Finite Group Databasebeta

← #30650700

Subgroups — #30650700

Order 16256.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550531C211yesmaximal, characteristic, complemented
2#22550246C211yesmaximal, characteristic, complemented
3#22550217C211yesmaximal, characteristic, complemented
4#22550206C211yesmaximal, characteristic, complemented
5#22550173C211yesmaximal, characteristic, complemented
6#22550120C211yesmaximal, characteristic, complemented
7#22550103C211yesmaximal, characteristic, complemented
8#22550081C211yesmaximal, characteristic, complemented
9#22550019C211yesmaximal, characteristic, complemented
10#22550019C211yesmaximal, characteristic, complemented
11#22549986C211yesmaximal, characteristic, complemented
12#22549948C211yesmaximal, characteristic, complemented
13#22549875C211yesmaximal, characteristic, complemented
14#22549870C211yesmaximal, characteristic, complemented
15#553×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523248C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523236C2211yescharacteristic
20C2×#14523233C2211yescharacteristic
21C2×#14523232C2211yescharacteristic
22C22×C127⋊C8C2211yescharacteristic
23#16861209C2211yescharacteristic
24#16861191—21no—
25#16861191—21no—
26#16861188C2211yescharacteristic
27#16861188C2211yescharacteristic
28#16861180—21no—
29C4×D508C2211yescharacteristic
30C4⋊C4×D127—21no—
31#16861142C2211yescharacteristic
32#16861140C2211yescharacteristic
33#16861140C2211yescharacteristic
34C4⋊(C127⋊C8)C2211yescharacteristic
35#16861132C2211yescharacteristic
36#16861132C2211yescharacteristic
37#16861132C2211yescharacteristic
38#16861132C2211yescharacteristic
39#16861132C2211yescharacteristic
40#16861132C2211yescharacteristic
41#16861130C2211yescharacteristic
42#16861129C2211yescharacteristic
43#16861129C2211yescharacteristic
44#16861129C2211yescharacteristic
45C22⋊(C127⋊C8)C2211yescharacteristic
46C22⋊(C127⋊C8)C2211yescharacteristic
47#16861120C2211yescharacteristic
48#126×C127—21no—
49#125×C127C2211yescharacteristic
50#123×C127—21no—
51#121×C127—21no—
52#121×C127C2211yescharacteristic
53#121×C127C2211yescharacteristic
54C4×D4×C127—21no—
55C4×D4×C127C2211yescharacteristic
56C4×D4×C127C2211yescharacteristic
57#117×C127C2211yescharacteristic
58C2×C4⋊C4×C127C2211yescharacteristic
59#14523248—22no—
60#14523248—22no—
61#14523248—41no—
62#14523248—41no—
63C2×Dic254C2311yescharacteristic
64C2×Dic254C2311yescharacteristic
65C2×D508C2311yescharacteristic
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C4×D127—21no—
69C2×C4×D127—21no—
70C2×C4×D127—21no—
71#14523236—41no—
72#14523236—41no—
73#14523233—22no—
74#14523232—41no—
75C2×C127⋊C8—41no—
76C2×C127⋊C8C2311yescharacteristic
77C2×C127⋊C8C2311yescharacteristic
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C4⋊Dic127—21no—
81C4⋊Dic127—21no—
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85C22⋊Dic127—21no—
86C22⋊Dic127—21no—
87(C2×D127)⋊C4—21no—
88C2×D4×C127—21no—
89C2×D4×C127—21no—
90C2×D4×C127D411yescharacteristic
91C2×D4×C127C2311yescharacteristic
92C22×C4×C127—21no—
93C22×C4×C127—21no—
94C22×C4×C127C2311yescharacteristic
95C22×C4×C127C2311yescharacteristic
96C4⋊C4×C127—21no—
97C4⋊C4×C127—21no—
98C4⋊C4×C127—21no—
99C4⋊C4×C127D411yescharacteristic
100C4⋊C4×C127C2311yescharacteristic
101C4⋊C4×C127C2311yescharacteristic
102C4⋊C4×C127C2311yescharacteristic
103C22⋊C4×C127—21no—
104C22⋊C4×C127—21no—
105C22⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C22⋊C4×C127D411yescharacteristic
108C22⋊C4×C127D411yescharacteristic
109C42×C127—21no—
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112Dic254—21no—
113Dic254—21no—
114Dic254—21no—
115Dic254—21no—
116D508—21no—
117D508#4112yes—
118C22⋊D127—42no—
119C22⋊D127—42no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C4×D127—42no—
124C4×D127—42no—
125C4×D127—42no—
126C127⋊C8—21no—
127C127⋊C8—21no—
128C127⋊C8—21no—
129C127⋊C8—21no—
130C23×C127—21no—
131C23×C127C2×D411yescharacteristic
132C23×C127C2×D411yescharacteristic
133D4×C127—21no—
134D4×C127—21no—
135D4×C127—41no—
136D4×C127—42no—
137D4×C127—81no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—81no—
146C2×C4×C127—81no—
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2×D411yescharacteristic
150C2×C4×C127C2×D411yescharacteristic
151C2×C4×C127C2411yescharacteristic
152C22×C127—21no—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—21no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127C22×D411yescharacteristic
162C2×D127—22no—
163C2×D127—41no—
164C2×D127—41no—
165C4×C127—21no—
166C4×C127—21no—
167C4×C127—21no—
168C4×C127—21no—
169C4×C127—41no—
170C4×C127—41no—
171C4×C127—41no—
172C4×C127C2×#4111yescharacteristic
173C4×C127H(5,𝔽2)11yescharacteristic
174Dic127—41no—
175Dic127—41no—
176Dic127—41no—
177C2×C127—41no—
178C2×C127—41no—
179C2×C127—81no—
180C2×C127#54611yescharacteristic
181C2×C127#57511yescharacteristic
182C2×C127#57711yescharacteristic
183D127—42no—
184#3307—1271nomaximal, Sylow
185C127#330711yescharacteristic, complemented, Sylow
186#553D12711yescharacteristic, complemented
187#547—1271no—
188#491—1271no—
189#485—1271no—
190#483—1271no—
191#478—1271no—
192#468—1271no—
193#465—1271no—
194#461—1271no—
195#449—1271no—
196#449—1271no—
197#442—1271no—
198#435—1271no—
199#417—1271no—
200#416—1271no—
201C2×#41—1271no—
202C2×#41—1271no—
203C2×SD16—1271no—
204C2×SD16—1271no—
205C2×D8—1271no—
206C2×M16—1271no—
207C22×C8—1271no—
208C4⋊Q8—1271no—
209#126—21no—
210#125C2×D12711yescharacteristic
211#123—21no—
212#123—2541no—
213#123—2541no—
214C22⋊Q8—1271no—
215C22⋊Q8—1271no—
216#121—21no—
217#121—2541no—
218#121C2×D12711yescharacteristic
219#121C2×D12711yescharacteristic
220C4×D4—21no—
221C4×D4—1271no—
222C4×D4C2×D12711yescharacteristic
223C4×D4C2×D12711yescharacteristic
224#117C2×D12711yescharacteristic
225C2×C4⋊C4—2541no—
226C2×C4⋊C4C2×D12711yescharacteristic
227#107—1271no—
228#106—1271no—
229#106—1271no—
230C4⋊C8—1271no—
231#103—1271no—
232#103—1271no—
233#103—1271no—
234#103—1271no—
235#103—1271no—
236#103—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102—1271no—
241C22⋊C8—1271no—
242C22⋊C8—1271no—
243#97—1271no—
244#41—5081no—
245#41—5081no—
246#41—2542no—
247#41—2542no—
248C2×Q8—1271no—
249C2×Q8—1271no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—1271no—
253C2×D4—2541no—
254C2×D4—2541no—
255C2×D4C22⋊D12711yescharacteristic
256C2×D4C22×D12711yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—2541no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4C22×D12711yescharacteristic
263C22×C4C22×D12711yescharacteristic
264SD16—5081no—
265SD16—5081no—
266D8—2542no—
267M16—5081no—
268C2×C8—1271no—
269C2×C8—1271no—
270C2×C8—1271no—
271C2×C8—1271no—
272C2×C8—5081no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—1271no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4C22⋊D12711yescharacteristic
282C4⋊C4C22×D12711yescharacteristic
283C4⋊C4C22×D12711yescharacteristic
284C4⋊C4C22×D12711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4C22⋊D12711yescharacteristic
293C22⋊C4C22⋊D12711yescharacteristic
294C42—21no—
295C42C22×D12711yescharacteristic
296C23—21no—
297C23—2541no—
298C23C2×C22⋊D12711yescharacteristic
299C23C2×C22⋊D12711yescharacteristic
300Q8—2541no—
301Q8—2541no—
302Q8—2541no—
303Q8—2541no—
304D4—21no—
305D4—21no—
306D4—41no—
307D4—42no—
308D4—81no—
309D4—1272no—
310D4—5082no—
311D4—5082no—
312D4—2541no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—5082no—
323C2×C4—5082no—
324C2×C4—5082no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4C2×C22⋊D12711yescharacteristic
329C2×C4C2×C22⋊D12711yescharacteristic
330C2×C4C2×C22⋊D12711yescharacteristic
331C2×C4C2×C22⋊D12711yescharacteristic
332C2×C4C23×D12711yescharacteristic
333C8—2541no—
334C8—2541no—
335C8—2541no—
336C8—2541no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—5081no—
347C22—5081no—
348C22—2542no—
349C22C22×C22⋊D12711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—5081no—
358C4—5081no—
359C4—5081no—
360C4#41×D12711yescharacteristic
361C4#1686125511yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—5082no—
366C2#2255051411yescharacteristic
367C2#2255067411yescharacteristic
368C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#553×C127C2
socleC22×C127C22×D4
hypercenter#553D127
hypocenterC127#3307
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #22550531C2semidirect
2. #22550246C2semidirect
3. #22550217C2semidirect
4. #22550206C2semidirect
5. #22550173C2semidirect
6. #22550120C2semidirect
7. #22550103C2semidirect
8. #22550081C2semidirect
9. #22550019C2semidirect
10. #22550019C2semidirect
11. #22549986C2semidirect
12. #22549948C2semidirect
13. #22549875C2semidirect
14. #22549870C2semidirect
15. #553×C127C2semidirect
185. C127#3307semidirect
186. #553D127semidirect

Maximal subgroups

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

subgroupindex
#225505312
#225502462
#225502172
#225502062
#225501732
#225501202
#225501032
#225500812
#225500192
#225499862
#225499482
#225498752
#225498702
#553×C1272
#3307127