Finite Group Databasebeta

← #30639978

Subgroups — #30639978

Order 16256.

382 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 71 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×#16861124C212yesmaximal, complemented
2#22549756C212yesmaximal, complemented
3#22549748C211yesmaximal, characteristic, complemented
4#22549643C211yesmaximal, characteristic, complemented
5#534×C127C211yesmaximal, characteristic
6C23×Dic127C2211yescharacteristic
7C2×Dic127⋊C4—21no—
8C2×C22⋊Dic127—22no—
9C2×C22⋊Dic127C2212yes—
10C2×C22⋊Dic127C2211yescharacteristic
11#16861124—24no—
12C127⋊#95—21no—
13C127⋊#95—22no—
14C22×D4×C127C2212yes—
15#123×C127—22no—
16#121×C127—21no—
17#121×C127—21no—
18#120×C127—22no—
19C2×C22⋊C4×C127C2211yescharacteristic
20C22×Dic127—22no—
21C22×Dic127—21no—
22C22×Dic127—21no—
23C22×Dic127—22no—
24C22×Dic127—42no—
25Dic127⋊C4—42no—
26C22⋊Dic127—22no—
27C22⋊Dic127—21no—
28C22⋊Dic127—22no—
29C22⋊Dic127—44no—
30C22⋊Dic127D412yescomplemented
31C24×C127D412yes—
32C24×C127C2311yescharacteristic
33C2×D4×C127—22no—
34C2×D4×C127—22no—
35C2×D4×C127—22no—
36C2×D4×C127—22no—
37C2×D4×C127—22no—
38C2×D4×C127—22no—
39C2×D4×C127—21no—
40C2×D4×C127—21no—
41C22×C4×C127D412yes—
42C22×C4×C127D411yescharacteristic
43C22×C4×C127D411yescharacteristic
44C4⋊C4×C127—21no—
45C4⋊C4×C127—21no—
46C22⋊C4×C127—22no—
47C22⋊C4×C127—22no—
48C22⋊C4×C127—22no—
49C2×Dic127—24no—
50C2×Dic127—22no—
51C2×Dic127—22no—
52C2×Dic127—41no—
53C2×Dic127—41no—
54C2×Dic127—41no—
55C2×Dic127—42no—
56C2×Dic127—41no—
57C2×Dic127—42no—
58C2×Dic127—42no—
59C2×Dic127—42no—
60C23×C127—22no—
61C23×C127—22no—
62C23×C127—22no—
63C23×C127—41no—
64C23×C127—41no—
65C23×C127—42no—
66C23×C127—42no—
67C23×C127C2×D411yescharacteristic
68C23×C127C2×D412yes—
69C23×C127C2×D411yescharacteristic
70C23×C127#4111yescharacteristic
71C23×C127#4112yes—
72D4×C127—42no—
73D4×C127—42no—
74D4×C127—42no—
75D4×C127—42no—
76D4×C127—81no—
77D4×C127—81no—
78C2×C4×C127—21no—
79C2×C4×C127—21no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C22×C127—21no—
86C22×C127—22no—
87C22×C127—21no—
88C22×C127—22no—
89C22×C127—41no—
90C22×C127—41no—
91C22×C127—42no—
92C22×C127—44no—
93C22×C127—42no—
94C22×C127—42no—
95C22×C127—42no—
96C22×C127—82no—
97C22×C127#12011yescharacteristic
98C22×C127#12111yescharacteristic
99C22×C127#12112yes—
100C22×C127#12312yes—
101C22×C127#12411yescharacteristic
102C4×C127—42no—
103C4×C127—41no—
104C4×C127—41no—
105Dic127—42no—
106Dic127—42no—
107Dic127—82no—
108C2×C127—21no—
109C2×C127—22no—
110C2×C127—22no—
111C2×C127—21no—
112C2×C127—82no—
113C2×C127#39411yescharacteristic
114C2×C127B(4,𝔽2)11yescharacteristic
115C2×C127#45811yescharacteristic
116#2044—1271nomaximal, Sylow
117C127#204411yescharacteristic, complemented, Sylow
118#534D12711yescharacteristic
119C2×#99—1272no—
120#386—1272no—
121#385—1271no—
122#342—1271no—
123C22×D4C2×D12712yes—
124C23×C4—1271no—
125#123—22no—
126#121—21no—
127#121—21no—
128#120—22no—
129C2×C4⋊C4—2541no—
130C2×C22⋊C4—1271no—
131C2×C22⋊C4—1272no—
132C2×C22⋊C4—2542no—
133C2×C22⋊C4C2×D12711yescharacteristic
134#99—2544no—
135#95—2541no—
136#95—2542no—
137C24C22⋊D12712yes—
138C24C22×D12711yescharacteristic
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—21no—
146C2×D4—21no—
147C22×C4—5082no—
148C22×C4—2542no—
149C22×C4—2541no—
150C22×C4—2542no—
151C22×C4—2541no—
152C22×C4C22⋊D12712yes—
153C22×C4C22⋊D12711yescharacteristic
154C22×C4C22⋊D12711yescharacteristic
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—5082no—
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—22no—
167C23—22no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—42no—
172C23—42no—
173C23D4×D12711yescharacteristic
174C23C2×C22⋊D12712yes—
175C23C2×C22⋊D12711yescharacteristic
176C23#1452324711yescharacteristic
177C23#1452324712yes—
178D4—42no—
179D4—42no—
180D4—42no—
181D4—42no—
182D4—81no—
183D4—81no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—5082no—
192C2×C4—5082no—
193C2×C4—5082no—
194C2×C4—5081no—
195C2×C4—5081no—
196C2×C4—5082no—
197C2×C4—5081no—
198C2×C4—5081no—
199C2×C4—2544no—
200C2×C4—2542no—
201C2×C4—2542no—
202C22—21no—
203C22—22no—
204C22—21no—
205C22—22no—
206C22—41no—
207C22—42no—
208C22—41no—
209C22—44no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—82no—
214C22#1686117811yescharacteristic
215C22#1686118011yescharacteristic
216C22#1686118112yes—
217C22#1686119012yes—
218C22#1686119611yescharacteristic
219C4—42no—
220C4—41no—
221C4—41no—
222C4—10162no—
223C4—5082no—
224C4—5082no—
225C2—22no—
226C2—21no—
227C2—22no—
228C2—21no—
229C2—82no—
230C2#2254979011yescharacteristic
231C2#2255006311yescharacteristic
232C2#2255006611yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC24×C127C23
FrattiniC24C22×D127
Fitting, prime radical#534×C127C2
socleC22×C127#120
hypercenter#534D127
hypocenterC127#2044
LCSG ▷ C24×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#16861124C2semidirect
2. #22549756C2semidirect
3. #22549748C2semidirect
4. #22549643C2semidirect
30. C22⋊Dic127D4semidirect
117. C127#2044semidirect

Maximal subgroups

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

subgroupindex
C2×#168611242
#225497562
#225497482
#225496432
#534×C1272
#2044127