Finite Group Databasebeta

← #30650575

Subgroups — #30650575

Order 16256.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550573C211yesmaximal, characteristic, complemented
2#22550523C211yesmaximal, characteristic, complemented
3#22550229C211yesmaximal, characteristic, complemented
4#22550213C211yesmaximal, characteristic, complemented
5#22550197C211yesmaximal, characteristic, complemented
6#22550179C211yesmaximal, characteristic, complemented
7#22550104C211yesmaximal, characteristic, complemented
8#22550089C211yesmaximal, characteristic, complemented
9#22550019C211yesmaximal, characteristic, complemented
10#22550008C211yesmaximal, characteristic, complemented
11#22549956C211yesmaximal, characteristic, complemented
12C8×C22⋊D127C211yesmaximal, characteristic, complemented
13#22549873C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#466×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×D4×D127C2211yescharacteristic
18C2×#14523236C2211yescharacteristic, complemented
19C2×#14523233C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic
21#16861202—21no—
22#16861200C2211yescharacteristic, complemented
23#16861196C2211yescharacteristic
24#16861191—21no—
25#16861190—21no—
26#16861189C2211yescharacteristic
27#16861180—21no—
28#16861180—21no—
29#16861180C2211yescharacteristic
30#16861177—21no—
31D4×Dic127C2211yescharacteristic, complemented
32#16861171C2211yescharacteristic
33C4×C22⋊D127C2211yescharacteristic
34(C22⋊D127)⋊C4—21no—
35#16861168C2211yescharacteristic, complemented
36C4⋊C4×D127C2211yescharacteristic
37C22⋊C4×D127—21no—
38#16861142C2211yescharacteristic, complemented
39#16861140C2211yescharacteristic, complemented
40Dic127⋊C8C2211yescharacteristic, complemented
41#16861134C2211yescharacteristic, complemented
42#16861134C2211yescharacteristic, complemented
43#16861132C2211yescharacteristic, complemented
44#16861131C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861129C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic, complemented
50(C2×D127)⋊C8C2211yescharacteristic
51C8×Dic127C2211yescharacteristic, complemented
52C2×D8×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic, complemented
54#121×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic, complemented
56#107×C127C2211yescharacteristic, complemented
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60#14523248—22no—
61#14523248—41no—
62C2×Dic254C2311yescharacteristic
63C2×D508C2311yescharacteristic
64C2×C22⋊D127—21no—
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127D411yescharacteristic
68D4×D127—22no—
69D4×D127—41no—
70C22×Dic127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74C2×C4×D127C2311yescharacteristic
75#14523236—41no—
76#14523233—22no—
77C2×C127⋊C8C2311yescharacteristic
78C2×C127⋊C8C2311yescharacteristic
79C8×D127—41no—
80C4⋊Dic127—21no—
81C4⋊Dic127—21no—
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4D411yescharacteristic
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92C22⋊Dic127D411yescharacteristic
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4D411yescharacteristic
96C4×Dic127—21no—
97C4×Dic127C2311yescharacteristic
98C2×D4×C127—21no—
99C2×D4×C127—21no—
100C2×D4×C127C2311yescharacteristic
101C2×D4×C127C2311yescharacteristic
102C22×C4×C127C2311yescharacteristic
103D8×C127—41no—
104C2×C8×C127—41no—
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C4⋊C4×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C22⋊C4×C127—21no—
110C22⋊C4×C127—21no—
111C22×D127—21no—
112C22×D127—42no—
113C22×D127—42no—
114C22×D127—81no—
115C22×D127C2×D411yescharacteristic
116Dic254—21no—
117Dic254—21no—
118D508—21no—
119D508#4112yes—
120C22⋊D127—22no—
121C22⋊D127—42no—
122C22⋊D127—42no—
123C22⋊D127—42no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—81no—
128C2×Dic127C2×D411yescharacteristic
129C2×Dic127C2×D411yescharacteristic
130C2×Dic127C2×D411yescharacteristic
131C4×D127—21no—
132C4×D127—21no—
133C4×D127—42no—
134C4×D127—81no—
135C4×D127—81no—
136C127⋊C8—21no—
137C127⋊C8—21no—
138C23×C127—21no—
139C23×C127—21no—
140C23×C127C2×D411yescharacteristic
141D4×C127—21no—
142D4×C127—21no—
143D4×C127—21no—
144D4×C127—21no—
145D4×C127—81no—
146D4×C127—81no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—21no—
151C2×C4×C127C2×D411yescharacteristic
152C2×C4×C127C2411yescharacteristic
153C8×C127—21no—
154C8×C127—21no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—21no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127C22×D411yescharacteristic
165C2×D127—21no—
166C2×D127—21no—
167C2×D127—22no—
168C2×D127—41no—
169C2×D127—41no—
170C2×D127—84no—
171C2×D127#13512yes—
172C4×C127—21no—
173C4×C127—41no—
174C4×C127—41no—
175C4×C127C2×#4111yescharacteristic
176C4×C127H(5,𝔽2)11yescharacteristic
177Dic127—21no—
178Dic127—21no—
179Dic127—21no—
180Dic127—41no—
181Dic127—41no—
182Dic127—41no—
183C2×C127—41no—
184C2×C127—81no—
185C2×C127—81no—
186C2×C127#54611yescharacteristic
187C2×C127C2×#13511yescharacteristic
188C2×C127#57611yescharacteristic
189D127—22no—
190D127—42no—
191#3297—1271nomaximal, Sylow
192C127#329711yescharacteristic, complemented, Sylow
193#553—1271no—
194#546—1271no—
195#488—1271no—
196#484—1271no—
197#482—1271no—
198#478—1271no—
199#466D12711yescharacteristic, complemented
200#465—1271no—
201#463—1271no—
202#449—1271no—
203#447—1271no—
204C4×D8—1271no—
205C8×D4—1271no—
206#416—1271no—
207C2×#102—1271no—
208C2×#41—1271no—
209C22×D4—1271no—
210C2×SD16—1271no—
211C2×D8—1271no—
212C2×D8C2×D12711yescharacteristic
213C22×C8—1271no—
214C22×C8C2×D12711yescharacteristic, complemented
215#126—2541no—
216#125—1271no—
217#124—1271no—
218#123—2541no—
219#123—2541no—
220C22⋊Q8—1271no—
221#121—1271no—
222#121—2541no—
223#121—2541no—
224#121C2×D12711yescharacteristic
225#121C2×D12711yescharacteristic, complemented
226#120—2541no—
227C4×D4—1271no—
228C4×D4—1271no—
229C4×D4—1271no—
230C4×D4—2541no—
231#117—1271no—
232C2×C4⋊C4—1271no—
233C2×C22⋊C4—2541no—
234#107—1271no—
235#107C2×D12711yescharacteristic, complemented
236#106—1271no—
237C4⋊C8—1271no—
238#103—1271no—
239#103—1271no—
240#103—1271no—
241#102—1271no—
242#102—1271no—
243#102—1271no—
244#102—1271no—
245#102—1271no—
246#102C2×D12711yescharacteristic
247#102C2×D12711yescharacteristic, complemented
248C22⋊C8—1271no—
249C22⋊C8—1271no—
250C4×C8—1271no—
251C24—2541no—
252#41—5081no—
253#41—2542no—
254C2×Q8—1271no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—1271no—
258C2×D4—1271no—
259C2×D4—5081no—
260C2×D4—2541no—
261C2×D4—2542no—
262C2×D4—2541no—
263C2×D4—2541no—
264C2×D4C22×D12711yescharacteristic
265C2×D4C22×D12711yescharacteristic
266C22×C4—1271no—
267C22×C4—2541no—
268C22×C4—2541no—
269C22×C4—2541no—
270C22×C4—2541no—
271C22×C4C22×D12711yescharacteristic
272SD16—5081no—
273D8—41no—
274D8—2542no—
275C2×C8—41no—
276C2×C8—1271no—
277C2×C8—1271no—
278C2×C8—5081no—
279C2×C8C22×D12711yescharacteristic
280C2×C8C22×D12711yescharacteristic
281C4⋊C4—1271no—
282C4⋊C4—1271no—
283C4⋊C4—1271no—
284C4⋊C4—2541no—
285C4⋊C4—2541no—
286C4⋊C4—2541no—
287C4⋊C4—2541no—
288C4⋊C4C22×D12711yescharacteristic
289C4⋊C4C22×D12711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—1271no—
293C22⋊C4—1271no—
294C22⋊C4—2541no—
295C22⋊C4—2541no—
296C22⋊C4—2541no—
297C22⋊C4—2541no—
298C22⋊C4—2541no—
299C22⋊C4—2541no—
300C22⋊C4—2541no—
301C42—1271no—
302C42—2541no—
303C23—21no—
304C23—21no—
305C23—1271no—
306C23—10161no—
307C23—5082no—
308C23—5082no—
309C23—2541no—
310C23D4×D12711yescharacteristic
311Q8—2541no—
312Q8—2541no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—81no—
318D4—81no—
319D4—1272no—
320D4—5082no—
321D4—5082no—
322D4—5082no—
323D4—2542no—
324D4—2541no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—1271no—
330C2×C4—1271no—
331C2×C4—1271no—
332C2×C4—10161no—
333C2×C4—10161no—
334C2×C4—10161no—
335C2×C4—5082no—
336C2×C4—2541no—
337C2×C4—2541no—
338C2×C4—2541no—
339C2×C4—2541no—
340C2×C4—2541no—
341C2×C4D4×D12711yescharacteristic
342C2×C4C23×D12711yescharacteristic
343C8—21no—
344C8—21no—
345C8—2541no—
346C8—2541no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—1272no—
357C22—10164no—
358C22—5081no—
359C22—5081no—
360C22—2542no—
361C22—2541no—
362C22—2541no—
363C22C2×D4×D12711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—5081no—
368C4—5081no—
369C4—5081no—
370C4—2541no—
371C4—2541no—
372C4—2541no—
373C4#41×D12711yescharacteristic
374C4#1686125411yescharacteristic
375C2—41no—
376C2—81no—
377C2—81no—
378C2—5082no—
379C2—2542no—
380C2#2255051511yescharacteristic
381C2#135×D12711yescharacteristic
382C2#2255067711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#466×C127C2
socleC22×C127C22×D4
hypercenter#466D127
hypocenterC127#3297
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #22550573C2semidirect
2. #22550523C2semidirect
3. #22550229C2semidirect
4. #22550213C2semidirect
5. #22550197C2semidirect
6. #22550179C2semidirect
7. #22550104C2semidirect
8. #22550089C2semidirect
9. #22550019C2semidirect
10. #22550008C2semidirect
11. #22549956C2semidirect
12. C8×C22⋊D127C2semidirect
13. #22549873C2semidirect
14. #102×D127C2semidirect
15. #466×C127C2semidirect
18. C2×#14523236C22semidirect
22. #16861200C22semidirect
31. D4×Dic127C22semidirect
35. #16861168C22semidirect
38. #16861142C22semidirect
39. #16861140C22semidirect
40. Dic127⋊C8C22semidirect
41. #16861134C22semidirect
42. #16861134C22semidirect
43. #16861132C22semidirect
49. C22⋊(C127⋊C8)C22semidirect
51. C8×Dic127C22semidirect
53. C22×C8×C127C22semidirect
55. #121×C127C22semidirect
56. #107×C127C22semidirect
58. #102×C127C22semidirect
192. C127#3297semidirect
199. #466D127semidirect
214. C22×C8C2×D127semidirect
225. #121C2×D127semidirect
235. #107C2×D127semidirect
247. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505732
#225505232
#225502292
#225502132
#225501972
#225501792
#225501042
#225500892
#225500192
#225500082
#225499562
C8×C22⋊D1272
#225498732
#102×D1272
#466×C1272
#3297127