Finite Group Databasebeta

← #30639138

Subgroups — #30639138

Order 16256.

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

normal subgroups only

This group has 578 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
1C2×#16861177C211yesmaximal, characteristic
2C2×#16861124C211yesmaximal, characteristic, complemented
3#22549731C212yesmaximal, complemented
4#22549643C212yesmaximal, complemented
5#518×C127C211yesmaximal, characteristic, complemented
6C24×D127C2211yescharacteristic
7C22×C22⋊D127—21no—
8#16861177—22no—
9#16861177C414yescomplemented
10C2×C22⋊Dic127C2212yes—
11C2×C22⋊Dic127C2211yescharacteristic
12C2×(C2×D127)⋊C4—21no—
13C2×(C2×D127)⋊C4—24no—
14#16861124—22no—
15C22×D4×C127C2211yescharacteristic
16C4×D4×C127—24no—
17#117×C127—21no—
18C2×C22⋊C4×C127—21no—
19C2×C22⋊C4×C127C2212yes—
20C23×D127—21no—
21C23×D127—22no—
22C23×D127—21no—
23C23×D127—22no—
24C23×D127—44no—
25C23×D127C2×C412yescomplemented
26C2×C22⋊D127—24no—
27C2×C22⋊D127—44no—
28C22×Dic127—21no—
29C22×Dic127—22no—
30C22⋊Dic127—21no—
31C22⋊Dic127—22no—
32C22⋊Dic127—22no—
33C22⋊Dic127C2×C412yescomplemented
34(C2×D127)⋊C4—24no—
35(C2×D127)⋊C4—48no—
36C24×C127D411yescharacteristic
37C24×C127C2311yescharacteristic
38C2×D4×C127—22no—
39C2×D4×C127—22no—
40C2×D4×C127C2×C412yes—
41C2×D4×C127D412yes—
42C22×C4×C127—22no—
43C22×C4×C127D411yescharacteristic
44C22×C4×C127D414yescomplemented
45C4⋊C4×C127—22no—
46C22⋊C4×C127—22no—
47C22⋊C4×C127—22no—
48C22⋊C4×C127—22no—
49C42×C127—22no—
50C22×D127—28no—
51C22×D127—24no—
52C22×D127—22no—
53C22×D127—24no—
54C22×D127—44no—
55C22×D127—44no—
56C22×D127—48no—
57C22×D127—44no—
58C22×D127—42no—
59C22×D127—41no—
60C22×D127—42no—
61C22×D127—41no—
62C22⋊D127—48no—
63C2×Dic127—22no—
64C2×Dic127—42no—
65C2×Dic127—44no—
66C2×Dic127—42no—
67C23×C127—21no—
68C23×C127—21no—
69C23×C127—42no—
70C23×C127—42no—
71C23×C127C22⋊C412yescomplemented
72C23×C127C22×C411yescharacteristic
73C23×C127C2×D411yescharacteristic
74C23×C127C2×D412yes—
75C23×C127C2×D411yescharacteristic
76C23×C127#4112yes—
77D4×C127—24no—
78D4×C127—24no—
79C2×C4×C127—21no—
80C2×C4×C127—21no—
81C2×C4×C127—24no—
82C2×C4×C127—44no—
83C2×C4×C127—82no—
84C2×C4×C127C22⋊C412yescomplemented
85C22×C127—24no—
86C22×C127—22no—
87C22×C127—22no—
88C22×C127—41no—
89C22×C127—41no—
90C22×C127—42no—
91C22×C127—42no—
92C22×C127—42no—
93C22×C127—81no—
94C22×C127C2×C22⋊C411yescharacteristic
95C22×C127C4×D412yes—
96C22×C127#12011yescharacteristic
97C22×C127#12112yes—
98C22×C127#12311yescharacteristic
99C2×D127—24no—
100C2×D127—28no—
101C2×D127—42no—
102C2×D127—48no—
103C2×D127—44no—
104C2×D127—44no—
105C2×D127—42no—
106C2×D127—44no—
107C4×C127—22no—
108C4×C127—44no—
109Dic127—42no—
110Dic127—82no—
111C2×C127—21no—
112C2×C127—22no—
113C2×C127—22no—
114C2×C127—21no—
115C2×C127—42no—
116C2×C127#38611yescharacteristic
117C2×C127B(4,𝔽2)12yes—
118D127—44no—
119#1889—1271nomaximal, Sylow
120C127#188911yescharacteristic, complemented, Sylow
121C2×#120—1271no—
122#518D12711yescharacteristic, complemented
123C2×#99—1271no—
124#379—1272no—
125#342—1272no—
126C25—1271no—
127C22×D4—2541no—
128C22×D4C2×D12711yescharacteristic
129#120—1274no—
130#120—2542no—
131C4×D4—24no—
132#117—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—1271no—
135C2×C22⋊C4—1272no—
136C2×C22⋊C4—2544no—
137C2×C22⋊C4—2541no—
138C2×C22⋊C4C2×D12712yes—
139#99—2542no—
140C24—1272no—
141C24—5084no—
142C24—2541no—
143C24—2542no—
144C24—2542no—
145C24—2541no—
146C24C22⋊D12711yescharacteristic
147C24C22×D12711yescharacteristic
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—5084no—
151C2×D4—2544no—
152C2×D4C4×D12712yes—
153C2×D4D50812yes—
154C22×C4—22no—
155C22×C4—2542no—
156C22×C4—2541no—
157C22×C4C22⋊D12711yescharacteristic
158C22×C4C22⋊D12714yescomplemented
159C4⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—1272no—
164C22⋊C4—5088no—
165C22⋊C4—2542no—
166C22⋊C4—2542no—
167C22⋊C4—2541no—
168C22⋊C4—2544no—
169C42—22no—
170C23—21no—
171C23—21no—
172C23—42no—
173C23—42no—
174C23—5084no—
175C23—5082no—
176C23—5084no—
177C23—5088no—
178C23—5084no—
179C23—5081no—
180C23—5081no—
181C23—5082no—
182C23—2548no—
183C23—2544no—
184C23—2542no—
185C23—2544no—
186C23(C2×D127)⋊C412yescomplemented
187C23C2×C4×D12711yescharacteristic
188C23C2×C22⋊D12711yescharacteristic
189C23C2×C22⋊D12712yes—
190C23C2×D50811yescharacteristic
191C23#1452324812yes—
192D4—24no—
193D4—24no—
194D4—5088no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—44no—
199C2×C4—82no—
200C2×C4—5084no—
201C2×C4—5082no—
202C2×C4—5082no—
203C2×C4—2542no—
204C2×C4(C2×D127)⋊C412yescomplemented
205C22—24no—
206C22—22no—
207C22—22no—
208C22—41no—
209C22—41no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—81no—
214C22—5088no—
215C22—5082no—
216C22—5084no—
217C22—5084no—
218C22—5084no—
219C22—5082no—
220C22—2544no—
221C22—2548no—
222C22C2×(C2×D127)⋊C411yescharacteristic
223C22C4×C22⋊D12712yes—
224C22#1686117811yescharacteristic
225C22#1686118312yes—
226C22#1686119211yescharacteristic
227C4—22no—
228C4—44no—
229C4—10162no—
230C4—5082no—
231C2—22no—
232C2—21no—
233C2—21no—
234C2—22no—
235C2—42no—
236C2—5084no—
237C2#2254975511yescharacteristic
238C2#2255006312yes—
subgroupG/H
centerC22#16861178
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#518×C127C2
socleC22×C127#120
prime radicalC2×#16861177C2
cosocleC24×C127C23
hypercenter#518D127
hypocenterC127#1889
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C2×#16861124C2semidirect
3. #22549731C2semidirect
4. #22549643C2semidirect
5. #518×C127C2semidirect
9. #16861177C4semidirect
25. C23×D127C2×C4semidirect
33. C22⋊Dic127C2×C4semidirect
44. C22×C4×C127D4semidirect
71. C23×C127C22⋊C4semidirect
84. C2×C4×C127C22⋊C4semidirect
120. C127#1889semidirect
122. #518D127semidirect
158. C22×C4C22⋊D127semidirect
186. C23(C2×D127)⋊C4semidirect
204. C2×C4(C2×D127)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#168611772
C2×#168611242
#225497312
#225496432
#518×C1272
#1889127