Finite Group Databasebeta

← #30640702

Subgroups — #30640702

Order 16256.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#22550678C211yesmaximal, characteristic
2#22550316C211yesmaximal, characteristic, complemented
3C2×#16861154C211yesmaximal, characteristic, complemented
4#22550152C211yesmaximal, characteristic, complemented
5#22549688C211yesmaximal, characteristic, complemented
6#22549659C211yesmaximal, characteristic, complemented
7#359×C127C211yesmaximal, characteristic, complemented
8#16861255—21no—
9#16861242—21no—
10#16861236—22no—
11#16861234—21no—
12#16861232—21no—
13C2×Dic508C2211yescharacteristic, complemented
14C2×D1016C2211yescharacteristic
15D8×D127—21no—
16#16861215C2211yescharacteristic
17#16861155—21no—
18#16861154—22no—
19#16861153—21no—
20#16861149C2211yescharacteristic, complemented
21#16861127—21no—
22#135×C127C2211yescharacteristic
23C2×C16×C127C2211yescharacteristic, complemented
24#108×C127C2211yescharacteristic, complemented
25#14523249—22no—
26#14523248—41no—
27#14523248D411yescharacteristic
28C2×Dic254—21no—
29C2×D508—21no—
30C2×D508—41no—
31D4×D127—22no—
32D4×D127—42no—
33Dic508—21no—
34Dic508—21no—
35Dic508—41no—
36#14523237—41no—
37#14523235—22no—
38D1016—21no—
39D1016D412yes—
40#14523233—22no—
41#14523232D411yescharacteristic
42#14523231—21no—
43C8×D127—21no—
44C127⋊C16—21no—
45#41×C127—21no—
46Q16×C127D411yescharacteristic
47SD16×C127—21no—
48D8×C127D411yescharacteristic
49M16×C127—21no—
50C2×C8×C127C2311yescharacteristic
51C16×C127—21no—
52C22×D127—42no—
53C22×D127—41no—
54Dic254—41no—
55Dic254—41no—
56Dic254D811yescharacteristic
57D508—22no—
58D508—42no—
59D508—41no—
60D508D811yescharacteristic
61C22⋊D127—21no—
62C22⋊D127—42no—
63C2×Dic127—41no—
64C4×D127—21no—
65C4×D127—42no—
66C127⋊C8—21no—
67Q8×C127—21no—
68D4×C127—21no—
69D4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—41no—
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—41no—
76C22×C127AGL(1,ℤ/8)11yescharacteristic
77C2×D127—21no—
78C2×D127—42no—
79C2×D127—162no—
80C2×D127—161no—
81C4×C127—41no—
82C4×C127#12011yescharacteristic
83C4×C127C2×D811yescharacteristic
84Dic127—21no—
85Dic127—81no—
86C2×C127—21no—
87C2×C127—81no—
88C2×C127#44711yescharacteristic
89D127—41no—
90D127—82no—
91#2194—1271nomaximal, Sylow
92C127#219411yescharacteristic, complemented, Sylow
93#576—1271no—
94#510—1271no—
95C2×SD32—1271no—
96#473—1271no—
97#361—1271no—
98#359D12711yescharacteristic, complemented
99#350—1271no—
100H(5,𝔽2)—2541no—
101#137—2541no—
102AGL(1,ℤ/8)—2541no—
103AGL(1,ℤ/8)—2542no—
104#135—2541no—
105#135C2×D12711yescharacteristic
106C2×Q16—1271no—
107C2×D8—1271no—
108C2×D8—2541no—
109#131—1271no—
110Q32—2541no—
111SD32—2542no—
112SD32—2541no—
113M32—1271no—
114C2×C16C2×D12711yescharacteristic, complemented
115#108C2×D12711yescharacteristic, complemented
116#101—2541no—
117#41—21no—
118#41—1271no—
119#41—5081no—
120#41—2542no—
121C2×Q8—2541no—
122C2×D4—5081no—
123C2×D4—5082no—
124C2×D4—2542no—
125C2×D4—2541no—
126Q16—5081no—
127Q16—2541no—
128Q16—2541no—
129Q16D50811yescharacteristic
130SD16—21no—
131SD16—5081no—
132SD16—2542no—
133D8—1272no—
134D8—2542no—
135D8—2541no—
136D8D50811yescharacteristic
137M16—21no—
138M16—1271no—
139M16—2541no—
140C2×C8—2541no—
141C2×C8C22×D12711yescharacteristic
142C16—21no—
143C16—2541no—
144C23—5082no—
145C23—5081no—
146Q8—21no—
147Q8—1271no—
148Q8—5081no—
149Q8—5081no—
150D4—21no—
151D4—41no—
152D4—1271no—
153D4—5082no—
154D4—5082no—
155D4—5081no—
156D4—2541no—
157D4—2542no—
158C2×C4—41no—
159C2×C4—5082no—
160C2×C4—5081no—
161C2×C4—2541no—
162C2×C4D4×D12711yescharacteristic
163C8—41no—
164C8—2541no—
165C8D4×D12711yescharacteristic
166C8C2×D50811yescharacteristic
167C22—41no—
168C22—20321no—
169C22—20322no—
170C22—5082no—
171C22—2541no—
172C22#1686123611yescharacteristic
173C4—41no—
174C4—10161no—
175C4—2541no—
176C4#1686117911yescharacteristic
177C4D8×D12711yescharacteristic
178C2—21no—
179C2—81no—
180C2—10162no—
181C2—5081no—
182C2#2255001111yescharacteristic
subgroupG/H
centerC2#22550011
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#359×C127C2
socleC2×C127#447
prime radical#22550678C2
hypercenter#359D127
hypocenterC127#2194
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #359

Complements

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

NH ≅ G/Ntype
2. #22550316C2semidirect
3. C2×#16861154C2semidirect
4. #22550152C2semidirect
5. #22549688C2semidirect
6. #22549659C2semidirect
7. #359×C127C2semidirect
13. C2×Dic508C22semidirect
20. #16861149C22semidirect
23. C2×C16×C127C22semidirect
24. #108×C127C22semidirect
92. C127#2194semidirect
98. #359D127semidirect
114. C2×C16C2×D127semidirect
115. #108C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506782
#225503162
C2×#168611542
#225501522
#225496882
#225496592
#359×C1272
#2194127