Finite Group Databasebeta

← #30648385

Subgroups — #30648385

Order 16256.

460 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×Dic508C211yesmaximal, characteristic, complemented
2#22550447C212yesmaximal
3#22550134C214yesmaximal, complemented
4#22550117C214yesmaximal, complemented
5#22549914C211yesmaximal, characteristic, complemented
6#22549886C212yesmaximal, complemented
7C2×#131×C127C211yesmaximal, characteristic
8C22×Dic254C2212yes—
9C2×Dic508—24no—
10C2×Dic508C2212yes—
11C2×Dic508C2212yes—
12#16861207—22no—
13#16861196—22no—
14#16861190—24no—
15#16861188C2218yes—
16#16861168C2212yes—
17#16861143C2212yes—
18#16861141C2212yes—
19#16861134C2218yes—
20C2×#41×C127C2211yescharacteristic
21#131×C127—24no—
22C2×M16×C127C2212yes—
23C2×M16×C127C2211yescharacteristic
24C22×C8×C127C2212yes—
25C22×C8×C127C2211yescharacteristic
26C2×Dic254—22no—
27C2×Dic254—24no—
28C2×Dic254—24no—
29C2×Dic254C2314yes—
30C22×Dic127—22no—
31Dic508—24no—
32Dic508—24no—
33C4⋊Dic127C2314yes—
34Dic127⋊C4—28no—
35C22⋊Dic127—28no—
36C22⋊Dic127—22no—
37C4×Dic127—22no—
38#41×C127—22no—
39#41×C127—22no—
40C2×Q8×C127D411yescharacteristic
41C2×D4×C127D412yes—
42C2×D4×C127D411yescharacteristic
43C22×C4×C127C2312yes—
44C22×C4×C127C2311yescharacteristic
45M16×C127—22no—
46M16×C127—24no—
47C2×C8×C127—24no—
48C2×C8×C127D414yes—
49C2×C8×C127C2312yes—
50C2×C8×C127C2311yescharacteristic
51C2×C8×C127C2311yescharacteristic
52Dic254—24no—
53Dic254—24no—
54Dic254—44no—
55C2×Dic127—24no—
56C2×Dic127—24no—
57C2×Dic127—44no—
58C23×C127C2×D412yes—
59C23×C127C2×D411yescharacteristic
60Q8×C127—22no—
61D4×C127—24no—
62D4×C127—22no—
63C2×C4×C127—22no—
64C2×C4×C127—22no—
65C2×C4×C127C2×D412yes—
66C2×C4×C127C2×D412yes—
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127#4112yes—
69C2×C4×C127C2411yescharacteristic
70C8×C127—22no—
71C8×C127C2×D412yes—
72C8×C127C2×D412yes—
73C22×C127—22no—
74C22×C127—22no—
75C22×C127—24no—
76C22×C127#12112yes—
77C22×C127C22×D411yescharacteristic
78C4×C127—22no—
79C4×C127#12112yes—
80C4×C127C22×D411yescharacteristic
81C4×C127C2×#4111yescharacteristic
82Dic127—44no—
83Dic127—44no—
84C2×C127—22no—
85C2×C127—42no—
86C2×C127C2×#12111yescharacteristic
87C2×C127#57812yes—
88#3059—1271nomaximal, Sylow
89C127#305911yescharacteristic, complemented, Sylow
90C22×Q16—1271no—
91C2×#131D12711yescharacteristic
92#536—1272no—
93#470—1274no—
94#467—1274no—
95#427—1271no—
96#419—1272no—
97C2×#41C2×D12711yescharacteristic
98C22×Q8—1272no—
99C2×Q16—1272no—
100C2×Q16—1272no—
101C2×Q16—2544no—
102#131—24no—
103C2×M16C2×D12711yescharacteristic
104C2×M16C2×D12712yes—
105C22×C8C2×D12712yes—
106C22×C8C2×D12711yescharacteristic
107C4⋊Q8—2542no—
108#124—2542no—
109#123—2544no—
110C22⋊Q8—1278no—
111#117—1272no—
112#107—1272no—
113#106—1272no—
114#103—1278no—
115#41—22no—
116#41—22no—
117C2×Q8—1274no—
118C2×Q8—2544no—
119C2×Q8—2544no—
120C2×Q8—2542no—
121C2×Q8D50811yescharacteristic
122C2×D4D50812yes—
123C2×D4D50811yescharacteristic
124C22×C4—2542no—
125C22×C4C22×D12712yes—
126C22×C4C22×D12711yescharacteristic
127Q16—2544no—
128Q16—2544no—
129M16—22no—
130M16—24no—
131C2×C8—24no—
132C2×C8C22⋊D12714yes—
133C2×C8C22×D12712yes—
134C2×C8C22×D12711yescharacteristic
135C2×C8C22×D12711yescharacteristic
136C4⋊C4—1274no—
137C4⋊C4—2548no—
138C22⋊C4—2542no—
139C22⋊C4—2548no—
140C42—2542no—
141C23C2×D50812yes—
142C23C2×D50811yescharacteristic
143Q8—22no—
144Q8—5084no—
145Q8—2544no—
146Q8—2544no—
147D4—24no—
148D4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—5084no—
152C2×C4—2544no—
153C2×C4—2544no—
154C2×C4C2×C22⋊D12712yes—
155C2×C4C2×D50812yes—
156C2×C4C2×D50811yescharacteristic
157C2×C4#1452324812yes—
158C2×C4C23×D12711yescharacteristic
159C8—22no—
160C8C2×C22⋊D12712yes—
161C8C2×C22⋊D12712yes—
162C22—22no—
163C22—22no—
164C22—24no—
165C22#1686118312yes—
166C22C22×D50811yescharacteristic
167C4—22no—
168C4—5084no—
169C4—5084no—
170C4#1686118312yes—
171C4C22×C22⋊D12711yescharacteristic
172C4C2×#1452324811yescharacteristic
173C2—22no—
174C2—42no—
175C2C2×#1686118311yescharacteristic
176C2#2255069312yes—
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#131×C127C2
socleC22×C127C22×D4
prime radicalC2×#41×C127C22
hypercenterC2×#131D127
hypocenterC127#3059
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C22×Dic508C2semidirect
3. #22550134C2semidirect
4. #22550117C2semidirect
5. #22549914C2semidirect
6. #22549886C2semidirect
89. C127#3059semidirect

Maximal subgroups

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

subgroupindex
C22×Dic5082
#225504472
#225501342
#225501172
#225499142
#225498862
C2×#131×C1272
#3059127