Finite Group Databasebeta

← #30639273

Subgroups — #30639273

Order 16256.

380 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 93 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#22550351C211yesmaximal, characteristic, complemented
2C2×#16861124C212yesmaximal, complemented
3C2×#16861124C211yesmaximal, characteristic, complemented
4#22549642C212yesmaximal, complemented
5#534×C127C211yesmaximal, characteristic
6D4×Dic127—24no—
7#16861168—21no—
8C2×C22⋊Dic127—21no—
9C2×C22⋊Dic127C2212yes—
10C2×C22⋊Dic127C2212yes—
11#16861124—24no—
12#16861124—22no—
13C22×D4×C127C2211yescharacteristic
14C22×D4×C127C2211yescharacteristic
15#123×C127—21no—
16#123×C127C412yescomplemented
17#121×C127—22no—
18#120×C127—21no—
19#120×C127C412yescomplemented
20C2×C22⋊C4×C127C2211yescharacteristic
21C22×Dic127—22no—
22C22×Dic127—22no—
23C22×Dic127—22no—
24C4⋊Dic127—22no—
25C22⋊Dic127—22no—
26C22⋊Dic127—22no—
27C22⋊Dic127—22no—
28C22⋊Dic127—22no—
29C22⋊Dic127D414yescomplemented
30C4×Dic127—22no—
31C24×C127C2×C411yescharacteristic, complemented
32C24×C127D411yescharacteristic
33C24×C127C2311yescharacteristic
34C2×D4×C127—22no—
35C2×D4×C127—22no—
36C2×D4×C127—21no—
37C2×D4×C127—21no—
38C2×D4×C127—22no—
39C2×D4×C127—22no—
40C2×D4×C127—22no—
41C2×D4×C127C2×C412yes—
42C2×D4×C127D412yes—
43C22×C4×C127—21no—
44C22×C4×C127C2×C411yescharacteristic, complemented
45C22×C4×C127D411yescharacteristic
46C4⋊C4×C127—22no—
47C22⋊C4×C127—22no—
48C22⋊C4×C127—21no—
49C22⋊C4×C127—22no—
50C22⋊C4×C127C2×C412yescomplemented
51C2×Dic127—24no—
52C2×Dic127—42no—
53C2×Dic127—82no—
54C2×Dic127—82no—
55C2×Dic127—82no—
56C23×C127—21no—
57C23×C127—21no—
58C23×C127—21no—
59C23×C127—22no—
60C23×C127—81no—
61C23×C127—81no—
62C23×C127—81no—
63C23×C127C22⋊C412yes—
64C23×C127C22×C411yescharacteristic
65C23×C127C2×D412yes—
66C23×C127C2×D411yescharacteristic
67C23×C127C2×D411yescharacteristic
68C23×C127#4112yes—
69D4×C127—24no—
70D4×C127—24no—
71D4×C127—41no—
72D4×C127—42no—
73D4×C127—41no—
74D4×C127—82no—
75C2×C4×C127—21no—
76C2×C4×C127—21no—
77C2×C4×C127—22no—
78C2×C4×C127—21no—
79C2×C4×C127—22no—
80C2×C4×C127—81no—
81C2×C4×C127C22⋊C412yes—
82C22×C127—22no—
83C22×C127—41no—
84C22×C127—41no—
85C22×C127—41no—
86C22×C127—42no—
87C22×C127—42no—
88C22×C127—42no—
89C22×C127—42no—
90C22×C127—42no—
91C22×C127—41no—
92C22×C127—81no—
93C22×C127—82no—
94C22×C127C2×C22⋊C411yescharacteristic
95C22×C127C4×D412yes—
96C22×C127#12011yescharacteristic
97C22×C127#12112yes—
98C22×C127#12311yescharacteristic
99C4×C127—22no—
100C4×C127—42no—
101C4×C127—41no—
102Dic127—44no—
103Dic127—82no—
104C2×C127—21no—
105C2×C127—22no—
106C2×C127—21no—
107C2×C127—22no—
108C2×C127—21no—
109C2×C127—42no—
110C2×C127—81no—
111C2×C127#38611yescharacteristic
112#1913—1271nomaximal, Sylow
113C127#191311yescharacteristic, complemented, Sylow
114#534D12711yescharacteristic
115#518—1271no—
116C2×#99—1272no—
117C2×#99—1271no—
118#342—1272no—
119C22×D4C2×D12711yescharacteristic
120C22×D4C2×D12711yescharacteristic
121#123—21no—
122#123Dic12712yescomplemented
123#121—22no—
124#120—21no—
125#120Dic12712yescomplemented
126C4×D4—2544no—
127#117—2541no—
128C2×C22⋊C4—1272no—
129C2×C22⋊C4—1272no—
130C2×C22⋊C4—2541no—
131C2×C22⋊C4C2×D12711yescharacteristic
132#99—2544no—
133#99—2542no—
134C24C2×Dic12711yescharacteristic, complemented
135C24C22⋊D12711yescharacteristic
136C24C22×D12711yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4C2×Dic12712yes—
145C2×D4C22⋊D12712yes—
146C22×C4—21no—
147C22×C4—2542no—
148C22×C4—2542no—
149C22×C4—2542no—
150C22×C4C2×Dic12711yescharacteristic, complemented
151C22×C4C22⋊D12711yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—2542no—
154C22⋊C4—22no—
155C22⋊C4—21no—
156C22⋊C4—22no—
157C22⋊C4—1274no—
158C22⋊C4—2542no—
159C22⋊C4—2542no—
160C22⋊C4—2542no—
161C22⋊C4—2542no—
162C22⋊C4C2×Dic12712yescomplemented
163C42—2542no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—22no—
168C23—81no—
169C23—81no—
170C23—81no—
171C23C22⋊Dic12712yes—
172C23C22×Dic12711yescharacteristic
173C23D4×D12712yes—
174C23C2×C22⋊D12711yescharacteristic
175C23C2×C22⋊D12711yescharacteristic
176C23#1452324712yes—
177D4—24no—
178D4—24no—
179D4—41no—
180D4—41no—
181D4—42no—
182D4—82no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—81no—
189C2×C4—10162no—
190C2×C4—10162no—
191C2×C4—10162no—
192C2×C4—5082no—
193C2×C4—2544no—
194C2×C4C22⋊Dic12712yes—
195C22—22no—
196C22—41no—
197C22—42no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—81no—
206C22—82no—
207C22C2×C22⋊Dic12711yescharacteristic
208C22D4×Dic12712yes—
209C22#1686117711yescharacteristic
210C22#1686118112yes—
211C22#1686119011yescharacteristic
212C4—22no—
213C4—42no—
214C4—41no—
215C4—10162no—
216C4—5084no—
217C2—21no—
218C2—22no—
219C2—21no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—81no—
224C2#2254975611yescharacteristic
subgroupG/H
centerC2#22549756
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting, prime radical#534×C127C2
socleC2×C127#386
cosocleC24×C127C23
hypercenter#534D127
hypocenterC127#1913
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #22550351C2semidirect
2. C2×#16861124C2semidirect
3. C2×#16861124C2semidirect
4. #22549642C2semidirect
16. #123×C127C4semidirect
19. #120×C127C4semidirect
29. C22⋊Dic127D4semidirect
31. C24×C127C2×C4semidirect
44. C22×C4×C127C2×C4semidirect
50. C22⋊C4×C127C2×C4semidirect
113. C127#1913semidirect
122. #123Dic127semidirect
125. #120Dic127semidirect
134. C24C2×Dic127semidirect
150. C22×C4C2×Dic127semidirect
162. C22⋊C4C2×Dic127semidirect

Maximal subgroups

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

subgroupindex
#225503512
C2×#168611242
#225496422
#534×C1272
#1913127