Finite Group Databasebeta

← #30650639

Subgroups — #30650639

Order 16256.

398 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#22550576C211yesmaximal, characteristic, complemented
2#22550534C211yesmaximal, characteristic, complemented
3#22550230C211yesmaximal, characteristic, complemented
4#22550216C211yesmaximal, characteristic, complemented
5#22550193C211yesmaximal, characteristic, complemented
6#22550184C211yesmaximal, characteristic, complemented
7#22550109C211yesmaximal, characteristic, complemented
8#22550102C211yesmaximal, characteristic, complemented
9#22550018C211yesmaximal, characteristic, complemented
10#22550013C211yesmaximal, characteristic, complemented
11C4×#14523236C211yesmaximal, characteristic, complemented
12#22549941C211yesmaximal, characteristic, complemented
13#22549869C211yesmaximal, characteristic, complemented
14#103×D127C211yesmaximal, characteristic, complemented
15#463×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×C8×D127C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic
21#16861202—21no—
22#16861201C2211yescharacteristic
23#16861195—21no—
24#16861192—21no—
25#16861186—21no—
26#16861185C2211yescharacteristic
27#16861182—21no—
28#16861182—21no—
29#16861180C2211yescharacteristic
30#16861180C2211yescharacteristic
31#16861180C2211yescharacteristic
32C4×Dic254C2211yescharacteristic
33C4×D508C2211yescharacteristic
34C4×C22⋊D127—21no—
35(C22⋊D127)⋊C4—21no—
36#16861167C2211yescharacteristic
37C4⋊C4×D127C2211yescharacteristic
38#16861143C2211yescharacteristic
39#16861141C2211yescharacteristic
40#16861140C2211yescharacteristic
41#16861134C2211yescharacteristic
42#16861134C2211yescharacteristic
43#16861133C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861132C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C4×C127⋊C8C2211yescharacteristic
52C2×SD16×C127C2211yescharacteristic
53C2×D8×C127C2211yescharacteristic
54#124×C127C2211yescharacteristic
55C4×D4×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#14523247—22no—
60#14523247—41no—
61C2×Dic254—21no—
62C2×Dic254C2311yescharacteristic
63Q8×D127—22no—
64Q8×D127—41no—
65C2×D508D411yescharacteristic
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C22×Dic127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127C2311yescharacteristic
74C2×C4×D127C2311yescharacteristic
75#14523236—22no—
76C2×C127⋊C8C2311yescharacteristic
77C2×C127⋊C8C2311yescharacteristic
78C8×D127—41no—
79C8×D127—41no—
80C4⋊Dic127D411yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89C22⋊Dic127—21no—
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4D411yescharacteristic
95(C2×D127)⋊C4D411yescharacteristic
96C4×Dic127—21no—
97C4×Dic127—21no—
98C2×Q8×C127C2311yescharacteristic
99C2×D4×C127C2311yescharacteristic
100C2×D4×C127C2311yescharacteristic
101C22×C4×C127—21no—
102SD16×C127—41no—
103D8×C127—41no—
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C4⋊C4×C127C2311yescharacteristic
107C22⋊C4×C127—21no—
108C22⋊C4×C127—21no—
109C22⋊C4×C127—21no—
110C42×C127C2311yescharacteristic
111C22×D127C2×D411yescharacteristic
112C22×D127C2×D411yescharacteristic
113Dic254—21no—
114Dic254—42no—
115Dic254#4112yes—
116D508—22no—
117C22⋊D127—42no—
118C22⋊D127—42no—
119C22⋊D127—42no—
120C22⋊D127—81no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—42no—
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C4×D127—21no—
129C4×D127—21no—
130C4×D127—21no—
131C4×D127—21no—
132C4×D127—42no—
133C4×D127—81no—
134C127⋊C8—21no—
135C127⋊C8—21no—
136C23×C127—21no—
137C23×C127—21no—
138Q8×C127—21no—
139Q8×C127—21no—
140D4×C127—21no—
141D4×C127—21no—
142D4×C127—21no—
143D4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—81no—
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2411yescharacteristic
150C8×C127—21no—
151C8×C127—21no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127C22×D411yescharacteristic
159C2×D127—21no—
160C2×D127—21no—
161C2×D127—21no—
162C2×D127—21no—
163C2×D127—21no—
164C2×D127#13512yes—
165C4×C127—21no—
166C4×C127—21no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127C2×#4111yescharacteristic
170C4×C127H(5,𝔽2)11yescharacteristic
171Dic127—21no—
172Dic127—21no—
173Dic127—22no—
174Dic127—41no—
175Dic127—41no—
176Dic127—41no—
177C2×C127—81no—
178C2×C127—81no—
179C2×C127#54611yescharacteristic
180C2×C127C2×#13511yescharacteristic
181C2×C127#57711yescharacteristic
182D127—22no—
183D127—41no—
184#3301—1271nomaximal, Sylow
185C127#330111yescharacteristic, complemented, Sylow
186#553—1271no—
187#548—1271no—
188#488—1271no—
189#485—1271no—
190#481—1271no—
191#479—1271no—
192#466—1271no—
193#465—1271no—
194#463D12711yescharacteristic, complemented
195#449—1271no—
196#448—1271no—
197C4×SD16—1271no—
198C8×D4—1271no—
199#416—1271no—
200C2×#103—1271no—
201C2×#41—1271no—
202C22×Q8—1271no—
203C2×SD16—1271no—
204C2×SD16C2×D12711yescharacteristic
205C2×D8C2×D12711yescharacteristic
206C22×C8—1271no—
207C22×C8—1271no—
208#126—2541no—
209#125—1271no—
210#124—2541no—
211#124C2×D12711yescharacteristic
212#123—2541no—
213C22⋊Q8—1271no—
214C22⋊Q8—2541no—
215#121—1271no—
216#121—1271no—
217#121—1271no—
218#121—2541no—
219#121—2541no—
220C4×Q8—1271no—
221C4×D4—1271no—
222C4×D4—2541no—
223C4×D4—2541no—
224C4×D4C2×D12711yescharacteristic
225#117—1271no—
226C2×C4⋊C4—1271no—
227#107—1271no—
228#106—1271no—
229#106—1271no—
230C4⋊C8C2×D12711yescharacteristic
231#103—1271no—
232#103—1271no—
233#103—1271no—
234#103—1271no—
235#103—1271no—
236#103C2×D12711yescharacteristic
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102C2×D12711yescharacteristic
241C22⋊C8—1271no—
242C22⋊C8—1271no—
243C4×C8—1271no—
244#41—5081no—
245#41—2542no—
246C2×Q8—1271no—
247C2×Q8—5081no—
248C2×Q8—2541no—
249C2×Q8—2542no—
250C2×Q8C22×D12711yescharacteristic
251C2×D4—1271no—
252C2×D4—2541no—
253C2×D4—2541no—
254C2×D4—2541no—
255C2×D4—2541no—
256C2×D4C22×D12711yescharacteristic
257C2×D4C22×D12711yescharacteristic
258C22×C4—21no—
259C22×C4—1271no—
260C22×C4—1271no—
261C22×C4—2541no—
262C22×C4—2541no—
263C22×C4—2541no—
264SD16—41no—
265SD16—2542no—
266D8—41no—
267C2×C8—1271no—
268C2×C8—1271no—
269C2×C8—5081no—
270C2×C8—5081no—
271C2×C8C22×D12711yescharacteristic
272C2×C8C22×D12711yescharacteristic
273C4⋊C4—1271no—
274C4⋊C4—1271no—
275C4⋊C4—1271no—
276C4⋊C4—1271no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4—2541no—
282C4⋊C4C22×D12711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—1271no—
287C22⋊C4—1271no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C42—2541no—
294C42—2541no—
295C42C22×D12711yescharacteristic
296C23—21no—
297C23—21no—
298C23—1271no—
299C23—1271no—
300Q8—21no—
301Q8—21no—
302Q8—1272no—
303Q8—5082no—
304Q8—2541no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—10161no—
310D4—5082no—
311D4—5082no—
312D4—5082no—
313D4—2542no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—81no—
317C2×C4—1271no—
318C2×C4—1271no—
319C2×C4—10161no—
320C2×C4—5082no—
321C2×C4—5082no—
322C2×C4—2541no—
323C2×C4—2541no—
324C2×C4—2541no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4—2541no—
330C2×C4D4×D12711yescharacteristic
331C2×C4D4×D12711yescharacteristic
332C2×C4C23×D12711yescharacteristic
333C8—21no—
334C8—21no—
335C8—2541no—
336C8—2541no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—1272no—
344C22—2541no—
345C22—2541no—
346C22—2541no—
347C22—2541no—
348C22—2541no—
349C22C2×D4×D12711yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—5081no—
355C4—5081no—
356C4—5081no—
357C4—2541no—
358C4—2541no—
359C4—2542no—
360C4#41×D12711yescharacteristic
361C4#1686125411yescharacteristic
362C2—81no—
363C2—81no—
364C2—5081no—
365C2—2542no—
366C2#2255052311yescharacteristic
367C2#135×D12711yescharacteristic
368C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#463×C127C2
socleC22×C127C22×D4
hypercenter#463D127
hypocenterC127#3301
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #22550576C2semidirect
2. #22550534C2semidirect
3. #22550230C2semidirect
4. #22550216C2semidirect
5. #22550193C2semidirect
6. #22550184C2semidirect
7. #22550109C2semidirect
8. #22550102C2semidirect
9. #22550018C2semidirect
10. #22550013C2semidirect
11. C4×#14523236C2semidirect
12. #22549941C2semidirect
13. #22549869C2semidirect
14. #103×D127C2semidirect
15. #463×C127C2semidirect
185. C127#3301semidirect
194. #463D127semidirect

Maximal subgroups

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

subgroupindex
#225505762
#225505342
#225502302
#225502162
#225501932
#225501842
#225501092
#225501022
#225500182
#225500132
C4×#145232362
#225499412
#225498692
#103×D1272
#463×C1272
#3301127