Finite Group Databasebeta

← #30639186

Subgroups — #30639186

Order 16256.

378 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 89 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×#16861192C211yesmaximal, characteristic
2C2×#16861168C211yesmaximal, characteristic, complemented
3C2×#16861123C212yesmaximal
4#22549642C212yesmaximal, complemented
5#381×C127C211yesmaximal, characteristic, complemented
6C22×C22⋊D127C2211yescharacteristic
7#16861192—22no—
8#16861192—21no—
9#16861192—21no—
10#16861168—24no—
11C2×C4⋊Dic127—21no—
12C2×Dic127⋊C4C412yescomplemented
13C2×C22⋊Dic127C2211yescharacteristic
14C2×C22⋊Dic127C2212yes—
15C2×(C2×D127)⋊C4C412yescomplemented
16C2×C4×Dic127—21no—
17#16861123—24no—
18C23×C4×C127C2211yescharacteristic
19C2×C22⋊C4×C127C2212yes—
20#95×C127—22no—
21C23×D127C2×C411yescharacteristic
22C2×C22⋊D127—21no—
23C2×C22⋊D127—21no—
24C2×C22⋊D127—22no—
25C2×C22⋊D127—22no—
26C22×Dic127—22no—
27C22×Dic127C2×C411yescharacteristic
28C22×Dic127C2×C412yes—
29C4⋊Dic127—24no—
30Dic127⋊C4—22no—
31Dic127⋊C4—22no—
32C22⋊Dic127—21no—
33C22⋊Dic127—21no—
34C22⋊Dic127—22no—
35C22⋊Dic127D414yes—
36(C2×D127)⋊C4—22no—
37(C2×D127)⋊C4—22no—
38C4×Dic127—24no—
39C24×C127C2311yescharacteristic
40C22×C4×C127—22no—
41C22×C4×C127—22no—
42C22×C4×C127—22no—
43C22×C4×C127C2×C412yes—
44C22×C4×C127D412yes—
45C22×C4×C127D411yescharacteristic
46C22×C4×C127D411yescharacteristic
47C22⋊C4×C127—22no—
48C22⋊C4×C127—22no—
49C22×D127—21no—
50C22×D127—22no—
51C22×D127—42no—
52C22⋊D127—41no—
53C22⋊D127—42no—
54C22⋊D127—41no—
55C2×Dic127—22no—
56C2×Dic127—21no—
57C2×Dic127—22no—
58C2×Dic127—24no—
59C2×Dic127—44no—
60C2×Dic127—42no—
61C23×C127—42no—
62C23×C127C22×C411yescharacteristic
63C23×C127C2×D412yes—
64C23×C127C2×D411yescharacteristic
65C23×C127C2×D411yescharacteristic
66C23×C127#4112yes—
67C2×C4×C127—24no—
68C2×C4×C127—21no—
69C2×C4×C127—21no—
70C2×C4×C127—24no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—44no—
74C2×C4×C127—42no—
75C2×C4×C127C22⋊C412yes—
76C2×C4×C127C22⋊C412yes—
77C22×C127—22no—
78C22×C127—22no—
79C22×C127—22no—
80C22×C127—42no—
81C22×C127—42no—
82C22×C127C2×C22⋊C411yescharacteristic
83C22×C127C4×D412yes—
84C22×C127#12011yescharacteristic
85C22×C127#12112yes—
86C22×C127#12311yescharacteristic
87C2×D127—41no—
88C2×D127—42no—
89C2×D127—82no—
90C4×C127—22no—
91C4×C127—22no—
92C4×C127—82no—
93Dic127—44no—
94Dic127—41no—
95Dic127—42no—
96C2×C127—21no—
97C2×C127—22no—
98C2×C127—22no—
99C2×C127—21no—
100C2×C127#38611yescharacteristic
101C2×C127#41012yes—
102D127—81no—
103#1899—1271nomaximal, Sylow
104C127#189911yescharacteristic, complemented, Sylow
105C2×#123—1271no—
106C2×#117—1271no—
107C2×#99—1272no—
108#381D12711yescharacteristic, complemented
109#342—1272no—
110C22×D4—1271no—
111C23×C4C2×D12711yescharacteristic
112#123—2542no—
113#123—2541no—
114#123—2541no—
115#117—2544no—
116C2×C4⋊C4—1272no—
117C2×C4⋊C4—2541no—
118C2×C22⋊C4—1272no—
119C2×C22⋊C4—1271no—
120C2×C22⋊C4—1272no—
121C2×C22⋊C4C2×D12712yes—
122C2×C42—2541no—
123#99—2544no—
124#95—22no—
125C24—1271no—
126C24C22×D12711yescharacteristic
127C2×D4—2542no—
128C2×D4—2541no—
129C2×D4—2541no—
130C2×D4—2542no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—1272no—
135C22×C4—1271no—
136C22×C4—2542no—
137C22×C4C4×D12712yes—
138C22×C4C22⋊D12712yes—
139C22×C4D50811yescharacteristic
140C22×C4D50811yescharacteristic
141C4⋊C4—2544no—
142C4⋊C4—2542no—
143C4⋊C4—2542no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—1274no—
147C22⋊C4—2542no—
148C22⋊C4—2541no—
149C22⋊C4—2541no—
150C22⋊C4—2542no—
151C22⋊C4—2542no—
152C42—2544no—
153C23—42no—
154C23—5082no—
155C23—2542no—
156C23—2541no—
157C23C2×C4×D12711yescharacteristic
158C23D4×D12712yes—
159C23C2×C22⋊D12711yescharacteristic
160C23C2×D50811yescharacteristic
161C23#1452324712yes—
162D4—5081no—
163D4—5081no—
164D4—5082no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—24no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—24no—
171C2×C4—44no—
172C2×C4—42no—
173C2×C4—5084no—
174C2×C4—5082no—
175C2×C4—2542no—
176C2×C4—2541no—
177C2×C4—2542no—
178C2×C4—2544no—
179C2×C4(C2×D127)⋊C412yes—
180C2×C4(C2×D127)⋊C412yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—10162no—
187C22—5082no—
188C22—5081no—
189C22C2×(C2×D127)⋊C411yescharacteristic
190C22(C22⋊D127)⋊C412yes—
191C22#1686117911yescharacteristic
192C22#1686118112yes—
193C22#1686119311yescharacteristic
194C4—22no—
195C4—22no—
196C4—82no—
197C4—5084no—
198C4—5082no—
199C4—5081no—
200C2—22no—
201C2—21no—
202C2—22no—
203C2—21no—
204C2—10161no—
205C2#2254975311yescharacteristic
206C2#2254984912yes—
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#381×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×C22⋊D127C22
cosocleC24×C127C23
hypercenter#381D127
hypocenterC127#1899
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#16861168C2semidirect
4. #22549642C2semidirect
5. #381×C127C2semidirect
12. C2×Dic127⋊C4C4semidirect
15. C2×(C2×D127)⋊C4C4semidirect
104. C127#1899semidirect
108. #381D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611922
C2×#168611682
C2×#168611232
#225496422
#381×C1272
#1899127