Finite Group Databasebeta

← #30639982

Subgroups — #30639982

Order 16256.

382 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 67 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#22550438C211yesmaximal, characteristic
2C2×#16861123C212yesmaximal, complemented
3#22549753C212yesmaximal, complemented
4#22549748C211yesmaximal, characteristic, complemented
5#342×C127C211yesmaximal, characteristic, complemented
6C22×C22⋊D127C2212yes—
7C23×Dic127C2211yescharacteristic
8#16861193—22no—
9#16861183—21no—
10#16861183—21no—
11#16861179—22no—
12C2×Dic127⋊C4—21no—
13C2×C22⋊Dic127C2211yescharacteristic
14C2×(C2×D127)⋊C4—22no—
15#16861123—24no—
16C127⋊#95—22no—
17C127⋊#95—21no—
18C2×C22⋊C4×C127C2212yes—
19C2×C22⋊C4×C127C2211yescharacteristic
20C23×D127D412yes—
21C2×D508—21no—
22C2×D508—21no—
23C2×C22⋊D127—22no—
24C2×C22⋊D127—22no—
25C2×C22⋊D127—22no—
26C2×C22⋊D127—22no—
27C2×C22⋊D127—22no—
28C2×C22⋊D127—22no—
29C22×Dic127—22no—
30C22×Dic127—21no—
31C22×Dic127—21no—
32C22×Dic127—42no—
33C22×Dic127D412yes—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36Dic127⋊C4—42no—
37C22⋊Dic127—21no—
38C22⋊Dic127D412yescomplemented
39(C2×D127)⋊C4—22no—
40(C2×D127)⋊C4—22no—
41(C2×D127)⋊C4—44no—
42C24×C127C2311yescharacteristic
43C22×C4×C127—22no—
44C22×C4×C127D411yescharacteristic
45C22×C4×C127D411yescharacteristic
46C22⋊C4×C127—22no—
47C22⋊C4×C127—22no—
48C22⋊C4×C127—22no—
49C22×D127—22no—
50C22×D127—22no—
51C22×D127—22no—
52C22×D127—42no—
53C22×D127—42no—
54D508—81no—
55D508—81no—
56C22⋊D127—42no—
57C22⋊D127—42no—
58C22⋊D127—42no—
59C22⋊D127—42no—
60C2×Dic127—24no—
61C2×Dic127—22no—
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127—41no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C2×Dic127—42no—
70C2×Dic127—41no—
71C23×C127—41no—
72C23×C127—41no—
73C23×C127C2×D412yes—
74C23×C127C2×D411yescharacteristic
75C23×C127C2×D411yescharacteristic
76C23×C127#4112yes—
77C23×C127#4111yescharacteristic
78C2×C4×C127—21no—
79C2×C4×C127—21no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127—42no—
83C2×C4×C127—42no—
84C2×C4×C127—42no—
85C22×C127—21no—
86C22×C127—22no—
87C22×C127—21no—
88C22×C127—22no—
89C22×C127—41no—
90C22×C127—42no—
91C22×C127—41no—
92C22×C127#12011yescharacteristic
93C22×C127#12111yescharacteristic
94C22×C127#12112yes—
95C22×C127#12312yes—
96C22×C127#12411yescharacteristic
97C2×D127—44no—
98C2×D127—42no—
99C2×D127—42no—
100C2×D127—42no—
101C2×D127—82no—
102C4×C127—41no—
103C4×C127—41no—
104C4×C127—82no—
105Dic127—42no—
106Dic127—42no—
107Dic127—42no—
108C2×C127—21no—
109C2×C127—22no—
110C2×C127—22no—
111C2×C127—21no—
112C2×C127#39411yescharacteristic
113C2×C127B(4,𝔽2)11yescharacteristic
114C2×C127#45811yescharacteristic
115D127—82no—
116#2044—1271nomaximal, Sylow
117C127#204411yescharacteristic, complemented, Sylow
118#534—1271no—
119C2×#99—1272no—
120#386—1272no—
121#385—1271no—
122#342D12711yescharacteristic, complemented
123C22×D4—1272no—
124C23×C4—1271no—
125#123—2542no—
126#121—2541no—
127#121—2541no—
128#120—2542no—
129C2×C4⋊C4—2541no—
130C2×C22⋊C4—1271no—
131C2×C22⋊C4—2542no—
132C2×C22⋊C4C2×D12712yes—
133C2×C22⋊C4C2×D12711yescharacteristic
134#99—2544no—
135#95—2542no—
136#95—2541no—
137C24—1272no—
138C24C22×D12711yescharacteristic
139C2×D4—2541no—
140C2×D4—2541no—
141C2×D4—2542no—
142C2×D4—2542no—
143C2×D4—2542no—
144C2×D4—2542no—
145C2×D4—2542no—
146C2×D4—2542no—
147C22×C4—22no—
148C22×C4—1272no—
149C22×C4—5082no—
150C22×C4—2542no—
151C22×C4—2541no—
152C22×C4—2541no—
153C22×C4D50811yescharacteristic
154C22×C4D50811yescharacteristic
155C4⋊C4—5082no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—1272no—
162C22⋊C4—5084no—
163C22⋊C4—2542no—
164C22⋊C4—2542no—
165C22⋊C4—2541no—
166C23—41no—
167C23—41no—
168C23—5082no—
169C23—5082no—
170C23—2542no—
171C23—2542no—
172C23—2542no—
173C23D4×D12712yes—
174C23D4×D12711yescharacteristic
175C23C2×D50811yescharacteristic
176C23#1452324812yes—
177C23#1452324911yescharacteristic
178D4—10161no—
179D4—10161no—
180D4—5082no—
181D4—5082no—
182D4—5082no—
183D4—5082no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—5082no—
194C2×C4—5081no—
195C2×C4—5081no—
196C2×C4—2544no—
197C2×C4—2542no—
198C2×C4—2542no—
199C2×C4—2542no—
200C2×C4—2542no—
201C2×C4—2542no—
202C22—21no—
203C22—22no—
204C22—21no—
205C22—22no—
206C22—41no—
207C22—42no—
208C22—41no—
209C22—10162no—
210C22—5084no—
211C22—5082no—
212C22—5082no—
213C22—5082no—
214C22#1686117911yescharacteristic
215C22#1686118212yes—
216C22#1686118411yescharacteristic
217C22#1686119412yes—
218C22#1686119711yescharacteristic
219C4—41no—
220C4—41no—
221C4—82no—
222C4—5082no—
223C4—5082no—
224C4—5082no—
225C2—22no—
226C2—21no—
227C2—22no—
228C2—21no—
229C2—10162no—
230C2#2254979411yescharacteristic
231C2#2255006411yescharacteristic
232C2#2255006711yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC24×C127C23
FrattiniC24C22×D127
Fitting#342×C127C2
socleC22×C127#120
prime radical#22550438C2
hypercenter#342D127
hypocenterC127#2044
LCSG ▷ C24×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#16861123C2semidirect
3. #22549753C2semidirect
4. #22549748C2semidirect
5. #342×C127C2semidirect
38. C22⋊Dic127D4semidirect
117. C127#2044semidirect
122. #342D127semidirect

Maximal subgroups

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

subgroupindex
#225504382
C2×#168611232
#225497532
#225497482
#342×C1272
#2044127