Finite Group Databasebeta

← #30641989

Subgroups — #30641989

Order 16256.

560 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861168C211yesmaximal, characteristic, complemented
2#22549815C211yesmaximal, characteristic, complemented
3#22549812C211yesmaximal, characteristic, complemented
4#22549806C211yesmaximal, characteristic, complemented
5#22549804C212yesmaximal, complemented
6#22549801C211yesmaximal, characteristic, complemented
7#22549790C212yesmaximal
8C4×Dic127⋊C4C212yesmaximal, complemented
9C4×C22⋊Dic127C212yesmaximal, complemented
10C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
11C2×C4×D4×C127C211yesmaximal, characteristic
12#16861168—24no—
13C2×C4⋊Dic127C2211yescharacteristic
14C2×C4⋊Dic127C2211yescharacteristic
15C2×Dic127⋊C4C2212yes—
16C2×Dic127⋊C4C2212yes—
17C2×C22⋊Dic127C2212yes—
18C2×C22⋊Dic127C2212yes—
19C2×C22⋊Dic127C2212yes—
20C2×C22⋊Dic127C2212yes—
21C2×C4×Dic127C2212yes—
22C2×C4×Dic127C2212yes—
23C127⋊#95C2212yes—
24C127⋊#95C2212yes—
25C127⋊#95C2211yescharacteristic
26C127⋊#95C2212yes—
27C127⋊#95C2212yes—
28C127⋊#95C2211yescharacteristic
29C22×D4×C127C2211yescharacteristic
30C23×C4×C127C2211yescharacteristic
31C23×C4×C127C2211yescharacteristic
32C4×D4×C127—24no—
33C2×C4⋊C4×C127C2211yescharacteristic
34C2×C22⋊C4×C127C2211yescharacteristic
35C2×C22⋊C4×C127C2211yescharacteristic
36C2×C42×C127C2211yescharacteristic
37C22×Dic127C2312yes—
38C22×Dic127C2312yes—
39C22×Dic127C2312yes—
40C22×Dic127C2312yes—
41C4⋊Dic127—22no—
42C4⋊Dic127—22no—
43Dic127⋊C4—24no—
44Dic127⋊C4—24no—
45C22⋊Dic127—24no—
46C22⋊Dic127—24no—
47C22⋊Dic127—24no—
48C22⋊Dic127—24no—
49C4×Dic127—24no—
50C4×Dic127—24no—
51C24×C127C2311yescharacteristic
52C24×C127C2311yescharacteristic
53C2×D4×C127—22no—
54C2×D4×C127—22no—
55C2×D4×C127—22no—
56C22×C4×C127—24no—
57C22×C4×C127—22no—
58C22×C4×C127—24no—
59C22×C4×C127D414yes—
60C22×C4×C127C2311yescharacteristic
61C22×C4×C127C2311yescharacteristic
62C22×C4×C127C2311yescharacteristic
63C22×C4×C127C2311yescharacteristic
64C22×C4×C127C2311yescharacteristic
65C4⋊C4×C127—22no—
66C22⋊C4×C127—22no—
67C22⋊C4×C127—22no—
68C42×C127—22no—
69C2×Dic127—22no—
70C2×Dic127—22no—
71C2×Dic127—22no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C2×Dic127—22no—
80C2×Dic127#4114yes—
81C23×C127—21no—
82C23×C127—22no—
83C23×C127—22no—
84C23×C127—21no—
85C23×C127—21no—
86C23×C127—21no—
87C23×C127—22no—
88C23×C127—22no—
89C23×C127C2×D412yes—
90C23×C127#4112yes—
91C23×C127C2411yescharacteristic
92D4×C127—28no—
93C2×C4×C127—21no—
94C2×C4×C127—22no—
95C2×C4×C127—21no—
96C2×C4×C127—28no—
97C2×C4×C127—21no—
98C2×C4×C127—21no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—44no—
102C2×C4×C127C2×D412yes—
103C2×C4×C127C2×D412yes—
104C2×C4×C127#4114yes—
105C2×C4×C127#4112yes—
106C22×C127—22no—
107C22×C127—22no—
108C22×C127—22no—
109C22×C127—24no—
110C22×C127—22no—
111C22×C127—41no—
112C22×C127—42no—
113C22×C127—42no—
114C22×C127—42no—
115C22×C127—41no—
116C22×C127C22×D411yescharacteristic
117C22×C127C2×#4112yes—
118C22×C127C2×#4112yes—
119C22×C127C2×#4111yescharacteristic
120C22×C127C2×#4111yescharacteristic
121C4×C127—22no—
122C4×C127—22no—
123C4×C127—41no—
124C4×C127—41no—
125C4×C127#12314yes—
126Dic127—24no—
127Dic127—42no—
128Dic127—42no—
129Dic127—42no—
130C2×C127—42no—
131C2×C127—42no—
132C2×C127C2×#12311yescharacteristic
133C2×C127#52511yescharacteristic
134C2×C127#52912yes—
135C2×C127#52912yes—
136C2×C127#53211yescharacteristic
137#2395—1271nomaximal, Sylow
138C127#239511yescharacteristic, complemented, Sylow
139C2×C4×D4D12711yescharacteristic
140C2×#117—1271no—
141#400—1271no—
142#399—1271no—
143#398—1271no—
144#397—1271no—
145#397—1272no—
146#394—1272no—
147C4×C4⋊C4—1272no—
148C4×C22⋊C4—1272no—
149C4×C22⋊C4—1271no—
150C22×D4C2×D12711yescharacteristic
151C23×C4C2×D12711yescharacteristic
152C23×C4C2×D12711yescharacteristic
153C4×D4—24no—
154#117—2544no—
155C2×C4⋊C4—1271no—
156C2×C4⋊C4—1271no—
157C2×C4⋊C4—1272no—
158C2×C4⋊C4—1272no—
159C2×C4⋊C4C2×D12711yescharacteristic
160C2×C22⋊C4—1272no—
161C2×C22⋊C4—1272no—
162C2×C22⋊C4—1272no—
163C2×C22⋊C4—1272no—
164C2×C22⋊C4C2×D12711yescharacteristic
165C2×C22⋊C4C2×D12711yescharacteristic
166C2×C42—1272no—
167C2×C42—1272no—
168C2×C42C2×D12711yescharacteristic
169#95—1272no—
170#95—1271no—
171#95—1272no—
172#95—1271no—
173#95—1272no—
174#95—1272no—
175C24C22×D12711yescharacteristic
176C24C22×D12711yescharacteristic
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—22no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—1272no—
184C22×C4—1272no—
185C22×C4—1272no—
186C22×C4—1272no—
187C22×C4C22⋊D12714yes—
188C22×C4C22×D12711yescharacteristic
189C22×C4C22×D12711yescharacteristic
190C22×C4C22×D12711yescharacteristic
191C22×C4C22×D12711yescharacteristic
192C22×C4C22×D12711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—2544no—
195C4⋊C4—2542no—
196C4⋊C4—2542no—
197C4⋊C4—2544no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—2544no—
201C22⋊C4—2544no—
202C22⋊C4—2544no—
203C22⋊C4—2544no—
204C42—22no—
205C42—2544no—
206C42—2544no—
207C23—21no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23C2×C22⋊D12712yes—
216C23#1452324812yes—
217C23C23×D12711yescharacteristic
218D4—28no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—28no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—44no—
228C2×C4—1274no—
229C2×C4—2542no—
230C2×C4—2542no—
231C2×C4—2542no—
232C2×C4—2542no—
233C2×C4—2542no—
234C2×C4—2542no—
235C2×C4—2542no—
236C2×C4—2542no—
237C2×C4—2542no—
238C2×C4—2542no—
239C2×C4—2542no—
240C2×C4C2×C22⋊D12712yes—
241C2×C4C2×C22⋊D12712yes—
242C2×C4#1452324714yes—
243C2×C4#1452324812yes—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—24no—
248C22—22no—
249C22—41no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—41no—
254C22C22×C22⋊D12711yescharacteristic
255C22#41×D12712yes—
256C22C2×#1452324712yes—
257C22C2×#1452324811yescharacteristic
258C22C2×#1452324811yescharacteristic
259C4—22no—
260C4—22no—
261C4—41no—
262C4—41no—
263C4—5082no—
264C4—5082no—
265C4—5082no—
266C4—2544no—
267C4#1686119014yes—
268C2—42no—
269C2—42no—
270C2C2×#1686119011yescharacteristic
271C2#2255038711yescharacteristic
272C2#2255040612yes—
273C2#2255040812yes—
274C2#2255042211yescharacteristic
subgroupG/H
centerC22×C4C22×D127
commutator, socle, cosocleC23×C127C24
FrattiniC23C23×D127
FittingC2×C4×D4×C127C2
prime radicalC22×D4×C127C22
hypercenterC2×C4×D4D127
hypocenterC127#2395
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16861168C2semidirect
2. #22549815C2semidirect
3. #22549812C2semidirect
4. #22549806C2semidirect
5. #22549804C2semidirect
6. #22549801C2semidirect
8. C4×Dic127⋊C4C2semidirect
9. C4×C22⋊Dic127C2semidirect
10. C4×C22⋊Dic127C2semidirect
138. C127#2395semidirect

Maximal subgroups

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

subgroupindex
C2×#168611682
#225498152
#225498122
#225498062
#225498042
#225498012
#225497902
C4×Dic127⋊C42
C4×C22⋊Dic1272
C2×C4×D4×C1272
#2395127