Finite Group Databasebeta

← #30650135

Subgroups — #30650135

Order 16256.

358 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#22550497C211yesmaximal, characteristic, complemented
2#22550180C211yesmaximal, characteristic, complemented
3#22550173C211yesmaximal, characteristic, complemented
4#22550166C211yesmaximal, characteristic, complemented
5#22550156C211yesmaximal, characteristic, complemented
6#22550127C211yesmaximal, characteristic, complemented
7#22550120C211yesmaximal, characteristic, complemented
8#22550110C211yesmaximal, characteristic, complemented
9#22550103C211yesmaximal, characteristic, complemented
10#22550091C211yesmaximal, characteristic, complemented
11#22550091C211yesmaximal, characteristic, complemented
12#22550073C211yesmaximal, characteristic, complemented
13#22550068C211yesmaximal, characteristic, complemented
14#22549903C211yesmaximal, characteristic, complemented
15#540×C127C211yesmaximal, characteristic, complemented
16C2×#14523236C2211yescharacteristic
17C2×#14523236C2211yescharacteristic
18C2×#14523233C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20C2×#14523232C2211yescharacteristic
21C22×C127⋊C8C2211yescharacteristic
22#16861209C2211yescharacteristic
23#16861204—21no—
24#16861197C2211yescharacteristic
25#16861195—21no—
26#16861191—21no—
27#16861188C2211yescharacteristic
28#16861187—21no—
29#16861183C2211yescharacteristic
30C4×Dic254C2211yescharacteristic
31C4×D508C2211yescharacteristic
32#16861142C2211yescharacteristic
33#16861142C2211yescharacteristic
34#16861140C2211yescharacteristic
35#16861140C2211yescharacteristic
36C4⋊(C127⋊C8)C2211yescharacteristic
37C4⋊(C127⋊C8)C2211yescharacteristic
38C4⋊(C127⋊C8)C2211yescharacteristic
39C4⋊(C127⋊C8)C2211yescharacteristic
40#16861132C2211yescharacteristic
41#16861132C2211yescharacteristic
42#16861130C2211yescharacteristic
43#16861130C2211yescharacteristic
44#16861130C2211yescharacteristic
45#16861130C2211yescharacteristic
46#16861129C2211yescharacteristic
47#16861129C2211yescharacteristic
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
59C2×Dic254—21no—
60C2×Dic254C2311yescharacteristic
61C2×D508C2311yescharacteristic
62C2×C22⋊D127—21no—
63C2×C4×D127—21no—
64#14523236—41no—
65#14523236—41no—
66#14523233—41no—
67#14523233—41no—
68#14523232—41no—
69C2×C127⋊C8—41no—
70C2×C127⋊C8C2311yescharacteristic
71C2×C127⋊C8C2311yescharacteristic
72C2×C127⋊C8C2311yescharacteristic
73C2×C127⋊C8C2311yescharacteristic
74C4⋊Dic127—21no—
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77Dic127⋊C4—21no—
78Dic127⋊C4—21no—
79C22⋊Dic127—21no—
80(C2×D127)⋊C4—21no—
81(C2×D127)⋊C4—21no—
82(C2×D127)⋊C4—21no—
83(C2×D127)⋊C4—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—21no—
111Dic254—81no—
112D508—21no—
113D508—21no—
114C22⋊D127—81no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C4×D127—81no—
119C127⋊C8—21no—
120C127⋊C8—21no—
121C127⋊C8—21no—
122C127⋊C8—21no—
123C23×C127—21no—
124C23×C127—21no—
125C23×C127C2×D411yescharacteristic
126D4×C127—21no—
127D4×C127—21no—
128D4×C127—21no—
129D4×C127—21no—
130D4×C127—81no—
131D4×C127—81no—
132D4×C127—81no—
133D4×C127—81no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—81no—
139C2×C4×C127—81no—
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2411yescharacteristic
146C22×C127—21no—
147C22×C127—21no—
148C22×C127—21no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127C22×D411yescharacteristic
156C2×D127—41no—
157C2×D127—41no—
158C2×D127—41no—
159C4×C127—21no—
160C4×C127—21no—
161C4×C127—21no—
162C4×C127—21no—
163C4×C127—21no—
164C4×C127—41no—
165C4×C127—41no—
166C4×C127H(5,𝔽2)11yescharacteristic
167C4×C1272−1+411yescharacteristic
168Dic127—41no—
169Dic127—41no—
170Dic127—41no—
171C2×C127—41no—
172C2×C127—81no—
173C2×C127—81no—
174C2×C127#53711yescharacteristic
175C2×C127#57611yescharacteristic
176C2×C127#57711yescharacteristic
177D127—81no—
178#3257—1271nomaximal, Sylow
179C127#325711yescharacteristic, complemented, Sylow
180#542—1271no—
181#540D12711yescharacteristic, complemented
182#479—1271no—
183#478—1271no—
184#476—1271no—
185#474—1271no—
186#469—1271no—
187#468—1271no—
188#466—1271no—
189#465—1271no—
190#463—1271no—
191#463—1271no—
192#460—1271no—
193#459—1271no—
194#424—1271no—
195C2×SD16—1271no—
196C2×SD16—1271no—
197C2×D8—1271no—
198C2×D8—1271no—
199C2×M16—1271no—
200C22×C8—1271no—
201C4⋊Q8—1271no—
202#127C2×D12711yescharacteristic
203#126—2541no—
204#125C2×D12711yescharacteristic
205#124—1271no—
206#124—2541no—
207#123—21no—
208#123—21no—
209#123—2541no—
210C22⋊Q8—1271no—
211C22⋊Q8—2541no—
212#121—21no—
213#121—21no—
214#121—1271no—
215#121C2×D12711yescharacteristic
216#121C2×D12711yescharacteristic
217C4×Q8—1271no—
218C4×D4—1271no—
219C4×D4C2×D12711yescharacteristic
220C4×D4C2×D12711yescharacteristic
221#117C2×D12711yescharacteristic
222#107—1271no—
223#107—1271no—
224#106—1271no—
225#106—1271no—
226C4⋊C8—1271no—
227C4⋊C8—1271no—
228C4⋊C8—1271no—
229C4⋊C8—1271no—
230#103—1271no—
231#103—1271no—
232#102—1271no—
233#102—1271no—
234#102—1271no—
235#102—1271no—
236#102—1271no—
237#102—1271no—
238C2×Q8—1271no—
239C2×Q8—2541no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—1271no—
245C2×D4—2541no—
246C2×D4C22×D12711yescharacteristic
247C2×D4C22×D12711yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—2541no—
251C22×C4C22×D12711yescharacteristic
252SD16—5081no—
253SD16—5081no—
254D8—5081no—
255D8—5081no—
256M16—5081no—
257C2×C8—1271no—
258C2×C8—1271no—
259C2×C8—1271no—
260C2×C8—1271no—
261C2×C8—5081no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—1271no—
265C4⋊C4—1271no—
266C4⋊C4—2541no—
267C4⋊C4—2541no—
268C4⋊C4—2541no—
269C4⋊C4C22⋊D12711yescharacteristic
270C4⋊C4C22⋊D12711yescharacteristic
271C4⋊C4C22×D12711yescharacteristic
272C4⋊C4C22×D12711yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4C22⋊D12711yescharacteristic
283C22⋊C4C22⋊D12711yescharacteristic
284C42—2541no—
285C42C22×D12711yescharacteristic
286C42C22×D12711yescharacteristic
287C23—21no—
288C23—21no—
289C23—2541no—
290C23C2×C22⋊D12711yescharacteristic
291Q8—10161no—
292Q8—2541no—
293Q8—2541no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—81no—
299D4—81no—
300D4—81no—
301D4—81no—
302D4—10161no—
303D4—2541no—
304D4—2541no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—10161no—
312C2×C4—2541no—
313C2×C4—2541no—
314C2×C4—2541no—
315C2×C4C2×C22⋊D12711yescharacteristic
316C2×C4C2×C22⋊D12711yescharacteristic
317C2×C4C2×C22⋊D12711yescharacteristic
318C2×C4C2×C22⋊D12711yescharacteristic
319C2×C4C2×C22⋊D12711yescharacteristic
320C2×C4C23×D12711yescharacteristic
321C8—2541no—
322C8—2541no—
323C8—2541no—
324C8—2541no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—5081no—
335C22—5081no—
336C22—5081no—
337C22C22×C22⋊D12711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—5081no—
346C4—5081no—
347C4—5081no—
348C4#1686125511yescharacteristic
349C4#1686125611yescharacteristic
350C2—41no—
351C2—81no—
352C2—81no—
353C2—10161no—
354C2#2255045411yescharacteristic
355C2#2255067911yescharacteristic
356C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#540×C127C2
socleC22×C127C22×D4
hypercenter#540D127
hypocenterC127#3257
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. #22550497C2semidirect
2. #22550180C2semidirect
3. #22550173C2semidirect
4. #22550166C2semidirect
5. #22550156C2semidirect
6. #22550127C2semidirect
7. #22550120C2semidirect
8. #22550110C2semidirect
9. #22550103C2semidirect
10. #22550091C2semidirect
11. #22550091C2semidirect
12. #22550073C2semidirect
13. #22550068C2semidirect
14. #22549903C2semidirect
15. #540×C127C2semidirect
179. C127#3257semidirect
181. #540D127semidirect

Maximal subgroups

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

subgroupindex
#225504972
#225501802
#225501732
#225501662
#225501562
#225501272
#225501202
#225501102
#225501032
#225500912
#225500732
#225500682
#225499032
#540×C1272
#3257127