Finite Group Databasebeta

← #30639854

Subgroups — #30639854

Order 16256.

638 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16861177C212yesmaximal, complemented
2C2×#16861124C212yesmaximal, complemented
3#22549731C211yesmaximal, characteristic, complemented
4#22549643C211yesmaximal, characteristic, complemented
5#534×C127C211yesmaximal, characteristic, complemented
6C24×D127C2211yescharacteristic
7C22×C22⋊D127—22no—
8#16861177—24no—
9#16861177—24no—
10C2×C22⋊Dic127C2211yescharacteristic
11C2×C22⋊Dic127C2212yes—
12C2×(C2×D127)⋊C4—22no—
13C2×(C2×D127)⋊C4—22no—
14#16861124—24no—
15C22×D4×C127C2212yes—
16#123×C127—22no—
17#121×C127—22no—
18#120×C127—22no—
19C2×C22⋊C4×C127C2211yescharacteristic
20C23×D127—22no—
21C23×D127—21no—
22C23×D127—22no—
23C23×D127—21no—
24C23×D127—44no—
25C23×D127D412yescomplemented
26C2×C22⋊D127—28no—
27C2×C22⋊D127—48no—
28C22×Dic127—22no—
29C22×Dic127—21no—
30C22⋊Dic127—22no—
31C22⋊Dic127—22no—
32C22⋊Dic127D414yescomplemented
33(C2×D127)⋊C4—28no—
34(C2×D127)⋊C4—44no—
35C24×C127D412yescomplemented
36C24×C127C2311yescharacteristic
37C2×D4×C127—22no—
38C2×D4×C127—22no—
39C2×D4×C127—22no—
40C2×D4×C127—22no—
41C2×D4×C127—22no—
42C2×D4×C127—22no—
43C2×D4×C127—22no—
44C22×C4×C127D412yescomplemented
45C22×C4×C127D412yescomplemented
46C4⋊C4×C127—22no—
47C22⋊C4×C127—24no—
48C22⋊C4×C127—22no—
49C22×D127—24no—
50C22×D127—28no—
51C22×D127—24no—
52C22×D127—22no—
53C22×D127—44no—
54C22×D127—48no—
55C22×D127—44no—
56C22×D127—42no—
57C22×D127—44no—
58C22×D127—41no—
59C22×D127—42no—
60C22×D127—41no—
61C22⋊D127—416no—
62C2×Dic127—24no—
63C2×Dic127—44no—
64C2×Dic127—42no—
65C2×Dic127—41no—
66C23×C127—22no—
67C23×C127—22no—
68C23×C127—22no—
69C23×C127—42no—
70C23×C127—44no—
71C23×C127C2×D411yescharacteristic
72C23×C127C2×D412yes—
73C23×C127C2×D412yes—
74C23×C127C2×D411yescharacteristic
75C23×C127#4111yescharacteristic
76D4×C127—42no—
77D4×C127—42no—
78D4×C127—42no—
79D4×C127—42no—
80D4×C127—82no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127—42no—
86C22×C127—22no—
87C22×C127—24no—
88C22×C127—41no—
89C22×C127—42no—
90C22×C127—41no—
91C22×C127—44no—
92C22×C127—42no—
93C22×C127—44no—
94C22×C127—82no—
95C22×C127#12012yes—
96C22×C127#12011yescharacteristic
97C22×C127#12111yescharacteristic
98C22×C127#12112yes—
99C22×C127#12711yescharacteristic
100C2×D127—28no—
101C2×D127—24no—
102C2×D127—44no—
103C2×D127—44no—
104C2×D127—42no—
105C2×D127—48no—
106C2×D127—42no—
107C2×D127—44no—
108C4×C127—42no—
109C4×C127—42no—
110Dic127—44no—
111Dic127—81no—
112C2×C127—22no—
113C2×C127—21no—
114C2×C127—21no—
115C2×C127—22no—
116C2×C127—82no—
117C2×C127#39211yescharacteristic
118C2×C127B(4,𝔽2)12yes—
119D127—44no—
120#2021—1271nomaximal, Sylow
121C127#202111yescharacteristic, complemented, Sylow
122#534D12711yescharacteristic, complemented
123C2×#120—1272no—
124C2×#99—1272no—
125#379—1271no—
126#342—1271no—
127C25—1271no—
128C22×D4—2542no—
129C22×D4C2×D12712yes—
130#123—22no—
131#121—22no—
132#120—22no—
133#120—2544no—
134#120—2544no—
135C2×C22⋊C4—1272no—
136C2×C22⋊C4—1271no—
137C2×C22⋊C4—2542no—
138C2×C22⋊C4—2542no—
139C2×C22⋊C4C2×D12711yescharacteristic
140#99—2544no—
141C24—1272no—
142C24—5084no—
143C24—2542no—
144C24—2541no—
145C24—2541no—
146C24—2542no—
147C24C22⋊D12712yescomplemented
148C24C22×D12711yescharacteristic
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—5088no—
157C2×D4—2548no—
158C22×C4—2541no—
159C22×C4—2542no—
160C22×C4C22⋊D12712yescomplemented
161C22×C4C22⋊D12712yescomplemented
162C4⋊C4—22no—
163C22⋊C4—24no—
164C22⋊C4—22no—
165C22⋊C4—1274no—
166C22⋊C4—5084no—
167C22⋊C4—2548no—
168C22⋊C4—2542no—
169C22⋊C4—2542no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—42no—
174C23—44no—
175C23—5082no—
176C23—5084no—
177C23—5088no—
178C23—5084no—
179C23—5084no—
180C23—5081no—
181C23—5081no—
182C23—5082no—
183C23—2544no—
184C23—2548no—
185C23—2544no—
186C23—2542no—
187C23D4×D12712yes—
188C23D4×D12711yescharacteristic
189C23C2×C22⋊D12711yescharacteristic
190C23C2×C22⋊D12712yes—
191C23#1452324711yescharacteristic
192D4—42no—
193D4—42no—
194D4—42no—
195D4—42no—
196D4—82no—
197D4—50816no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—5082no—
204C2×C4—5081no—
205C2×C4—5084no—
206C2×C4—2544no—
207C22—22no—
208C22—24no—
209C22—42no—
210C22—41no—
211C22—41no—
212C22—44no—
213C22—42no—
214C22—44no—
215C22—82no—
216C22—5084no—
217C22—5084no—
218C22—5088no—
219C22—5082no—
220C22—5082no—
221C22—5084no—
222C22—2548no—
223C22—2544no—
224C22#1686117712yes—
225C22#1686117811yescharacteristic
226C22#1686118011yescharacteristic
227C22#1686118112yes—
228C22#1686120511yescharacteristic
229C4—42no—
230C4—42no—
231C4—10161no—
232C4—5084no—
233C2—21no—
234C2—22no—
235C2—22no—
236C2—21no—
237C2—82no—
238C2—5084no—
239C2#2254978411yescharacteristic
240C2#2255006312yes—
subgroupG/H
centerC22#16861178
commutator, cosocleC24×C127C23
FrattiniC24C22×D127
Fitting#534×C127C2
socleC22×C127#120
hypercenter#534D127
hypocenterC127#2021
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×#16861177C2semidirect
2. C2×#16861124C2semidirect
3. #22549731C2semidirect
4. #22549643C2semidirect
5. #534×C127C2semidirect
25. C23×D127D4semidirect
32. C22⋊Dic127D4semidirect
35. C24×C127D4semidirect
44. C22×C4×C127D4semidirect
45. C22×C4×C127D4semidirect
121. C127#2021semidirect
122. #534D127semidirect
147. C24C22⋊D127semidirect
160. C22×C4C22⋊D127semidirect
161. C22×C4C22⋊D127semidirect

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
#534×C1272
#2021127