Finite Group Databasebeta

← #30650614

Subgroups — #30650614

Order 16256.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550564C211yesmaximal, characteristic, complemented
2#22550538C211yesmaximal, characteristic
3#22550233C211yesmaximal, characteristic, complemented
4#22550220C211yesmaximal, characteristic, complemented
5#22550191C211yesmaximal, characteristic, complemented
6#22550118C211yesmaximal, characteristic, complemented
7#22550106C211yesmaximal, characteristic, complemented
8#22550099C211yesmaximal, characteristic, complemented
9#22550041C211yesmaximal, characteristic, complemented
10#22550033C211yesmaximal, characteristic, complemented
11C4×#14523238C211yesmaximal, characteristic, complemented
12#22549941C211yesmaximal, characteristic, complemented
13#22549871C211yesmaximal, characteristic, complemented
14#103×D127C211yesmaximal, characteristic, complemented
15#479×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×Dic508C2211yescharacteristic
19C2×#14523238C2211yescharacteristic
20C2×#14523237C2211yescharacteristic
21C2×C8×D127C2211yescharacteristic
22C2×C8×D127C2211yescharacteristic
23#16861204—21no—
24#16861200—21no—
25#16861197—21no—
26#16861197C2211yescharacteristic
27#16861194—21no—
28#16861189—21no—
29#16861189C2211yescharacteristic
30#16861189C2211yescharacteristic
31#16861189C2211yescharacteristic
32#16861184—21no—
33#16861184C2211yescharacteristic
34C4×Dic254C2211yescharacteristic
35C4×D508—21no—
36C4×D508C2211yescharacteristic
37#16861167—21no—
38#16861167C2211yescharacteristic
39C4⋊C4×D127C2211yescharacteristic
40#16861142C2211yescharacteristic
41#16861134C2211yescharacteristic
42#16861133C2211yescharacteristic
43#16861132C2211yescharacteristic
44#16861132C2211yescharacteristic
45#16861132C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C4×C127⋊C8C2211yescharacteristic
52#125×C127C2211yescharacteristic
53C4×Q8×C127C2211yescharacteristic
54#107×C127C2211yescharacteristic
55#106×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic
59#14523249—22no—
60#14523249—41no—
61C2×Dic254—21no—
62C2×Dic254C2311yescharacteristic
63C2×Dic254C2311yescharacteristic
64Q8×D127—22no—
65Q8×D127—41no—
66C2×D508—21no—
67C2×D508—21no—
68C2×D508D411yescharacteristic
69C2×D508C2311yescharacteristic
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127C2311yescharacteristic
74C2×C4×D127C2311yescharacteristic
75Dic508—41no—
76#14523238—22no—
77#14523237—41no—
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C8×D127—41no—
81C8×D127—41no—
82C4⋊Dic127—21no—
83C4⋊Dic127—21no—
84C4⋊Dic127—21no—
85C4⋊Dic127D411yescharacteristic
86C4⋊Dic127C2311yescharacteristic
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4D411yescharacteristic
97(C2×D127)⋊C4D411yescharacteristic
98C4×Dic127—21no—
99C4×Dic127—21no—
100C2×Q8×C127C2311yescharacteristic
101C2×C8×C127C2311yescharacteristic
102C2×C8×C127C2311yescharacteristic
103C4⋊C4×C127—21no—
104C4⋊C4×C127—21no—
105C4⋊C4×C127—21no—
106C4⋊C4×C127C2311yescharacteristic
107C4⋊C4×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C42×C127—21no—
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112C22×D127C2×D411yescharacteristic
113C22×D127C2×D411yescharacteristic
114Dic254—21no—
115Dic254—21no—
116Dic254—21no—
117Dic254—21no—
118Dic254—42no—
119D508—22no—
120D508—21no—
121D508—21no—
122D508—42no—
123D508—42no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C4×D127—21no—
130C4×D127—21no—
131C4×D127—21no—
132C4×D127—21no—
133C4×D127—42no—
134C4×D127—42no—
135C4×D127—42no—
136C127⋊C8—21no—
137C127⋊C8—21no—
138Q8×C127—21no—
139Q8×C127#4112yes—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2411yescharacteristic
147C8×C127—21no—
148C8×C127—21no—
149C22×C127C22×D411yescharacteristic
150C2×D127—21no—
151C2×D127—21no—
152C2×D127—21no—
153C2×D127—21no—
154C2×D127—21no—
155C2×D127—41no—
156C2×D127—41no—
157C2×D127—41no—
158C2×D127#13512yes—
159C4×C127—22no—
160C4×C127—21no—
161C4×C127—21no—
162C4×C127—41no—
163C4×C127—41no—
164C4×C127—41no—
165C4×C127C2×#4111yescharacteristic
166C4×C127H(5,𝔽2)11yescharacteristic
167Dic127—21no—
168Dic127—21no—
169Dic127—41no—
170Dic127—41no—
171Dic127—41no—
172C2×C127#54611yescharacteristic
173C2×C127C2×#13511yescharacteristic
174C2×C127#57811yescharacteristic
175D127—22no—
176D127—41no—
177D127—81no—
178#3300—1271nomaximal, Sylow
179C127#330011yescharacteristic, complemented, Sylow
180#552—1271no—
181#548—1271no—
182#488—1271no—
183#485—1271no—
184#480—1271no—
185#479D12711yescharacteristic, complemented
186#467—1271no—
187#465—1271no—
188#464—1271no—
189#452—1271no—
190#451—1271no—
191C4×Q16—1271no—
192C8×D4—1271no—
193#416—1271no—
194C2×#103—1271no—
195C2×#41—1271no—
196C22×Q8—1271no—
197C2×Q16—1271no—
198C2×Q16—1271no—
199C2×SD16—1271no—
200C22×C8—1271no—
201C22×C8—1271no—
202#126—2541no—
203#125—2541no—
204#125C2×D12711yescharacteristic
205#124—1271no—
206#124—2541no—
207#123—2541no—
208C22⋊Q8—1271no—
209C22⋊Q8—1271no—
210C22⋊Q8—1271no—
211C22⋊Q8—2541no—
212#121—1271no—
213#121—2541no—
214C4×Q8—1271no—
215C4×Q8C2×D12711yescharacteristic
216C4×D4—1271no—
217C4×D4—2541no—
218#117—1271no—
219#117—2541no—
220C2×C4⋊C4—1271no—
221#107—1271no—
222#107C2×D12711yescharacteristic
223#106C2×D12711yescharacteristic
224C4⋊C8C2×D12711yescharacteristic
225#103—1271no—
226#103—1271no—
227#103—1271no—
228#103—1271no—
229#103—1271no—
230#103C2×D12711yescharacteristic
231#103C2×D12711yescharacteristic
232#102—1271no—
233#102—1271no—
234#102—1271no—
235C22⋊C8—1271no—
236C22⋊C8—1271no—
237C4×C8—1271no—
238#41—5081no—
239#41—2542no—
240C2×Q8—1271no—
241C2×Q8—1271no—
242C2×Q8—5081no—
243C2×Q8—2541no—
244C2×Q8—2542no—
245C2×Q8C22×D12711yescharacteristic
246C2×D4—1271no—
247C2×D4—1271no—
248C2×D4—2541no—
249C2×D4—2541no—
250C22×C4—1271no—
251C22×C4—1271no—
252C22×C4—2541no—
253C22×C4—2541no—
254C22×C4—2541no—
255Q16—5081no—
256Q16—2542no—
257SD16—5081no—
258C2×C8—1271no—
259C2×C8—1271no—
260C2×C8—5081no—
261C2×C8—5081no—
262C2×C8C22×D12711yescharacteristic
263C2×C8C22×D12711yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—1271no—
268C4⋊C4—1271no—
269C4⋊C4—2541no—
270C4⋊C4—2541no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4—2541no—
274C4⋊C4C22×D12711yescharacteristic
275C4⋊C4C22×D12711yescharacteristic
276C4⋊C4C22×D12711yescharacteristic
277C22⋊C4—1271no—
278C22⋊C4—1271no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C42—21no—
287C42—2541no—
288C42—2541no—
289C42C22×D12711yescharacteristic
290C23—1271no—
291C23—1271no—
292C23—2541no—
293Q8—21no—
294Q8—5082no—
295Q8—2541no—
296Q8—2541no—
297Q8—2541no—
298Q8—2541no—
299Q8#1452324912yes—
300D4—5082no—
301D4—5082no—
302D4—2542no—
303D4—2541no—
304D4—2541no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—1271no—
310C2×C4—1271no—
311C2×C4—5082no—
312C2×C4—5082no—
313C2×C4—5082no—
314C2×C4—2541no—
315C2×C4—2541no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4—2541no—
320C2×C4—2541no—
321C2×C4D4×D12711yescharacteristic
322C2×C4D4×D12711yescharacteristic
323C2×C4C23×D12711yescharacteristic
324C8—21no—
325C8—21no—
326C8—2541no—
327C8—2541no—
328C22—1272no—
329C22—5081no—
330C22—5081no—
331C22—5081no—
332C22—2541no—
333C22—2541no—
334C22—2541no—
335C22—2541no—
336C22—2541no—
337C22C2×D4×D12711yescharacteristic
338C4—21no—
339C4—21no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—5081no—
345C4—5081no—
346C4—5081no—
347C4—2541no—
348C4—2541no—
349C4C2×#1452324911yescharacteristic
350C4#1686125511yescharacteristic
351C2—10161no—
352C2—5081no—
353C2—2542no—
354C2#2255051711yescharacteristic
355C2#135×D12711yescharacteristic
356C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#479×C127C2
socleC22×C127C22×D4
prime radical#22550538C2
hypercenter#479D127
hypocenterC127#3300
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
1. #22550564C2semidirect
3. #22550233C2semidirect
4. #22550220C2semidirect
5. #22550191C2semidirect
6. #22550118C2semidirect
7. #22550106C2semidirect
8. #22550099C2semidirect
9. #22550041C2semidirect
10. #22550033C2semidirect
11. C4×#14523238C2semidirect
12. #22549941C2semidirect
13. #22549871C2semidirect
14. #103×D127C2semidirect
15. #479×C127C2semidirect
179. C127#3300semidirect
185. #479D127semidirect

Maximal subgroups

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

subgroupindex
#225505642
#225505382
#225502332
#225502202
#225501912
#225501182
#225501062
#225500992
#225500412
#225500332
C4×#145232382
#225499412
#225498712
#103×D1272
#479×C1272
#3300127