Finite Group Databasebeta

← #30651814

Subgroups — #30651814

Order 16256.

380 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550611C211yesmaximal, characteristic, complemented
2#22550266C211yesmaximal, characteristic, complemented
3#22550263C211yesmaximal, characteristic, complemented
4#22550258C211yesmaximal, characteristic, complemented
5#22550098C211yesmaximal, characteristic, complemented
6#22550095C211yesmaximal, characteristic, complemented
7#22550091C211yesmaximal, characteristic, complemented
8#22550088C211yesmaximal, characteristic, complemented
9#22550071C211yesmaximal, characteristic, complemented
10#22550068C211yesmaximal, characteristic, complemented
11#22550007C211yesmaximal, characteristic, complemented
12#22549976C211yesmaximal, characteristic, complemented
13#22549974C211yesmaximal, characteristic, complemented
14C4×#14523233C211yesmaximal, characteristic, complemented
15#559×C127C211yesmaximal, characteristic, complemented
16C2×#14523238C2211yescharacteristic
17C2×#14523236C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523235C2211yescharacteristic
20C2×#14523235C2211yescharacteristic
21C2×#14523233C2211yescharacteristic
22C2×#14523233C2211yescharacteristic
23C2×#14523233C2211yescharacteristic
24C2×#14523233C2211yescharacteristic
25#16861206C2211yescharacteristic
26#16861198—21no—
27#16861197C2211yescharacteristic
28#16861197C2211yescharacteristic
29#16861194—21no—
30#16861194—21no—
31C4×Dic254C2211yescharacteristic
32C4×D508C2211yescharacteristic
33C4×D508C2211yescharacteristic
34#16861171—21no—
35#16861142C2211yescharacteristic
36C4⋊(C127⋊C8)C2211yescharacteristic
37C4⋊(C127⋊C8)C2211yescharacteristic
38C4⋊(C127⋊C8)C2211yescharacteristic
39#16861133C2211yescharacteristic
40#16861132C2211yescharacteristic
41#16861130C2211yescharacteristic
42#16861130C2211yescharacteristic
43#16861129C2211yescharacteristic
44#16861129C2211yescharacteristic
45#16861120C2211yescharacteristic
46#16861120C2211yescharacteristic
47C4×C127⋊C8C2211yescharacteristic
48#127×C127C2211yescharacteristic
49#124×C127—21no—
50#124×C127C2211yescharacteristic
51#124×C127C2211yescharacteristic
52#123×C127—21no—
53#123×C127—21no—
54C4×Q8×C127C2211yescharacteristic
55C4×Q8×C127C2211yescharacteristic
56C4×D4×C127—21no—
57C4×D4×C127C2211yescharacteristic
58C4×D4×C127C2211yescharacteristic
59C2×Dic254C2311yescharacteristic
60C2×D508—21no—
61C2×D508—21no—
62C2×D508C2311yescharacteristic
63C2×D508C2311yescharacteristic
64C2×C4×D127—21no—
65C2×C4×D127—21no—
66#14523238—41no—
67#14523236—22no—
68#14523236—41no—
69#14523235—22no—
70#14523235—41no—
71#14523233—22no—
72#14523233—41no—
73#14523233—41no—
74#14523233—41no—
75C2×C127⋊C8C2311yescharacteristic
76C2×C127⋊C8C2311yescharacteristic
77C2×C127⋊C8C2311yescharacteristic
78C2×C127⋊C8C2311yescharacteristic
79C4⋊Dic127C2311yescharacteristic
80Dic127⋊C4—21no—
81(C2×D127)⋊C4—21no—
82(C2×D127)⋊C4—21no—
83(C2×D127)⋊C4—21no—
84(C2×D127)⋊C4—21no—
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87C4×Dic127—21no—
88C2×Q8×C127D411yescharacteristic
89C2×Q8×C127C2311yescharacteristic
90C2×D4×C127—21no—
91C2×D4×C127—21no—
92C2×D4×C127C2311yescharacteristic
93C2×D4×C127C2311yescharacteristic
94C22×C4×C127—21no—
95C22×C4×C127—21no—
96C4⋊C4×C127—21no—
97C4⋊C4×C127D411yescharacteristic
98C4⋊C4×C127D411yescharacteristic
99C4⋊C4×C127D411yescharacteristic
100C4⋊C4×C127C2311yescharacteristic
101C22⋊C4×C127—21no—
102C22⋊C4×C127—21no—
103C22⋊C4×C127—21no—
104C22⋊C4×C127—21no—
105C22⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C42×C127—21no—
108C42×C127C2311yescharacteristic
109C42×C127C2311yescharacteristic
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112C22×D127—21no—
113Dic254—21no—
114Dic254—21no—
115D508—21no—
116D508—21no—
117D508—21no—
118D508—21no—
119D508—42no—
120D508—81no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C4×D127—81no—
124C4×D127—81no—
125C127⋊C8—21no—
126C127⋊C8—21no—
127C127⋊C8—21no—
128C127⋊C8#4112yes—
129C23×C127—21no—
130C23×C127—21no—
131Q8×C127—21no—
132Q8×C127—21no—
133Q8×C127—41no—
134D4×C127—21no—
135D4×C127—21no—
136D4×C127—21no—
137D4×C127—21no—
138D4×C127—42no—
139D4×C127—81no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—81no—
143C2×C4×C127—81no—
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2×D411yescharacteristic
150C2×C4×C127C2411yescharacteristic
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127C22×D411yescharacteristic
158C2×D127—41no—
159C2×D127—41no—
160C2×D127—41no—
161C2×D127—41no—
162C2×D127—41no—
163C2×D127—41no—
164C4×C127—21no—
165C4×C127—21no—
166C4×C127—21no—
167C4×C127—21no—
168C4×C127—21no—
169C4×C127—21no—
170C4×C127—41no—
171C4×C127—41no—
172C4×C127C2×#4111yescharacteristic
173C4×C127H(5,𝔽2)11yescharacteristic
174Dic127—41no—
175Dic127—41no—
176C2×C127—81no—
177C2×C127—81no—
178C2×C127#55011yescharacteristic
179C2×C127#57511yescharacteristic
180C2×C127#57611yescharacteristic
181D127—81no—
182D127—81no—
183#3403—1271nomaximal, Sylow
184C127#340311yescharacteristic, complemented, Sylow
185#559—1271no—
186#559D12711yescharacteristic, complemented
187#496—1271no—
188#496—1271no—
189#495—1271no—
190#464—1271no—
191#464—1271no—
192#463—1271no—
193#463—1271no—
194#459—1271no—
195#459—1271no—
196#446—1271no—
197#440—1271no—
198#440—1271no—
199C4×D8—1271no—
200C2×Q16—1271no—
201C2×SD16—1271no—
202C2×SD16—1271no—
203C2×SD16—1271no—
204C2×SD16—1271no—
205C2×D8—1271no—
206C2×D8—1271no—
207C2×D8—1271no—
208C2×D8—1271no—
209#127—1271no—
210#127C2×D12711yescharacteristic
211#124—21no—
212#124—1271no—
213#124—1271no—
214#124—2541no—
215#124C2×D12711yescharacteristic
216#124C2×D12711yescharacteristic
217#123—21no—
218#123—21no—
219#123—2541no—
220#123—2541no—
221C4×Q8—1271no—
222C4×Q8C2×D12711yescharacteristic
223C4×Q8C2×D12711yescharacteristic
224C4×D4—21no—
225C4×D4—1271no—
226C4×D4—1271no—
227C4×D4—2541no—
228C4×D4C2×D12711yescharacteristic
229C4×D4C2×D12711yescharacteristic
230#107—1271no—
231C4⋊C8—1271no—
232C4⋊C8—1271no—
233C4⋊C8—1271no—
234#103—1271no—
235#103—1271no—
236#102—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#97—1271no—
241#97—1271no—
242C4×C8—1271no—
243C2×Q8—1271no—
244C2×Q8C22⋊D12711yescharacteristic
245C2×Q8C22×D12711yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—1271no—
249C2×D4—1271no—
250C2×D4—2541no—
251C2×D4—2541no—
252C2×D4C22×D12711yescharacteristic
253C2×D4C22×D12711yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—2541no—
257C22×C4—2541no—
258Q16—5081no—
259SD16—5081no—
260SD16—5081no—
261SD16—2542no—
262SD16—2542no—
263D8—5081no—
264D8—5081no—
265D8—5081no—
266D8—2542no—
267C2×C8—1271no—
268C2×C8—1271no—
269C2×C8—1271no—
270C2×C8—1271no—
271C4⋊C4—21no—
272C4⋊C4—1271no—
273C4⋊C4—2541no—
274C4⋊C4C22⋊D12711yescharacteristic
275C4⋊C4C22⋊D12711yescharacteristic
276C4⋊C4C22⋊D12711yescharacteristic
277C4⋊C4C22×D12711yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C42—21no—
291C42—2541no—
292C42C22×D12711yescharacteristic
293C42C22×D12711yescharacteristic
294C42C22×D12711yescharacteristic
295C23—21no—
296C23—21no—
297C23—2541no—
298C23—2541no—
299Q8—21no—
300Q8—21no—
301Q8—41no—
302Q8—2541no—
303Q8—2541no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—10161no—
311D4—5082no—
312D4—2541no—
313D4—2541no—
314D4—2541no—
315D4—2541no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—10161no—
321C2×C4—10161no—
322C2×C4—2541no—
323C2×C4—2541no—
324C2×C4C2×C22⋊D12711yescharacteristic
325C2×C4C2×C22⋊D12711yescharacteristic
326C2×C4C2×C22⋊D12711yescharacteristic
327C2×C4C2×C22⋊D12711yescharacteristic
328C2×C4C2×C22⋊D12711yescharacteristic
329C2×C4C2×C22⋊D12711yescharacteristic
330C2×C4C23×D12711yescharacteristic
331C8—1272no—
332C8—2541no—
333C8—2541no—
334C8—2541no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—5081no—
342C22—5081no—
343C22—5081no—
344C22—5081no—
345C22—5081no—
346C22—5081no—
347C22C22×C22⋊D12711yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—5081no—
357C4—5081no—
358C4#41×D12711yescharacteristic
359C4#1686125511yescharacteristic
360C2—81no—
361C2—81no—
362C2—10161no—
363C2—10161no—
364C2#2255054511yescharacteristic
365C2#2255067411yescharacteristic
366C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#559×C127C2
socleC22×C127C22×D4
hypercenter#559D127
hypocenterC127#3403
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #22550611C2semidirect
2. #22550266C2semidirect
3. #22550263C2semidirect
4. #22550258C2semidirect
5. #22550098C2semidirect
6. #22550095C2semidirect
7. #22550091C2semidirect
8. #22550088C2semidirect
9. #22550071C2semidirect
10. #22550068C2semidirect
11. #22550007C2semidirect
12. #22549976C2semidirect
13. #22549974C2semidirect
14. C4×#14523233C2semidirect
15. #559×C127C2semidirect
184. C127#3403semidirect
186. #559D127semidirect

Maximal subgroups

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

subgroupindex
#225506112
#225502662
#225502632
#225502582
#225500982
#225500952
#225500912
#225500882
#225500712
#225500682
#225500072
#225499762
#225499742
C4×#145232332
#559×C1272
#3403127