Finite Group Databasebeta

← #30650168

Subgroups — #30650168

Order 16256.

370 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#22550482C211yesmaximal, characteristic, complemented
2#22550477C211yesmaximal, characteristic, complemented
3#22550179C211yesmaximal, characteristic, complemented
4#22550175C211yesmaximal, characteristic, complemented
5#22550173C211yesmaximal, characteristic, complemented
6#22550130C211yesmaximal, characteristic, complemented
7#22550128C211yesmaximal, characteristic, complemented
8#22550113C211yesmaximal, characteristic, complemented
9#22550111C211yesmaximal, characteristic, complemented
10#22550072C211yesmaximal, characteristic, complemented
11#22550071C211yesmaximal, characteristic, complemented
12#22550070C211yesmaximal, characteristic, complemented
13#22550070C211yesmaximal, characteristic, complemented
14#22549901C211yesmaximal, characteristic, complemented
15#478×C127C211yesmaximal, characteristic, complemented
16C2×D1016C2211yescharacteristic
17C2×D1016C2211yescharacteristic
18C2×#14523233C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20C2×#14523231C2211yescharacteristic
21C2×C8×D127C2211yescharacteristic
22#16861206C2211yescharacteristic
23#16861205C2211yescharacteristic
24#16861200C2211yescharacteristic
25#16861194—21no—
26#16861194—21no—
27#16861192—21no—
28#16861191—21no—
29#16861184—21no—
30#16861184—21no—
31#16861184C2211yescharacteristic
32#16861184C2211yescharacteristic
33#16861184C2211yescharacteristic
34#16861183—21no—
35#16861182—21no—
36#16861180C2211yescharacteristic
37C4×D508C2211yescharacteristic
38#16861171C2211yescharacteristic
39#16861171C2211yescharacteristic
40#16861166C2211yescharacteristic
41#16861142C2211yescharacteristic
42#16861140C2211yescharacteristic
43C4⋊(C127⋊C8)C2211yescharacteristic
44Dic127⋊C8C2211yescharacteristic
45Dic127⋊C8C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861131C2211yescharacteristic
48#16861129C2211yescharacteristic
49#16861129C2211yescharacteristic
50#16861129C2211yescharacteristic
51#16861129C2211yescharacteristic
52#125×C127C2211yescharacteristic
53C4×D4×C127C2211yescharacteristic
54#107×C127C2211yescharacteristic
55#106×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59C2×D508—21no—
60C2×D508—21no—
61C2×D508—21no—
62C2×D508—21no—
63C2×D508—21no—
64C2×D508C2311yescharacteristic
65C2×D508C2311yescharacteristic
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127C2311yescharacteristic
74D1016—41no—
75D1016—41no—
76#14523233—41no—
77#14523233—41no—
78#14523231—41no—
79C2×C127⋊C8C2311yescharacteristic
80C2×C127⋊C8C2311yescharacteristic
81C8×D127—41no—
82C4⋊Dic127—21no—
83C4⋊Dic127C2311yescharacteristic
84Dic127⋊C4—21no—
85Dic127⋊C4D411yescharacteristic
86Dic127⋊C4D411yescharacteristic
87C22⋊Dic127—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4D411yescharacteristic
95(C2×D127)⋊C4D411yescharacteristic
96C4×Dic127C2311yescharacteristic
97C2×D4×C127C2311yescharacteristic
98C22×C4×C127—21no—
99C2×C8×C127C2311yescharacteristic
100C2×C8×C127C2311yescharacteristic
101C4⋊C4×C127—21no—
102C4⋊C4×C127—21no—
103C4⋊C4×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C22⋊C4×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127—21no—
110C22×D127—21no—
111C22×D127C2×D411yescharacteristic
112D508—21no—
113D508—21no—
114D508—21no—
115D508—21no—
116D508—21no—
117D508—21no—
118D508—81no—
119D508—81no—
120D508—81no—
121D508—81no—
122D508—81no—
123C22⋊D127—81no—
124C22⋊D127—81no—
125C22⋊D127—81no—
126C2×Dic127—21no—
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C2×Dic127C2×D411yescharacteristic
130C4×D127—21no—
131C4×D127—21no—
132C4×D127—81no—
133C4×D127—81no—
134C4×D127—81no—
135C127⋊C8—21no—
136C127⋊C8—21no—
137C23×C127—21no—
138D4×C127—21no—
139D4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—81no—
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2411yescharacteristic
147C8×C127—21no—
148C8×C127—21no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127C22×D411yescharacteristic
153C2×D127—21no—
154C2×D127—21no—
155C2×D127—21no—
156C2×D127—41no—
157C2×D127—41no—
158C2×D127—41no—
159C2×D127—41no—
160C2×D127—41no—
161C2×D127—41no—
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—41no—
165C4×C127—21no—
166C4×C127—21no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127—41no—
170C4×C127H(5,𝔽2)11yescharacteristic
171C4×C1272−1+411yescharacteristic
172Dic127—21no—
173Dic127—21no—
174Dic127—21no—
175Dic127—41no—
176C2×C127—81no—
177C2×C127#53711yescharacteristic
178C2×C127#57611yescharacteristic
179C2×C127#57611yescharacteristic
180D127—41no—
181D127—81no—
182D127—81no—
183D127—81no—
184#3259—1271nomaximal, Sylow
185C127#325911yescharacteristic, complemented, Sylow
186#540—1271no—
187#540—1271no—
188#478—1271no—
189#478—1271no—
190#478—1271no—
191#478D12711yescharacteristic, complemented
192#469—1271no—
193#469—1271no—
194#466—1271no—
195#466—1271no—
196#459—1271no—
197#459—1271no—
198#459—1271no—
199#459—1271no—
200#424—1271no—
201C2×D8—1271no—
202C2×D8—1271no—
203C2×D8—1271no—
204C2×D8—1271no—
205C2×M16—1271no—
206C22×C8—1271no—
207#127—1271no—
208#127—1271no—
209#125—1271no—
210#125C2×D12711yescharacteristic
211#123—2541no—
212#123—2541no—
213#123—2541no—
214#123—2541no—
215#121—1271no—
216#121—1271no—
217#121—1271no—
218#121—1271no—
219#121—2541no—
220#121—2541no—
221#121—2541no—
222#121—2541no—
223C4×D4—1271no—
224C4×D4—1271no—
225C4×D4—1271no—
226C4×D4C2×D12711yescharacteristic
227#117—1271no—
228#107—1271no—
229#107C2×D12711yescharacteristic
230#106—1271no—
231#106C2×D12711yescharacteristic
232C4⋊C8—1271no—
233C4⋊C8—1271no—
234C4⋊C8—1271no—
235C4⋊C8C2×D12711yescharacteristic
236#102—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102—1271no—
241#102—1271no—
242#102C2×D12711yescharacteristic
243#102C2×D12711yescharacteristic
244C2×D4—1271no—
245C2×D4—1271no—
246C2×D4—1271no—
247C2×D4—2541no—
248C2×D4—2541no—
249C2×D4—2541no—
250C2×D4—2541no—
251C2×D4—2541no—
252C2×D4—2541no—
253C2×D4—2541no—
254C2×D4—2541no—
255C2×D4C22×D12711yescharacteristic
256C22×C4—21no—
257C22×C4—1271no—
258C22×C4—2541no—
259C22×C4—2541no—
260C22×C4—2541no—
261D8—5081no—
262D8—5081no—
263D8—5081no—
264D8—5081no—
265M16—5081no—
266C2×C8—1271no—
267C2×C8—1271no—
268C2×C8—5081no—
269C2×C8C22×D12711yescharacteristic
270C2×C8C22×D12711yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—1271no—
274C4⋊C4—1271no—
275C4⋊C4—1271no—
276C4⋊C4—2541no—
277C4⋊C4—2541no—
278C4⋊C4C22×D12711yescharacteristic
279C4⋊C4C22×D12711yescharacteristic
280C4⋊C4C22×D12711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—1271no—
283C22⋊C4—1271no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C42—1271no—
292C42C22×D12711yescharacteristic
293C23—21no—
294C23—1271no—
295C23—2541no—
296C23—2541no—
297C23—2541no—
298D4—21no—
299D4—21no—
300D4—10161no—
301D4—10161no—
302D4—10161no—
303D4—10161no—
304D4—10161no—
305D4—10161no—
306D4—10161no—
307D4—10161no—
308D4—2541no—
309D4—2541no—
310D4—2541no—
311D4—2541no—
312D4—2541no—
313D4—2541no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—81no—
318C2×C4—1271no—
319C2×C4—1271no—
320C2×C4—1271no—
321C2×C4—10161no—
322C2×C4—10161no—
323C2×C4—10161no—
324C2×C4—2541no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4D4×D12711yescharacteristic
328C2×C4D4×D12711yescharacteristic
329C2×C4C23×D12711yescharacteristic
330C8—21no—
331C8—21no—
332C8—2541no—
333C8—2541no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—5081no—
338C22—5081no—
339C22—5081no—
340C22—5081no—
341C22—5081no—
342C22—5081no—
343C22—5081no—
344C22—5081no—
345C22—5081no—
346C22—2541no—
347C22—2541no—
348C22—2541no—
349C22C2×D4×D12711yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—5081no—
356C4—2541no—
357C4—2541no—
358C4—2541no—
359C4#1686125511yescharacteristic
360C4#1686125711yescharacteristic
361C2—81no—
362C2—10161no—
363C2—10161no—
364C2—10161no—
365C2—5081no—
366C2#2255045611yescharacteristic
367C2#2255067811yescharacteristic
368C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#478×C127C2
socleC22×C127C22×D4
hypercenter#478D127
hypocenterC127#3259
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #22550482C2semidirect
2. #22550477C2semidirect
3. #22550179C2semidirect
4. #22550175C2semidirect
5. #22550173C2semidirect
6. #22550130C2semidirect
7. #22550128C2semidirect
8. #22550113C2semidirect
9. #22550111C2semidirect
10. #22550072C2semidirect
11. #22550071C2semidirect
12. #22550070C2semidirect
13. #22550070C2semidirect
14. #22549901C2semidirect
15. #478×C127C2semidirect
185. C127#3259semidirect
191. #478D127semidirect

Maximal subgroups

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

subgroupindex
#225504822
#225504772
#225501792
#225501752
#225501732
#225501302
#225501282
#225501132
#225501112
#225500722
#225500712
#225500702
#225499012
#478×C1272
#3259127