Finite Group Databasebeta

← #30638014

Subgroups — #30638014

Order 16256.

240 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 61 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#22550272C211yesmaximal, characteristic, complemented
2#22550156C211yesmaximal, characteristic, complemented
3#22550081C211yesmaximal, characteristic, complemented
4#22549622C211yesmaximal, characteristic, complemented
5D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
7#547×C127C211yesmaximal, characteristic
8C2×#14523236—21no—
9#16861209C2211yescharacteristic
10#16861142—21no—
11#16861140—22no—
12C4⋊(C127⋊C8)C2211yescharacteristic
13C4⋊(C127⋊C8)C2211yescharacteristic
14#16861132—21no—
15#16861130—21no—
16C4×C127⋊C8C2211yescharacteristic
17C2×#41×C127—21no—
18C4⋊Q8×C127C2211yescharacteristic
19#123×C127—22no—
20C22⋊Q8×C127—21no—
21#121×C127—21no—
22C4×D4×C127C2211yescharacteristic
23C4×D4×C127C2211yescharacteristic
24C2×C4⋊C4×C127—21no—
25C2×Dic254—21no—
26#14523236—42no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32#41×C127—42no—
33#41×C127—42no—
34C2×Q8×C127—21no—
35C2×D4×C127—22no—
36C2×D4×C127D411yescharacteristic
37C2×D4×C127D411yescharacteristic
38C22×C4×C127—22no—
39C22×C4×C127—21no—
40C22×C4×C127—21no—
41C4⋊C4×C127—22no—
42C4⋊C4×C127—21no—
43C4⋊C4×C127D411yescharacteristic
44C4⋊C4×C127D411yescharacteristic
45C4⋊C4×C127D412yes—
46C22⋊C4×C127—22no—
47C22⋊C4×C127—21no—
48C22⋊C4×C127—21no—
49C42×C127C2311yescharacteristic
50Dic254—41no—
51Dic254—41no—
52C2×Dic127—41no—
53C127⋊C8—22no—
54C127⋊C8—41no—
55C127⋊C8—41no—
56C23×C127—21no—
57C23×C127—21no—
58Q8×C127—41no—
59Q8×C127—41no—
60D4×C127—21no—
61D4×C127—21no—
62D4×C127—21no—
63D4×C127—44no—
64D4×C127SD1612yescomplemented
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—22no—
68C2×C4×C127—44no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127—41no—
72C2×C4×C127—41no—
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C22×C127—22no—
77C22×C127—41no—
78C22×C127—41no—
79C22×C127—41no—
80C22×C127—41no—
81C22×C127—41no—
82C22×C127#12011yescharacteristic
83C4×C127—21no—
84C4×C127—41no—
85C4×C127—41no—
86C4×C127—42no—
87C4×C127C2×SD1611yescharacteristic
88C4×C127#13511yescharacteristic
89C4×C127AGL(1,ℤ/8)11yescharacteristic
90C4×C127AGL(1,ℤ/8)11yescharacteristic
91Dic127—81no—
92C2×C127—42no—
93C2×C127—81no—
94C2×C127#44911yescharacteristic
95C2×C127#45011yescharacteristic
96C2×C127#45611yescharacteristic
97#1654—1271nomaximal, Sylow
98C127#165411yescharacteristic, complemented, Sylow
99#547D12711yescharacteristic
100#498—1271no—
101#474—1271no—
102#461—1271no—
103#332—1271no—
104D4⋊C8—1271no—
105D4⋊C8—1271no—
106C2×#41—21no—
107C2×SD16—2541no—
108C4⋊Q8—1271no—
109C4⋊Q8C2×D12711yescharacteristic
110#123—22no—
111C22⋊Q8—21no—
112#121—21no—
113C4×D4C2×D12711yescharacteristic
114C4×D4C2×D12711yescharacteristic
115C2×C4⋊C4—21no—
116#107—2541no—
117#106—2542no—
118C4⋊C8—1271no—
119C4⋊C8—1271no—
120#103—2541no—
121#102—2541no—
122C4×C8—1271no—
123#41—42no—
124#41—42no—
125C2×Q8—21no—
126C2×Q8—2541no—
127C2×D4—22no—
128C2×D4C22⋊D12711yescharacteristic
129C2×D4C22⋊D12711yescharacteristic
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133SD16—5082no—
134C2×C8—2541no—
135C2×C8—2541no—
136C2×C8—2541no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—2541no—
140C4⋊C4—2541no—
141C4⋊C4C22⋊D12711yescharacteristic
142C4⋊C4C22⋊D12712yes—
143C4⋊C4C22⋊D12711yescharacteristic
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C42C22×D12711yescharacteristic
148C23—21no—
149C23—21no—
150Q8—41no—
151Q8—41no—
152Q8—5081no—
153Q8—5081no—
154D4—21no—
155D4—21no—
156D4—21no—
157D4—44no—
158D4#1452323612yescomplemented
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—44no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—5081no—
168C2×C4C2×C22⋊D12711yescharacteristic
169C2×C4C2×C22⋊D12711yescharacteristic
170C2×C4C2×C22⋊D12711yescharacteristic
171C8—5081no—
172C8—5081no—
173C8—2542no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22#1686117811yescharacteristic
181C4—21no—
182C4—41no—
183C4—41no—
184C4—42no—
185C4—10161no—
186C4C2×#1452323611yescharacteristic
187C4#1686122911yescharacteristic
188C4#1686123411yescharacteristic
189C4#1686123511yescharacteristic
190C2—42no—
191C2—81no—
192C2#2255002111yescharacteristic
193C2#2255002711yescharacteristic
194C2#2255005911yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting, prime radical#547×C127C2
socleC22×C127#120
hypercenter#547D127
hypocenterC127#1654
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #22550272C2semidirect
2. #22550156C2semidirect
3. #22550081C2semidirect
4. #22549622C2semidirect
5. D4⋊(C127⋊C8)C2semidirect
6. D4⋊(C127⋊C8)C2semidirect
64. D4×C127SD16semidirect
98. C127#1654semidirect
158. D4#14523236semidirect

Maximal subgroups

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

subgroupindex
#225502722
#225501562
#225500812
#225496222
D4⋊(C127⋊C8)2
#547×C1272
#1654127