Finite Group Databasebeta

← #30649019

Subgroups — #30649019

Order 16256.

382 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550480C211yesmaximal, characteristic
2#22550352C211yesmaximal, characteristic, complemented
3#22550199C211yesmaximal, characteristic, complemented
4#22550198C211yesmaximal, characteristic, complemented
5#22550184C211yesmaximal, characteristic, complemented
6#22550183C211yesmaximal, characteristic, complemented
7#22550073C211yesmaximal, characteristic, complemented
8#22550013C211yesmaximal, characteristic, complemented
9#22550013C211yesmaximal, characteristic, complemented
10#22549977C211yesmaximal, characteristic, complemented
11#22549973C211yesmaximal, characteristic, complemented
12#22549967C211yesmaximal, characteristic, complemented
13SD16×Dic127C211yesmaximal, characteristic, complemented
14#22549936C211yesmaximal, characteristic, complemented
15#460×C127C211yesmaximal, characteristic, complemented
16C2×Q8×D127C2211yescharacteristic
17C2×#14523236C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19#16861201C2211yescharacteristic
20#16861190—21no—
21#16861190—21no—
22#16861181—21no—
23#16861181—21no—
24#16861180C2211yescharacteristic
25#16861180C2211yescharacteristic
26Q8×Dic127C2211yescharacteristic
27C4×Dic254—21no—
28C4×Dic254C2211yescharacteristic
29#16861174—21no—
30#16861174C2211yescharacteristic
31D4×Dic127C2211yescharacteristic
32D4×Dic127C2211yescharacteristic
33#16861167—21no—
34#16861167C2211yescharacteristic
35#16861166—21no—
36#16861166C2211yescharacteristic
37#16861143C2211yescharacteristic
38#16861142C2211yescharacteristic
39#16861141C2211yescharacteristic
40#16861140C2211yescharacteristic
41C4⋊(C127⋊C8)C2211yescharacteristic
42#16861134C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861133C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic
48(C2×D127)⋊C8C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50#16861119C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C2×SD16×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54#127×C127C2211yescharacteristic
55C4×Q8×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59C2×Dic254—21no—
60C2×Dic254C2311yescharacteristic
61Q8×D127—22no—
62Q8×D127—22no—
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C22×Dic127—21no—
66C22×Dic127—21no—
67C2×C4×D127—21no—
68C2×C4×D127C2311yescharacteristic
69#14523236—22no—
70#14523236—22no—
71C2×C127⋊C8C2311yescharacteristic
72C2×C127⋊C8C2311yescharacteristic
73C4⋊Dic127—21no—
74C4⋊Dic127C2311yescharacteristic
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77Dic127⋊C4—21no—
78Dic127⋊C4—21no—
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81Dic127⋊C4D411yescharacteristic
82Dic127⋊C4D411yescharacteristic
83C22⋊Dic127—21no—
84C22⋊Dic127—21no—
85C22⋊Dic127—21no—
86C22⋊Dic127—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4D411yescharacteristic
89(C2×D127)⋊C4D411yescharacteristic
90C4×Dic127—21no—
91C4×Dic127—21no—
92C4×Dic127—21no—
93C4×Dic127—21no—
94C4×Dic127C2311yescharacteristic
95C2×Q8×C127C2311yescharacteristic
96C2×D4×C127—21no—
97C2×D4×C127—21no—
98C2×D4×C127C2311yescharacteristic
99C2×D4×C127C2311yescharacteristic
100SD16×C127—22no—
101SD16×C127—22no—
102C2×C8×C127C2311yescharacteristic
103C2×C8×C127C2311yescharacteristic
104C4⋊C4×C127—21no—
105C4⋊C4×C127C2311yescharacteristic
106C42×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127C2×D411yescharacteristic
109Dic254—24no—
110Dic254—21no—
111Dic254#4112yes—
112C22⋊D127—81no—
113C22⋊D127—81no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—81no—
119C2×Dic127—81no—
120C2×Dic127C2×D411yescharacteristic
121C2×Dic127C2×D411yescharacteristic
122C2×Dic127C2×D411yescharacteristic
123C4×D127—24no—
124C4×D127—21no—
125C4×D127—21no—
126C127⋊C8—21no—
127C127⋊C8—21no—
128C23×C127—21no—
129C23×C127—21no—
130Q8×C127—21no—
131Q8×C127#4112yes—
132D4×C127—21no—
133D4×C127—21no—
134D4×C127—21no—
135D4×C127—21no—
136D4×C127—81no—
137D4×C127—81no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2411yescharacteristic
143C8×C127—21no—
144C8×C127—21no—
145C22×C127—41no—
146C22×C127—41no—
147C22×C127—41no—
148C22×C127—41no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127C22×D411yescharacteristic
152C2×D127—21no—
153C2×D127—21no—
154C2×D127—21no—
155C4×C127—22no—
156C4×C127—21no—
157C4×C127—21no—
158C4×C127—41no—
159C4×C127C2×#4111yescharacteristic
160C4×C127C2×#4111yescharacteristic
161Dic127—22no—
162Dic127—21no—
163Dic127—21no—
164Dic127—21no—
165Dic127—41no—
166Dic127—41no—
167Dic127—41no—
168C2×C127—81no—
169C2×C127—81no—
170C2×C127#52511yescharacteristic
171C2×C127#57711yescharacteristic
172C2×C127#57711yescharacteristic
173D127—41no—
174#3126—1271nomaximal, Sylow
175C127#312611yescharacteristic, complemented, Sylow
176#540—1271no—
177#519—1271no—
178#482—1271no—
179#482—1271no—
180#479—1271no—
181#479—1271no—
182#460—1271no—
183#460D12711yescharacteristic, complemented
184#448—1271no—
185#448—1271no—
186#440—1271no—
187#440—1271no—
188C4×SD16—1271no—
189C4×SD16—1271no—
190#433—1271no—
191C22×Q8—1271no—
192C2×SD16—1271no—
193C2×SD16—1271no—
194C2×SD16C2×D12711yescharacteristic
195C2×SD16C2×D12711yescharacteristic
196#127C2×D12711yescharacteristic
197#125—1271no—
198#123—2541no—
199#123—2541no—
200#121—1271no—
201#121—1271no—
202#121—2541no—
203#121—2541no—
204C4×Q8—1271no—
205C4×Q8—1271no—
206C4×Q8—1271no—
207C4×Q8—2541no—
208C4×Q8—2541no—
209C4×Q8C2×D12711yescharacteristic
210C4×D4—1271no—
211C4×D4—1271no—
212#117—1271no—
213#117—1271no—
214#117—2541no—
215#117—2541no—
216#107—1271no—
217#107—1271no—
218#106—1271no—
219#106—1271no—
220C4⋊C8—1271no—
221C4⋊C8C2×D12711yescharacteristic
222#103—1271no—
223#103—1271no—
224#103—1271no—
225#103—1271no—
226#102—1271no—
227#102—1271no—
228#102C2×D12711yescharacteristic
229#102C2×D12711yescharacteristic
230C22⋊C8—1271no—
231C22⋊C8—1271no—
232#97—1271no—
233C4×C8—1271no—
234C2×Q8—1271no—
235C2×Q8—2541no—
236C2×Q8—2542no—
237C2×Q8—2542no—
238C2×Q8C22×D12711yescharacteristic
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—2541no—
242C2×D4—2541no—
243C2×D4C22×D12711yescharacteristic
244C2×D4C22×D12711yescharacteristic
245C22×C4—1271no—
246C22×C4—2541no—
247C22×C4—2541no—
248C22×C4—2541no—
249SD16—22no—
250SD16—22no—
251SD16—2542no—
252SD16—2542no—
253C2×C8—1271no—
254C2×C8—1271no—
255C2×C8C22×D12711yescharacteristic
256C2×C8C22×D12711yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—1271no—
259C4⋊C4—1271no—
260C4⋊C4—1271no—
261C4⋊C4—1271no—
262C4⋊C4—1271no—
263C4⋊C4—2541no—
264C4⋊C4—2541no—
265C4⋊C4—2541no—
266C4⋊C4—2541no—
267C4⋊C4—2541no—
268C4⋊C4C22×D12711yescharacteristic
269C22⋊C4—1271no—
270C22⋊C4—1271no—
271C22⋊C4—2541no—
272C22⋊C4—2541no—
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C42—21no—
277C42—1271no—
278C42—2541no—
279C42—2541no—
280C42—2541no—
281C42—2541no—
282C42C22×D12711yescharacteristic
283C23—21no—
284C23—21no—
285C23—1271no—
286Q8—21no—
287Q8—1272no—
288Q8—2544no—
289Q8—2541no—
290Q8#1452324712yes—
291D4—21no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—81no—
296D4—81no—
297D4—10161no—
298D4—10161no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—1271no—
302C2×C4—1271no—
303C2×C4—1271no—
304C2×C4—10161no—
305C2×C4—10161no—
306C2×C4—2544no—
307C2×C4—2541no—
308C2×C4—2541no—
309C2×C4—2541no—
310C2×C4—2541no—
311C2×C4—2541no—
312C2×C4—2541no—
313C2×C4D4×D12711yescharacteristic
314C2×C4D4×D12711yescharacteristic
315C2×C4C23×D12711yescharacteristic
316C8—21no—
317C8—21no—
318C8—2541no—
319C8—2541no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—2541no—
327C22—2541no—
328C22—2541no—
329C22C2×D4×D12711yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—41no—
334C4—5081no—
335C4—5081no—
336C4—5081no—
337C4—2541no—
338C4—2542no—
339C4—2541no—
340C4—2541no—
341C4#41×D12711yescharacteristic
342C4C2×#1452324711yescharacteristic
343C2—81no—
344C2—81no—
345C2—5081no—
346C2#2255038611yescharacteristic
347C2#2255068211yescharacteristic
348C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#460×C127C2
socleC22×C127C22×D4
prime radical#22550480C2
hypercenter#460D127
hypocenterC127#3126
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #22550352C2semidirect
3. #22550199C2semidirect
4. #22550198C2semidirect
5. #22550184C2semidirect
6. #22550183C2semidirect
7. #22550073C2semidirect
8. #22550013C2semidirect
9. #22550013C2semidirect
10. #22549977C2semidirect
11. #22549973C2semidirect
12. #22549967C2semidirect
13. SD16×Dic127C2semidirect
14. #22549936C2semidirect
15. #460×C127C2semidirect
175. C127#3126semidirect
183. #460D127semidirect

Maximal subgroups

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

subgroupindex
#225504802
#225503522
#225501992
#225501982
#225501842
#225501832
#225500732
#225500132
#225499772
#225499732
#225499672
SD16×Dic1272
#225499362
#460×C1272
#3126127