Finite Group Databasebeta

← #30651269

Subgroups — #30651269

Order 16256.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550596C211yesmaximal, characteristic, complemented
2#22550286C211yesmaximal, characteristic, complemented
3#22550217C211yesmaximal, characteristic, complemented
4#22550206C211yesmaximal, characteristic, complemented
5#22550197C211yesmaximal, characteristic, complemented
6#22550197C211yesmaximal, characteristic, complemented
7#22550180C211yesmaximal, characteristic, complemented
8#22550166C211yesmaximal, characteristic, complemented
9#22550120C211yesmaximal, characteristic, complemented
10#22550103C211yesmaximal, characteristic, complemented
11#22549976C211yesmaximal, characteristic, complemented
12#22549948C211yesmaximal, characteristic, complemented
13#22549916C211yesmaximal, characteristic, complemented
14#22549912C211yesmaximal, characteristic, complemented
15#553×C127C211yesmaximal, characteristic
16C2×#14523236C2211yescharacteristic
17C2×#14523232C2211yescharacteristic
18C22×C127⋊C8C2211yescharacteristic
19#16861209C2211yescharacteristic
20#16861202—21no—
21#16861202—21no—
22#16861196—21no—
23#16861188C2211yescharacteristic
24#16861188C2211yescharacteristic
25C4×Dic254C2211yescharacteristic
26#16861174—21no—
27#16861168C2211yescharacteristic
28#16861168C2211yescharacteristic
29#16861142C2211yescharacteristic
30#16861142C2211yescharacteristic
31#16861142C2211yescharacteristic
32#16861142C2211yescharacteristic
33#16861140C2211yescharacteristic
34#16861140C2211yescharacteristic
35#16861140C2211yescharacteristic
36#16861140C2211yescharacteristic
37#16861140C2211yescharacteristic
38C4⋊(C127⋊C8)C2211yescharacteristic
39#16861132C2211yescharacteristic
40#16861132C2211yescharacteristic
41#16861132C2211yescharacteristic
42#16861130C2211yescharacteristic
43#16861130C2211yescharacteristic
44#16861130C2211yescharacteristic
45C22⋊(C127⋊C8)C2211yescharacteristic
46C22⋊(C127⋊C8)C2211yescharacteristic
47#16861120C2211yescharacteristic
48#126×C127—21no—
49#125×C127C2211yescharacteristic
50#123×C127—21no—
51#121×C127—21no—
52#121×C127C2211yescharacteristic
53#121×C127C2211yescharacteristic
54C4×D4×C127—21no—
55C4×D4×C127C2211yescharacteristic
56C4×D4×C127C2211yescharacteristic
57#117×C127C2211yescharacteristic
58C2×C4⋊C4×C127C2211yescharacteristic
59C2×Dic254—21no—
60C2×Dic254C2311yescharacteristic
61#14523236—22no—
62#14523232—22no—
63C2×C127⋊C8—22no—
64C2×C127⋊C8C2311yescharacteristic
65C2×C127⋊C8C2311yescharacteristic
66C2×C127⋊C8C2311yescharacteristic
67C2×C127⋊C8C2311yescharacteristic
68C4⋊Dic127—21no—
69C4⋊Dic127C2311yescharacteristic
70C4⋊Dic127C2311yescharacteristic
71C4⋊Dic127C2311yescharacteristic
72Dic127⋊C4—21no—
73Dic127⋊C4—21no—
74Dic127⋊C4—21no—
75C22⋊Dic127—21no—
76C22⋊Dic127—21no—
77C22⋊Dic127—21no—
78C22⋊Dic127—21no—
79C4×Dic127—21no—
80C4×Dic127—21no—
81C4×Dic127—21no—
82C2×D4×C127—21no—
83C2×D4×C127—21no—
84C2×D4×C127D411yescharacteristic
85C2×D4×C127C2311yescharacteristic
86C22×C4×C127—21no—
87C22×C4×C127—21no—
88C22×C4×C127C2311yescharacteristic
89C22×C4×C127C2311yescharacteristic
90C4⋊C4×C127—21no—
91C4⋊C4×C127—21no—
92C4⋊C4×C127—21no—
93C4⋊C4×C127D411yescharacteristic
94C4⋊C4×C127C2311yescharacteristic
95C4⋊C4×C127C2311yescharacteristic
96C4⋊C4×C127C2311yescharacteristic
97C22⋊C4×C127—21no—
98C22⋊C4×C127—21no—
99C22⋊C4×C127—21no—
100C22⋊C4×C127—21no—
101C22⋊C4×C127D411yescharacteristic
102C22⋊C4×C127D411yescharacteristic
103C42×C127—21no—
104C42×C127C2311yescharacteristic
105Dic254—21no—
106Dic254—21no—
107Dic254—42no—
108C2×Dic127—21no—
109C2×Dic127—21no—
110C2×Dic127—21no—
111C2×Dic127—21no—
112C127⋊C8—21no—
113C127⋊C8—21no—
114C127⋊C8—21no—
115C127⋊C8#4112yes—
116C23×C127—21no—
117C23×C127C2×D411yescharacteristic
118C23×C127C2×D411yescharacteristic
119D4×C127—21no—
120D4×C127—21no—
121D4×C127—41no—
122D4×C127—42no—
123D4×C127—81no—
124C2×C4×C127—21no—
125C2×C4×C127—21no—
126C2×C4×C127—21no—
127C2×C4×C127—21no—
128C2×C4×C127—21no—
129C2×C4×C127—21no—
130C2×C4×C127—21no—
131C2×C4×C127—81no—
132C2×C4×C127—81no—
133C2×C4×C127C2×D411yescharacteristic
134C2×C4×C127C2×D411yescharacteristic
135C2×C4×C127C2×D411yescharacteristic
136C2×C4×C127C2×D411yescharacteristic
137C2×C4×C127C2411yescharacteristic
138C22×C127—21no—
139C22×C127—21no—
140C22×C127—21no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—21no—
144C22×C127—41no—
145C22×C127—41no—
146C22×C127—41no—
147C22×C127C22×D411yescharacteristic
148C4×C127—21no—
149C4×C127—21no—
150C4×C127—21no—
151C4×C127—21no—
152C4×C127—41no—
153C4×C127—41no—
154C4×C127—41no—
155C4×C127C2×#4111yescharacteristic
156C4×C1272−1+411yescharacteristic
157Dic127—41no—
158Dic127—41no—
159Dic127—41no—
160Dic127—41no—
161C2×C127—41no—
162C2×C127—41no—
163C2×C127—81no—
164C2×C127#54711yescharacteristic
165C2×C127#57511yescharacteristic
166C2×C127#57711yescharacteristic
167#3350—1271nomaximal, Sylow
168C127#335011yescharacteristic, complemented, Sylow
169#556—1271no—
170#553D12711yescharacteristic
171#501—1271no—
172#485—1271no—
173#483—1271no—
174#482—1271no—
175#482—1271no—
176#479—1271no—
177#476—1271no—
178#468—1271no—
179#465—1271no—
180#440—1271no—
181#435—1271no—
182#428—1271no—
183#427—1271no—
184C2×SD16—1271no—
185C2×M16—1271no—
186C22×C8—1271no—
187C4⋊Q8—1271no—
188#126—21no—
189#126—2541no—
190#126—2541no—
191#125C2×D12711yescharacteristic
192#124—2541no—
193#123—21no—
194C22⋊Q8—1271no—
195C22⋊Q8—1271no—
196#121—21no—
197#121C2×D12711yescharacteristic
198#121C2×D12711yescharacteristic
199C4×Q8—1271no—
200C4×Q8—2541no—
201C4×D4—21no—
202C4×D4C2×D12711yescharacteristic
203C4×D4C2×D12711yescharacteristic
204#117—1271no—
205#117—1271no—
206#117C2×D12711yescharacteristic
207C2×C4⋊C4C2×D12711yescharacteristic
208#107—1271no—
209#107—1271no—
210#107—1271no—
211#107—1271no—
212#106—1271no—
213#106—1271no—
214#106—1271no—
215#106—1271no—
216#106—1271no—
217C4⋊C8—1271no—
218#103—1271no—
219#103—1271no—
220#103—1271no—
221#102—1271no—
222#102—1271no—
223#102—1271no—
224C22⋊C8—1271no—
225C22⋊C8—1271no—
226#97—1271no—
227C2×Q8—1271no—
228C2×Q8—2541no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4C22⋊D12711yescharacteristic
232C2×D4C22×D12711yescharacteristic
233C22×C4—21no—
234C22×C4—21no—
235C22×C4C22×D12711yescharacteristic
236C22×C4C22×D12711yescharacteristic
237SD16—2542no—
238M16—2542no—
239C2×C8—1271no—
240C2×C8—1271no—
241C2×C8—1271no—
242C2×C8—1271no—
243C2×C8—2542no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—1271no—
248C4⋊C4—1271no—
249C4⋊C4—1271no—
250C4⋊C4—2541no—
251C4⋊C4—2541no—
252C4⋊C4—2541no—
253C4⋊C4—2541no—
254C4⋊C4C22⋊D12711yescharacteristic
255C4⋊C4C22×D12711yescharacteristic
256C4⋊C4C22×D12711yescharacteristic
257C4⋊C4C22×D12711yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—2541no—
263C22⋊C4—2541no—
264C22⋊C4—2541no—
265C22⋊C4—2541no—
266C22⋊C4C22⋊D12711yescharacteristic
267C22⋊C4C22⋊D12711yescharacteristic
268C42—21no—
269C42—2541no—
270C42—2541no—
271C42—2541no—
272C42C22×D12711yescharacteristic
273C23—21no—
274C23C2×C22⋊D12711yescharacteristic
275C23C2×C22⋊D12711yescharacteristic
276Q8—5082no—
277Q8—2541no—
278Q8—2541no—
279D4—21no—
280D4—21no—
281D4—41no—
282D4—42no—
283D4—81no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—2541no—
294C2×C4—2541no—
295C2×C4—2541no—
296C2×C4—2541no—
297C2×C4C2×C22⋊D12711yescharacteristic
298C2×C4C2×C22⋊D12711yescharacteristic
299C2×C4C2×C22⋊D12711yescharacteristic
300C2×C4C2×C22⋊D12711yescharacteristic
301C2×C4C23×D12711yescharacteristic
302C8—1272no—
303C8—2541no—
304C8—2541no—
305C8—2541no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22C22×C22⋊D12711yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—5081no—
324C4—5081no—
325C4—5081no—
326C4—5081no—
327C4#41×D12711yescharacteristic
328C4#1686125611yescharacteristic
329C2—41no—
330C2—41no—
331C2—81no—
332C2#2255052511yescharacteristic
333C2#2255067411yescharacteristic
334C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#553×C127C2
socleC22×C127C22×D4
hypercenter#553D127
hypocenterC127#3350
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #22550596C2semidirect
2. #22550286C2semidirect
3. #22550217C2semidirect
4. #22550206C2semidirect
5. #22550197C2semidirect
6. #22550197C2semidirect
7. #22550180C2semidirect
8. #22550166C2semidirect
9. #22550120C2semidirect
10. #22550103C2semidirect
11. #22549976C2semidirect
12. #22549948C2semidirect
13. #22549916C2semidirect
14. #22549912C2semidirect
168. C127#3350semidirect

Maximal subgroups

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

subgroupindex
#225505962
#225502862
#225502172
#225502062
#225501972
#225501802
#225501662
#225501202
#225501032
#225499762
#225499482
#225499162
#225499122
#553×C1272
#3350127