Finite Group Databasebeta

← #30637996

Subgroups — #30637996

Order 16256.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×D508C211yesmaximal, characteristic
2#22550274C211yesmaximal, characteristic, complemented
3#22550161C211yesmaximal, characteristic, complemented
4#22550082C211yesmaximal, characteristic
5#22549621C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic
7Q8⋊C8×C127C211yesmaximal, characteristic, complemented
8C2×Q8×D127—21no—
9C2×#14523237—21no—
10#16861209—21no—
11#16861209C2211yescharacteristic
12#16861209C2211yescharacteristic
13#16861189—22no—
14#16861189—21no—
15C4×D508—21no—
16C4×D508C2211yescharacteristic
17#16861141—22no—
18#16861140—21no—
19C4⋊(C127⋊C8)C2211yescharacteristic
20#16861134—21no—
21#16861133—21no—
22C4×Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25C2×Dic254—22no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28Q8×D127—42no—
29Q8×D127—42no—
30C2×D508D411yescharacteristic
31C2×C4×D127—22no—
32C2×C4×D127—21no—
33#14523237—42no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—22no—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127D412yes—
40C4⋊Dic127D411yescharacteristic
41(C2×D127)⋊C4—22no—
42(C2×D127)⋊C4—21no—
43C2×Q8×C127D411yescharacteristic
44C2×C8×C127—21no—
45C2×C8×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127D411yescharacteristic
48C42×C127—21no—
49C42×C127C2311yescharacteristic
50C22×D127—21no—
51Dic254—44no—
52Dic254—41no—
53Dic254—41no—
54Dic254—41no—
55Dic254—41no—
56D508—21no—
57D508SD1612yes—
58C2×Dic127—22no—
59C2×Dic127—21no—
60C2×Dic127—41no—
61C4×D127—44no—
62C4×D127—41no—
63C4×D127—41no—
64C127⋊C8—41no—
65Q8×C127—21no—
66Q8×C127SD1612yescomplemented
67C2×C4×C127—21no—
68C2×C4×C127—21no—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—22no—
73C8×C127—41no—
74C22×C127#12011yescharacteristic
75C2×D127—22no—
76C2×D127—41no—
77C2×D127—41no—
78C4×C127—22no—
79C4×C127—21no—
80C4×C127—41no—
81C4×C127C2×SD1611yescharacteristic
82C4×C127C2×SD1611yescharacteristic
83C4×C127AGL(1,ℤ/8)11yescharacteristic
84C4×C127#13711yescharacteristic
85Dic127—42no—
86Dic127—41no—
87Dic127—81no—
88C2×C127#44811yescharacteristic
89C2×C127#45011yescharacteristic
90C2×C127#45611yescharacteristic
91D127—42no—
92#1651—1271nomaximal, Sylow
93C127#165111yescharacteristic, complemented, Sylow
94D4×Q8—1271no—
95#498—1271no—
96#475—1271no—
97#461—1271no—
98#332—1271no—
99Q8⋊C8D12711yescharacteristic, complemented
100D4⋊C8—1271no—
101C22×Q8—2541no—
102C2×SD16—2541no—
103C4⋊Q8—1271no—
104C4⋊Q8—1271no—
105C4⋊Q8—2541no—
106C22⋊Q8—2542no—
107C22⋊Q8—2541no—
108C4×Q8C2×D12711yescharacteristic
109C4×D4—1271no—
110C4×D4—2541no—
111#106—2542no—
112#106—2541no—
113C4⋊C8—1271no—
114C4⋊C8C2×D12711yescharacteristic
115#103—2541no—
116#103—2541no—
117C4×C8C2×D12711yescharacteristic
118C2×Q8—5082no—
119C2×Q8—5082no—
120C2×Q8—2542no—
121C2×Q8—2541no—
122C2×Q8—2541no—
123C2×Q8D50811yescharacteristic
124C2×D4—1271no—
125C22×C4—2542no—
126C22×C4—2541no—
127SD16—5082no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—2541no—
131C4⋊C4—21no—
132C4⋊C4—1271no—
133C4⋊C4—1272no—
134C4⋊C4—2542no—
135C4⋊C4—2541no—
136C4⋊C4—2541no—
137C4⋊C4—2541no—
138C4⋊C4D50811yescharacteristic
139C22⋊C4—2542no—
140C22⋊C4—2541no—
141C42—21no—
142C42C22×D12711yescharacteristic
143C23—2541no—
144Q8—21no—
145Q8—5084no—
146Q8—5081no—
147Q8—5081no—
148Q8—5081no—
149Q8—5081no—
150Q8#1452323712yescomplemented
151D4—1272no—
152D4—2541no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—5084no—
156C2×C4—5081no—
157C2×C4—5081no—
158C2×C4—5081no—
159C2×C4—2542no—
160C2×C4—2541no—
161C2×C4D4×D12711yescharacteristic
162C2×C4D4×D12711yescharacteristic
163C2×C4C2×D50811yescharacteristic
164C8—22no—
165C8—41no—
166C8—5081no—
167C22—5081no—
168C22—5081no—
169C22—2542no—
170C22#1686117911yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—10161no—
175C4—5082no—
176C4—5081no—
177C4SD16×D12711yescharacteristic
178C4C2×#1452323711yescharacteristic
179C4#1686123711yescharacteristic
180C4#1686124111yescharacteristic
181C2—5082no—
182C2#2255001611yescharacteristic
183C2#2255002911yescharacteristic
184C2#2255006011yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingQ8⋊C8×C127C2
socleC22×C127#120
prime radicalC2×D508D4
hypercenterQ8⋊C8D127
hypocenterC127#1651
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #22550274C2semidirect
3. #22550161C2semidirect
5. #22549621C2semidirect
7. Q8⋊C8×C127C2semidirect
66. Q8×C127SD16semidirect
93. C127#1651semidirect
99. Q8⋊C8D127semidirect
150. Q8#14523237semidirect

Maximal subgroups

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

subgroupindex
Q8×D5082
#225502742
#225501612
#225500822
#225496212
#225496052
Q8⋊C8×C1272
#1651127