Finite Group Databasebeta

← #30640691

Subgroups — #30640691

Order 16256.

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

normal subgroups only

This group has 210 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#22550690C211yesmaximal, characteristic, complemented
2C2×#16861153C211yesmaximal, characteristic, complemented
3#22550143C211yesmaximal, characteristic, complemented
4#22549693C211yesmaximal, characteristic, complemented
5#22549683C211yesmaximal, characteristic, complemented
6#22549661C211yesmaximal, characteristic, complemented
7#509×C127C211yesmaximal, characteristic, complemented
8#16861256—21no—
9#16861239—21no—
10#16861238—22no—
11#16861235—21no—
12#16861228—21no—
13C2×Dic508C2211yescharacteristic
14Q16×D127—21no—
15#16861215C2211yescharacteristic
16#16861153—22no—
17C2×C127⋊C16C2211yescharacteristic
18#16861145C2211yescharacteristic
19#16861126—21no—
20#135×C127C2211yescharacteristic
21C2×D8×C127C2211yescharacteristic
22SD32×C127—21no—
23D16×C127—21no—
24M32×C127C2211yescharacteristic
25#14523248—41no—
26#14523248D411yescharacteristic
27#14523247—22no—
28#14523247—42no—
29C2×Dic254—21no—
30C2×Dic254—41no—
31Q8×D127—22no—
32Dic508—21no—
33Dic508D412yes—
34#14523238—22no—
35#14523236—22no—
36#14523236—41no—
37#14523233—41no—
38#14523232—21no—
39#14523232D411yescharacteristic
40#14523231—21no—
41C8×D127—21no—
42C127⋊C16—21no—
43#41×C127—21no—
44C2×D4×C127—21no—
45Q16×C127D411yescharacteristic
46SD16×C127—21no—
47D8×C127—21no—
48D8×C127—21no—
49D8×C127D411yescharacteristic
50C2×C8×C127C2311yescharacteristic
51C16×C127—21no—
52Dic254—22no—
53Dic254—42no—
54Dic254—41no—
55Dic254D811yescharacteristic
56D508D811yescharacteristic
57C22⋊D127—21no—
58C22⋊D127—42no—
59C2×Dic127—42no—
60C2×Dic127—41no—
61C4×D127—21no—
62C4×D127—42no—
63C127⋊C8—21no—
64C127⋊C8—41no—
65C23×C127—41no—
66Q8×C127—21no—
67D4×C127—21no—
68D4×C127—41no—
69D4×C127—41no—
70D4×C127—41no—
71C2×C4×C127—41no—
72C2×C4×C127C2×D411yescharacteristic
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—41no—
76C22×C127—81no—
77C22×C127—161no—
78C22×C127AGL(1,ℤ/8)11yescharacteristic
79C2×D127—21no—
80C4×C127—41no—
81C4×C127#12011yescharacteristic
82C4×C127C2×D811yescharacteristic
83Dic127—21no—
84Dic127—42no—
85C2×C127—21no—
86C2×C127—81no—
87C2×C127—161no—
88C2×C127#44711yescharacteristic
89D127—41no—
90#2193—1271nomaximal, Sylow
91C127#219311yescharacteristic, complemented, Sylow
92#578—1271no—
93#509D12711yescharacteristic, complemented
94C2×SD32—1271no—
95#472—1271no—
96#362—1271no—
97#359—1271no—
98#350—1271no—
992−1+4—2541no—
100#137—2541no—
101#137—2542no—
102AGL(1,ℤ/8)—2541no—
103#135—2541no—
104#135C2×D12711yescharacteristic
105C2×Q16—1271no—
106C2×Q16—2541no—
107C2×D8C2×D12711yescharacteristic
108#131—1271no—
109SD32—21no—
110SD32—2542no—
111D16—21no—
112M32C2×D12711yescharacteristic
113C2×C16—1271no—
114#108—1271no—
115#100—2541no—
116#41—21no—
117#41—1271no—
118#41—5081no—
119#41—5082no—
120#41—2542no—
121C2×Q8—5081no—
122C2×Q8—2542no—
123C2×Q8—2541no—
124C2×D4—21no—
125Q16—1272no—
126Q16—2542no—
127Q16—2541no—
128Q16C22⋊D12711yescharacteristic
129SD16—21no—
130SD16—5081no—
131SD16—2542no—
132D8—21no—
133D8—21no—
134D8—5081no—
135D8C22⋊D12711yescharacteristic
136M16—1271no—
137M16—2541no—
138M16—2541no—
139C2×C8—2541no—
140C2×C8C22×D12711yescharacteristic
141C16—21no—
142C16—2541no—
143C23—41no—
144Q8—21no—
145Q8—1271no—
146Q8—5082no—
147Q8—5081no—
148Q8—2542no—
149D4—21no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—1271no—
154D4—5082no—
155D4—2541no—
156C2×C4—41no—
157C2×C4—5082no—
158C2×C4—5081no—
159C2×C4—5082no—
160C2×C4—2541no—
161C2×C4D4×D12711yescharacteristic
162C8—5081no—
163C8—2541no—
164C8D4×D12711yescharacteristic
165C8C2×C22⋊D12711yescharacteristic
166C22—41no—
167C22—81no—
168C22—161no—
169C22—2541no—
170C22#1686123311yescharacteristic
171C4—41no—
172C4—5082no—
173C4—2541no—
174C4#1686117711yescharacteristic
175C4D8×D12711yescharacteristic
176C2—21no—
177C2—81no—
178C2—161no—
179C2—5081no—
180C2#2255000811yescharacteristic
subgroupG/H
centerC2#22550008
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#509×C127C2
socleC2×C127#447
hypercenter#509D127
hypocenterC127#2193
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #22550690C2semidirect
2. C2×#16861153C2semidirect
3. #22550143C2semidirect
4. #22549693C2semidirect
5. #22549683C2semidirect
6. #22549661C2semidirect
7. #509×C127C2semidirect
91. C127#2193semidirect
93. #509D127semidirect

Maximal subgroups

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

subgroupindex
#225506902
C2×#168611532
#225501432
#225496932
#225496832
#225496612
#509×C1272
#2193127