Finite Group Databasebeta

← #30637943

Subgroups — #30637943

Order 16256.

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

normal subgroups only

This group has 208 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#22550583C211yesmaximal, characteristic
2#22550228C211yesmaximal, characteristic, complemented
3#22550083C211yesmaximal, characteristic, complemented
4#22549624C211yesmaximal, characteristic, complemented
5#22549606C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic
7#474×C127C211yesmaximal, characteristic
8C2×#14523238—21no—
9#16861209C2211yescharacteristic
10#16861200—21no—
11#16861188—21no—
12#16861186—22no—
13C4×Dic254C2211yescharacteristic
14D4×Dic127—21no—
15C2×C4⋊Dic127—21no—
16C4⋊(C127⋊C8)C2211yescharacteristic
17#16861133—21no—
18#16861132—22no—
19C4×C127⋊C8C2211yescharacteristic
20C4⋊Q8×C127C2211yescharacteristic
21C4×D4×C127C2211yescharacteristic
22#107×C127—21no—
23C4⋊C8×C127C2211yescharacteristic
24#102×C127—21no—
25C2×Dic254D411yescharacteristic
26C2×Dic254D412yes—
27C22×Dic127—22no—
28#14523238—42no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C4⋊Dic127—22no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127D411yescharacteristic
35Dic127⋊C4—22no—
36Dic127⋊C4—21no—
37C22⋊Dic127—22no—
38C4×Dic127—21no—
39C2×Q8×C127—21no—
40C2×D4×C127D411yescharacteristic
41C22×C4×C127—21no—
42C2×C8×C127—21no—
43C4⋊C4×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127D411yescharacteristic
46C22⋊C4×C127—21no—
47C42×C127C2311yescharacteristic
48Dic254—21no—
49Dic254—21no—
50Dic254—42no—
51C2×Dic127—21no—
52C2×Dic127—21no—
53C2×Dic127—22no—
54C2×Dic127—82no—
55C127⋊C8—41no—
56C127⋊C8—41no—
57C23×C127—21no—
58Q8×C127—41no—
59Q8×C127—41no—
60D4×C127—21no—
61D4×C127Q1612yescomplemented
62C2×C4×C127—21no—
63C2×C4×C127—41no—
64C2×C4×C127—41no—
65C2×C4×C127—41no—
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C8×C127—41no—
70C22×C127—22no—
71C22×C127—41no—
72C22×C127—41no—
73C22×C127#12011yescharacteristic
74C4×C127—21no—
75C4×C127—41no—
76C4×C127—81no—
77C4×C127C2×Q1611yescharacteristic
78C4×C127#13511yescharacteristic
79C4×C127AGL(1,ℤ/8)11yescharacteristic
80C4×C127#13711yescharacteristic
81Dic127—41no—
82Dic127—41no—
83Dic127—42no—
84C2×C127—42no—
85C2×C127#44911yescharacteristic
86C2×C127#45111yescharacteristic
87C2×C127#45511yescharacteristic
88#1644—1271nomaximal, Sylow
89C127#164411yescharacteristic, complemented, Sylow
90#554—1271no—
91#487—1271no—
92#474D12711yescharacteristic
93#462—1271no—
94#333—1271no—
95Q8⋊C8—1271no—
96D4⋊C8—1271no—
97C2×Q16—2541no—
98C4⋊Q8—1271no—
99C4⋊Q8C2×D12711yescharacteristic
100#125—2541no—
101C22⋊Q8—2542no—
102C22⋊Q8—2541no—
103C4×Q8—1271no—
104C4×D4—2541no—
105C4×D4C2×D12711yescharacteristic
106C2×C4⋊C4—2541no—
107#107—21no—
108C4⋊C8—1271no—
109C4⋊C8C2×D12711yescharacteristic
110#103—2542no—
111#103—2541no—
112#102—21no—
113C4×C8—1271no—
114C2×Q8—21no—
115C2×Q8—1272no—
116C2×Q8—1271no—
117C2×D4C22⋊D12711yescharacteristic
118C22×C4—21no—
119C22×C4—2542no—
120Q16—5082no—
121C2×C8—21no—
122C2×C8—2541no—
123C2×C8—2541no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—1271no—
127C4⋊C4—2542no—
128C4⋊C4—2542no—
129C4⋊C4—2541no—
130C4⋊C4—2541no—
131C4⋊C4—2541no—
132C4⋊C4C22⋊D12711yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—2542no—
135C42—2541no—
136C42C22×D12711yescharacteristic
137C23—21no—
138Q8—41no—
139Q8—41no—
140Q8—5082no—
141Q8—2541no—
142Q8—2541no—
143D4—21no—
144D4#1452323812yescomplemented
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—10162no—
150C2×C4—2541no—
151C2×C4—2541no—
152C2×C4—2542no—
153C2×C4D4×D12711yescharacteristic
154C2×C4D4×D12711yescharacteristic
155C2×C4C2×C22⋊D12711yescharacteristic
156C8—41no—
157C8—5081no—
158C8—5081no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22#1686117711yescharacteristic
163C4—21no—
164C4—41no—
165C4—81no—
166C4—5081no—
167C4—5081no—
168C4—5082no—
169C4C2×#1452323811yescharacteristic
170C4#1686122811yescharacteristic
171C4#1686123311yescharacteristic
172C4#1686123911yescharacteristic
173C2—42no—
174C2#2255001811yescharacteristic
175C2#2255003111yescharacteristic
176C2#2255005211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#474×C127C2
socleC22×C127#120
prime radicalC2×D4×C127D4
hypercenter#474D127
hypocenterC127#1644
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #22550228C2semidirect
3. #22550083C2semidirect
4. #22549624C2semidirect
5. #22549606C2semidirect
61. D4×C127Q16semidirect
89. C127#1644semidirect
144. D4#14523238semidirect

Maximal subgroups

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

subgroupindex
#225505832
#225502282
#225500832
#225496242
#225496062
D4⋊(C127⋊C8)2
#474×C1272
#1644127