Finite Group Databasebeta

← #30650722

Subgroups — #30650722

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#22550527C211yesmaximal, characteristic, complemented
3#22550234C211yesmaximal, characteristic, complemented
4#22550211C211yesmaximal, characteristic, complemented
5#22550188C211yesmaximal, characteristic, complemented
6#22550169C211yesmaximal, characteristic, complemented
7#22550122C211yesmaximal, characteristic, complemented
8#22550111C211yesmaximal, characteristic, complemented
9#22550090C211yesmaximal, characteristic, complemented
10#22550021C211yesmaximal, characteristic, complemented
11#22550016C211yesmaximal, characteristic, complemented
12#22549979C211yesmaximal, characteristic, complemented
13#22549945C211yesmaximal, characteristic, complemented
14#103×D127C211yesmaximal, characteristic, complemented
15#417×C127C211yesmaximal, characteristic, complemented
16C2×#14523248—21no—
17C2×Q8×D127C2211yescharacteristic
18C2×#14523237C2211yescharacteristic
19C2×#14523235C2211yescharacteristic
20C2×#14523233C2211yescharacteristic
21C2×C8×D127C2211yescharacteristic
22#16861208C2211yescharacteristic
23#16861196—21no—
24#16861196C2211yescharacteristic
25#16861194—21no—
26#16861192—21no—
27#16861188—21no—
28#16861188C2211yescharacteristic
29#16861187—21no—
30#16861184C2211yescharacteristic
31#16861183C2211yescharacteristic
32#16861180—21no—
33#16861180C2211yescharacteristic
34Q8×Dic127C2211yescharacteristic
35#16861171C2211yescharacteristic
36C4×C22⋊D127—21no—
37C4×C22⋊D127C2211yescharacteristic
38C4⋊C4×D127C2211yescharacteristic
39#16861143C2211yescharacteristic
40#16861142C2211yescharacteristic
41#16861142C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861132C2211yescharacteristic
46#16861132C2211yescharacteristic
47#16861131C2211yescharacteristic
48#16861130C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51#16861119C2211yescharacteristic
52C2×#41×C127C2211yescharacteristic
53C2×M16×C127C2211yescharacteristic
54C2×C4⋊C4×C127C2211yescharacteristic
55#103×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#14523248—44no—
60C2×Dic254—21no—
61C2×Dic254—21no—
62C2×Dic254C2311yescharacteristic
63Q8×D127—22no—
64Q8×D127—41no—
65C2×D508—21no—
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127D411yescharacteristic
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127C2311yescharacteristic
74#14523237—41no—
75#14523235—22no—
76#14523233—41no—
77C2×C127⋊C8C2311yescharacteristic
78C2×C127⋊C8C2311yescharacteristic
79C8×D127—41no—
80C4⋊Dic127—21no—
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⋊Dic127D411yescharacteristic
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4D411yescharacteristic
96C4×Dic127—21no—
97C4×Dic127C2311yescharacteristic
98#41×C127—22no—
99#41×C127—41no—
100C2×Q8×C127C2311yescharacteristic
101C2×D4×C127—21no—
102C2×D4×C127C2311yescharacteristic
103C22×C4×C127—21no—
104C22×C4×C127—21no—
105C22×C4×C127C2311yescharacteristic
106M16×C127—41no—
107C2×C8×C127C2311yescharacteristic
108C2×C8×C127C2311yescharacteristic
109C4⋊C4×C127—21no—
110C4⋊C4×C127C2311yescharacteristic
111C4⋊C4×C127C2311yescharacteristic
112C22×D127—21no—
113C22×D127C2×D411yescharacteristic
114Dic254—21no—
115Dic254—21no—
116Dic254—42no—
117Dic254—42no—
118D508—21no—
119D508—21no—
120D508—42no—
121C22⋊D127—22no—
122C22⋊D127—42no—
123C22⋊D127—42no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C2×Dic127C2×D411yescharacteristic
130C4×D127—21no—
131C4×D127—21no—
132C4×D127—42no—
133C4×D127—42no—
134C4×D127—42no—
135C127⋊C8—21no—
136C127⋊C8—21no—
137C23×C127—21no—
138C23×C127C2×D411yescharacteristic
139Q8×C127—21no—
140Q8×C127#4112yescomplemented
141D4×C127—21no—
142D4×C127—21no—
143D4×C127—42no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—42no—
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×C127C22×D411yescharacteristic
162C2×D127—21no—
163C2×D127—21no—
164C2×D127—41no—
165C2×D127—41no—
166C2×D127—41no—
167C2×D127AGL(1,ℤ/8)12yes—
168C4×C127—21no—
169C4×C127—22no—
170C4×C127—41no—
171C4×C127—41no—
172C4×C127C2×#4111yescharacteristic
173C4×C127H(5,𝔽2)11yescharacteristic
174Dic127—21no—
175Dic127—21no—
176Dic127—21no—
177Dic127—41no—
178Dic127—41no—
179Dic127—41no—
180C2×C127—41no—
181C2×C127—81no—
182C2×C127#54611yescharacteristic
183C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C127#57811yescharacteristic
185D127—22no—
186D127—81no—
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#449—1271no—
199#448—1271no—
200#440—1271no—
201#435—1271no—
202#417D12711yescharacteristic, complemented
203C2×#103—1271no—
204C2×#41—2541no—
205C2×#41C2×D12711yescharacteristic
206C22×Q8—1271no—
207C2×SD16—1271no—
208C2×SD16—1271no—
209C2×D8—1271no—
210C2×M16C2×D12711yescharacteristic
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—1271no—
223#121—2541no—
224C4×Q8—1271no—
225C4×D4—1271no—
226C4×D4—1271no—
227C4×D4—2541no—
228C2×C4⋊C4—1271no—
229C2×C4⋊C4C2×D12711yescharacteristic
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#103C2×D12711yescharacteristic
239#103C2×D12711yescharacteristic
240#102—1271no—
241#102—1271no—
242#102C2×D12711yescharacteristic
243#102C2×D12711yescharacteristic
244C22⋊C8—1271no—
245C22⋊C8—1271no—
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—1271no—
258C2×D4—1271no—
259C2×D4—2541no—
260C2×D4—2541no—
261C2×D4—2541no—
262C2×D4C22×D12711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—1271no—
266C22×C4—2541no—
267C22×C4—2541no—
268C22×C4—2541no—
269C22×C4C22×D12711yescharacteristic
270SD16—5081no—
271SD16—2542no—
272D8—5081no—
273M16—41no—
274C2×C8—1271no—
275C2×C8—1271no—
276C2×C8—5081no—
277C2×C8C22×D12711yescharacteristic
278C2×C8C22×D12711yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—1271no—
281C4⋊C4—1271no—
282C4⋊C4—1271no—
283C4⋊C4—2541no—
284C4⋊C4—2541no—
285C4⋊C4—2541no—
286C4⋊C4—2541no—
287C4⋊C4C22×D12711yescharacteristic
288C4⋊C4C22×D12711yescharacteristic
289C22⋊C4—1271no—
290C22⋊C4—1271no—
291C22⋊C4—2541no—
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—1271no—
302C23—2541no—
303C23D4×D12711yescharacteristic
304Q8—21no—
305Q8—5082no—
306Q8—5082no—
307Q8—2541no—
308Q8—2541no—
309Q8#1452324812yescomplemented
310D4—21no—
311D4—21no—
312D4—42no—
313D4—5082no—
314D4—5082no—
315D4—5082no—
316D4—2542no—
317D4—2541no—
318D4—2541no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—1271no—
327C2×C4—1271no—
328C2×C4—1271no—
329C2×C4—5082no—
330C2×C4—5082no—
331C2×C4—5082no—
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—1272no—
350C22—5081no—
351C22—5081no—
352C22—5081no—
353C22—2541no—
354C22—2541no—
355C22C2×D4×D12711yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—5081no—
361C4—5081no—
362C4—5081no—
363C4—2541no—
364C4—2541no—
365C4—2541no—
366C4C2×#1452324811yescharacteristic
367C4#1686125411yescharacteristic
368C2—41no—
369C2—81no—
370C2—10161no—
371C2—2542no—
372C2#2255051611yescharacteristic
373C2AGL(1,ℤ/8)×D12711yescharacteristic
374C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#417×C127C2
socleC22×C127C22×D4
hypercenter#417D127
hypocenterC127#3308
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #22550533C2semidirect
2. #22550527C2semidirect
3. #22550234C2semidirect
4. #22550211C2semidirect
5. #22550188C2semidirect
6. #22550169C2semidirect
7. #22550122C2semidirect
8. #22550111C2semidirect
9. #22550090C2semidirect
10. #22550021C2semidirect
11. #22550016C2semidirect
12. #22549979C2semidirect
13. #22549945C2semidirect
14. #103×D127C2semidirect
15. #417×C127C2semidirect
140. Q8×C127#41semidirect
188. C127#3308semidirect
202. #417D127semidirect
309. Q8#14523248semidirect

Maximal subgroups

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

subgroupindex
#225505332
#225505272
#225502342
#225502112
#225501882
#225501692
#225501222
#225501112
#225500902
#225500212
#225500162
#225499792
#225499452
#103×D1272
#417×C1272
#3308127