Finite Group Databasebeta

← #30640376

Subgroups — #30640376

Order 16256.

280 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 73 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×#16861126C211yesmaximal, characteristic, complemented
2C2×#16861124C211yesmaximal, characteristic, complemented
3#22549666C212yesmaximal, complemented
4#22549663C212yesmaximal, complemented
5#534×C127C211yesmaximal, characteristic
6C2×#14523232—21no—
7C2×C22⋊Dic127—21no—
8#16861126—21no—
9#16861126C2212yes—
10#16861124—21no—
11#16861124C2212yes—
12C22×D4×C127C411yescharacteristic, complemented
13C22×D4×C127C2211yescharacteristic
14#123×C127—21no—
15#123×C127C411yescharacteristic, complemented
16#123×C127C2211yescharacteristic
17#121×C127—21no—
18#121×C127—21no—
19#120×C127—21no—
20#120×C127C411yescharacteristic, complemented
21#120×C127C2211yescharacteristic
22C2×C22⋊C4×C127C411yescharacteristic, complemented
23C22×Dic127—41no—
24#14523232—22no—
25#14523232—41no—
26C2×C127⋊C8—41no—
27C22⋊Dic127—22no—
28C22⋊Dic127—41no—
29C24×C127C2×C411yescharacteristic
30C24×C127C2×C411yescharacteristic, complemented
31C24×C127D411yescharacteristic
32C2×D4×C127—21no—
33C2×D4×C127—21no—
34C2×D4×C127—21no—
35C2×D4×C127—21no—
36C2×D4×C127—41no—
37C2×D4×C127—41no—
38C2×D4×C127—41no—
39C2×D4×C127—41no—
40C2×D4×C127C2×C411yescharacteristic
41C2×D4×C127D411yescharacteristic
42C2×D4×C127D411yescharacteristic
43C2×D4×C127C2311yescharacteristic
44C22×C4×C127—21no—
45C22×C4×C127C2×C411yescharacteristic, complemented
46C22×C4×C127D411yescharacteristic
47C4⋊C4×C127—21no—
48C4⋊C4×C127—21no—
49C22⋊C4×C127—21no—
50C22⋊C4×C127—21no—
51C22⋊C4×C127—21no—
52C22⋊C4×C127—21no—
53C22⋊C4×C127—21no—
54C22⋊C4×C127C2×C411yescharacteristic, complemented
55C22⋊C4×C127C2×C411yescharacteristic, complemented
56C2×Dic127—42no—
57C2×Dic127—82no—
58C127⋊C8—42no—
59C23×C127—21no—
60C23×C127—21no—
61C23×C127—21no—
62C23×C127—21no—
63C23×C127—21no—
64C23×C127—41no—
65C23×C127—81no—
66C23×C127—81no—
67C23×C127—81no—
68C23×C127C22⋊C411yescharacteristic
69C23×C127C22⋊C411yescharacteristic
70C23×C127C22⋊C411yescharacteristic
71C23×C127C22×C411yescharacteristic
72C23×C127C2×D411yescharacteristic
73D4×C127—41no—
74D4×C127—41no—
75D4×C127—41no—
76D4×C127—41no—
77D4×C127—42no—
78D4×C127—41no—
79D4×C127—41no—
80D4×C127—81no—
81D4×C127—81no—
82C2×C4×C127—21no—
83C2×C4×C127—21no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—41no—
88C2×C4×C127—81no—
89C2×C4×C127C22⋊C411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C22×C127—21no—
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127—41no—
96C22×C127—41no—
97C22×C127—41no—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127—42no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127—41no—
104C22×C127—81no—
105C22×C127—81no—
106C22×C127—81no—
107C22×C127—82no—
108C22×C127#9911yescharacteristic
109C22×C127#9911yescharacteristic
110C22×C127C2×C22⋊C411yescharacteristic
111C4×C127—21no—
112C4×C127—21no—
113C4×C127—41no—
114C4×C127—41no—
115C4×C127—41no—
116Dic127—82no—
117C2×C127—21no—
118C2×C127—21no—
119C2×C127—21no—
120C2×C127—41no—
121C2×C127—41no—
122C2×C127—41no—
123C2×C127—41no—
124C2×C127—81no—
125C2×C127C2×#9911yescharacteristic
126#2121—1271nomaximal, Sylow
127C127#212111yescharacteristic, complemented, Sylow
128#534D12711yescharacteristic
129C2×#100—1271no—
130C2×#99—1271no—
131#352—1272no—
132AΣL(1,𝔽16)—1272no—
133C22×D4Dic12711yescharacteristic, complemented
134C22×D4C2×D12711yescharacteristic
135C2×M16—2541no—
136#123—21no—
137#123Dic12711yescharacteristic, complemented
138#123C2×D12711yescharacteristic
139#121—21no—
140#121—21no—
141#120—21no—
142#120Dic12711yescharacteristic, complemented
143#120C2×D12711yescharacteristic
144C2×C22⋊C4—2541no—
145C2×C22⋊C4Dic12711yescharacteristic, complemented
146#100—1272no—
147#100—2541no—
148#99—1272no—
149#99—2541no—
150C24C2×Dic12711yescharacteristic
151C24C2×Dic12711yescharacteristic, complemented
152C24C22⋊D12711yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4C2×Dic12711yescharacteristic
162C2×D4C22⋊D12711yescharacteristic
163C2×D4C22⋊D12711yescharacteristic
164C2×D4C22×D12711yescharacteristic
165C22×C4—21no—
166C22×C4—5081no—
167C22×C4C2×Dic12711yescharacteristic, complemented
168C22×C4C22⋊D12711yescharacteristic
169M16—5081no—
170M16—2542no—
171C2×C8—5081no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—5081no—
180C22⋊C4—2542no—
181C22⋊C4C2×Dic12711yescharacteristic, complemented
182C22⋊C4C2×Dic12711yescharacteristic, complemented
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—41no—
189C23—81no—
190C23—81no—
191C23—81no—
192C23C22⋊Dic12711yescharacteristic
193C23C22⋊Dic12711yescharacteristic
194C23C22⋊Dic12711yescharacteristic
195C23C22×Dic12711yescharacteristic
196C23C2×C22⋊D12711yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—42no—
202D4—41no—
203D4—41no—
204D4—81no—
205D4—81no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—81no—
213C2×C4—10162no—
214C2×C4—5082no—
215C2×C4C22⋊Dic12711yescharacteristic
216C2×C4C2×C22⋊D12711yescharacteristic
217C8—5082no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—42no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—82no—
235C22#1686112411yescharacteristic
236C22#1686112411yescharacteristic
237C22C2×C22⋊Dic12711yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—10162no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2C2×#1686112411yescharacteristic
subgroupG/H
centerC2C2×#16861124
commutatorC23×C127C22×C4
FrattiniC2×D4C22×D127
Fitting, prime radical#534×C127C2
socleC2×C127C2×#99
cosocleC2×D4×C127C23
hypercenter#534D127
hypocenterC127#2121
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#16861126C2semidirect
2. C2×#16861124C2semidirect
3. #22549666C2semidirect
4. #22549663C2semidirect
12. C22×D4×C127C4semidirect
15. #123×C127C4semidirect
20. #120×C127C4semidirect
22. C2×C22⋊C4×C127C4semidirect
30. C24×C127C2×C4semidirect
45. C22×C4×C127C2×C4semidirect
54. C22⋊C4×C127C2×C4semidirect
55. C22⋊C4×C127C2×C4semidirect
127. C127#2121semidirect
133. C22×D4Dic127semidirect
137. #123Dic127semidirect
142. #120Dic127semidirect
145. C2×C22⋊C4Dic127semidirect
151. C24C2×Dic127semidirect
167. C22×C4C2×Dic127semidirect
181. C22⋊C4C2×Dic127semidirect
182. C22⋊C4C2×Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611262
C2×#168611242
#225496662
#225496632
#534×C1272
#2121127