Finite Group Databasebeta

← #30648381

Subgroups — #30648381

Order 16256.

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

normal subgroups only

This group has 460 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
2#22550450C212yesmaximal
3#22550131C214yesmaximal, complemented
4#22550114C214yesmaximal, complemented
5#22549914C211yesmaximal, characteristic, complemented
6#22549885C212yesmaximal, complemented
7C22×Q16×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861215—24no—
10C2×#14523232C2211yescharacteristic
11C2×#14523231C2212yes—
12C2×C8×D127C2212yes—
13#16861207—22no—
14#16861198—22no—
15#16861192—24no—
16#16861185C2218yes—
17#16861168C2212yes—
18#16861143C2212yes—
19#16861141C2212yes—
20#16861133C2218yes—
21C22×Q8×C127C2212yes—
22C2×Q16×C127—24no—
23C2×Q16×C127C2212yes—
24C2×Q16×C127C2212yes—
25C22×C8×C127C2211yescharacteristic
26#14523248—22no—
27#14523248—22no—
28C2×Dic254D411yescharacteristic
29C2×D508D411yescharacteristic
30C2×C22⋊D127D412yes—
31C2×C4×D127C2312yes—
32#14523232—22no—
33#14523231—24no—
34C2×C127⋊C8C2312yes—
35C8×D127—24no—
36C4⋊Dic127C2314yes—
37Dic127⋊C4—28no—
38C22⋊Dic127—22no—
39(C2×D127)⋊C4—28no—
40C4×Dic127—22no—
41C2×Q8×C127—22no—
42C2×Q8×C127—24no—
43C2×Q8×C127—24no—
44C2×Q8×C127C2314yes—
45C22×C4×C127—22no—
46C22×C4×C127C2311yescharacteristic
47Q16×C127—24no—
48Q16×C127—24no—
49C2×C8×C127D414yes—
50C2×C8×C127C2311yescharacteristic
51C2×C8×C127C2311yescharacteristic
52C22×D127C2×D412yes—
53Dic254—22no—
54D508—22no—
55C22⋊D127—24no—
56C2×Dic127—24no—
57C2×Dic127C2×D412yes—
58C4×D127—22no—
59C4×D127—22no—
60C127⋊C8—22no—
61C23×C127C2×D411yescharacteristic
62Q8×C127—24no—
63Q8×C127—24no—
64Q8×C127—44no—
65C2×C4×C127—24no—
66C2×C4×C127—44no—
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D412yes—
69C2×C4×C127#4112yes—
70C2×C4×C127C2411yescharacteristic
71C8×C127C2×D412yes—
72C8×C127C2×D412yes—
73C22×C127—22no—
74C22×C127#12112yes—
75C22×C127C22×D411yescharacteristic
76C2×D127—22no—
77C2×D127—24no—
78C4×C127—44no—
79C4×C127#12112yes—
80C4×C127C22×D411yescharacteristic
81C4×C127C2×#4111yescharacteristic
82Dic127—22no—
83Dic127—44no—
84C2×C127—22no—
85C2×C127C2×#12111yescharacteristic
86C2×C127#57812yes—
87D127—42no—
88#3059—1271nomaximal, Sylow
89C127#305911yescharacteristic, complemented, Sylow
90C22×Q16D12711yescharacteristic, complemented
91C2×#131—1271no—
92#536—1272no—
93#470—1274no—
94#467—1274no—
95#427—1271no—
96#419—1272no—
97C2×#41—1271no—
98C22×Q8C2×D12712yes—
99C2×Q16—24no—
100C2×Q16C2×D12712yes—
101C2×Q16C2×D12712yes—
102#131—2544no—
103C2×M16—1271no—
104C2×M16—1272no—
105C22×C8—1272no—
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—2542no—
116#41—2542no—
117C2×Q8—22no—
118C2×Q8—24no—
119C2×Q8—24no—
120C2×Q8—1271no—
121C2×Q8C22×D12714yes—
122C2×D4—1272no—
123C2×D4—1271no—
124C22×C4—22no—
125C22×C4—1272no—
126C22×C4C22×D12711yescharacteristic
127Q16—24no—
128Q16—24no—
129M16—2542no—
130M16—2544no—
131C2×C8—1272no—
132C2×C8—2544no—
133C2×C8C22⋊D12714yes—
134C2×C8C22×D12711yescharacteristic
135C2×C8C22×D12711yescharacteristic
136C4⋊C4—1274no—
137C4⋊C4—2548no—
138C22⋊C4—2548no—
139C22⋊C4—2542no—
140C42—2542no—
141C23—1272no—
142C23D4×D12711yescharacteristic
143Q8—24no—
144Q8—24no—
145Q8—44no—
146Q8—2542no—
147D4—2544no—
148D4—2542no—
149C2×C4—24no—
150C2×C4—44no—
151C2×C4—1272no—
152C2×C4—2544no—
153C2×C4—2542no—
154C2×C4—2542no—
155C2×C4D4×D12711yescharacteristic
156C2×C4C2×C22⋊D12712yes—
157C2×C4#1452324712yes—
158C2×C4C23×D12711yescharacteristic
159C8—2542no—
160C8C2×C22⋊D12712yes—
161C8C2×C22⋊D12712yes—
162C22—22no—
163C22—2542no—
164C22—2544no—
165C22#1686118012yes—
166C22C2×D4×D12711yescharacteristic
167C4—44no—
168C4—5084no—
169C4—2542no—
170C4#1686118012yes—
171C4C22×C22⋊D12711yescharacteristic
172C4C2×#1452324711yescharacteristic
173C2—22no—
174C2—5082no—
175C2C2×#1686118011yescharacteristic
176C2#2255068912yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×Q16×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523248C22
hypercenterC22×Q16D127
hypocenterC127#3059
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
3. #22550131C2semidirect
4. #22550114C2semidirect
5. #22549914C2semidirect
6. #22549885C2semidirect
7. C22×Q16×C127C2semidirect
89. C127#3059semidirect
90. C22×Q16D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612152
#225504502
#225501312
#225501142
#225499142
#225498852
C22×Q16×C1272
#3059127