Finite Group Databasebeta

← #30641968

Subgroups — #30641968

Order 16256.

696 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D127C212yesmaximal, complemented
2C2×#16861166C211yesmaximal, characteristic, complemented
3#22549807C211yesmaximal, characteristic, complemented
4#22549802C211yesmaximal, characteristic, complemented
5#22549799C212yesmaximal, complemented
6#22549792C212yesmaximal, complemented
7#22549790C211yesmaximal, characteristic, complemented
8#22549784C211yesmaximal, characteristic
9C4×C4⋊Dic127C211yesmaximal, characteristic, complemented
10C4×C22⋊Dic127C212yesmaximal, complemented
11C2×C4×D4×C127C211yesmaximal, characteristic, complemented
12C22×C22⋊D127C2212yes—
13C22×C4×D127C2211yescharacteristic
14C4×C22⋊D127—28no—
15#16861166—24no—
16C2×C4⋊Dic127C2212yes—
17C2×Dic127⋊C4C2211yescharacteristic
18C2×Dic127⋊C4C2211yescharacteristic
19C2×Dic127⋊C4C2212yes—
20C2×C22⋊Dic127C2212yes—
21C2×C22⋊Dic127C2212yes—
22C2×C22⋊Dic127C2212yes—
23C2×(C2×D127)⋊C4C2211yescharacteristic
24C2×(C2×D127)⋊C4C2211yescharacteristic
25C2×(C2×D127)⋊C4C2212yes—
26C2×C4×Dic127C2211yescharacteristic
27C2×C4×Dic127C2212yes—
28C127⋊#95C2211yescharacteristic
29C127⋊#95C2211yescharacteristic
30C127⋊#95C2212yes—
31C127⋊#95C2212yes—
32C22×D4×C127C2211yescharacteristic
33C23×C4×C127C2212yes—
34C4×D4×C127—24no—
35C2×C4⋊C4×C127C2211yescharacteristic
36C2×C22⋊C4×C127C2212yes—
37C2×C42×C127C2211yescharacteristic
38C23×D127C2311yescharacteristic
39C2×C22⋊D127—24no—
40C2×C22⋊D127—24no—
41C2×C22⋊D127—24no—
42C22×Dic127C2311yescharacteristic
43C22×Dic127C2311yescharacteristic
44C22×Dic127C2312yes—
45C22×Dic127C2311yescharacteristic
46C22×Dic127C2312yes—
47C2×C4×D127—22no—
48C2×C4×D127—22no—
49C2×C4×D127D414yescomplemented
50C4⋊Dic127—24no—
51Dic127⋊C4—24no—
52Dic127⋊C4—22no—
53Dic127⋊C4—22no—
54C22⋊Dic127—24no—
55C22⋊Dic127—24no—
56C22⋊Dic127—24no—
57(C2×D127)⋊C4—24no—
58(C2×D127)⋊C4—22no—
59(C2×D127)⋊C4—22no—
60C4×Dic127—22no—
61C4×Dic127—24no—
62C24×C127C2312yes—
63C2×D4×C127—24no—
64C2×D4×C127—22no—
65C22×C4×C127—24no—
66C22×C4×C127—24no—
67C22×C4×C127—24no—
68C22×C4×C127C2312yes—
69C22×C4×C127C2311yescharacteristic
70C22×C4×C127C2311yescharacteristic
71C22×C4×C127C2311yescharacteristic
72C4⋊C4×C127—22no—
73C22⋊C4×C127—24no—
74C42×C127D414yescomplemented
75C22×D127—21no—
76C22×D127—21no—
77C22×D127—22no—
78C22×D127—22no—
79C22×D127C2×D412yes—
80C22⋊D127—216no—
81C2×Dic127—22no—
82C2×Dic127—21no—
83C2×Dic127—21no—
84C2×Dic127—22no—
85C2×Dic127—22no—
86C2×Dic127—21no—
87C2×Dic127—21no—
88C2×Dic127—22no—
89C2×Dic127—22no—
90C2×Dic127—22no—
91C2×Dic127—22no—
92C2×Dic127C2×D412yes—
93C2×Dic127#4114yes—
94C4×D127—28no—
95C23×C127—22no—
96C23×C127—22no—
97C23×C127—22no—
98C23×C127—22no—
99C23×C127—22no—
100C23×C127—22no—
101C23×C127#4114yes—
102C23×C127C2411yescharacteristic
103D4×C127—28no—
104C2×C4×C127—22no—
105C2×C4×C127—22no—
106C2×C4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—48no—
110C2×C4×C127C2×D412yes—
111C2×C4×C127C2×D412yes—
112C2×C4×C127C2×D412yes—
113C2×C4×C127C2×D412yes—
114C2×C4×C127#4112yes—
115C22×C127—24no—
116C22×C127—24no—
117C22×C127—24no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127—42no—
121C22×C127—42no—
122C22×C127C22×D411yescharacteristic
123C22×C127C22×D411yescharacteristic
124C22×C127C2×#4112yes—
125C22×C127C2×#4111yescharacteristic
126C22×C127C2×#4112yes—
127C2×D127—22no—
128C2×D127—24no—
129C2×D127—41no—
130C2×D127—42no—
131C2×D127—41no—
132C4×C127—22no—
133C4×C127—22no—
134C4×C127—42no—
135C4×C127#12114yes—
136Dic127—22no—
137Dic127—24no—
138Dic127—42no—
139Dic127—41no—
140Dic127—41no—
141C2×C127—44no—
142C2×C127C2×#12111yescharacteristic
143C2×C127#52512yes—
144C2×C127#52511yescharacteristic
145C2×C127#52912yes—
146C2×C127#53211yescharacteristic
147D127—42no—
148#2392—1271nomaximal, Sylow
149C127#239211yescharacteristic, complemented, Sylow
150C2×C4×D4—1272no—
151C2×C4×D4D12711yescharacteristic, complemented
152C2×#117—1271no—
153#398—1271no—
154#397—1271no—
155#396—1272no—
156#394—1271no—
157#394—1272no—
158#392—1271no—
159C4×C4⋊C4—1271no—
160C4×C22⋊C4—1272no—
161C22×D4—1272no—
162C22×D4C2×D12711yescharacteristic
163C23×C4—1271no—
164C23×C4C2×D12712yes—
165C4×D4—24no—
166C4×D4—2548no—
167#117—2544no—
168C2×C4⋊C4—1271no—
169C2×C4⋊C4—1271no—
170C2×C4⋊C4—1272no—
171C2×C4⋊C4—1272no—
172C2×C4⋊C4C2×D12711yescharacteristic
173C2×C22⋊C4—1272no—
174C2×C22⋊C4—1272no—
175C2×C22⋊C4—1272no—
176C2×C22⋊C4—1271no—
177C2×C22⋊C4—1271no—
178C2×C22⋊C4—1272no—
179C2×C22⋊C4C2×D12712yes—
180C2×C42—1271no—
181C2×C42—1272no—
182C2×C42C2×D12711yescharacteristic
183#95—1272no—
184#95—1272no—
185#95—1271no—
186#95—1271no—
187C24—1271no—
188C24C22×D12712yes—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—2544no—
192C2×D4—2544no—
193C2×D4—2544no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—1272no—
198C22×C4—1271no—
199C22×C4—1274no—
200C22×C4—1272no—
201C22×C4—1271no—
202C22×C4—1271no—
203C22×C4—2542no—
204C22×C4—2542no—
205C22×C4C22×D12712yes—
206C22×C4C22×D12711yescharacteristic
207C22×C4C22×D12711yescharacteristic
208C22×C4C22×D12711yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—2544no—
211C4⋊C4—2544no—
212C4⋊C4—2542no—
213C4⋊C4—2542no—
214C22⋊C4—24no—
215C22⋊C4—2544no—
216C22⋊C4—2544no—
217C22⋊C4—2544no—
218C22⋊C4—2544no—
219C22⋊C4—2542no—
220C22⋊C4—2542no—
221C42—2544no—
222C42—2542no—
223C42C22⋊D12714yescomplemented
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—1272no—
231C23—2541no—
232C23—2542no—
233C23—2541no—
234C23—2542no—
235C23#1452324814yes—
236C23C23×D12711yescharacteristic
237D4—28no—
238D4—25416no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—48no—
245C2×C4—1272no—
246C2×C4—1274no—
247C2×C4—2542no—
248C2×C4—2541no—
249C2×C4—2541no—
250C2×C4—2542no—
251C2×C4—2542no—
252C2×C4—2542no—
253C2×C4—2542no—
254C2×C4—2542no—
255C2×C4—2541no—
256C2×C4—2541no—
257C2×C4—2542no—
258C2×C4—2548no—
259C2×C4D4×D12712yes—
260C2×C4C2×C22⋊D12712yes—
261C2×C4C2×C22⋊D12712yes—
262C2×C4C2×C22⋊D12712yes—
263C2×C4#1452324712yes—
264C22—24no—
265C22—24no—
266C22—24no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—5081no—
272C22—5082no—
273C22—5081no—
274C22—2542no—
275C22—2544no—
276C22C2×D4×D12711yescharacteristic
277C22C22×C22⋊D12711yescharacteristic
278C22#41×D12712yes—
279C22C2×#1452324711yescharacteristic
280C22C2×#1452324812yes—
281C4—22no—
282C4—22no—
283C4—42no—
284C4—5082no—
285C4—5081no—
286C4—5081no—
287C4—2542no—
288C4—2544no—
289C4#1686118014yes—
290C2—44no—
291C2—5082no—
292C2C2×#1686118011yescharacteristic
293C2#2255038512yes—
294C2#2255038711yescharacteristic
295C2#2255040612yes—
296C2#2255042211yescharacteristic
subgroupG/H
centerC22×C4C22×D127
commutator, socle, cosocleC23×C127C24
FrattiniC23C23×D127
FittingC2×C4×D4×C127C2
prime radical#22549784C2
hypercenterC2×C4×D4D127
hypocenterC127#2392
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×C4×C22⋊D127C2semidirect
2. C2×#16861166C2semidirect
3. #22549807C2semidirect
4. #22549802C2semidirect
5. #22549799C2semidirect
6. #22549792C2semidirect
7. #22549790C2semidirect
9. C4×C4⋊Dic127C2semidirect
10. C4×C22⋊Dic127C2semidirect
11. C2×C4×D4×C127C2semidirect
49. C2×C4×D127D4semidirect
74. C42×C127D4semidirect
149. C127#2392semidirect
151. C2×C4×D4D127semidirect
223. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1272
C2×#168611662
#225498072
#225498022
#225497992
#225497922
#225497902
#225497842
C4×C4⋊Dic1272
C4×C22⋊Dic1272
C2×C4×D4×C1272
#2392127