Finite Group Databasebeta

← #30639894

Subgroups — #30639894

Order 16256.

466 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 69 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#22549732C212yesmaximal
3#22549643C211yesmaximal, characteristic, complemented
4#22549642C212yesmaximal, complemented
5#379×C127C211yesmaximal, characteristic
6#16861188—22no—
7#16861186—24no—
8C2×C22⋊Dic127—24no—
9C2×C22⋊Dic127C2212yes—
10C2×C22⋊Dic127C2212yes—
11C25×C127C2211yescharacteristic
12C2×C22⋊C4×C127—22no—
13C2×C22⋊C4×C127C2211yescharacteristic
14C2×C22⋊C4×C127C2211yescharacteristic
15C2×Dic254—22no—
16C22×Dic127—22no—
17C22×Dic127D414yes—
18C4⋊Dic127—22no—
19Dic127⋊C4—24no—
20C22⋊Dic127—22no—
21C22⋊Dic127—22no—
22C22⋊Dic127—22no—
23C22⋊Dic127—22no—
24C22⋊Dic127—48no—
25C24×C127—21no—
26C24×C127—22no—
27C24×C127—22no—
28C24×C127—21no—
29C24×C127—44no—
30C24×C127Q812yescomplemented
31C24×C127C2311yescharacteristic
32C22×C4×C127—21no—
33C22×C4×C127D412yes—
34C22⋊C4×C127—22no—
35C22⋊C4×C127—22no—
36C22⋊C4×C127—44no—
37Dic254—82no—
38C2×Dic127—24no—
39C2×Dic127—44no—
40C2×Dic127—44no—
41C2×Dic127—42no—
42C23×C127—24no—
43C23×C127—28no—
44C23×C127—24no—
45C23×C127—22no—
46C23×C127—44no—
47C23×C127—42no—
48C23×C127—42no—
49C23×C127—44no—
50C23×C127—48no—
51C23×C127—44no—
52C23×C127—41no—
53C23×C127—42no—
54C23×C127—41no—
55C23×C127C2×D412yes—
56C23×C127C2×D411yescharacteristic
57C23×C127C2×Q811yescharacteristic
58C23×C127#4112yes—
59C23×C127#4111yescharacteristic
60C2×C4×C127—22no—
61C2×C4×C127—42no—
62C2×C4×C127—42no—
63C2×C4×C127—41no—
64C22×C127—22no—
65C22×C127—24no—
66C22×C127—24no—
67C22×C127—28no—
68C22×C127—41no—
69C22×C127—42no—
70C22×C127—41no—
71C22×C127—48no—
72C22×C127—44no—
73C22×C127—44no—
74C22×C127—42no—
75C22×C127—42no—
76C22×C127—44no—
77C22×C127#12011yescharacteristic
78C22×C127C22⋊Q812yes—
79C22×C127C22⋊Q811yescharacteristic
80C22×C127#12412yes—
81C22×C127#12411yescharacteristic
82C4×C127—42no—
83C4×C127—81no—
84Dic127—44no—
85Dic127—82no—
86C2×C127—22no—
87C2×C127—21no—
88C2×C127—22no—
89C2×C127—21no—
90C2×C127—44no—
91C2×C127#39311yescharacteristic
92C2×C127B(4,𝔽2)12yes—
93#2029—1271nomaximal, Sylow
94C127#202911yescharacteristic, complemented, Sylow
95#543—1271no—
96#379—1272no—
97#379D12711yescharacteristic
98#342—1272no—
99#342—1271no—
100C25C2×D12711yescharacteristic
101C22⋊Q8—2544no—
102C22⋊Q8—2542no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—1272no—
105C2×C22⋊C4—1272no—
106C2×C22⋊C4—2544no—
107C2×C22⋊C4C2×D12711yescharacteristic
108C2×C22⋊C4C2×D12711yescharacteristic
109C24—22no—
110C24—21no—
111C24—22no—
112C24—21no—
113C24—44no—
114C24Dic25412yescomplemented
115C24C22×D12711yescharacteristic
116C2×Q8—2542no—
117C22×C4—21no—
118C22×C4—1274no—
119C22×C4—2542no—
120C22×C4C22⋊D12712yes—
121C4⋊C4—2544no—
122C4⋊C4—2542no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—44no—
126C22⋊C4—5088no—
127C22⋊C4—2542no—
128C22⋊C4—2542no—
129C22⋊C4—2542no—
130C22⋊C4—2542no—
131C23—24no—
132C23—28no—
133C23—24no—
134C23—22no—
135C23—44no—
136C23—42no—
137C23—42no—
138C23—44no—
139C23—48no—
140C23—44no—
141C23—42no—
142C23—41no—
143C23—41no—
144C23D4×D12712yes—
145C23C2×C22⋊D12711yescharacteristic
146C23C2×Dic25411yescharacteristic
147C23#1452324712yes—
148C23#1452324811yescharacteristic
149Q8—10162no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—5084no—
155C2×C4—5082no—
156C2×C4—5084no—
157C2×C4—2544no—
158C22—22no—
159C22—24no—
160C22—28no—
161C22—24no—
162C22—41no—
163C22—42no—
164C22—41no—
165C22—44no—
166C22—48no—
167C22—44no—
168C22—42no—
169C22—42no—
170C22—44no—
171C22#1686117711yescharacteristic
172C22#1686118612yes—
173C22#1686118811yescharacteristic
174C22#1686119512yes—
175C22#1686119611yescharacteristic
176C4—42no—
177C4—81no—
178C4—10162no—
179C4—5084no—
180C2—21no—
181C2—22no—
182C2—22no—
183C2—21no—
184C2—44no—
185C2#2254978811yescharacteristic
186C2#2255006212yes—
subgroupG/H
centerC22#16861177
commutator, cosocleC24×C127C23
FrattiniC24C22×D127
Fitting#379×C127C2
socleC22×C127#120
prime radicalC25×C127C22
hypercenter#379D127
hypocenterC127#2029
LCSG ▷ C24×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #379

Complements

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

NH ≅ G/Ntype
1. #22550502C2semidirect
3. #22549643C2semidirect
4. #22549642C2semidirect
30. C24×C127Q8semidirect
94. C127#2029semidirect
114. C24Dic254semidirect

Maximal subgroups

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

subgroupindex
#225505022
#225497322
#225496432
#225496422
#379×C1272
#2029127