Finite Group Databasebeta

← #30638803

Subgroups — #30638803

Order 16256.

504 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C2×C22⋊C4×D127C211yesmaximal, characteristic
2#22549731C212yesmaximal
3#22549643C211yesmaximal, characteristic, complemented
4#22549642C212yesmaximal, complemented
5#342×C127C211yesmaximal, characteristic, complemented
6C24×D127C2211yescharacteristic
7C22×C4×D127C412yescomplemented
8C2×C22⋊Dic127C2212yes—
9C2×C22⋊Dic127C2211yescharacteristic
10C2×(C2×D127)⋊C4—24no—
11C2×(C2×D127)⋊C4C412yescomplemented
12C22⋊C4×D127—24no—
13C2×C22⋊C4×C127C2212yes—
14C2×C22⋊C4×C127C2211yescharacteristic
15C23×D127—21no—
16C23×D127—22no—
17C23×D127—21no—
18C23×D127—44no—
19C23×D127C2×C411yescharacteristic
20C23×D127C2×C411yescharacteristic
21C23×D127D414yes—
22C22×Dic127—22no—
23C22×Dic127C2×C412yes—
24C2×C4×D127—22no—
25C2×C4×D127—22no—
26C2×C4×D127—44no—
27C22⋊Dic127—22no—
28C22⋊Dic127—22no—
29C22⋊Dic127—22no—
30(C2×D127)⋊C4—24no—
31(C2×D127)⋊C4—48no—
32C24×C127C2311yescharacteristic
33C22×C4×C127—22no—
34C22×C4×C127C2×C412yes—
35C22⋊C4×C127—22no—
36C22⋊C4×C127—22no—
37C22⋊C4×C127—22no—
38C22×D127—21no—
39C22×D127—21no—
40C22×D127—24no—
41C22×D127—21no—
42C22×D127—21no—
43C22×D127—24no—
44C22×D127—24no—
45C22×D127—28no—
46C22×D127—42no—
47C22×D127—44no—
48C22×D127—44no—
49C22×D127—48no—
50C22×D127—44no—
51C22×D127C22⋊C412yes—
52C22×D127C22⋊C412yes—
53C2×Dic127—22no—
54C2×Dic127—42no—
55C2×Dic127—44no—
56C2×Dic127—42no—
57C4×D127—44no—
58C4×D127—44no—
59C23×C127—42no—
60C23×C127C22×C411yescharacteristic
61C23×C127C2×D412yes—
62C23×C127#4111yescharacteristic
63C23×C127#4112yes—
64C23×C127#4111yescharacteristic
65C2×C4×C127—22no—
66C2×C4×C127—42no—
67C2×C4×C127—44no—
68C2×C4×C127—42no—
69C22×C127—22no—
70C22×C127—24no—
71C22×C127—41no—
72C22×C127—42no—
73C22×C127—41no—
74C22×C127C2×C22⋊C411yescharacteristic
75C22×C127#11711yescharacteristic
76C22×C127#11711yescharacteristic
77C22×C127#12312yes—
78C22×C127#12312yes—
79C2×D127—24no—
80C2×D127—24no—
81C2×D127—22no—
82C2×D127—22no—
83C2×D127—22no—
84C2×D127—24no—
85C2×D127—48no—
86C2×D127—48no—
87C2×D127—42no—
88C2×D127—42no—
89C2×D127#9914yes—
90C4×C127—42no—
91C4×C127—82no—
92Dic127—42no—
93Dic127—82no—
94C2×C127—22no—
95C2×C127—22no—
96C2×C127—21no—
97C2×C127—21no—
98C2×C127#38111yescharacteristic
99C2×C127C2×#9912yes—
100D127—24no—
101D127—42no—
102#1819—1271nomaximal, Sylow
103C127#181911yescharacteristic, complemented, Sylow
104C22×C22⋊C4—1271no—
105#379—1272no—
106#342—1272no—
107#342—1271no—
108#342D12711yescharacteristic, complemented
109C25—1271no—
110C23×C4—1272no—
111C2×C22⋊C4—1271no—
112C2×C22⋊C4—1272no—
113C2×C22⋊C4—1272no—
114C2×C22⋊C4—2544no—
115C2×C22⋊C4—2544no—
116C2×C22⋊C4C2×D12712yes—
117C2×C22⋊C4C2×D12711yescharacteristic
118C24—1271no—
119C24—1274no—
120C24—1271no—
121C24—5084no—
122C24—2541no—
123C24—2542no—
124C24—2541no—
125C24C22×D12711yescharacteristic
126C22×C4—22no—
127C22×C4—1272no—
128C22×C4—5084no—
129C22×C4—2542no—
130C22×C4—2542no—
131C22×C4—2542no—
132C22×C4C4×D12712yes—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—5088no—
137C22⋊C4—2544no—
138C22⋊C4—2542no—
139C22⋊C4—2542no—
140C22⋊C4—2542no—
141C23—42no—
142C23—1272no—
143C23—1272no—
144C23—5082no—
145C23—5084no—
146C23—5084no—
147C23—5088no—
148C23—5084no—
149C23—2548no—
150C23—2544no—
151C23—2544no—
152C23—2544no—
153C23—2541no—
154C23—2541no—
155C23—2541no—
156C23—2541no—
157C23C2×C4×D12711yescharacteristic
158C23D4×D12712yes—
159C23#1452324711yescharacteristic
160C23#1452324812yes—
161C23#1452324911yescharacteristic
162C2×C4—22no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4—42no—
166C2×C4—5084no—
167C2×C4—5084no—
168C2×C4—5084no—
169C2×C4—5082no—
170C2×C4—5082no—
171C2×C4—2542no—
172C22—22no—
173C22—24no—
174C22—41no—
175C22—42no—
176C22—41no—
177C22—1274no—
178C22—5082no—
179C22—5082no—
180C22—5088no—
181C22—5088no—
182C22—2544no—
183C22—2544no—
184C22—2544no—
185C22—2542no—
186C22—2542no—
187C22—2542no—
188C22C22⋊C4×D12711yescharacteristic
189C22#1686116611yescharacteristic
190C22#1686116711yescharacteristic
191C22#1686119112yes—
192C22#1686119412yes—
193C4—42no—
194C4—82no—
195C4—10162no—
196C4—5082no—
197C2—22no—
198C2—22no—
199C2—21no—
200C2—21no—
201C2—5082no—
202C2—2544no—
203C2#2254973811yescharacteristic
204C2#99×D12712yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#342×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC24×D127C22
cosocleC24×C127C23
hypercenter#342D127
hypocenterC127#1819
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. #22549643C2semidirect
4. #22549642C2semidirect
5. #342×C127C2semidirect
7. C22×C4×D127C4semidirect
11. C2×(C2×D127)⋊C4C4semidirect
103. C127#1819semidirect
108. #342D127semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D1272
#225497312
#225496432
#225496422
#342×C1272
#1819127