Finite Group Databasebeta

← #30641945

Subgroups — #30641945

Order 16256.

944 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523248C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D127C212yesmaximal, complemented
3#22549796C211yesmaximal, characteristic, complemented
4#22549793C211yesmaximal, characteristic, complemented
5#22549791C212yesmaximal, complemented
6#22549788C211yesmaximal, characteristic, complemented
7#22549785C211yesmaximal, characteristic, complemented
8#22549784C212yesmaximal, complemented
9C4×(C2×D127)⋊C4C212yesmaximal, complemented
10C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
11C2×C4×D4×C127C211yesmaximal, characteristic, complemented
12C2×#14523248—28no—
13C2×#14523248—24no—
14C22×Dic254C2211yescharacteristic
15C22×D508C2211yescharacteristic
16C22×C22⋊D127C2212yes—
17C22×C22⋊D127C2212yes—
18C22×C4×D127C2212yes—
19C4×C22⋊D127—28no—
20C2×Dic127⋊C4C2212yes—
21C2×C22⋊Dic127C2212yes—
22C2×C22⋊Dic127C2212yes—
23C2×(C2×D127)⋊C4C2212yes—
24C2×(C2×D127)⋊C4C2212yes—
25C2×(C2×D127)⋊C4C2212yes—
26C2×C4×Dic127C2212yes—
27C127⋊#95C2212yes—
28C127⋊#95C2211yescharacteristic
29C127⋊#95C2212yes—
30C127⋊#95C2211yescharacteristic
31C22×D4×C127C2211yescharacteristic
32C23×C4×C127C2211yescharacteristic
33C23×C4×C127C2211yescharacteristic
34C4×D4×C127—24no—
35C2×C4⋊C4×C127C2211yescharacteristic
36C2×C22⋊C4×C127C2211yescharacteristic
37C2×C22⋊C4×C127C2211yescharacteristic
38C2×C42×C127C2211yescharacteristic
39C23×D127C2312yes—
40#14523248—232no—
41C2×Dic254—24no—
42C2×Dic254—22no—
43C2×D508—24no—
44C2×D508—22no—
45C2×C22⋊D127—24no—
46C2×C22⋊D127—24no—
47C2×C22⋊D127—24no—
48C2×C22⋊D127—24no—
49C2×C22⋊D127—24no—
50C2×C22⋊D127—24no—
51C22×Dic127C2312yes—
52C22×Dic127C2312yes—
53C22×Dic127C2312yes—
54C2×C4×D127—24no—
55C2×C4×D127—24no—
56C2×C4×D127D418yescomplemented
57Dic127⋊C4—24no—
58C22⋊Dic127—24no—
59C22⋊Dic127—24no—
60(C2×D127)⋊C4—24no—
61(C2×D127)⋊C4—24no—
62(C2×D127)⋊C4—24no—
63C4×Dic127—24no—
64C24×C127C2311yescharacteristic
65C24×C127C2311yescharacteristic
66C2×D4×C127—22no—
67C2×D4×C127—22no—
68C2×D4×C127—22no—
69C22×C4×C127—24no—
70C22×C4×C127—22no—
71C22×C4×C127—24no—
72C22×C4×C127D414yescomplemented
73C22×C4×C127C2311yescharacteristic
74C22×C4×C127C2311yescharacteristic
75C22×C4×C127C2311yescharacteristic
76C22×C4×C127C2311yescharacteristic
77C22×C4×C127C2311yescharacteristic
78C4⋊C4×C127—22no—
79C22⋊C4×C127—22no—
80C22⋊C4×C127—22no—
81C42×C127—22no—
82C22×D127—22no—
83C22×D127—22no—
84C22×D127—22no—
85C22×D127—22no—
86C22×D127—22no—
87C22×D127—22no—
88C22×D127C2×D414yes—
89Dic254—28no—
90D508—28no—
91C22⋊D127—216no—
92C22⋊D127—216no—
93C2×Dic127—22no—
94C2×Dic127—22no—
95C2×Dic127—22no—
96C2×Dic127—22no—
97C2×Dic127—22no—
98C2×Dic127—22no—
99C2×Dic127—22no—
100C2×Dic127C2×D414yes—
101C2×Dic127#4114yescomplemented
102C4×D127—216no—
103C23×C127—21no—
104C23×C127—22no—
105C23×C127—22no—
106C23×C127—21no—
107C23×C127—21no—
108C23×C127—21no—
109C23×C127—22no—
110C23×C127—22no—
111C23×C127C2×D412yes—
112C23×C127#4112yescomplemented
113C23×C127C2411yescharacteristic
114D4×C127—28no—
115C2×C4×C127—21no—
116C2×C4×C127—22no—
117C2×C4×C127—21no—
118C2×C4×C127—28no—
119C2×C4×C127—21no—
120C2×C4×C127—21no—
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C2×C4×C127—44no—
124C2×C4×C127C2×D414yes—
125C2×C4×C127C2×D412yes—
126C2×C4×C127C2×D412yes—
127C2×C4×C127#4112yescomplemented
128C22×C127—22no—
129C22×C127—22no—
130C22×C127—22no—
131C22×C127—24no—
132C22×C127—22no—
133C22×C127—41no—
134C22×C127—42no—
135C22×C127—41no—
136C22×C127—42no—
137C22×C127—42no—
138C22×C127C22×D412yes—
139C22×C127C22×D411yescharacteristic
140C22×C127C2×#4112yes—
141C22×C127C2×#4111yescharacteristic
142C22×C127C2×#4111yescharacteristic
143C2×D127—24no—
144C2×D127—24no—
145C2×D127—24no—
146C2×D127—42no—
147C2×D127—42no—
148C2×D127—42no—
149C2×D127—42no—
150C4×C127—22no—
151C4×C127—22no—
152C4×C127—41no—
153C4×C127—41no—
154C4×C127#12014yes—
155Dic127—24no—
156Dic127—24no—
157Dic127—42no—
158C2×C127—42no—
159C2×C127—42no—
160C2×C127C2×#12011yescharacteristic
161C2×C127#52512yes—
162C2×C127#52511yescharacteristic
163C2×C127#53211yescharacteristic
164C2×C127#53212yes—
165D127—44no—
166#2388—1271nomaximal, Sylow
167C127#238811yescharacteristic, complemented, Sylow
168C22×#41—1271no—
169C2×C4×D4—1272no—
170C2×C4×D4D12711yescharacteristic, complemented
171#395—1271no—
172#394—1271no—
173#394—1272no—
174#393—1271no—
175#392—1272no—
176#392—1271no—
177C4×C22⋊C4—1271no—
178C4×C22⋊C4—1272no—
179C2×#41—2548no—
180C2×#41—2544no—
181C22×Q8—1271no—
182C22×D4—1272no—
183C22×D4—1272no—
184C22×D4—1271no—
185C22×D4C2×D12711yescharacteristic
186C23×C4—1272no—
187C23×C4C2×D12711yescharacteristic
188C23×C4C2×D12711yescharacteristic
189C4×D4—24no—
190C4×D4—2548no—
191C2×C4⋊C4—1272no—
192C2×C4⋊C4C2×D12711yescharacteristic
193C2×C22⋊C4—1272no—
194C2×C22⋊C4—1272no—
195C2×C22⋊C4—1272no—
196C2×C22⋊C4—1272no—
197C2×C22⋊C4—1272no—
198C2×C22⋊C4C2×D12711yescharacteristic
199C2×C22⋊C4C2×D12711yescharacteristic
200C2×C42—1272no—
201C2×C42C2×D12711yescharacteristic
202#95—1271no—
203#95—1272no—
204#95—1272no—
205#95—1271no—
206C24—1272no—
207C24C22×D12711yescharacteristic
208C24C22×D12711yescharacteristic
209#41—25432no—
210C2×Q8—2544no—
211C2×Q8—2542no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—2544no—
216C2×D4—2544no—
217C2×D4—2544no—
218C2×D4—2544no—
219C2×D4—2544no—
220C2×D4—2544no—
221C2×D4—2544no—
222C2×D4—2542no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—1272no—
227C22×C4—1272no—
228C22×C4—1278no—
229C22×C4—1272no—
230C22×C4—2544no—
231C22×C4—2544no—
232C22×C4C22⋊D12714yescomplemented
233C22×C4C22×D12711yescharacteristic
234C22×C4C22×D12711yescharacteristic
235C22×C4C22×D12711yescharacteristic
236C22×C4C22×D12711yescharacteristic
237C22×C4C22×D12711yescharacteristic
238C4⋊C4—22no—
239C4⋊C4—2544no—
240C22⋊C4—22no—
241C22⋊C4—22no—
242C22⋊C4—2544no—
243C22⋊C4—2544no—
244C22⋊C4—2544no—
245C22⋊C4—2544no—
246C22⋊C4—2544no—
247C42—22no—
248C42—2544no—
249C23—22no—
250C23—22no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—1274no—
258C23—2542no—
259C23—2542no—
260C23—2542no—
261C23—2542no—
262C23—2542no—
263C23—2542no—
264C23C2×C22⋊D12712yes—
265C23#1452324812yescomplemented
266C23C23×D12711yescharacteristic
267Q8—2548no—
268D4—28no—
269D4—2548no—
270D4—25416no—
271D4—25416no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—28no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—44no—
281C2×C4—1274no—
282C2×C4—1274no—
283C2×C4—2542no—
284C2×C4—2542no—
285C2×C4—2542no—
286C2×C4—2542no—
287C2×C4—2542no—
288C2×C4—2542no—
289C2×C4—2542no—
290C2×C4—25416no—
291C2×C4D4×D12714yes—
292C2×C4C2×C22⋊D12712yes—
293C2×C4C2×C22⋊D12712yes—
294C2×C4#1452324812yescomplemented
295C22—22no—
296C22—24no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—41no—
301C22—42no—
302C22—41no—
303C22—42no—
304C22—42no—
305C22—5082no—
306C22—5082no—
307C22—5082no—
308C22—5082no—
309C22—2544no—
310C22—2544no—
311C22—2544no—
312C22C2×D4×D12712yes—
313C22C22×C22⋊D12711yescharacteristic
314C22#41×D12712yes—
315C22C2×#1452324811yescharacteristic
316C22C2×#1452324811yescharacteristic
317C4—22no—
318C4—22no—
319C4—41no—
320C4—41no—
321C4—5082no—
322C4—2544no—
323C4—2544no—
324C4#1686117714yes—
325C2—42no—
326C2—42no—
327C2—5084no—
328C2C2×#1686117711yescharacteristic
329C2#2255038512yes—
330C2#2255038711yescharacteristic
331C2#2255042211yescharacteristic
332C2#2255042312yes—
subgroupG/H
centerC22×C4C22×D127
commutator, socle, cosocleC23×C127C24
FrattiniC23C23×D127
FittingC2×C4×D4×C127C2
hypercenterC2×C4×D4D127
hypocenterC127#2388
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. C22×#14523248C2semidirect
2. C2×C4×C22⋊D127C2semidirect
3. #22549796C2semidirect
4. #22549793C2semidirect
5. #22549791C2semidirect
6. #22549788C2semidirect
7. #22549785C2semidirect
8. #22549784C2semidirect
9. C4×(C2×D127)⋊C4C2semidirect
10. C4×C22⋊Dic127C2semidirect
11. C2×C4×D4×C127C2semidirect
56. C2×C4×D127D4semidirect
72. C22×C4×C127D4semidirect
101. C2×Dic127#41semidirect
112. C23×C127#41semidirect
127. C2×C4×C127#41semidirect
167. C127#2388semidirect
170. C2×C4×D4D127semidirect
232. C22×C4C22⋊D127semidirect
265. C23#14523248semidirect
294. C2×C4#14523248semidirect

Maximal subgroups

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

subgroupindex
C22×#145232482
C2×C4×C22⋊D1272
#225497962
#225497932
#225497912
#225497882
#225497852
#225497842
C4×(C2×D127)⋊C42
C4×C22⋊Dic1272
C2×C4×D4×C1272
#2388127