Finite Group Databasebeta

← #30637860

Subgroups — #30637860

Order 16256.

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

normal subgroups only

This group has 256 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#22550547C211yesmaximal, characteristic
2#22550249C211yesmaximal, characteristic, complemented
3#22550172C211yesmaximal, characteristic, complemented
4#22550166C211yesmaximal, characteristic, complemented
5#22549619C211yesmaximal, characteristic, complemented
6#22549608C211yesmaximal, characteristic, complemented
7D4⋊C8×C127C211yesmaximal, characteristic, complemented
8C2×#14523248—21no—
9C2×#14523237—22no—
10#16861209C2211yescharacteristic
11#16861206C2211yescharacteristic
12#16861205—21no—
13#16861183—21no—
14#16861182—22no—
15C4×Dic254C2211yescharacteristic
16#16861171—21no—
17#16861143—21no—
18#16861140—21no—
19C4⋊(C127⋊C8)C2211yescharacteristic
20#16861134—21no—
21#16861130—21no—
22C4×D4×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25#14523248—42no—
26#14523248—42no—
27C2×Dic254—21no—
28C2×Dic254D411yescharacteristic
29C2×D508—21no—
30C2×D508—21no—
31C2×D508D412yes—
32C2×C22⋊D127—22no—
33C2×C22⋊D127—22no—
34C2×C4×D127—22no—
35#14523237—44no—
36C2×C127⋊C8—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127D411yescharacteristic
40Dic127⋊C4—21no—
41(C2×D127)⋊C4—22no—
42C4×Dic127—21no—
43C2×D4×C127D411yescharacteristic
44C22×C4×C127—21no—
45C2×C8×C127—21no—
46C2×C8×C127—21no—
47C4⋊C4×C127D411yescharacteristic
48C22⋊C4×C127—21no—
49C42×C127C2311yescharacteristic
50C22×D127—22no—
51Dic254—21no—
52Dic254—42no—
53Dic254SD1612yescomplemented
54D508—42no—
55D508—41no—
56D508—41no—
57D508—41no—
58D508—41no—
59C22⋊D127—44no—
60C22⋊D127—82no—
61C2×Dic127—21no—
62C2×Dic127—21no—
63C2×Dic127—41no—
64C4×D127—44no—
65C127⋊C8—41no—
66C23×C127—21no—
67D4×C127—21no—
68D4×C127—21no—
69C2×C4×C127—21no—
70C2×C4×C127—41no—
71C2×C4×C127—41no—
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C8×C127—22no—
76C8×C127—41no—
77C22×C127—41no—
78C22×C127—41no—
79C22×C127—41no—
80C22×C127#12011yescharacteristic
81C2×D127—42no—
82C2×D127—42no—
83C2×D127—42no—
84C4×C127—21no—
85C4×C127—41no—
86C4×C127C2×SD1611yescharacteristic
87C4×C127#13511yescharacteristic
88C4×C127AGL(1,ℤ/8)11yescharacteristic
89C4×C127#13711yescharacteristic
90Dic127—22no—
91Dic127—41no—
92Dic127—81no—
93C2×C127—81no—
94C2×C127#45011yescharacteristic
95C2×C127#45211yescharacteristic
96C2×C127#45311yescharacteristic
97D127—82no—
98#1631—1271nomaximal, Sylow
99C127#163111yescharacteristic, complemented, Sylow
100#550—1271no—
101#492—1271no—
102#477—1271no—
103#476—1271no—
104#331—1271no—
105Q8⋊C8—1271no—
106D4⋊C8D12711yescharacteristic, complemented
107C2×#41—2541no—
108C2×SD16—2542no—
109C4⋊Q8—1271no—
110#127—1271no—
111#127—2541no—
112#121—2542no—
113#121—2541no—
114C4×Q8—1271no—
115C4×D4—2541no—
116C4×D4C2×D12711yescharacteristic
117#107—2541no—
118#106—2541no—
119C4⋊C8—1271no—
120C4⋊C8C2×D12711yescharacteristic
121#103—2541no—
122#102—2541no—
123C4×C8C2×D12711yescharacteristic
124#41—5082no—
125#41—5082no—
126C2×Q8—1271no—
127C2×Q8—2541no—
128C2×D4—1272no—
129C2×D4—2542no—
130C2×D4—2542no—
131C2×D4—2541no—
132C2×D4—2541no—
133C2×D4D50811yescharacteristic
134C22×C4—21no—
135C22×C4—2542no—
136SD16—5084no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—2541no—
140C4⋊C4—1271no—
141C4⋊C4—2541no—
142C4⋊C4—2541no—
143C4⋊C4—2541no—
144C4⋊C4D50811yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—2542no—
147C42—2541no—
148C42C22×D12711yescharacteristic
149C23—21no—
150C23—2542no—
151Q8—1272no—
152Q8—5082no—
153Q8—2541no—
154D4—21no—
155D4—21no—
156D4—10162no—
157D4—5082no—
158D4—5084no—
159D4—5081no—
160D4—5081no—
161D4—5081no—
162D4—5081no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—5084no—
167C2×C4—5081no—
168C2×C4—2541no—
169C2×C4—2541no—
170C2×C4D4×D12711yescharacteristic
171C2×C4D4×D12711yescharacteristic
172C2×C4C2×D50811yescharacteristic
173C8—22no—
174C8—41no—
175C8—5081no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—5082no—
180C22—5082no—
181C22—5082no—
182C22#1686117911yescharacteristic
183C4—21no—
184C4—41no—
185C4—10161no—
186C4—5081no—
187C4—2542no—
188C4SD16×D12711yescharacteristic
189C4#1686123111yescharacteristic
190C4#1686123311yescharacteristic
191C4#1686124211yescharacteristic
192C2—81no—
193C2—10162no—
194C2#2255002811yescharacteristic
195C2#2255004011yescharacteristic
196C2#2255004511yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingD4⋊C8×C127C2
socleC22×C127#120
prime radical#22550547C2
hypercenterD4⋊C8D127
hypocenterC127#1631
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #22550249C2semidirect
3. #22550172C2semidirect
4. #22550166C2semidirect
5. #22549619C2semidirect
6. #22549608C2semidirect
7. D4⋊C8×C127C2semidirect
53. Dic254SD16semidirect
99. C127#1631semidirect
106. D4⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
#225505472
#225502492
#225501722
#225501662
#225496192
#225496082
D4⋊C8×C1272
#1631127