Finite Group Databasebeta

← #30640286

Subgroups — #30640286

Order 16256.

210 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 63 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#22549924C211yesmaximal, characteristic, complemented
2#22549902C211yesmaximal, characteristic, complemented
3#22549894C211yesmaximal, characteristic
4#22549892C211yesmaximal, characteristic
5#22549842C211yesmaximal, characteristic
6#22549632C211yesmaximal, characteristic, complemented
7#424×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861168C2211yescharacteristic
12#16861144—22no—
13Dic127⋊C8—22no—
14#16861137—22no—
15#16861135—22no—
16(C2×D127)⋊C8—22no—
17C2×M16×C127C2211yescharacteristic
18C22×C8×C127C2211yescharacteristic
19#117×C127C2211yescharacteristic
20C4⋊C8×C127—22no—
21#14523248—21no—
22#14523248—21no—
23#14523248—21no—
24#14523248—21no—
25C2×Dic254Q811yescharacteristic
26C2×D508Q811yescharacteristic
27C2×C22⋊D127—21no—
28C2×C4×D127—21no—
29#14523232—21no—
30#14523232—21no—
31#14523232—21no—
32#14523232—21no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8D412yes—
35C4⋊Dic127—21no—
36C22⋊Dic127—21no—
37C4×Dic127D412yes—
38C22×C4×C127C2311yescharacteristic
39M16×C127—21no—
40M16×C127—21no—
41C2×C8×C127—21no—
42C2×C8×C127—21no—
43C2×C8×C127—21no—
44C2×C8×C127—21no—
45C4⋊C4×C127—21no—
46C22⋊C4×C127—21no—
47C42×C127D412yes—
48C22×D127—21no—
49Dic254—21no—
50Dic254—21no—
51D508—21no—
52D508—21no—
53C22⋊D127—41no—
54C22⋊D127—41no—
55C2×Dic127—21no—
56C2×Dic127—22no—
57C4×D127—41no—
58C4×D127—41no—
59C127⋊C8—22no—
60C127⋊C8—41no—
61C23×C127#4111yescharacteristic
62C2×C4×C127—22no—
63C2×C4×C127C2×D411yescharacteristic
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127C2×Q811yescharacteristic
67C2×C4×C127#4111yescharacteristic
68C2×C4×C127#4111yescharacteristic
69C8×C127—41no—
70C8×C127—41no—
71C22×C127—41no—
72C22×C127C22⋊Q811yescharacteristic
73C22×C127#12311yescharacteristic
74C22×C127#12311yescharacteristic
75C2×D127—42no—
76C2×D127—41no—
77C4×C127—42no—
78C4×C127C22⋊Q811yescharacteristic
79C4×C127C22⋊Q811yescharacteristic
80C4×C127#12311yescharacteristic
81C4×C127#12711yescharacteristic
82Dic127—42no—
83Dic127—41no—
84C2×C127—21no—
85C2×C127—21no—
86C2×C127—21no—
87C2×C127#39911yescharacteristic
88D127—81no—
89#2098—1271nomaximal, Sylow
90C127#209811yescharacteristic, complemented, Sylow
91#430—1271no—
92#424—1271no—
93#424D12711yescharacteristic, complemented
94#421—1271no—
95#421—1271no—
96#408—1271no—
97H(3,ℤ/4)—1271no—
98C2×#41—1271no—
99C2×M16—1271no—
100C2×M16—1271no—
101C2×M16C2×D12711yescharacteristic
102C22×C8C2×D12711yescharacteristic
103#117—1271no—
104#117C2×D12711yescharacteristic
105#108—2542no—
106C4⋊C8—22no—
107C4⋊C8—2542no—
108#104—2542no—
109#104—2542no—
110C22⋊C8—2542no—
111#41—2541no—
112#41—2541no—
113#41—2541no—
114#41—2541no—
115C2×Q8—1271no—
116C2×D4—1271no—
117C2×D4—2541no—
118C22×C4—2541no—
119C22×C4C22×D12711yescharacteristic
120M16—21no—
121M16—21no—
122M16—2541no—
123M16—2541no—
124M16—2541no—
125M16—2541no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—1272no—
131C2×C8—2541no—
132C4⋊C4—21no—
133C4⋊C4—2541no—
134C22⋊C4—21no—
135C22⋊C4—2541no—
136C42—1272no—
137C42C22⋊D12712yes—
138C23—2541no—
139C23#1452324811yescharacteristic
140Q8—2541no—
141Q8—2541no—
142D4—5081no—
143D4—5081no—
144D4—2541no—
145D4—2541no—
146C2×C4—22no—
147C2×C4—5081no—
148C2×C4—5081no—
149C2×C4—2541no—
150C2×C4—2542no—
151C2×C4D4×D12711yescharacteristic
152C2×C4D4×D12711yescharacteristic
153C2×C4C2×C22⋊D12711yescharacteristic
154C2×C4Q8×D12711yescharacteristic
155C2×C4#1452324811yescharacteristic
156C2×C4#1452324911yescharacteristic
157C8—41no—
158C8—41no—
159C8—5081no—
160C8—2542no—
161C22—41no—
162C22—5082no—
163C22—5081no—
164C22#1686118711yescharacteristic
165C22#1686119211yescharacteristic
166C22#1686119411yescharacteristic
167C4—42no—
168C4—5082no—
169C4—5081no—
170C4#1686118511yescharacteristic
171C4#1686118711yescharacteristic
172C4#1686119411yescharacteristic
173C4#1686120511yescharacteristic
174C2—21no—
175C2—21no—
176C2—21no—
177C2—10161no—
178C2#2254981111yescharacteristic
subgroupG/H
centerC4#16861205
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#424×C127C2
socleC2×C127#399
prime radicalC2×#14523248C22
hypercenter#424D127
hypocenterC127#2098
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #22549924C2semidirect
2. #22549902C2semidirect
6. #22549632C2semidirect
7. #424×C127C2semidirect
90. C127#2098semidirect
93. #424D127semidirect

Maximal subgroups

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

subgroupindex
#225499242
#225499022
#225498942
#225498922
#225498422
#225496322
#424×C1272
#2098127