637895

Subgroups — #30637895

Order 16256.

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

normal subgroups only

This group has 228 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#22550228C211yesmaximal, characteristic, complemented
2#22550171C211yesmaximal, characteristic, complemented
3#22550166C211yesmaximal, characteristic
4#22549622C211yesmaximal, characteristic, complemented
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic
7D4×Q8×C127C211yesmaximal, characteristic
8#16861209C2211yescharacteristic
9#16861142—21no—
10#16861140—21no—
11C4⋊(C127⋊C8)C2211yescharacteristic
12C4⋊(C127⋊C8)C2211yescharacteristic
13#16861133—21no—
14#16861132—22no—
15#16861130—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—
26C2×C127⋊C8—21no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8—21no—
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31C2×Q8×C127—22no—
32C2×Q8×C127—21no—
33C2×Q8×C127—42no—
34C2×Q8×C127—42no—
35C2×Q8×C127D411yescharacteristic
36C2×D4×C127D411yescharacteristic
37C22×C4×C127—22no—
38C22×C4×C127—21no—
39C4⋊C4×C127—22no—
40C4⋊C4×C127—21no—
41C4⋊C4×C127—21no—
42C4⋊C4×C127D411yescharacteristic
43C4⋊C4×C127D411yescharacteristic
44C4⋊C4×C127D412yes—
45C22⋊C4×C127—22no—
46C22⋊C4×C127—21no—
47C42×C127—21no—
48C42×C127C2311yescharacteristic
49Dic254—42no—
50C2×Dic127—41no—
51C127⋊C8—41no—
52C127⋊C8—41no—
53C127⋊C8—41no—
54C23×C127—21no—
55Q8×C127—21no—
56Q8×C127—44no—
57Q8×C127—41no—
58Q8×C127—41no—
59Q8×C127SD1612yes—
60D4×C127—21no—
61D4×C127Q1612yes—
62C2×C4×C127—21no—
63C2×C4×C127—21no—
64C2×C4×C127—22no—
65C2×C4×C127—21no—
66C2×C4×C127—44no—
67C2×C4×C127—41no—
68C2×C4×C127—41no—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C22×C127—22no—
73C22×C127—41no—
74C22×C127—41no—
75C22×C127#12011yescharacteristic
76C4×C127—22no—
77C4×C127—21no—
78C4×C127—42no—
79C4×C127—41no—
80C4×C127—41no—
81C4×C127C2×SD1611yescharacteristic
82C4×C127C2×Q1611yescharacteristic
83C4×C127AGL(1,ℤ/8)11yescharacteristic
84C4×C127#13711yescharacteristic
85Dic127—81no—
86C2×C127—42no—
87C2×C127#45011yescharacteristic
88C2×C127#45111yescharacteristic
89C2×C127#45411yescharacteristic
90#1637—1271nomaximal, Sylow
91C127#163711yescharacteristic, complemented, Sylow
92D4×Q8D12711yescharacteristic
93#487—1271no—
94#477—1271no—
95#476—1271no—
96#332—1271no—
97Q8⋊C8—1271no—
98D4⋊C8—1271no—
99C22×Q8—21no—
100C4⋊Q8—21no—
101C4⋊Q8—1271no—
102C4⋊Q8C2×D12711yescharacteristic
103C22⋊Q8—21no—
104C22⋊Q8—22no—
105C4×Q8C2×D12711yescharacteristic
106C4×D4—21no—
107C4×D4C2×D12711yescharacteristic
108#107—2541no—
109#106—2541no—
110C4⋊C8—1271no—
111C4⋊C8—1271no—
112#103—2541no—
113#103—2542no—
114#102—2541no—
115C4×C8—1271no—
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—2541no—
121C2×Q8C22⋊D12711yescharacteristic
122C2×D4C22⋊D12711yescharacteristic
123C22×C4—22no—
124C22×C4—21no—
125C2×C8—2541no—
126C2×C8—2541no—
127C2×C8—2541no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—2541no—
132C4⋊C4—2541no—
133C4⋊C4C22⋊D12712yes—
134C4⋊C4C22⋊D12711yescharacteristic
135C4⋊C4C22⋊D12711yescharacteristic
136C22⋊C4—22no—
137C22⋊C4—21no—
138C42—21no—
139C42C22×D12711yescharacteristic
140C23—21no—
141Q8—21no—
142Q8—44no—
143Q8—41no—
144Q8—41no—
145Q8—5082no—
146Q8#1452323612yes—
147D4—21no—
148D4#1452323812yes—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—44no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—5081no—
157C2×C4C2×C22⋊D12711yescharacteristic
158C2×C4C2×C22⋊D12711yescharacteristic
159C2×C4C2×C22⋊D12711yescharacteristic
160C8—5081no—
161C8—5081no—
162C8—5081no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#1686117811yescharacteristic
167C4—22no—
168C4—21no—
169C4—42no—
170C4—41no—
171C4—41no—
172C4—10161no—
173C4C2×#1452323611yescharacteristic
174C4C2×#1452323811yescharacteristic
175C4#1686123511yescharacteristic
176C4#1686124011yescharacteristic
177C2—42no—
178C2#2255002711yescharacteristic
179C2#2255003211yescharacteristic
180C2#2255004911yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingD4×Q8×C127C2
socleC22×C127#120
prime radicalC2×D4×C127D4
hypercenterD4×Q8D127
hypocenterC127#1637
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. #22550228C2semidirect
2. #22550171C2semidirect
4. #22549622C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
91. C127#1637semidirect

Maximal subgroups

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

subgroupindex
#225502282
#225501712
#225501662
#225496222
Q8⋊(C127⋊C8)2
D4⋊(C127⋊C8)2
D4×Q8×C1272
#1637127