Finite Group Databasebeta

← #30641423

Subgroups — #30641423

Order 16256.

668 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861183C211yesmaximal, characteristic
2C2×C4×D508C211yesmaximal, characteristic, complemented
3#22549778C211yesmaximal, characteristic, complemented
4#22549776C212yesmaximal, complemented
5#22549774C211yesmaximal, characteristic, complemented
6#22549766C212yesmaximal, complemented
7#22549752C212yesmaximal, complemented
8#22549747C212yesmaximal, complemented
9#22549736C211yesmaximal, characteristic, complemented
10C4×C4⋊Dic127C211yesmaximal, characteristic, complemented
11C2×#117×C127C211yesmaximal, characteristic, complemented
12C22×D508C2211yescharacteristic
13C22×C22⋊D127C2212yes—
14C22×C4×D127C2212yes—
15#16861183C418yescomplemented
16C4×D508—24no—
17C2×C4⋊Dic127C2211yescharacteristic
18C2×C4⋊Dic127C2211yescharacteristic
19C2×C4⋊Dic127C2211yescharacteristic
20C2×Dic127⋊C4C2212yes—
21C2×C22⋊Dic127C2212yes—
22C2×(C2×D127)⋊C4C2212yes—
23C2×(C2×D127)⋊C4C2212yes—
24C2×(C2×D127)⋊C4C2212yes—
25C2×(C2×D127)⋊C4C2212yes—
26C2×C4×Dic127C2212yes—
27C127⋊#95C2212yes—
28C127⋊#95C2212yes—
29C127⋊#95C2212yes—
30C23×C4×C127C2211yescharacteristic
31#117×C127—24no—
32C2×C4⋊C4×C127C2211yescharacteristic
33C2×C4⋊C4×C127C2211yescharacteristic
34C2×C22⋊C4×C127C2211yescharacteristic
35C2×C22⋊C4×C127C2211yescharacteristic
36C2×C42×C127C2211yescharacteristic
37C2×C42×C127C2211yescharacteristic
38C23×D127C2312yes—
39C2×D508—22no—
40C2×D508—22no—
41C2×D508C2×C414yes—
42C2×C22⋊D127—42no—
43C2×C22⋊D127—42no—
44C2×C22⋊D127C2×C418yescomplemented
45C22×Dic127C2312yes—
46C22×Dic127C2312yes—
47C22×Dic127C2312yes—
48C2×C4×D127—24no—
49C2×C4×D127—42no—
50C2×C4×D127—42no—
51C4⋊Dic127—22no—
52C4⋊Dic127—22no—
53C4⋊Dic127C2×C414yescomplemented
54Dic127⋊C4—42no—
55C22⋊Dic127—42no—
56(C2×D127)⋊C4—24no—
57(C2×D127)⋊C4—42no—
58(C2×D127)⋊C4—42no—
59(C2×D127)⋊C4C2×C418yescomplemented
60C4×Dic127—42no—
61C24×C127C2311yescharacteristic
62C22×C4×C127—22no—
63C22×C4×C127—22no—
64C22×C4×C127C2×C414yescomplemented
65C22×C4×C127C2311yescharacteristic
66C22×C4×C127C2311yescharacteristic
67C22×C4×C127C2311yescharacteristic
68C22×C4×C127C2311yescharacteristic
69C22×C4×C127C2311yescharacteristic
70C22×C4×C127C2311yescharacteristic
71C4⋊C4×C127—22no—
72C4⋊C4×C127—22no—
73C22⋊C4×C127—22no—
74C22⋊C4×C127—22no—
75C42×C127—22no—
76C42×C127D414yescomplemented
77C22×D127—22no—
78C22×D127—22no—
79C22×D127—22no—
80C22×D127—22no—
81C22×D127—22no—
82C22×D127—22no—
83C22×D127C22×C414yes—
84D508—28no—
85C22⋊D127—48no—
86C2×Dic127—22no—
87C2×Dic127—22no—
88C2×Dic127—22no—
89C2×Dic127—22no—
90C2×Dic127—22no—
91C2×Dic127—22no—
92C2×Dic127—22no—
93C2×Dic127—22no—
94C2×Dic127C22×C414yes—
95C4×D127—48no—
96C23×C127—21no—
97C23×C127—21no—
98C23×C127—21no—
99C23×C127—21no—
100C23×C127—21no—
101C23×C127—21no—
102C23×C127C22×C412yes—
103C23×C127C2411yescharacteristic
104C2×C4×C127—21no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127—28no—
111C2×C4×C127—21no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127C22×C412yes—
118C2×C4×C127C22×C412yes—
119C2×C4×C127C2×D412yes—
120C2×C4×C127C2×D412yes—
121C2×C4×C127C2×D412yes—
122C2×C4×C127#4112yes—
123C22×C127—22no—
124C22×C127—22no—
125C22×C127—22no—
126C22×C127—41no—
127C22×C127—41no—
128C22×C127—41no—
129C22×C127—41no—
130C22×C127C23×C411yescharacteristic
131C22×C127C22×D411yescharacteristic
132C22×C127C2×#4111yescharacteristic
133C22×C127H(5,𝔽2)11yescharacteristic
134C22×C127H(5,𝔽2)11yescharacteristic
135C22×C127H(5,𝔽2)11yescharacteristic
136C22×C1272−1+411yescharacteristic
137C2×D127—24no—
138C2×D127—24no—
139C2×D127—24no—
140C2×D127—42no—
141C2×D127—42no—
142C2×D127—42no—
143C2×D127—42no—
144C4×C127—22no—
145C4×C127—22no—
146C4×C127—22no—
147C4×C127—41no—
148C4×C127—41no—
149C4×C127C4×D414yes—
150Dic127—24no—
151Dic127—42no—
152Dic127—42no—
153C2×C127—42no—
154C2×C127C2×C4×D411yescharacteristic
155C2×C127#51811yescharacteristic
156C2×C127#52011yescharacteristic
157C2×C127#53511yescharacteristic
158C2×C127#53711yescharacteristic
159C2×C127#54011yescharacteristic
160C2×C127#54211yescharacteristic
161D127—44no—
162#2324—1271nomaximal, Sylow
163C127#232411yescharacteristic, complemented, Sylow
164C2×#121—1271no—
165C2×C4×D4—1271no—
166C2×#117D12711yescharacteristic, complemented
167#390—1272no—
168#390—1271no—
169#389—1271no—
170#388—1272no—
171#386—1272no—
172#385—1272no—
173#380—1271no—
174C4×C4⋊C4—1271no—
175C22×D4—1272no—
176C22×D4—1271no—
177C23×C4—1272no—
178C23×C4C2×D12711yescharacteristic
179#121—1278no—
180C4×D4—2544no—
181#117—24no—
182C2×C4⋊C4—1271no—
183C2×C4⋊C4—1272no—
184C2×C4⋊C4—1271no—
185C2×C4⋊C4—1271no—
186C2×C4⋊C4C2×D12711yescharacteristic
187C2×C4⋊C4C2×D12711yescharacteristic
188C2×C22⋊C4—1272no—
189C2×C22⋊C4—1272no—
190C2×C22⋊C4—1272no—
191C2×C22⋊C4—1272no—
192C2×C22⋊C4—1272no—
193C2×C22⋊C4C2×D12711yescharacteristic
194C2×C22⋊C4C2×D12711yescharacteristic
195C2×C42—1272no—
196C2×C42C2×D12711yescharacteristic
197C2×C42C2×D12711yescharacteristic
198#95—1272no—
199#95—1272no—
200#95—1272no—
201C24—1272no—
202C24C22×D12711yescharacteristic
203C2×D4—1278no—
204C2×D4—1274no—
205C2×D4—5082no—
206C2×D4—5082no—
207C2×D4—2542no—
208C2×D4—2542no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—1272no—
212C22×C4—1272no—
213C22×C4—1272no—
214C22×C4—5082no—
215C22×C4—5082no—
216C22×C4—2544no—
217C22×C4C4×D12714yescomplemented
218C22×C4C22×D12711yescharacteristic
219C22×C4C22×D12711yescharacteristic
220C22×C4C22×D12711yescharacteristic
221C22×C4C22×D12711yescharacteristic
222C22×C4C22×D12711yescharacteristic
223C22×C4C22×D12711yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—1274no—
227C4⋊C4—5082no—
228C4⋊C4—2542no—
229C4⋊C4—2542no—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—1278no—
233C22⋊C4—5082no—
234C22⋊C4—5082no—
235C22⋊C4—5082no—
236C22⋊C4—2544no—
237C42—22no—
238C42—5082no—
239C42C22⋊D12714yescomplemented
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—1274no—
247C23—2542no—
248C23—2542no—
249C23—2542no—
250C23—2542no—
251C23—2542no—
252C23—2542no—
253C23C2×C4×D12712yes—
254C23C23×D12711yescharacteristic
255D4—5088no—
256D4—2548no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—28no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—1274no—
271C2×C4—5088no—
272C2×C4—2542no—
273C2×C4—2542no—
274C2×C4—2542no—
275C2×C4—2542no—
276C2×C4—2542no—
277C2×C4—2542no—
278C2×C4—2542no—
279C2×C4—2542no—
280C2×C4C2×C4×D12712yes—
281C2×C4C2×C4×D12712yes—
282C2×C4C2×C22⋊D12712yes—
283C2×C4C2×C22⋊D12712yes—
284C2×C4C2×C22⋊D12712yes—
285C2×C4#1452324812yes—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—5082no—
294C22—5082no—
295C22—5082no—
296C22—5082no—
297C22—2544no—
298C22—2544no—
299C22—2544no—
300C22C22×C4×D12711yescharacteristic
301C22C22×C22⋊D12711yescharacteristic
302C22C2×#1452324811yescharacteristic
303C22#1686125511yescharacteristic
304C22#1686125511yescharacteristic
305C22#1686125511yescharacteristic
306C22#1686125611yescharacteristic
307C4—22no—
308C4—22no—
309C4—22no—
310C4—41no—
311C4—41no—
312C4—5082no—
313C4—5082no—
314C4—2544no—
315C4C4×C22⋊D12714yes—
316C2—42no—
317C2—5084no—
318C2C2×C4×C22⋊D12711yescharacteristic
319C2#2255035011yescharacteristic
320C2#2255035611yescharacteristic
321C2#2255044011yescharacteristic
322C2#2255045411yescharacteristic
323C2#2255048111yescharacteristic
324C2#2255049711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC2×#117×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861183C2
hypercenterC2×#117D127
hypocenterC127#2324
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×C4×D508C2semidirect
3. #22549778C2semidirect
4. #22549776C2semidirect
5. #22549774C2semidirect
6. #22549766C2semidirect
7. #22549752C2semidirect
8. #22549747C2semidirect
9. #22549736C2semidirect
10. C4×C4⋊Dic127C2semidirect
11. C2×#117×C127C2semidirect
15. #16861183C4semidirect
44. C2×C22⋊D127C2×C4semidirect
53. C4⋊Dic127C2×C4semidirect
59. (C2×D127)⋊C4C2×C4semidirect
64. C22×C4×C127C2×C4semidirect
76. C42×C127D4semidirect
163. C127#2324semidirect
166. C2×#117D127semidirect
217. C22×C4C4×D127semidirect
239. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611832
C2×C4×D5082
#225497782
#225497762
#225497742
#225497662
#225497522
#225497472
#225497362
C4×C4⋊Dic1272
C2×#117×C1272
#2324127