Finite Group Databasebeta

← #30637995

Subgroups — #30637995

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
1D4×Dic254C211yesmaximal, characteristic
2#22550274C211yesmaximal, characteristic, complemented
3#22550162C211yesmaximal, characteristic, complemented
4#22550081C211yesmaximal, characteristic
5#22549623C211yesmaximal, characteristic, complemented
6#22549608C211yesmaximal, characteristic, complemented
7D4⋊C8×C127C211yesmaximal, characteristic
8C22×Dic254—21no—
9C2×#14523236—21no—
10#16861209C2211yescharacteristic
11#16861209C2211yescharacteristic
12#16861208—21no—
13#16861188—21no—
14#16861186—22no—
15C4×Dic254C2211yescharacteristic
16D4×Dic127—21no—
17#16861141—22no—
18#16861141—21no—
19C4⋊(C127⋊C8)C2211yescharacteristic
20#16861134—21no—
21#16861132—21no—
22C4×D4×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25C2×Dic254—22no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—42no—
29C2×Dic254—42no—
30C2×Dic254D411yescharacteristic
31C22×Dic127—22no—
32#14523236—42no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127D412yes—
38C4⋊Dic127D411yescharacteristic
39Dic127⋊C4—22no—
40Dic127⋊C4—21no—
41C22⋊Dic127—22no—
42C4×Dic127—21no—
43C2×D4×C127D411yescharacteristic
44C22×C4×C127—21no—
45C2×C8×C127—21no—
46C2×C8×C127—21no—
47C4⋊C4×C127D411yescharacteristic
48C22⋊C4×C127—21no—
49C42×C127C2311yescharacteristic
50Dic254—21no—
51Dic254—44no—
52Dic254—41no—
53Dic254—41no—
54Dic254—41no—
55Dic254—41no—
56Dic254SD1612yescomplemented
57C2×Dic127—21no—
58C2×Dic127—21no—
59C2×Dic127—22no—
60C2×Dic127—44no—
61C2×Dic127—41no—
62C127⋊C8—41no—
63C23×C127—21no—
64D4×C127—21no—
65D4×C127SD1612yes—
66C2×C4×C127—21no—
67C2×C4×C127—41no—
68C2×C4×C127—41no—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—22no—
73C8×C127—41no—
74C22×C127—22no—
75C22×C127—41no—
76C22×C127—41no—
77C22×C127#12011yescharacteristic
78C4×C127—21no—
79C4×C127—41no—
80C4×C127C2×SD1611yescharacteristic
81C4×C127C2×SD1611yescharacteristic
82C4×C127AGL(1,ℤ/8)11yescharacteristic
83C4×C127#13711yescharacteristic
84Dic127—22no—
85Dic127—42no—
86Dic127—41no—
87Dic127—81no—
88C2×C127—42no—
89C2×C127#44811yescharacteristic
90C2×C127#45011yescharacteristic
91C2×C127#45611yescharacteristic
92#1651—1271nomaximal, Sylow
93C127#165111yescharacteristic, complemented, Sylow
94D4×Q8—1271no—
95#498—1271no—
96#475—1271no—
97#461—1271no—
98#332—1271no—
99Q8⋊C8—1271no—
100D4⋊C8D12711yescharacteristic
101C22×Q8—2541no—
102C2×SD16—2541no—
103C4⋊Q8—1271no—
104C4⋊Q8—1271no—
105C4⋊Q8—2541no—
106C22⋊Q8—2542no—
107C22⋊Q8—2541no—
108C4×Q8—1271no—
109C4×D4—2541no—
110C4×D4C2×D12711yescharacteristic
111#106—2541no—
112#106—2542no—
113C4⋊C8—1271no—
114C4⋊C8C2×D12711yescharacteristic
115#103—2541no—
116#103—2541no—
117C4×C8C2×D12711yescharacteristic
118C2×Q8—1271no—
119C2×Q8—5082no—
120C2×Q8—5082no—
121C2×Q8—2542no—
122C2×Q8—2541no—
123C2×Q8—2541no—
124C2×D4D50811yescharacteristic
125C22×C4—21no—
126C22×C4—2542no—
127SD16—5082no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—2541no—
131C4⋊C4—1271no—
132C4⋊C4—1272no—
133C4⋊C4—2542no—
134C4⋊C4—2541no—
135C4⋊C4—2541no—
136C4⋊C4—2541no—
137C4⋊C4—2541no—
138C4⋊C4D50811yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—2542no—
141C42—2541no—
142C42C22×D12711yescharacteristic
143C23—21no—
144Q8—1272no—
145Q8—5084no—
146Q8—5081no—
147Q8—5081no—
148Q8—5081no—
149Q8—5081no—
150Q8—2541no—
151D4—21no—
152D4#1452323712yes—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—5081no—
157C2×C4—5084no—
158C2×C4—2541no—
159C2×C4—2542no—
160C2×C4—2541no—
161C2×C4D4×D12711yescharacteristic
162C2×C4D4×D12711yescharacteristic
163C2×C4C2×D50811yescharacteristic
164C8—22no—
165C8—41no—
166C8—5081no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1686117911yescharacteristic
171C4—21no—
172C4—41no—
173C4—10161no—
174C4—5082no—
175C4—5081no—
176C4—2542no—
177C4SD16×D12711yescharacteristic
178C4C2×#1452323711yescharacteristic
179C4#1686123311yescharacteristic
180C4#1686124211yescharacteristic
181C2—42no—
182C2#2255001711yescharacteristic
183C2#2255002811yescharacteristic
184C2#2255006011yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingD4⋊C8×C127C2
socleC22×C127#120
prime radicalC2×D4×C127D4
hypercenterD4⋊C8D127
hypocenterC127#1651
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #22550274C2semidirect
3. #22550162C2semidirect
5. #22549623C2semidirect
6. #22549608C2semidirect
56. Dic254SD16semidirect
93. C127#1651semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2542
#225502742
#225501622
#225500812
#225496232
#225496082
D4⋊C8×C1272
#1651127