Finite Group Databasebeta

← #30637898

Subgroups — #30637898

Order 16256.

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

normal subgroups only

This group has 228 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#22550227C211yesmaximal, characteristic, complemented
3#22550172C211yesmaximal, characteristic, complemented
4#22550166C211yesmaximal, characteristic
5#22549623C211yesmaximal, characteristic, complemented
6#22549608C211yesmaximal, characteristic, complemented
7D4⋊C8×C127C211yesmaximal, characteristic
8C22×Dic254—21no—
9#16861209C2211yescharacteristic
10#16861209C2211yescharacteristic
11#16861208—21no—
12#16861188—21no—
13#16861186—22no—
14C4×Dic254C2211yescharacteristic
15D4×Dic127—21no—
16#16861143—21no—
17#16861140—21no—
18C4⋊(C127⋊C8)C2211yescharacteristic
19#16861134—21no—
20#16861134—22no—
21#16861130—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—
32C2×C127⋊C8—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127D412yes—
37C4⋊Dic127D411yescharacteristic
38Dic127⋊C4—22no—
39Dic127⋊C4—21no—
40C22⋊Dic127—22no—
41C4×Dic127—21no—
42C2×D4×C127D411yescharacteristic
43C22×C4×C127—21no—
44C2×C8×C127—21no—
45C2×C8×C127—21no—
46C4⋊C4×C127D411yescharacteristic
47C22⋊C4×C127—21no—
48C42×C127C2311yescharacteristic
49Dic254—21no—
50Dic254—44no—
51Dic254—41no—
52Dic254—41no—
53Dic254—42no—
54Dic254SD1612yes—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C2×Dic127—22no—
58C2×Dic127—44no—
59C2×Dic127—41no—
60C127⋊C8—41no—
61C23×C127—21no—
62D4×C127—21no—
63D4×C127Q1612yes—
64C2×C4×C127—21no—
65C2×C4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C8×C127—41no—
71C8×C127—41no—
72C22×C127—22no—
73C22×C127—41no—
74C22×C127—41no—
75C22×C127#12011yescharacteristic
76C4×C127—21no—
77C4×C127—41no—
78C4×C127C2×SD1611yescharacteristic
79C4×C127C2×Q1611yescharacteristic
80C4×C127AGL(1,ℤ/8)11yescharacteristic
81C4×C127#13711yescharacteristic
82Dic127—22no—
83Dic127—42no—
84Dic127—41no—
85Dic127—81no—
86C2×C127—42no—
87C2×C127#45011yescharacteristic
88C2×C127#45111yescharacteristic
89C2×C127#45411yescharacteristic
90#1637—1271nomaximal, Sylow
91C127#163711yescharacteristic, complemented, Sylow
92D4×Q8—1271no—
93#487—1271no—
94#477—1271no—
95#476—1271no—
96#332—1271no—
97Q8⋊C8—1271no—
98D4⋊C8D12711yescharacteristic
99C22×Q8—2541no—
100C4⋊Q8—1271no—
101C4⋊Q8—1271no—
102C4⋊Q8—2541no—
103C22⋊Q8—2542no—
104C22⋊Q8—2541no—
105C4×Q8—1271no—
106C4×D4—2541no—
107C4×D4C2×D12711yescharacteristic
108#107—2541no—
109#106—2541no—
110C4⋊C8—1271no—
111C4⋊C8C2×D12711yescharacteristic
112#103—2542no—
113#103—2541no—
114#102—2541no—
115C4×C8C2×D12711yescharacteristic
116C2×Q8—1271no—
117C2×Q8—5082no—
118C2×Q8—5082no—
119C2×Q8—2542no—
120C2×Q8—2541no—
121C2×Q8—2541no—
122C2×D4D50811yescharacteristic
123C22×C4—21no—
124C22×C4—2542no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—2541no—
128C4⋊C4—1272no—
129C4⋊C4—1271no—
130C4⋊C4—2542no—
131C4⋊C4—2541no—
132C4⋊C4—2541no—
133C4⋊C4—2541no—
134C4⋊C4—2541no—
135C4⋊C4D50811yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—2542no—
138C42—2541no—
139C42C22×D12711yescharacteristic
140C23—21no—
141Q8—1272no—
142Q8—5082no—
143Q8—5084no—
144Q8—5081no—
145Q8—5081no—
146Q8—2541no—
147D4—21no—
148D4Dic50812yes—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—5081no—
153C2×C4—5084no—
154C2×C4—2541no—
155C2×C4—2542no—
156C2×C4—2541no—
157C2×C4D4×D12711yescharacteristic
158C2×C4D4×D12711yescharacteristic
159C2×C4C2×D50811yescharacteristic
160C8—41no—
161C8—41no—
162C8—5081no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#1686117911yescharacteristic
167C4—21no—
168C4—41no—
169C4—10161no—
170C4—5082no—
171C4—5081no—
172C4—2542no—
173C4SD16×D12711yescharacteristic
174C4C2×Dic50811yescharacteristic
175C4#1686123311yescharacteristic
176C4#1686124211yescharacteristic
177C2—42no—
178C2#2255002811yescharacteristic
179C2#2255003411yescharacteristic
180C2#2255005011yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingD4⋊C8×C127C2
socleC22×C127#120
prime radicalC2×D4×C127D4
hypercenterD4⋊C8D127
hypocenterC127#1637
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. #22550227C2semidirect
3. #22550172C2semidirect
5. #22549623C2semidirect
6. #22549608C2semidirect
91. C127#1637semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2542
#225502272
#225501722
#225501662
#225496232
#225496082
D4⋊C8×C1272
#1637127