Finite Group Databasebeta

← #30637931

Subgroups — #30637931

Order 16256.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#22550546C211yesmaximal, characteristic
2#22550222C211yesmaximal, characteristic, complemented
3#22550162C211yesmaximal, characteristic, complemented
4#22550082C211yesmaximal, characteristic, complemented
5#22549606C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#331×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9C2×#14523237—21no—
10#16861209C2211yescharacteristic
11#16861205—21no—
12#16861181—22no—
13#16861180—21no—
14C4×Dic254C2211yescharacteristic
15C4×D508C2211yescharacteristic
16D4×Dic127—21no—
17#16861141—21no—
18#16861134—21no—
19#16861134—21no—
20#16861130—22no—
21C4×C127⋊C8C2211yescharacteristic
22#127×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25#14523247—42no—
26#14523247—42no—
27C2×Dic254—21no—
28C2×Dic254D411yescharacteristic
29C2×D508D411yescharacteristic
30C2×C22⋊D127—22no—
31C2×C22⋊D127—22no—
32C22×Dic127—22no—
33C2×C4×D127—21no—
34#14523237—42no—
35C2×C127⋊C8—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127D411yescharacteristic
39C4⋊Dic127D411yescharacteristic
40Dic127⋊C4—21no—
41C22⋊Dic127—22no—
42(C2×D127)⋊C4—21no—
43C4×Dic127—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127D412yes—
47C2×C8×C127—21no—
48C2×C8×C127—21no—
49C42×C127C2311yescharacteristic
50C22×D127—21no—
51Dic254—21no—
52Dic254—41no—
53Dic254—41no—
54Dic254SD1612yescomplemented
55D508—21no—
56D508—21no—
57C22⋊D127—44no—
58C22⋊D127—82no—
59C2×Dic127—21no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C2×Dic127—44no—
63C2×Dic127—41no—
64C4×D127—41no—
65C4×D127—41no—
66C127⋊C8—41no—
67C23×C127—22no—
68D4×C127—42no—
69D4×C127—41no—
70D4×C127—41no—
71D4×C127—41no—
72D4×C127—41no—
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C8×C127—41no—
77C8×C127—41no—
78C22×C127—42no—
79C22×C127—42no—
80C22×C127—42no—
81C22×C127#12011yescharacteristic
82C2×D127—41no—
83C2×D127—41no—
84C2×D127—41no—
85C4×C127—21no—
86C4×C127C2×SD1611yescharacteristic
87C4×C127#13511yescharacteristic
88C4×C127AGL(1,ℤ/8)11yescharacteristic
89C4×C127AGL(1,ℤ/8)11yescharacteristic
90Dic127—22no—
91Dic127—41no—
92Dic127—41no—
93Dic127—81no—
94C2×C127—82no—
95C2×C127#44911yescharacteristic
96C2×C127#45011yescharacteristic
97C2×C127#45511yescharacteristic
98D127—81no—
99#1642—1271nomaximal, Sylow
100C127#164211yescharacteristic, complemented, Sylow
101#550—1271no—
102#486—1271no—
103#475—1271no—
104#461—1271no—
105#331D12711yescharacteristic, complemented
106Q8⋊C8—1271no—
107D4⋊C8—1271no—
108C2×#41—2541no—
109C2×SD16—2541no—
110C4⋊Q8—1271no—
111#127—2541no—
112#127C2×D12711yescharacteristic
113#121—2541no—
114#121—2542no—
115C4×Q8—1271no—
116C4×D4—1271no—
117C4×D4—2541no—
118#106—2541no—
119C4⋊C8C2×D12711yescharacteristic
120C4⋊C8C2×D12711yescharacteristic
121#103—2541no—
122#103—2541no—
123#102—2542no—
124C4×C8—1271no—
125#41—5082no—
126#41—5082no—
127C2×Q8—1271no—
128C2×Q8—2541no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—1271no—
132C2×D4—2542no—
133C2×D4—2542no—
134C2×D4D50812yes—
135C22×C4—2542no—
136C22×C4—2541no—
137SD16—5082no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—2541no—
141C4⋊C4—1271no—
142C4⋊C4—1271no—
143C4⋊C4—2541no—
144C4⋊C4—2541no—
145C4⋊C4—2541no—
146C22⋊C4—2542no—
147C22⋊C4—2541no—
148C42—2541no—
149C42C22×D12711yescharacteristic
150C23—22no—
151C23—2541no—
152Q8—1272no—
153Q8—5081no—
154Q8—5081no—
155Q8—2541no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—10162no—
162D4—5084no—
163D4—2541no—
164D4—2541no—
165C2×C4—5081no—
166C2×C4—5084no—
167C2×C4—5081no—
168C2×C4—5081no—
169C2×C4—2541no—
170C2×C4—2541no—
171C2×C4—2541no—
172C2×C4D4×D12711yescharacteristic
173C2×C4D4×D12711yescharacteristic
174C2×C4C2×D50811yescharacteristic
175C8—41no—
176C8—41no—
177C8—5081no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—5081no—
182C22—5081no—
183C22—5081no—
184C22#1686117911yescharacteristic
185C4—21no—
186C4—10161no—
187C4—5081no—
188C4—5081no—
189C4—2542no—
190C4SD16×D12711yescharacteristic
191C4#1686123011yescharacteristic
192C4#1686123311yescharacteristic
193C4#1686123311yescharacteristic
194C2—82no—
195C2—10161no—
196C2#2255002211yescharacteristic
197C2#2255002811yescharacteristic
198C2#2255005611yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#331×C127C2
socleC22×C127#120
prime radical#22550546C2
hypercenter#331D127
hypocenterC127#1642
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #22550222C2semidirect
3. #22550162C2semidirect
4. #22550082C2semidirect
5. #22549606C2semidirect
6. #22549603C2semidirect
7. #331×C127C2semidirect
54. Dic254SD16semidirect
100. C127#1642semidirect
105. #331D127semidirect

Maximal subgroups

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

subgroupindex
#225505462
#225502222
#225501622
#225500822
#225496062
#225496032
#331×C1272
#1642127