Finite Group Databasebeta

← #30637953

Subgroups — #30637953

Order 16256.

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

normal subgroups only

This group has 200 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×Dic254C211yesmaximal, characteristic
2#22550227C211yesmaximal, characteristic
3#22550162C211yesmaximal, characteristic
4#22550086C211yesmaximal, characteristic
5#22549624C211yesmaximal, characteristic
6#22549608C211yesmaximal, characteristic
7Q8⋊C8×C127C211yesmaximal, characteristic
8C2×#14523238—21no—
9#16861209—21no—
10#16861209C2211yescharacteristic
11#16861209C2211yescharacteristic
12#16861208—22no—
13#16861208—21no—
14Q8×Dic127—21no—
15C4×Dic254—21no—
16C4×Dic254C2211yescharacteristic
17#16861141—21no—
18C4⋊(C127⋊C8)C2211yescharacteristic
19#16861134—21no—
20#16861134—22no—
21#16861132—21no—
22C4×Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25C2×Dic254—22no—
26C2×Dic254—21no—
27C2×Dic254D411yescharacteristic
28C2×Dic254D412yes—
29#14523238—42no—
30C2×C127⋊C8—21no—
31C4⋊Dic127—22no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127D411yescharacteristic
37Dic127⋊C4—22no—
38Dic127⋊C4—21no—
39C4×Dic127—22no—
40C4×Dic127—21no—
41C2×Q8×C127D411yescharacteristic
42C2×C8×C127—21no—
43C2×C8×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127D411yescharacteristic
46C42×C127—21no—
47C42×C127C2311yescharacteristic
48Dic254—21no—
49Dic254—42no—
50Dic254—41no—
51Dic254—41no—
52Dic254—82no—
53Dic254Q1612yescomplemented
54C2×Dic127—21no—
55C2×Dic127—22no—
56C2×Dic127—21no—
57C2×Dic127—41no—
58C127⋊C8—41no—
59Q8×C127—21no—
60Q8×C127SD1612yes—
61C2×C4×C127—21no—
62C2×C4×C127—21no—
63C2×C4×C127C2×D411yescharacteristic
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127C2×D411yescharacteristic
66C8×C127—41no—
67C8×C127—41no—
68C22×C127#12011yescharacteristic
69C4×C127—22no—
70C4×C127—21no—
71C4×C127—41no—
72C4×C127C2×SD1611yescharacteristic
73C4×C127C2×Q1611yescharacteristic
74C4×C127#13711yescharacteristic
75C4×C127#13711yescharacteristic
76Dic127—22no—
77Dic127—42no—
78Dic127—41no—
79Dic127—81no—
80C2×C127#44811yescharacteristic
81C2×C127#45111yescharacteristic
82C2×C127#45511yescharacteristic
83#1645—1271nomaximal, Sylow
84C127#164511yescharacteristic, complemented, Sylow
85Q82—1271no—
86#487—1271no—
87#475—1271no—
88#462—1271no—
89#333—1271no—
90Q8⋊C8—1271no—
91Q8⋊C8D12711yescharacteristic
92C2×Q16—2541no—
93C4⋊Q8—1271no—
94C4⋊Q8—1271no—
95C4⋊Q8—2541no—
96C4⋊Q8—2542no—
97C4⋊Q8—2541no—
98C4×Q8—1271no—
99C4×Q8—2541no—
100C4×Q8—2541no—
101C4×Q8C2×D12711yescharacteristic
102#106—2541no—
103C4⋊C8—1271no—
104C4⋊C8C2×D12711yescharacteristic
105#103—2541no—
106#103—2542no—
107#103—2541no—
108C4×C8C2×D12711yescharacteristic
109C2×Q8—1271no—
110C2×Q8—1272no—
111C2×Q8—2542no—
112C2×Q8—2541no—
113C2×Q8D50811yescharacteristic
114Q16—5082no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—2541no—
118C4⋊C4—21no—
119C4⋊C4—1271no—
120C4⋊C4—2542no—
121C4⋊C4—2542no—
122C4⋊C4—2541no—
123C4⋊C4—2541no—
124C4⋊C4—2541no—
125C4⋊C4—2541no—
126C4⋊C4—2541no—
127C4⋊C4D50811yescharacteristic
128C42—21no—
129C42—2542no—
130C42—2541no—
131C42C22×D12711yescharacteristic
132Q8—21no—
133Q8—1272no—
134Q8—10162no—
135Q8—5082no—
136Q8—5081no—
137Q8—5081no—
138Q8—2541no—
139Q8#1452323712yes—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—5081no—
143C2×C4—2541no—
144C2×C4—2542no—
145C2×C4—2541no—
146C2×C4D4×D12711yescharacteristic
147C2×C4D4×D12711yescharacteristic
148C2×C4C2×D50811yescharacteristic
149C8—41no—
150C8—41no—
151C8—5081no—
152C22#1686117911yescharacteristic
153C4—22no—
154C4—21no—
155C4—41no—
156C4—10161no—
157C4—5081no—
158C4—5082no—
159C4—2542no—
160C4C2×#1452323711yescharacteristic
161C4Q16×D12711yescharacteristic
162C4#1686123811yescharacteristic
163C4#1686124211yescharacteristic
164C2#2255001711yescharacteristic
165C2#2255003311yescharacteristic
166C2#2255005511yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingQ8⋊C8×C127C2
socle, prime radicalC22×C127#120
hypercenterQ8⋊C8D127
hypocenterC127#1645
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
53. Dic254Q16semidirect
84. C127#1645semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2542
#225502272
#225501622
#225500862
#225496242
#225496082
Q8⋊C8×C1272
#1645127