Finite Group Databasebeta

← #30648374

Subgroups — #30648374

Order 16256.

524 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#16861215C211yesmaximal, characteristic, complemented
2#22550443C212yesmaximal, complemented
3#22550126C214yesmaximal, complemented
4#22550109C214yesmaximal, complemented
5#22549914C211yesmaximal, characteristic, complemented
6#22549881C212yesmaximal, complemented
7C22×D8×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861215—24no—
10C2×#14523232C2211yescharacteristic
11C2×#14523231C2212yescomplemented
12C2×C8×D127C2212yescomplemented
13#16861205—22no—
14#16861196—22no—
15#16861180C2218yescomplemented
16#16861178—24no—
17#16861168C2212yes—
18#16861143C2212yescomplemented
19#16861141C2212yescomplemented
20#16861130C2218yescomplemented
21C22×D4×C127C2212yes—
22C2×D8×C127—24no—
23C2×D8×C127C2212yes—
24C2×D8×C127C2212yes—
25C22×C8×C127C2211yescharacteristic
26#14523248—22no—
27#14523248—22no—
28C2×Dic254D411yescharacteristic
29C2×D508D411yescharacteristic
30C2×C22⋊D127—28no—
31C2×C22⋊D127D412yes—
32C2×C4×D127C2312yes—
33#14523232—22no—
34#14523231—24no—
35C2×C127⋊C8C2312yes—
36C8×D127—24no—
37C4⋊Dic127C2314yes—
38C22⋊Dic127—28no—
39C22⋊Dic127—22no—
40C4×Dic127—22no—
41C24×C127—22no—
42C2×D4×C127—24no—
43C2×D4×C127—22no—
44C2×D4×C127—24no—
45C2×D4×C127C2314yes—
46C22×C4×C127C2311yescharacteristic
47D8×C127—24no—
48D8×C127—24no—
49C2×C8×C127D414yescomplemented
50C2×C8×C127C2311yescharacteristic
51C2×C8×C127C2311yescharacteristic
52C22×D127C2×D412yes—
53Dic254—22no—
54D508—22no—
55C22⋊D127—24no—
56C22⋊D127—88no—
57C2×Dic127—24no—
58C2×Dic127C2×D412yes—
59C4×D127—22no—
60C4×D127—22no—
61C127⋊C8—22no—
62C23×C127—24no—
63C23×C127—44no—
64C23×C127—84no—
65C23×C127C2×D411yescharacteristic
66D4×C127—24no—
67D4×C127—24no—
68D4×C127—44no—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D412yes—
71C2×C4×C127#4112yes—
72C2×C4×C127C2411yescharacteristic
73C8×C127C2×D412yes—
74C8×C127C2×D412yes—
75C22×C127—22no—
76C22×C127—44no—
77C22×C127—48no—
78C22×C127—88no—
79C22×C127#12112yes—
80C22×C127C22×D411yescharacteristic
81C2×D127—22no—
82C2×D127—24no—
83C4×C127#12112yes—
84C4×C127C22×D411yescharacteristic
85C4×C127C2×#4111yescharacteristic
86Dic127—22no—
87Dic127—44no—
88C2×C127—22no—
89C2×C127—84no—
90C2×C127C2×#12111yescharacteristic
91C2×C127#57612yes—
92D127—42no—
93#3058—1271nomaximal, Sylow
94C127#305811yescharacteristic, complemented, Sylow
95C22×D8D12711yescharacteristic, complemented
96C2×#131—1271no—
97#535—1272no—
98#469—1274no—
99#466—1274no—
100#427—1271no—
101#418—1272no—
102C2×#41—1271no—
103C22×D4C2×D12712yes—
104C2×D8—24no—
105C2×D8C2×D12712yes—
106C2×D8C2×D12712yes—
107#131—2544no—
108C2×M16—1271no—
109C2×M16—1272no—
110C22×C8—1272no—
111C22×C8C2×D12711yescharacteristic
112#127—2542no—
113#124—2542no—
114#121—1278no—
115#120—2544no—
116#117—1272no—
117#107—1272no—
118#106—1272no—
119#102—1278no—
120C24—22no—
121#41—2542no—
122#41—2542no—
123C2×Q8—1271no—
124C2×D4—22no—
125C2×D4—24no—
126C2×D4—24no—
127C2×D4—1272no—
128C2×D4—1271no—
129C2×D4—2548no—
130C2×D4C22×D12714yes—
131C22×C4—1272no—
132C22×C4C22×D12711yescharacteristic
133D8—24no—
134D8—24no—
135M16—2542no—
136M16—2544no—
137C2×C8—1272no—
138C2×C8—2544no—
139C2×C8C22⋊D12714yescomplemented
140C2×C8C22×D12711yescharacteristic
141C2×C8C22×D12711yescharacteristic
142C4⋊C4—1274no—
143C22⋊C4—2548no—
144C22⋊C4—2542no—
145C42—2542no—
146C23—24no—
147C23—44no—
148C23—84no—
149C23—1272no—
150C23D4×D12711yescharacteristic
151Q8—2542no—
152D4—24no—
153D4—24no—
154D4—44no—
155D4—10168no—
156D4—2544no—
157D4—2542no—
158C2×C4—1272no—
159C2×C4—2544no—
160C2×C4—2542no—
161C2×C4—2542no—
162C2×C4D4×D12711yescharacteristic
163C2×C4C2×C22⋊D12712yes—
164C2×C4#1452324712yes—
165C2×C4C23×D12711yescharacteristic
166C8—2542no—
167C8C2×C22⋊D12712yes—
168C8C2×C22⋊D12712yes—
169C22—22no—
170C22—44no—
171C22—48no—
172C22—88no—
173C22—2542no—
174C22—2544no—
175C22#1686118012yes—
176C22C2×D4×D12711yescharacteristic
177C4—5084no—
178C4—2542no—
179C4#1686118012yes—
180C4C22×C22⋊D12711yescharacteristic
181C4C2×#1452324711yescharacteristic
182C2—22no—
183C2—84no—
184C2—5082no—
185C2C2×#1686118011yescharacteristic
186C2#2255067712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×D8×C127C2
socleC22×C127C22×D4
hypercenterC22×D8D127
hypocenterC127#3058
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16861215C2semidirect
2. #22550443C2semidirect
3. #22550126C2semidirect
4. #22550109C2semidirect
5. #22549914C2semidirect
6. #22549881C2semidirect
7. C22×D8×C127C2semidirect
11. C2×#14523231C22semidirect
12. C2×C8×D127C22semidirect
15. #16861180C22semidirect
18. #16861143C22semidirect
19. #16861141C22semidirect
20. #16861130C22semidirect
49. C2×C8×C127D4semidirect
94. C127#3058semidirect
95. C22×D8D127semidirect
139. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612152
#225504432
#225501262
#225501092
#225499142
#225498812
C22×D8×C1272
#3058127