Finite Group Databasebeta

← #30651821

Subgroups — #30651821

Order 16256.

380 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 113 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#22550608C212yesmaximal, complemented
2#22550262C212yesmaximal, complemented
3#22550100C212yesmaximal, complemented
4#22550088C212yesmaximal, complemented
5#22550069C212yesmaximal, complemented
6#22550006C211yesmaximal, characteristic, complemented
7#22549977C212yesmaximal, complemented
8C4×D1016C211yesmaximal, characteristic, complemented
9#495×C127C211yesmaximal, characteristic, complemented
10C2×#14523235C2212yes—
11C2×D1016C2211yescharacteristic
12C2×#14523233C2212yes—
13#16861205C2212yes—
14#16861198C2212yes—
15#16861195—22no—
16#16861193—22no—
17#16861191—22no—
18Q8×Dic127C2212yes—
19C4×Dic254C2211yescharacteristic
20C4×D508C2212yes—
21D4×Dic127C2212yes—
22(C22⋊D127)⋊C4—22no—
23#16861143C2211yescharacteristic
24C4⋊(C127⋊C8)C2211yescharacteristic
25Dic127⋊C8C2212yes—
26#16861133C2212yes—
27#16861131C2212yes—
28#16861130C2212yes—
29#16861119C2212yes—
30C2×Q16×C127C2211yescharacteristic
31C2×SD16×C127C2212yes—
32C2×D8×C127C2211yescharacteristic
33#124×C127C2212yes—
34C4×C8×C127C2211yescharacteristic
35C2×Dic254D411yescharacteristic
36C2×D508C2312yes—
37C2×C22⋊D127—22no—
38C2×C22⋊D127—22no—
39C22×Dic127—22no—
40C2×C4×D127—22no—
41#14523235—42no—
42D1016—22no—
43#14523233—42no—
44C2×C127⋊C8C2312yes—
45C4⋊Dic127—22no—
46C4⋊Dic127D411yescharacteristic
47C4⋊Dic127C2312yes—
48Dic127⋊C4D412yes—
49C22⋊Dic127—22no—
50(C2×D127)⋊C4—22no—
51(C2×D127)⋊C4—22no—
52(C2×D127)⋊C4—22no—
53C4×Dic127—22no—
54C4×Dic127C2312yes—
55C2×Q8×C127C2312yes—
56C2×D4×C127C2312yes—
57Q16×C127—41no—
58SD16×C127—24no—
59D8×C127—41no—
60C2×C8×C127C2311yescharacteristic
61C2×C8×C127C2311yescharacteristic
62C22⋊C4×C127—22no—
63C22⋊C4×C127—22no—
64C42×C127C2311yescharacteristic
65C22×D127—22no—
66Dic254—41no—
67D508—22no—
68D508—22no—
69C22⋊D127—44no—
70C22⋊D127—82no—
71C2×Dic127—22no—
72C2×Dic127—82no—
73C2×Dic127C2×D412yes—
74C2×Dic127C2×D412yes—
75C4×D127—82no—
76C127⋊C8—22no—
77C23×C127—22no—
78Q8×C127—22no—
79Q8×C127—22no—
80D4×C127—22no—
81D4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2411yescharacteristic
86C8×C127—21no—
87C8×C127#4112yes—
88C22×C127—42no—
89C22×C127—42no—
90C22×C127—42no—
91C22×C127C22×D411yescharacteristic
92C2×D127—42no—
93C2×D127—42no—
94C2×D127—42no—
95C4×C127—21no—
96C4×C127—21no—
97C4×C127—42no—
98C4×C127C2×#4111yescharacteristic
99C4×C127H(5,𝔽2)11yescharacteristic
100Dic127—22no—
101Dic127—22no—
102Dic127—42no—
103C2×C127—82no—
104C2×C127#55011yescharacteristic
105C2×C127#57511yescharacteristic
106C2×C127#57611yescharacteristic
107D127—82no—
108#3403—1271nomaximal, Sylow
109C127#340311yescharacteristic, complemented, Sylow
110#559—1272no—
111#496—1272no—
112#495D12711yescharacteristic, complemented
113#464—1272no—
114#463—1272no—
115#459—1272no—
116#446—1271no—
117#440—1272no—
118C4×D8—1271no—
119C2×Q16C2×D12711yescharacteristic
120C2×SD16—1272no—
121C2×SD16C2×D12712yes—
122C2×D8—1272no—
123C2×D8—1271no—
124C2×D8C2×D12711yescharacteristic
125#127—1272no—
126#124—1272no—
127#124—2542no—
128#124C2×D12712yes—
129#123—2542no—
130#123—2542no—
131C4×Q8—1271no—
132C4×Q8—1272no—
133C4×D4—1272no—
134C4×D4—1272no—
135C4×D4—2542no—
136#107—1271no—
137C4⋊C8—1272no—
138C4⋊C8—1271no—
139#103—1272no—
140#102—1272no—
141#102—1272no—
142#97—1272no—
143C4×C8C2×D12711yescharacteristic
144C2×Q8—1271no—
145C2×Q8C22×D12712yes—
146C2×D4—1272no—
147C2×D4—2542no—
148C2×D4—2542no—
149C2×D4C22×D12712yes—
150C22×C4—2542no—
151C22×C4—2542no—
152Q16—41no—
153SD16—24no—
154SD16—5082no—
155D8—41no—
156D8—5082no—
157D8—2542no—
158C2×C8—1272no—
159C2×C8C22×D12711yescharacteristic
160C2×C8C22×D12711yescharacteristic
161C4⋊C4—1272no—
162C4⋊C4—1271no—
163C4⋊C4—1272no—
164C4⋊C4—2542no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—2542no—
168C22⋊C4—2542no—
169C22⋊C4—2542no—
170C22⋊C4—2542no—
171C42—1272no—
172C42—2542no—
173C42C22×D12711yescharacteristic
174C23—22no—
175C23—2542no—
176Q8—22no—
177Q8—22no—
178Q8—5081no—
179D4—22no—
180D4—22no—
181D4—10162no—
182D4—5084no—
183D4—2542no—
184D4—2542no—
185C2×C4—22no—
186C2×C4—1272no—
187C2×C4—1272no—
188C2×C4—10162no—
189C2×C4—10162no—
190C2×C4—2542no—
191C2×C4D4×D12711yescharacteristic
192C2×C4D4×D12711yescharacteristic
193C2×C4C23×D12711yescharacteristic
194C8—21no—
195C8—2542no—
196C8#1452324712yes—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—5082no—
201C22—5082no—
202C22—5082no—
203C22C2×D4×D12711yescharacteristic
204C4—21no—
205C4—21no—
206C4—42no—
207C4—5082no—
208C4—2542no—
209C4—2542no—
210C4C2×#1452324711yescharacteristic
211C4#1686125411yescharacteristic
212C2—82no—
213C2—10162no—
214C2#2255054611yescharacteristic
215C2#2255067311yescharacteristic
216C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#495×C127C2
socleC22×C127C22×D4
hypercenter#495D127
hypocenterC127#3403
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #22550608C2semidirect
2. #22550262C2semidirect
3. #22550100C2semidirect
4. #22550088C2semidirect
5. #22550069C2semidirect
6. #22550006C2semidirect
7. #22549977C2semidirect
8. C4×D1016C2semidirect
9. #495×C127C2semidirect
109. C127#3403semidirect
112. #495D127semidirect

Maximal subgroups

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

subgroupindex
#225506082
#225502622
#225501002
#225500882
#225500692
#225500062
#225499772
C4×D10162
#495×C1272
#3403127