Finite Group Databasebeta

← #30637815

Subgroups — #30637815

Order 16256.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550591C211yesmaximal, characteristic, complemented
2#22550249C211yesmaximal, characteristic, complemented
3#22550072C211yesmaximal, characteristic, complemented
4#22550071C211yesmaximal, characteristic, complemented
5#22549624C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic, complemented
7D4⋊C8×C127C211yesmaximal, characteristic, complemented
8C2×#14523237—22no—
9C2×D1016—21no—
10C2×#14523233—21no—
11#16861209C2211yescharacteristic
12#16861206C2211yescharacteristic
13#16861195—22no—
14#16861183—21no—
15#16861180—21no—
16C4×D508C2211yescharacteristic
17#16861168—21no—
18#16861167—21no—
19C4⋊(C127⋊C8)C2211yescharacteristic
20#16861131—21no—
21#16861129—21no—
22C4×D4×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25C2×Dic254D412yes—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508D411yescharacteristic
30C2×C22⋊D127—22no—
31C2×C4×D127—21no—
32#14523237—44no—
33D1016—42no—
34#14523233—42no—
35C2×C127⋊C8—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127D411yescharacteristic
39C22⋊Dic127—22no—
40(C2×D127)⋊C4—22no—
41(C2×D127)⋊C4—21no—
42C4×Dic127—22no—
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—21no—
51C22×D127—41no—
52Dic254—42no—
53D508—21no—
54D508—21no—
55D508—42no—
56D508—41no—
57D508—41no—
58D508—41no—
59D508—41no—
60C22⋊D127—82no—
61C2×Dic127—21no—
62C2×Dic127—22no—
63C4×D127—41no—
64C4×D127—41no—
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—41no—
82C2×D127—41no—
83C2×D127—41no—
84C2×D127—81no—
85C2×D127—81no—
86C2×D127—81no—
87C4×C127—21no—
88C4×C127—41no—
89C4×C127#13511yescharacteristic
90C4×C127#13511yescharacteristic
91C4×C127AGL(1,ℤ/8)11yescharacteristic
92C4×C127AGL(1,ℤ/8)11yescharacteristic
93Dic127—41no—
94Dic127—42no—
95C2×C127—81no—
96C2×C127#44911yescharacteristic
97C2×C127#44911yescharacteristic
98C2×C127#45311yescharacteristic
99D127—81no—
100D127—161no—
101#1624—1271nomaximal, Sylow
102C127#162411yescharacteristic, complemented, Sylow
103#555—1271no—
104#492—1271no—
105#459—1271no—
106#459—1271no—
107#333—1271no—
108D4⋊C8—1271no—
109D4⋊C8D12711yescharacteristic, complemented
110C2×SD16—2542no—
111C2×D8—2541no—
112C2×D8—2541no—
113C4⋊Q8—1271no—
114#127—1271no—
115#124—2542no—
116#121—2541no—
117#121—2541no—
118C4×D4—1271no—
119C4×D4C2×D12711yescharacteristic
120#117—2541no—
121#117—2541no—
122C4⋊C8—1271no—
123C4⋊C8C2×D12711yescharacteristic
124#102—2541no—
125#102—2541no—
126C4×C8C2×D12711yescharacteristic
127C2×Q8—1272no—
128C2×D4—1271no—
129C2×D4—2542no—
130C2×D4—2541no—
131C2×D4—2541no—
132C2×D4—2541no—
133C2×D4D50811yescharacteristic
134C22×C4—21no—
135C22×C4—2541no—
136SD16—5084no—
137D8—5082no—
138D8—5082no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—2541no—
142C4⋊C4—1271no—
143C4⋊C4—2541no—
144C4⋊C4—2541no—
145C4⋊C4D50811yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—2542no—
148C22⋊C4—2542no—
149C22⋊C4—2541no—
150C42—2542no—
151C42C22×D12711yescharacteristic
152C23—21no—
153C23—5081no—
154C23—2541no—
155Q8—5082no—
156D4—21no—
157D4—21no—
158D4—10162no—
159D4—5082no—
160D4—5081no—
161D4—5081no—
162D4—5081no—
163D4—5081no—
164D4—2541no—
165D4—2541no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—5081no—
170C2×C4—5081no—
171C2×C4—2541no—
172C2×C4—2542no—
173C2×C4D4×D12711yescharacteristic
174C2×C4D4×D12711yescharacteristic
175C2×C4C2×D50811yescharacteristic
176C8—22no—
177C8—41no—
178C8—5081no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—10161no—
183C22—10161no—
184C22—10161no—
185C22—5081no—
186C22—5081no—
187C22—5081no—
188C22#1686117911yescharacteristic
189C4—21no—
190C4—41no—
191C4—5081no—
192C4—5082no—
193C4#1686123011yescharacteristic
194C4#1686123111yescharacteristic
195C4#1686123311yescharacteristic
196C4#1686123711yescharacteristic
197C2—81no—
198C2—20321no—
199C2—10161no—
200C2#2255002211yescharacteristic
201C2#2255002311yescharacteristic
202C2#2255004511yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingD4⋊C8×C127C2
socleC22×C127#120
hypercenterD4⋊C8D127
hypocenterC127#1624
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
1. #22550591C2semidirect
2. #22550249C2semidirect
3. #22550072C2semidirect
4. #22550071C2semidirect
5. #22549624C2semidirect
6. #22549605C2semidirect
7. D4⋊C8×C127C2semidirect
102. C127#1624semidirect
109. D4⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
#225505912
#225502492
#225500722
#225500712
#225496242
#225496052
D4⋊C8×C1272
#1624127