Finite Group Databasebeta

← #30637902

Subgroups — #30637902

Order 16256.

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

normal subgroups only

This group has 194 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#22550228C211yesmaximal, characteristic, complemented
2#22550171C211yesmaximal, characteristic, complemented
3#22550166C211yesmaximal, characteristic
4#22549622C211yesmaximal, characteristic, complemented
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic
7#556×C127C211yesmaximal, characteristic
8#16861209C2211yescharacteristic
9#16861142—21no—
10#16861140—21no—
11C4⋊(C127⋊C8)C2211yescharacteristic
12C4⋊(C127⋊C8)C2211yescharacteristic
13#16861133—21no—
14#16861132—22no—
15#16861130—21no—
16C4×C127⋊C8C2211yescharacteristic
17C4⋊Q8×C127C2211yescharacteristic
18#126×C127—22no—
19#124×C127—21no—
20C22⋊Q8×C127—21no—
21C4×Q8×C127—21no—
22C4×Q8×C127C2211yescharacteristic
23C4×D4×C127C2211yescharacteristic
24#117×C127—21no—
25C2×Dic254—21no—
26C2×C127⋊C8—21no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8—21no—
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31C2×Q8×C127—21no—
32C2×Q8×C127D411yescharacteristic
33C2×D4×C127D411yescharacteristic
34C22×C4×C127—21no—
35C4⋊C4×C127—22no—
36C4⋊C4×C127—21no—
37C4⋊C4×C127—21no—
38C4⋊C4×C127D411yescharacteristic
39C4⋊C4×C127D411yescharacteristic
40C4⋊C4×C127D412yes—
41C22⋊C4×C127—22no—
42C22⋊C4×C127—22no—
43C22⋊C4×C127—21no—
44C42×C127—22no—
45C42×C127—21no—
46C42×C127C2311yescharacteristic
47Dic254—42no—
48C2×Dic127—41no—
49C127⋊C8—41no—
50C127⋊C8—41no—
51C127⋊C8—41no—
52C23×C127—21no—
53Q8×C127—21no—
54Q8×C127—21no—
55Q8×C127—41no—
56Q8×C127—41no—
57D4×C127—21no—
58D4×C127—21no—
59C2×C4×C127—21no—
60C2×C4×C127—21no—
61C2×C4×C127—21no—
62C2×C4×C127—22no—
63C2×C4×C127—41no—
64C2×C4×C127—41no—
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C22×C127—41no—
69C22×C127—41no—
70C22×C127—41no—
71C22×C127#12011yescharacteristic
72C4×C127—21no—
73C4×C127—41no—
74C4×C127—41no—
75C4×C127—42no—
76C4×C127—41no—
77C4×C127#13511yescharacteristic
78C4×C127#13511yescharacteristic
79C4×C127#13711yescharacteristic
80C4×C127#13711yescharacteristic
81Dic127—81no—
82C2×C127—81no—
83C2×C127#45211yescharacteristic
84C2×C127#45211yescharacteristic
85C2×C127#45411yescharacteristic
86#1638—1271nomaximal, Sylow
87C127#163811yescharacteristic, complemented, Sylow
88#556D12711yescharacteristic
89#487—1271no—
90#477—1271no—
91#476—1271no—
92#332—1271no—
93Q8⋊C8—1271no—
94D4⋊C8—1271no—
95C4⋊Q8—1271no—
96C4⋊Q8C2×D12711yescharacteristic
97#126—22no—
98#124—21no—
99C22⋊Q8—21no—
100C4×Q8—21no—
101C4×Q8C2×D12711yescharacteristic
102C4×D4C2×D12711yescharacteristic
103#117—21no—
104#107—2541no—
105#106—2541no—
106C4⋊C8—1271no—
107C4⋊C8—1271no—
108#103—2541no—
109#103—2542no—
110#102—2541no—
111C4×C8—1271no—
112C2×Q8—21no—
113C2×Q8—2541no—
114C2×Q8C22⋊D12711yescharacteristic
115C2×D4C22⋊D12711yescharacteristic
116C22×C4—21no—
117C2×C8—2541no—
118C2×C8—2541no—
119C2×C8—2541no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—2541no—
124C4⋊C4—2541no—
125C4⋊C4C22⋊D12712yes—
126C4⋊C4C22⋊D12711yescharacteristic
127C4⋊C4C22⋊D12711yescharacteristic
128C22⋊C4—22no—
129C22⋊C4—22no—
130C22⋊C4—21no—
131C42—22no—
132C42—21no—
133C42C22×D12711yescharacteristic
134C23—21no—
135Q8—21no—
136Q8—21no—
137Q8—41no—
138Q8—41no—
139Q8—5082no—
140D4—21no—
141D4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—5081no—
149C2×C4C2×C22⋊D12711yescharacteristic
150C2×C4C2×C22⋊D12711yescharacteristic
151C2×C4C2×C22⋊D12711yescharacteristic
152C8—5081no—
153C8—5081no—
154C8—5081no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22#1686117811yescharacteristic
159C4—21no—
160C4—41no—
161C4—41no—
162C4—42no—
163C4—41no—
164C4—10161no—
165C4#1686122911yescharacteristic
166C4#1686122911yescharacteristic
167C4#1686123911yescharacteristic
168C4#1686123911yescharacteristic
169C2—81no—
170C2#2255003811yescharacteristic
171C2#2255003811yescharacteristic
172C2#2255004911yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#556×C127C2
socleC22×C127#120
prime radicalC2×D4×C127D4
hypercenter#556D127
hypocenterC127#1638
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #22550228C2semidirect
2. #22550171C2semidirect
4. #22549622C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
87. C127#1638semidirect

Maximal subgroups

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

subgroupindex
#225502282
#225501712
#225501662
#225496222
Q8⋊(C127⋊C8)2
D4⋊(C127⋊C8)2
#556×C1272
#1638127