Finite Group Databasebeta

← #398×D127 #30641959

Subgroups — #398×D127 #30641959

Order 16256.

692 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D127C211yesmaximal, characteristic
2C2×C4⋊C4×D127C212yesmaximal
3C2×C4⋊C4×D127C211yesmaximal, characteristic
4#22549810C212yesmaximal, complemented
5#22549809C212yesmaximal, complemented
6#22549808C211yesmaximal, characteristic, complemented
7#22549807C211yesmaximal, characteristic, complemented
8#22549806C211yesmaximal, characteristic, complemented
9#95×D127C212yesmaximal
10#95×D127C211yesmaximal, characteristic
11#398×C127C211yesmaximal, characteristic, complemented
12C22×C4×D127C2212yes—
13C22×C4×D127C2211yescharacteristic
14C22×C4×D127C2211yescharacteristic
15C22×C4×D127C2211yescharacteristic
16C22×C4×D127C2212yes—
17C2×C4⋊Dic127C2211yescharacteristic
18C2×C4⋊Dic127C2212yes—
19C2×C4⋊Dic127C2211yescharacteristic
20C2×Dic127⋊C4C2212yes—
21C2×Dic127⋊C4C2211yescharacteristic
22C2×Dic127⋊C4C2212yes—
23C2×Dic127⋊C4C2212yes—
24C2×Dic127⋊C4C2211yescharacteristic
25C4⋊C4×D127—24no—
26C4⋊C4×D127—28no—
27C4⋊C4×D127—24no—
28C127⋊#95C2211yescharacteristic
29C127⋊#95C2212yes—
30C127⋊#95C2212yes—
31C127⋊#95C2212yes—
32C127⋊#95C2212yes—
33C2×C4⋊C4×C127C2211yescharacteristic
34C2×C4⋊C4×C127C2211yescharacteristic
35C2×C4⋊C4×C127C2212yes—
36#95×C127C2211yescharacteristic
37#95×C127C2212yes—
38C23×D127C2311yescharacteristic
39C22×Dic127C2311yescharacteristic
40C22×Dic127C2312yes—
41C22×Dic127C2311yescharacteristic
42C22×Dic127C2311yescharacteristic
43C22×Dic127C2312yes—
44C2×C4×D127—22no—
45C2×C4×D127—24no—
46C2×C4×D127—22no—
47C2×C4×D127—24no—
48C2×C4×D127—24no—
49C2×C4×D127—24no—
50C2×C4×D127—24no—
51C2×C4×D127—22no—
52C2×C4×D127—24no—
53C2×C4×D127—24no—
54C2×C4×D127D414yes—
55C2×C4×D127D414yes—
56C2×C4×D127Q818yes—
57C4⋊Dic127—22no—
58C4⋊Dic127—24no—
59C4⋊Dic127—22no—
60Dic127⋊C4—22no—
61Dic127⋊C4—24no—
62Dic127⋊C4—24no—
63Dic127⋊C4—24no—
64Dic127⋊C4—22no—
65C22×C4×C127C2311yescharacteristic
66C22×C4×C127C2312yes—
67C22×C4×C127C2311yescharacteristic
68C22×C4×C127C2311yescharacteristic
69C22×C4×C127C2312yes—
70C4⋊C4×C127—24no—
71C4⋊C4×C127—22no—
72C4⋊C4×C127—22no—
73C22×D127C2×D412yes—
74C22×D127C2×D412yes—
75C22×D127C2×Q814yes—
76C22×D127#4112yes—
77C22×D127#4114yes—
78C2×Dic127—21no—
79C2×Dic127—22no—
80C2×Dic127—21no—
81C2×Dic127—22no—
82C2×Dic127—22no—
83C2×Dic127—22no—
84C2×Dic127—22no—
85C2×Dic127—21no—
86C2×Dic127—22no—
87C2×Dic127—22no—
88C2×Dic127C2×D412yes—
89C2×Dic127C2×D412yes—
90C2×Dic127C2×Q814yes—
91C4×D127—28no—
92C4×D127—216no—
93C4×D127—28no—
94C4×D127—48no—
95C4×D127—44no—
96C23×C127C2411yescharacteristic
97C2×C4×C127—21no—
98C2×C4×C127—22no—
99C2×C4×C127—21no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127—21no—
105C2×C4×C127—22no—
106C2×C4×C127—22no—
107C2×C4×C127C2×D412yes—
108C2×C4×C127C2×D412yes—
109C2×C4×C127C2×Q814yes—
110C22×C127C22×D411yescharacteristic
111C22×C127C22×D411yescharacteristic
112C22×C127C22×Q812yes—
113C22×C127C2×#4111yescharacteristic
114C22×C127C2×#4112yes—
115C2×D127#12114yes—
116C2×D127C22⋊Q818yes—
117C2×D127#12314yes—
118C2×D127#12518yes—
119C2×D127C4⋊Q814yes—
120C4×C127—22no—
121C4×C127—24no—
122C4×C127—22no—
123C4×C127—42no—
124C4×C127—41no—
125Dic127—22no—
126Dic127—24no—
127Dic127—22no—
128Dic127—42no—
129Dic127—41no—
130C2×C127C2×#12111yescharacteristic
131C2×C127C2×C22⋊Q812yes—
132C2×C127C2×#12311yescharacteristic
133C2×C127C2×#12512yes—
134C2×C127C2×C4⋊Q811yescharacteristic
135D127#39818yesdirect product factor
136C2×#398—1271nomaximal, Sylow
137C127C2×#39811yescharacteristic, complemented, Sylow
138C22×C4⋊C4—1271no—
139C22×C4⋊C4—1272no—
140C22×C4⋊C4—1271no—
141#398—1271no—
142#398—1271no—
143#398—1272no—
144#398—1271no—
145#398—1272no—
146#398D12711yescharacteristic, direct product factor
147C2×#95—1271no—
148C2×#95—1272no—
149C23×C4—1271no—
150C23×C4—1271no—
151C23×C4—1272no—
152C23×C4—1272no—
153C23×C4—1271no—
154C2×C4⋊C4—1272no—
155C2×C4⋊C4—1271no—
156C2×C4⋊C4—1272no—
157C2×C4⋊C4—1272no—
158C2×C4⋊C4—1271no—
159C2×C4⋊C4—1271no—
160C2×C4⋊C4—1272no—
161C2×C4⋊C4—1271no—
162C2×C4⋊C4—2544no—
163C2×C4⋊C4—2548no—
164C2×C4⋊C4—2544no—
165C2×C4⋊C4C2×D12711yescharacteristic
166C2×C4⋊C4C2×D12711yescharacteristic
167C2×C4⋊C4C2×D12712yes—
168#95—1271no—
169#95—1272no—
170#95—1272no—
171#95—1272no—
172#95—1272no—
173#95C2×D12712yes—
174#95C2×D12711yescharacteristic
175C24—1271no—
176C22×C4—1272no—
177C22×C4—1271no—
178C22×C4—1271no—
179C22×C4—1274no—
180C22×C4—1274no—
181C22×C4—1278no—
182C22×C4—1271no—
183C22×C4—1272no—
184C22×C4—2544no—
185C22×C4—2544no—
186C22×C4—2542no—
187C22×C4—2544no—
188C22×C4—2544no—
189C22×C4—2542no—
190C22×C4—2544no—
191C22×C4—2542no—
192C22×C4—2544no—
193C22×C4—2544no—
194C22×C4C22×D12711yescharacteristic
195C22×C4C22×D12712yes—
196C22×C4C22×D12711yescharacteristic
197C22×C4C22×D12711yescharacteristic
198C22×C4C22×D12712yes—
199C4⋊C4—22no—
200C4⋊C4—24no—
201C4⋊C4—22no—
202C4⋊C4—2544no—
203C4⋊C4—2544no—
204C4⋊C4—2542no—
205C4⋊C4—2542no—
206C4⋊C4—2544no—
207C4⋊C4—2544no—
208C4⋊C4—2542no—
209C4⋊C4—2542no—
210C23—1272no—
211C23—1272no—
212C23—1274no—
213C23—1272no—
214C23—1274no—
215C23C23×D12711yescharacteristic
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—1272no—
227C2×C4—1272no—
228C2×C4—1274no—
229C2×C4—5088no—
230C2×C4—5084no—
231C2×C4—2548no—
232C2×C4—25416no—
233C2×C4—2548no—
234C2×C4—2542no—
235C2×C4—2542no—
236C2×C4—2541no—
237C2×C4—2542no—
238C2×C4—2542no—
239C2×C4—2541no—
240C2×C4—2542no—
241C2×C4—2541no—
242C2×C4—2542no—
243C2×C4—2542no—
244C2×C4D4×D12712yes—
245C2×C4D4×D12712yes—
246C2×C4Q8×D12714yes—
247C22—1274no—
248C22—1278no—
249C22—1274no—
250C22—1278no—
251C22—1274no—
252C22C2×D4×D12711yescharacteristic
253C22C2×D4×D12711yescharacteristic
254C22C2×Q8×D12712yes—
255C22#41×D12711yescharacteristic
256C22#41×D12712yes—
257C4—22no—
258C4—24no—
259C4—22no—
260C4—42no—
261C4—41no—
262C4—5082no—
263C4—5081no—
264C4—2542no—
265C4—2544no—
266C4—2542no—
267C2—1278no—
268C2#121×D12711yescharacteristic
269C2C22⋊Q8×D12712yes—
270C2#123×D12711yescharacteristic
271C2#125×D12712yes—
272C2C4⋊Q8×D12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#398×C127C2
prime radicalC23×D127C23
hypercenter#398D127
hypocenterC127C2×#398
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
4. #22549810C2semidirect
5. #22549809C2semidirect
6. #22549808C2semidirect
7. #22549807C2semidirect
8. #22549806C2semidirect
11. #398×C127C2semidirect
135. D127#398direct (N × H)
137. C127C2×#398semidirect
146. #398D127direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1272
#225498102
#225498092
#225498082
#225498072
#225498062
#95×D1272
#398×C1272
C2×#398127