Finite Group Databasebeta

← #30640692

Subgroups — #30640692

Order 16256.

210 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550693C211yesmaximal, characteristic, complemented
2#22550312C211yesmaximal, characteristic, complemented
3C2×#16861153C211yesmaximal, characteristic, complemented
4#22549691C211yesmaximal, characteristic, complemented
5#22549682C211yesmaximal, characteristic, complemented
6#22549660C211yesmaximal, characteristic, complemented
7#472×C127C211yesmaximal, characteristic, complemented
8#16861256—21no—
9#16861242—22no—
10#16861242—21no—
11#16861231—21no—
12#16861231C2211yescharacteristic
13C2×Dic508—21no—
14C2×Dic508C2211yescharacteristic
15#16861153—22no—
16#16861153—21no—
17#16861150—21no—
18#16861149C2211yescharacteristic
19C2×C127⋊C16C2211yescharacteristic
20AGL(1,ℤ/8)×C127—21no—
21C2×D8×C127C2211yescharacteristic
22#131×C127C2211yescharacteristic
23#108×C127C2211yescharacteristic
24#100×C127—21no—
25#14523248—22no—
26#14523248—21no—
27#14523247—42no—
28#14523247—41no—
29C2×Dic254—22no—
30C2×Dic254—21no—
31Q8×D127—41no—
32Dic508—22no—
33Dic508—21no—
34Dic508D411yescharacteristic
35Dic508D412yes—
36#14523237—22no—
37#14523237—21no—
38D1016D411yescharacteristic
39C127⋊C16—21no—
40C127⋊C16—21no—
41#41×C127D411yescharacteristic
42C2×D4×C127—21no—
43SD16×C127—41no—
44D8×C127—21no—
45D8×C127—21no—
46D8×C127—41no—
47M16×C127—21no—
48M16×C127—21no—
49M16×C127D411yescharacteristic
50C2×C8×C127—21no—
51C2×C8×C127C2311yescharacteristic
52Dic254—22no—
53Dic254—21no—
54Dic254—42no—
55Dic254—41no—
56D508—21no—
57C22⋊D127—42no—
58C22⋊D127—41no—
59C2×Dic127—42no—
60C2×Dic127—41no—
61C4×D127—42no—
62C4×D127—41no—
63C23×C127—41no—
64Q8×C127D811yescharacteristic
65D4×C127—21no—
66D4×C127—41no—
67D4×C127—41no—
68D4×C127D811yescharacteristic
69C2×C4×C127—21no—
70C2×C4×C127C2×D411yescharacteristic
71C8×C127—21no—
72C8×C127—41no—
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—21no—
76C22×C127—81no—
77C22×C127—161no—
78C22×C127AGL(1,ℤ/8)11yescharacteristic
79C2×D127—41no—
80C4×C127—21no—
81C4×C127#12011yescharacteristic
82C4×C127C2×D811yescharacteristic
83Dic127—42no—
84Dic127—41no—
85C2×C127—21no—
86C2×C127—41no—
87C2×C127—161no—
88C2×C127#44711yescharacteristic
89D127—81no—
90#2193—1271nomaximal, Sylow
91C127#219311yescharacteristic, complemented, Sylow
92#578—1271no—
93#509—1271no—
94C2×SD32—1271no—
95#472D12711yescharacteristic, complemented
96#362—1271no—
97#359—1271no—
98#350—1271no—
992−1+4—2541no—
100#137—2542no—
101#137—2541no—
102AGL(1,ℤ/8)—21no—
103#135—1271no—
104#135—2541no—
105C2×Q16—1271no—
106C2×Q16—2541no—
107C2×D8C2×D12711yescharacteristic
108#131C2×D12711yescharacteristic
109SD32—2542no—
110SD32—2541no—
111D16—2541no—
112M32—1271no—
113C2×C16—1271no—
114#108C2×D12711yescharacteristic
115#100—21no—
116#41—5082no—
117#41—5081no—
118#41—2542no—
119#41—2541no—
120#41C22⋊D12711yescharacteristic
121C2×Q8—5081no—
122C2×Q8—2542no—
123C2×Q8—2541no—
124C2×D4—21no—
125Q16—1272no—
126Q16—1271no—
127Q16—2542no—
128Q16—2541no—
129SD16—41no—
130SD16—2542no—
131SD16—2541no—
132D8—21no—
133D8—21no—
134D8—41no—
135D8—1271no—
136M16—21no—
137M16—21no—
138M16C22⋊D12711yescharacteristic
139C2×C8—21no—
140C2×C8C22×D12711yescharacteristic
141C16—2541no—
142C16—2541no—
143C23—41no—
144Q8—5082no—
145Q8—5081no—
146Q8—2542no—
147Q8—2541no—
148Q8#1452323311yescharacteristic
149D4—21no—
150D4—41no—
151D4—41no—
152D4—5082no—
153D4—5081no—
154D4—2541no—
155D4#1452323311yescharacteristic
156C2×C4—21no—
157C2×C4—5082no—
158C2×C4—5082no—
159C2×C4—5081no—
160C2×C4—5081no—
161C2×C4C2×C22⋊D12711yescharacteristic
162C8—21no—
163C8—41no—
164C8D4×D12711yescharacteristic
165C8D4×D12711yescharacteristic
166C22—21no—
167C22—81no—
168C22—161no—
169C22—5081no—
170C22#1686123411yescharacteristic
171C4—21no—
172C4—5082no—
173C4—5081no—
174C4#1686117711yescharacteristic
175C4C2×#1452323311yescharacteristic
176C2—21no—
177C2—41no—
178C2—161no—
179C2—10161no—
180C2#2255000911yescharacteristic
subgroupG/H
centerC2#22550009
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#472×C127C2
socleC2×C127#447
hypercenter#472D127
hypocenterC127#2193
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #22550693C2semidirect
2. #22550312C2semidirect
3. C2×#16861153C2semidirect
4. #22549691C2semidirect
5. #22549682C2semidirect
6. #22549660C2semidirect
7. #472×C127C2semidirect
91. C127#2193semidirect
95. #472D127semidirect

Maximal subgroups

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

subgroupindex
#225506932
#225503122
C2×#168611532
#225496912
#225496822
#225496602
#472×C1272
#2193127