Finite Group Databasebeta

← #30637652

Subgroups — #30637652

Order 16256.

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

normal subgroups only

This group has 316 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#22550502C211yesmaximal, characteristic, complemented
2#22550027C212yesmaximal, complemented
3#22549612C212yesmaximal, complemented
4#22549600C211yesmaximal, characteristic, complemented
5#392×C127C211yesmaximal, characteristic
6C2×#14523236—22no—
7#16861188—21no—
8#16861188C2212yes—
9#16861186—22no—
10#16861186—22no—
11C2×C22⋊Dic127—21no—
12#16861130—22no—
13C22⋊(C127⋊C8)C2212yes—
14C22×D4×C127—21no—
15C22×D4×C127C2211yescharacteristic
16C2×C22⋊C4×C127—21no—
17C2×C22⋊C4×C127C2211yescharacteristic
18#95×C127C2211yescharacteristic
19C2×Dic254—21no—
20C2×Dic254D412yescomplemented
21C22×Dic127—21no—
22C22×Dic127—21no—
23#14523236—44no—
24C2×C127⋊C8—22no—
25C4⋊Dic127—21no—
26C4⋊Dic127D412yescomplemented
27Dic127⋊C4—22no—
28Dic127⋊C4—22no—
29C22⋊Dic127—22no—
30C22⋊Dic127—22no—
31C24×C127—21no—
32C24×C127D411yescharacteristic
33C2×D4×C127—22no—
34C2×D4×C127—42no—
35C2×D4×C127—42no—
36C2×D4×C127—42no—
37C2×D4×C127—42no—
38C22×C4×C127—21no—
39C22×C4×C127D411yescharacteristic
40C22×C4×C127C2311yescharacteristic
41C22⋊C4×C127—22no—
42C22⋊C4×C127—42no—
43Dic254—42no—
44Dic254—81no—
45C2×Dic127—22no—
46C2×Dic127—22no—
47C2×Dic127—81no—
48C2×Dic127—81no—
49C127⋊C8—42no—
50C23×C127—21no—
51C23×C127—21no—
52C23×C127—22no—
53C23×C127—41no—
54C23×C127—41no—
55C23×C127—41no—
56C23×C127—41no—
57C23×C127—41no—
58C23×C127—41no—
59C23×C127—41no—
60C23×C127—41no—
61C23×C127SD1612yescomplemented
62C23×C127C2×D411yescharacteristic
63D4×C127—24no—
64D4×C127—42no—
65D4×C127—48no—
66C2×C4×C127—22no—
67C2×C4×C127—21no—
68C2×C4×C127—41no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127—41no—
72C2×C4×C127C2×D412yes—
73C22×C127—21no—
74C22×C127—21no—
75C22×C127—22no—
76C22×C127—42no—
77C22×C127—42no—
78C22×C127—42no—
79C22×C127—41no—
80C22×C127—41no—
81C22×C127—42no—
82C22×C127—81no—
83C22×C127—81no—
84C22×C127—81no—
85C22×C127—81no—
86C22×C127—81no—
87C22×C127#12011yescharacteristic
88C22×C127C2×SD1611yescharacteristic
89C22×C127#13711yescharacteristic
90C4×C127—22no—
91C4×C127—42no—
92C4×C127—41no—
93Dic127—42no—
94Dic127—42no—
95C2×C127—21no—
96C2×C127—21no—
97C2×C127—42no—
98C2×C127—82no—
99C2×C127#44811yescharacteristic
100C2×C127#45311yescharacteristic
101C2×C127B(4,𝔽2)11yescharacteristic
102#1596—1271nomaximal, Sylow
103C127#159611yescharacteristic, complemented, Sylow
104#543—1271no—
105#450—1272no—
106#392D12711yescharacteristic
107#328—1272no—
108#323—1271no—
109C22×D4—21no—
110C22×D4C2×D12711yescharacteristic
111C2×SD16—2542no—
112C22⋊Q8—1272no—
113C22⋊Q8—2542no—
114C22⋊Q8—2542no—
115C22⋊Q8—2541no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—2541no—
118C2×C22⋊C4C2×D12711yescharacteristic
119#102—2542no—
120C22⋊C8—1272no—
121#95C2×D12711yescharacteristic
122C24—21no—
123C24C22⋊D12711yescharacteristic
124C2×Q8—1272no—
125C2×Q8—2541no—
126C2×D4—22no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4—42no—
131C22×C4—21no—
132C22×C4—2541no—
133C22×C4—2541no—
134C22×C4C22⋊D12711yescharacteristic
135C22×C4C22×D12711yescharacteristic
136SD16—5084no—
137C2×C8—2542no—
138C4⋊C4—1272no—
139C4⋊C4—2542no—
140C4⋊C4—2541no—
141C4⋊C4—2542no—
142C22⋊C4—22no—
143C22⋊C4—42no—
144C22⋊C4—2542no—
145C22⋊C4—2542no—
146C23—21no—
147C23—21no—
148C23—22no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23#1452323612yescomplemented
158C23C2×C22⋊D12711yescharacteristic
159Q8—10161no—
160Q8—5082no—
161D4—24no—
162D4—42no—
163D4—48no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—10161no—
171C2×C4—10161no—
172C2×C4—2542no—
173C2×C4—2542no—
174C2×C4D4×D12712yes—
175C8—5082no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—41no—
183C22—41no—
184C22—42no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22#1686117711yescharacteristic
191C22C2×#1452323611yescharacteristic
192C22#1686123911yescharacteristic
193C4—22no—
194C4—42no—
195C4—41no—
196C4—5082no—
197C4—5082no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—82no—
202C2#2255001411yescharacteristic
203C2#2255004311yescharacteristic
204C2#2255006211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radical#392×C127C2
socleC22×C127#120
hypercenter#392D127
hypocenterC127#1596
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #22550502C2semidirect
2. #22550027C2semidirect
3. #22549612C2semidirect
4. #22549600C2semidirect
20. C2×Dic254D4semidirect
26. C4⋊Dic127D4semidirect
61. C23×C127SD16semidirect
103. C127#1596semidirect
157. C23#14523236semidirect

Maximal subgroups

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

subgroupindex
#225505022
#225500272
#225496122
#225496002
#392×C1272
#1596127