Finite Group Databasebeta

← #30638015

Subgroups — #30638015

Order 16256.

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

normal subgroups only

This group has 240 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#22550529C211yesmaximal, characteristic
2#22550272C211yesmaximal, characteristic, complemented
3#22550157C211yesmaximal, characteristic, complemented
4#22550082C211yesmaximal, characteristic, complemented
5#22549603C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#332×C127C211yesmaximal, characteristic, complemented
8C2×#14523249—21no—
9C2×#14523237—21no—
10#16861209C2211yescharacteristic
11#16861194—22no—
12#16861185—21no—
13#16861184—21no—
14C4×D508C2211yescharacteristic
15C4×D508C2211yescharacteristic
16C4⋊C4×D127—21no—
17#16861143—21no—
18#16861140—22no—
19#16861134—21no—
20#16861131—21no—
21C4×C127⋊C8C2211yescharacteristic
22C4⋊Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25#14523249—42no—
26#14523249—42no—
27C2×Dic254—21no—
28C2×D508—22no—
29C2×D508D411yescharacteristic
30C2×D508D411yescharacteristic
31C2×C4×D127—22no—
32C2×C4×D127—21no—
33C2×C4×D127—21no—
34#14523237—42no—
35C2×C127⋊C8—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127D411yescharacteristic
39C4⋊Dic127D411yescharacteristic
40Dic127⋊C4—22no—
41(C2×D127)⋊C4—22no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44C2×Q8×C127—21no—
45C2×C8×C127—21no—
46C2×C8×C127—21no—
47C4⋊C4×C127—21no—
48C4⋊C4×C127D412yes—
49C42×C127C2311yescharacteristic
50C22×D127—21no—
51C22×D127—21no—
52Dic254—41no—
53Dic254—41no—
54D508—21no—
55D508—21no—
56D508—21no—
57D508—44no—
58D508SD1612yescomplemented
59C2×Dic127—21no—
60C2×Dic127—21no—
61C2×Dic127—41no—
62C4×D127—44no—
63C4×D127—41no—
64C4×D127—41no—
65C4×D127—41no—
66C4×D127—41no—
67C127⋊C8—22no—
68Q8×C127—41no—
69Q8×C127—41no—
70C2×C4×C127—22no—
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C8×C127—41no—
75C8×C127—41no—
76C22×C127#12011yescharacteristic
77C2×D127—22no—
78C2×D127—41no—
79C2×D127—41no—
80C2×D127—41no—
81C2×D127—41no—
82C2×D127—41no—
83C4×C127—21no—
84C4×C127—42no—
85C4×C127C2×SD1611yescharacteristic
86C4×C127#13511yescharacteristic
87C4×C127AGL(1,ℤ/8)11yescharacteristic
88C4×C127AGL(1,ℤ/8)11yescharacteristic
89Dic127—41no—
90Dic127—41no—
91Dic127—81no—
92C2×C127#44911yescharacteristic
93C2×C127#45011yescharacteristic
94C2×C127#45611yescharacteristic
95D127—42no—
96D127—81no—
97#1654—1271nomaximal, Sylow
98C127#165411yescharacteristic, complemented, Sylow
99#547—1271no—
100#498—1271no—
101#474—1271no—
102#461—1271no—
103#332D12711yescharacteristic, complemented
104D4⋊C8—1271no—
105D4⋊C8—1271no—
106C2×#41—2541no—
107C2×SD16—2541no—
108C4⋊Q8—1271no—
109C4⋊Q8C2×D12711yescharacteristic
110#123—2542no—
111C22⋊Q8—2541no—
112#121—2541no—
113C4×D4—1271no—
114C4×D4—1271no—
115C2×C4⋊C4—2541no—
116#107—2541no—
117#106—2542no—
118C4⋊C8C2×D12711yescharacteristic
119C4⋊C8C2×D12711yescharacteristic
120#103—2541no—
121#102—2541no—
122C4×C8—1271no—
123#41—5082no—
124#41—5082no—
125C2×Q8—21no—
126C2×Q8—2541no—
127C2×D4—1271no—
128C2×D4—1271no—
129C2×D4—2542no—
130C22×C4—2542no—
131C22×C4—2541no—
132C22×C4—2541no—
133SD16—5082no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—2541no—
137C4⋊C4—21no—
138C4⋊C4—1271no—
139C4⋊C4—1271no—
140C4⋊C4—2542no—
141C4⋊C4—2541no—
142C4⋊C4—2541no—
143C4⋊C4D50812yes—
144C22⋊C4—2542no—
145C22⋊C4—2541no—
146C22⋊C4—2541no—
147C42C22×D12711yescharacteristic
148C23—2541no—
149C23—2541no—
150Q8—41no—
151Q8—41no—
152Q8—5081no—
153Q8—5081no—
154D4—1272no—
155D4—5084no—
156D4—2541no—
157D4—2541no—
158D4—2541no—
159C2×C4—22no—
160C2×C4—5084no—
161C2×C4—5081no—
162C2×C4—5081no—
163C2×C4—5081no—
164C2×C4—5081no—
165C2×C4—5081no—
166C2×C4—2541no—
167C2×C4—2541no—
168C2×C4D4×D12711yescharacteristic
169C2×C4D4×D12711yescharacteristic
170C2×C4C2×D50811yescharacteristic
171C8—41no—
172C8—41no—
173C8—2542no—
174C22—5081no—
175C22—5081no—
176C22—5081no—
177C22—5081no—
178C22—5081no—
179C22—2542no—
180C22#1686117911yescharacteristic
181C4—21no—
182C4—42no—
183C4—10161no—
184C4—5081no—
185C4—5081no—
186C4SD16×D12711yescharacteristic
187C4#1686123211yescharacteristic
188C4#1686123311yescharacteristic
189C4#1686123611yescharacteristic
190C2—10161no—
191C2—5082no—
192C2#2255002411yescharacteristic
193C2#2255002811yescharacteristic
194C2#2255006111yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#332×C127C2
socleC22×C127#120
prime radical#22550529C2
hypercenter#332D127
hypocenterC127#1654
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #22550272C2semidirect
3. #22550157C2semidirect
4. #22550082C2semidirect
5. #22549603C2semidirect
6. #22549603C2semidirect
7. #332×C127C2semidirect
58. D508SD16semidirect
98. C127#1654semidirect
103. #332D127semidirect

Maximal subgroups

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

subgroupindex
#225505292
#225502722
#225501572
#225500822
#225496032
#332×C1272
#1654127