Finite Group Databasebeta

← #30648923

Subgroups — #30648923

Order 16256.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550356C211yesmaximal, characteristic, complemented
2#22550192C211yesmaximal, characteristic, complemented
3#22550187C211yesmaximal, characteristic, complemented
4#22550180C211yesmaximal, characteristic, complemented
5#22550173C211yesmaximal, characteristic, complemented
6#22550073C211yesmaximal, characteristic, complemented
7#22550068C211yesmaximal, characteristic, complemented
8#22550019C211yesmaximal, characteristic, complemented
9#22550019C211yesmaximal, characteristic, complemented
10#22549986C211yesmaximal, characteristic, complemented
11#22549976C211yesmaximal, characteristic, complemented
12C4×#14523236C211yesmaximal, characteristic, complemented
13C4×#14523233C211yesmaximal, characteristic, complemented
14#22549937C211yesmaximal, characteristic, complemented
15#540×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523236C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20C2×#14523233C2211yescharacteristic
21C4×Dic254C2211yescharacteristic
22C4×Dic254C2211yescharacteristic
23C4×D508C2211yescharacteristic
24C4×D508C2211yescharacteristic
25(C22⋊D127)⋊C4—21no—
26(C22⋊D127)⋊C4—21no—
27#16861167—21no—
28C2×C4⋊Dic127C2211yescharacteristic
29C4⋊C4×D127—21no—
30#16861142C2211yescharacteristic
31#16861142C2211yescharacteristic
32#16861140C2211yescharacteristic
33#16861140C2211yescharacteristic
34C4⋊(C127⋊C8)C2211yescharacteristic
35C4⋊(C127⋊C8)C2211yescharacteristic
36#16861132C2211yescharacteristic
37#16861132C2211yescharacteristic
38#16861130C2211yescharacteristic
39#16861130C2211yescharacteristic
40#16861130C2211yescharacteristic
41#16861130C2211yescharacteristic
42#16861129C2211yescharacteristic
43#16861129C2211yescharacteristic
44C22⋊(C127⋊C8)C2211yescharacteristic
45C22⋊(C127⋊C8)C2211yescharacteristic
46#16861120C2211yescharacteristic
47C4×C127⋊C8C2211yescharacteristic
48#127×C127C2211yescharacteristic
49#125×C127C2211yescharacteristic
50#123×C127—21no—
51#123×C127—21no—
52#121×C127—21no—
53#121×C127—21no—
54#121×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56C4×D4×C127C2211yescharacteristic
57C4×D4×C127C2211yescharacteristic
58#117×C127C2211yescharacteristic
59#14523248—22no—
60#14523248—22no—
61C2×Dic254C2311yescharacteristic
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64C22×Dic127—21no—
65C2×C4×D127—21no—
66C2×C4×D127—21no—
67C2×C4×D127—21no—
68#14523236—22no—
69#14523236—22no—
70#14523233—22no—
71#14523233—22no—
72C2×C127⋊C8C2311yescharacteristic
73C2×C127⋊C8C2311yescharacteristic
74C2×C127⋊C8C2311yescharacteristic
75C2×C127⋊C8C2311yescharacteristic
76C4⋊Dic127—21no—
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81(C2×D127)⋊C4—21no—
82(C2×D127)⋊C4—21no—
83C4×Dic127—21no—
84C4×Dic127—21no—
85C2×D4×C127—21no—
86C2×D4×C127—21no—
87C2×D4×C127—21no—
88C2×D4×C127—21no—
89C2×D4×C127C2311yescharacteristic
90C2×D4×C127C2311yescharacteristic
91C22×C4×C127—21no—
92C22×C4×C127—21no—
93C22×C4×C127C2311yescharacteristic
94C4⋊C4×C127—21no—
95C4⋊C4×C127—21no—
96C4⋊C4×C127D411yescharacteristic
97C4⋊C4×C127D411yescharacteristic
98C4⋊C4×C127C2311yescharacteristic
99C4⋊C4×C127C2311yescharacteristic
100C22⋊C4×C127—21no—
101C22⋊C4×C127—21no—
102C22⋊C4×C127—21no—
103C22⋊C4×C127—21no—
104C22⋊C4×C127D411yescharacteristic
105C22⋊C4×C127D411yescharacteristic
106C42×C127C2311yescharacteristic
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109Dic254—21no—
110Dic254#4112yes—
111D508—21no—
112D508#4112yes—
113C22⋊D127—24no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—81no—
118C4×D127—24no—
119C4×D127—81no—
120C4×D127—81no—
121C127⋊C8—21no—
122C127⋊C8—21no—
123C127⋊C8—21no—
124C127⋊C8—21no—
125C23×C127—21no—
126C23×C127—21no—
127C23×C127C2×D411yescharacteristic
128D4×C127—21no—
129D4×C127—21no—
130D4×C127—21no—
131D4×C127—21no—
132D4×C127—81no—
133D4×C127—81no—
134D4×C127—81no—
135D4×C127—81no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—81no—
141C2×C4×C127—81no—
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2411yescharacteristic
148C22×C127—21no—
149C22×C127—21no—
150C22×C127—21no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127C22×D411yescharacteristic
158C2×D127—22no—
159C2×D127—41no—
160C2×D127—41no—
161C4×C127—21no—
162C4×C127—21no—
163C4×C127—21no—
164C4×C127—21no—
165C4×C127—21no—
166C4×C127—41no—
167C4×C127—41no—
168C4×C127C2×#4111yescharacteristic
169C4×C127C2×#4111yescharacteristic
170Dic127—22no—
171Dic127—41no—
172Dic127—41no—
173C2×C127—41no—
174C2×C127—81no—
175C2×C127—81no—
176C2×C127#52511yescharacteristic
177C2×C127#57611yescharacteristic
178C2×C127#57711yescharacteristic
179D127—42no—
180#3119—1271nomaximal, Sylow
181C127#311911yescharacteristic, complemented, Sylow
182#540D12711yescharacteristic, complemented
183#520—1271no—
184#481—1271no—
185#480—1271no—
186#479—1271no—
187#478—1271no—
188#460—1271no—
189#459—1271no—
190#449—1271no—
191#449—1271no—
192#442—1271no—
193#440—1271no—
194C4×SD16—1271no—
195C4×D8—1271no—
196#433—1271no—
197C2×#41—1271no—
198C2×SD16—1271no—
199C2×SD16—1271no—
200C2×D8—1271no—
201C2×D8—1271no—
202#127C2×D12711yescharacteristic
203#125C2×D12711yescharacteristic
204#123—21no—
205#123—21no—
206#121—21no—
207#121—21no—
208#121C2×D12711yescharacteristic
209#121C2×D12711yescharacteristic
210C4×Q8—1271no—
211C4×Q8—1271no—
212C4×D4—1271no—
213C4×D4—1271no—
214C4×D4—2541no—
215C4×D4—2541no—
216C4×D4C2×D12711yescharacteristic
217C4×D4C2×D12711yescharacteristic
218#117—2541no—
219#117C2×D12711yescharacteristic
220C2×C4⋊C4—1271no—
221C2×C4⋊C4—2541no—
222#107—1271no—
223#107—1271no—
224#106—1271no—
225#106—1271no—
226C4⋊C8—1271no—
227C4⋊C8—1271no—
228#103—1271no—
229#103—1271no—
230#102—1271no—
231#102—1271no—
232#102—1271no—
233#102—1271no—
234#102—1271no—
235#102—1271no—
236C22⋊C8—1271no—
237C22⋊C8—1271no—
238#97—1271no—
239C4×C8—1271no—
240#41—2542no—
241#41—2542no—
242C2×Q8—1271no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—1271no—
248C2×D4—2541no—
249C2×D4C22×D12711yescharacteristic
250C2×D4C22×D12711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—2541no—
254C22×C4—2541no—
255C22×C4—2541no—
256C22×C4—2541no—
257C22×C4C22×D12711yescharacteristic
258SD16—2542no—
259SD16—2542no—
260D8—2542no—
261D8—2542no—
262C2×C8—1271no—
263C2×C8—1271no—
264C2×C8—1271no—
265C2×C8—1271no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—1271no—
269C4⋊C4—1271no—
270C4⋊C4—2541no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4C22⋊D12711yescharacteristic
274C4⋊C4C22⋊D12711yescharacteristic
275C4⋊C4C22×D12711yescharacteristic
276C4⋊C4C22×D12711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4C22⋊D12711yescharacteristic
284C22⋊C4C22⋊D12711yescharacteristic
285C42—2541no—
286C42—2541no—
287C42C22×D12711yescharacteristic
288C42C22×D12711yescharacteristic
289C23—21no—
290C23—21no—
291C23—2541no—
292C23C2×C22⋊D12711yescharacteristic
293Q8—1272no—
294Q8—2541no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—81no—
301D4—81no—
302D4—81no—
303D4—1272no—
304D4—2544no—
305D4—2541no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—10161no—
313C2×C4—10161no—
314C2×C4—10161no—
315C2×C4—2544no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4C2×C22⋊D12711yescharacteristic
320C2×C4C2×C22⋊D12711yescharacteristic
321C2×C4C2×C22⋊D12711yescharacteristic
322C2×C4C2×C22⋊D12711yescharacteristic
323C2×C4C2×C22⋊D12711yescharacteristic
324C2×C4C23×D12711yescharacteristic
325C8—2541no—
326C8—2541no—
327C8—2541no—
328C8—2541no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—5081no—
339C22—5081no—
340C22—2542no—
341C22C22×C22⋊D12711yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—5081no—
350C4—5081no—
351C4—2542no—
352C4#41×D12711yescharacteristic
353C4#41×D12711yescharacteristic
354C2—41no—
355C2—81no—
356C2—81no—
357C2—5082no—
358C2#2255038311yescharacteristic
359C2#2255067911yescharacteristic
360C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#540×C127C2
socleC22×C127C22×D4
hypercenter#540D127
hypocenterC127#3119
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #22550356C2semidirect
2. #22550192C2semidirect
3. #22550187C2semidirect
4. #22550180C2semidirect
5. #22550173C2semidirect
6. #22550073C2semidirect
7. #22550068C2semidirect
8. #22550019C2semidirect
9. #22550019C2semidirect
10. #22549986C2semidirect
11. #22549976C2semidirect
12. C4×#14523236C2semidirect
13. C4×#14523233C2semidirect
14. #22549937C2semidirect
15. #540×C127C2semidirect
181. C127#3119semidirect
182. #540D127semidirect

Maximal subgroups

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

subgroupindex
#225503562
#225501922
#225501872
#225501802
#225501732
#225500732
#225500682
#225500192
#225499862
#225499762
C4×#145232362
C4×#145232332
#225499372
#540×C1272
#3119127