Finite Group Databasebeta

← #30650714

Subgroups — #30650714

Order 16256.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550533C211yesmaximal, characteristic, complemented
2#22550526C211yesmaximal, characteristic, complemented
3#22550234C211yesmaximal, characteristic, complemented
4#22550212C211yesmaximal, characteristic, complemented
5#22550188C211yesmaximal, characteristic, complemented
6#22550169C211yesmaximal, characteristic, complemented
7#22550120C211yesmaximal, characteristic, complemented
8#22550109C211yesmaximal, characteristic, complemented
9#22550089C211yesmaximal, characteristic, complemented
10#22550013C211yesmaximal, characteristic, complemented
11#22549977C211yesmaximal, characteristic, complemented
12#22549950C211yesmaximal, characteristic, complemented
13#22549877C211yesmaximal, characteristic, complemented
14#103×D127C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16C2×#14523247—21no—
17C2×Q8×D127C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523232C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic
21#16861208C2211yescharacteristic
22#16861196—21no—
23#16861196C2211yescharacteristic
24#16861191—21no—
25#16861190—21no—
26#16861188—21no—
27#16861188C2211yescharacteristic
28#16861186—21no—
29#16861180—21no—
30#16861180C2211yescharacteristic
31#16861180C2211yescharacteristic
32Q8×Dic127C2211yescharacteristic
33D4×Dic127C2211yescharacteristic
34C4×C22⋊D127—21no—
35C4×C22⋊D127C2211yescharacteristic
36C2×C4⋊Dic127C2211yescharacteristic
37C4⋊C4×D127C2211yescharacteristic
38#16861143C2211yescharacteristic
39#16861143C2211yescharacteristic
40#16861142C2211yescharacteristic
41Dic127⋊C8C2211yescharacteristic
42#16861134C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861133C2211yescharacteristic
46#16861132C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49#16861130C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51#16861119C2211yescharacteristic
52C2×#41×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54C2×D8×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic
59#14523247—44no—
60C2×Dic254—21no—
61C2×Dic254—21no—
62C2×Dic254C2311yescharacteristic
63Q8×D127—22no—
64Q8×D127—41no—
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127D411yescharacteristic
68C22×Dic127—21no—
69C22×Dic127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127C2311yescharacteristic
73#14523236—41no—
74#14523232—41no—
75C2×C127⋊C8C2311yescharacteristic
76C2×C127⋊C8C2311yescharacteristic
77C8×D127—41no—
78C4⋊Dic127—21no—
79C4⋊Dic127—21no—
80C4⋊Dic127C2311yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4D411yescharacteristic
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92C22⋊Dic127—21no—
93C22⋊Dic127D411yescharacteristic
94(C2×D127)⋊C4D411yescharacteristic
95C4×Dic127—21no—
96C4×Dic127C2311yescharacteristic
97#41×C127—22no—
98#41×C127—41no—
99C2×Q8×C127C2311yescharacteristic
100C2×D4×C127—21no—
101C2×D4×C127—21no—
102C2×D4×C127C2311yescharacteristic
103C2×D4×C127C2311yescharacteristic
104C22×C4×C127—21no—
105C22×C4×C127C2311yescharacteristic
106SD16×C127—22no—
107D8×C127—41no—
108C2×C8×C127C2311yescharacteristic
109C2×C8×C127C2311yescharacteristic
110C4⋊C4×C127C2311yescharacteristic
111C22⋊C4×C127—21no—
112C22×D127C2×D411yescharacteristic
113Dic254—21no—
114Dic254—21no—
115Dic254—42no—
116Dic254—42no—
117C22⋊D127—22no—
118C22⋊D127—42no—
119C22⋊D127—42no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—42no—
125C2×Dic127—42no—
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C4×D127—21no—
130C4×D127—21no—
131C4×D127—42no—
132C4×D127—42no—
133C127⋊C8—21no—
134C127⋊C8—21no—
135C23×C127—21no—
136C23×C127—21no—
137C23×C127C2×D411yescharacteristic
138Q8×C127—21no—
139Q8×C127#4112yescomplemented
140D4×C127—21no—
141D4×C127—21no—
142D4×C127—21no—
143D4×C127—21no—
144D4×C127—42no—
145D4×C127—42no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—42no—
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×D127AGL(1,ℤ/8)12yes—
168C4×C127—21no—
169C4×C127—22no—
170C4×C127—41no—
171C4×C127C2×#4111yescharacteristic
172C4×C127H(5,𝔽2)11yescharacteristic
173Dic127—21no—
174Dic127—21no—
175Dic127—21no—
176Dic127—41no—
177Dic127—41no—
178Dic127—41no—
179Dic127—41no—
180C2×C127—41no—
181C2×C127—81no—
182C2×C127—81no—
183C2×C127#54611yescharacteristic
184C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C127#57811yescharacteristic
186D127—22no—
187#3308—1271nomaximal, Sylow
188C127#330811yescharacteristic, complemented, Sylow
189#548—1271no—
190#547—1271no—
191#489—1271no—
192#484—1271no—
193#480—1271no—
194#477—1271no—
195#468—1271no—
196#466—1271no—
197#463—1271no—
198#449D12711yescharacteristic, complemented
199#448—1271no—
200#440—1271no—
201#435—1271no—
202#417—1271no—
203C2×#103—1271no—
204C2×#41—2541no—
205C2×#41C2×D12711yescharacteristic
206C22×Q8—1271no—
207C2×SD16—1271no—
208C2×SD16C2×D12711yescharacteristic
209C2×D8C2×D12711yescharacteristic
210C2×M16—1271no—
211C22×C8—1271no—
212C4⋊Q8—1271no—
213#124—1271no—
214#124—2541no—
215#123—2541no—
216#123—2541no—
217C22⋊Q8—1271no—
218C22⋊Q8—2541no—
219C22⋊Q8—2541no—
220#121—1271no—
221#121—1271no—
222#121—2541no—
223#121C2×D12711yescharacteristic
224C4×Q8—1271no—
225C4×D4—1271no—
226C4×D4—1271no—
227C4×D4—2541no—
228C2×C4⋊C4—1271no—
229C2×C4⋊C4—1271no—
230#107—1271no—
231#107—1271no—
232#107—1271no—
233C4⋊C8—1271no—
234#103—1271no—
235#103—1271no—
236#103—1271no—
237#103—1271no—
238#103—1271no—
239#103C2×D12711yescharacteristic
240#102—1271no—
241#102—1271no—
242#102—1271no—
243#102C2×D12711yescharacteristic
244C22⋊C8—1271no—
245C22⋊C8C2×D12711yescharacteristic
246#97—1271no—
247#41—22no—
248#41—41no—
249#41—5084no—
250C2×Q8—1271no—
251C2×Q8—5081no—
252C2×Q8—2541no—
253C2×Q8—2541no—
254C2×Q8—2542no—
255C2×Q8C22×D12711yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—1271no—
259C2×D4—2541no—
260C2×D4—2541no—
261C2×D4C22×D12711yescharacteristic
262C2×D4C22×D12711yescharacteristic
263C22×C4—21no—
264C22×C4—1271no—
265C22×C4—2541no—
266C22×C4—2541no—
267C22×C4—2541no—
268C22×C4—2541no—
269C22×C4C22×D12711yescharacteristic
270SD16—22no—
271SD16—5081no—
272D8—41no—
273M16—5081no—
274C2×C8—1271no—
275C2×C8—1271no—
276C2×C8—5081no—
277C2×C8C22×D12711yescharacteristic
278C2×C8C22×D12711yescharacteristic
279C4⋊C4—1271no—
280C4⋊C4—1271no—
281C4⋊C4—1271no—
282C4⋊C4—1271no—
283C4⋊C4—2541no—
284C4⋊C4—2541no—
285C4⋊C4—2541no—
286C4⋊C4—2541no—
287C4⋊C4—2541no—
288C4⋊C4C22×D12711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—1271no—
291C22⋊C4—1271no—
292C22⋊C4—2541no—
293C22⋊C4—2541no—
294C22⋊C4—2541no—
295C22⋊C4—2541no—
296C22⋊C4—2541no—
297C22⋊C4—2541no—
298C42—1271no—
299C42—2541no—
300C23—21no—
301C23—21no—
302C23—1271no—
303C23D4×D12711yescharacteristic
304Q8—21no—
305Q8—5082no—
306Q8—5082no—
307Q8—2541no—
308Q8—2541no—
309Q8#1452324712yescomplemented
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—42no—
316D4—5082no—
317D4—5082no—
318D4—2542no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—1271no—
325C2×C4—1271no—
326C2×C4—1271no—
327C2×C4—5082no—
328C2×C4—5082no—
329C2×C4—5082no—
330C2×C4—5082no—
331C2×C4—2541no—
332C2×C4—2541no—
333C2×C4—2541no—
334C2×C4—2541no—
335C2×C4—2541no—
336C2×C4—2541no—
337C2×C4D4×D12711yescharacteristic
338C2×C4C23×D12711yescharacteristic
339C8—21no—
340C8—21no—
341C8—2541no—
342C8—2541no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—1272no—
353C22—2541no—
354C22—2541no—
355C22C2×D4×D12711yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—5081no—
360C4—5081no—
361C4—5081no—
362C4—5081no—
363C4—2541no—
364C4—2541no—
365C4—2541no—
366C4C2×#1452324711yescharacteristic
367C4#1686125411yescharacteristic
368C2—41no—
369C2—81no—
370C2—81no—
371C2—2542no—
372C2#2255051311yescharacteristic
373C2AGL(1,ℤ/8)×D12711yescharacteristic
374C2#2255069011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3308
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #22550533C2semidirect
2. #22550526C2semidirect
3. #22550234C2semidirect
4. #22550212C2semidirect
5. #22550188C2semidirect
6. #22550169C2semidirect
7. #22550120C2semidirect
8. #22550109C2semidirect
9. #22550089C2semidirect
10. #22550013C2semidirect
11. #22549977C2semidirect
12. #22549950C2semidirect
13. #22549877C2semidirect
14. #103×D127C2semidirect
15. #449×C127C2semidirect
139. Q8×C127#41semidirect
188. C127#3308semidirect
198. #449D127semidirect
309. Q8#14523247semidirect

Maximal subgroups

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

subgroupindex
#225505332
#225505262
#225502342
#225502122
#225501882
#225501692
#225501202
#225501092
#225500892
#225500132
#225499772
#225499502
#225498772
#103×D1272
#449×C1272
#3308127