Finite Group Databasebeta

← #30652842

Subgroups — #30652842

Order 16256.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550611C213yesmaximal, complemented
2#22550592C213yesmaximal, complemented
3#22550579C213yesmaximal, complemented
4#22550564C213yesmaximal, complemented
5#22550548C211yesmaximal, characteristic, complemented
6#22550538C211yesmaximal, characteristic, complemented
7#22550538C211yesmaximal, characteristic, complemented
8#22550500C213yesmaximal, complemented
9#22550482C213yesmaximal, complemented
10#22550413C213yesmaximal, complemented
11#22550411C213yesmaximal, complemented
12#22550358C211yesmaximal, characteristic, complemented
13#22550352C211yesmaximal, characteristic, complemented
14C4×#14523249C211yesmaximal, characteristic, complemented
15#557×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523249C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic
19#16861206C2213yes—
20#16861204C2216yes—
21#16861204C2213yes—
22#16861204C2211yescharacteristic
23#16861200C2213yes—
24#16861198C2213yes—
25#16861198C2216yes—
26#16861197C2213yes—
27#16861197C2213yes—
28#16861194C2216yes—
29#16861194C2213yes—
30#16861194C2213yes—
31#16861194C2216yes—
32#16861189C2213yes—
33#16861189C2213yes—
34#16861185C2213yes—
35#16861184C2213yes—
36#16861184C2213yes—
37#16861184C2213yes—
38#16861184C2213yes—
39#16861184C2216yes—
40#16861184C2213yes—
41Q8×Dic127C2211yescharacteristic
42C4×Dic254C2213yes—
43#16861174C2213yes—
44C4×D508C2213yes—
45C4×D508C2213yes—
46C4×D508C2213yes—
47C4×D508C2213yes—
48C4×D508C2213yes—
49#16861171C2216yes—
50#16861171C2213yes—
51#16861171C2213yes—
52#16861167C2213yes—
53#16861167C2216yes—
54#16861167C2213yes—
55#16861166C2213yes—
56#16861166C2213yes—
57C4⋊C4×D127C2213yes—
58C42×D127C2213yes—
59C4⋊Q8×C127C2213yes—
60#125×C127C2213yes—
61#125×C127C2213yes—
62C4×Q8×C127C2211yescharacteristic
63C4×Q8×C127C2211yescharacteristic
64C4×Q8×C127C2211yescharacteristic
65C4×Q8×C127C2213yes—
66#14523249—24no—
67#14523249—24no—
68C2×Dic254C2313yes—
69Q8×D127—24no—
70C2×D508C2313yes—
71C2×D508C2313yes—
72C2×D508C2313yes—
73C2×D508C2313yes—
74C2×D508C2316yes—
75C2×D508C2313yes—
76C2×C4×D127C2313yes—
77C2×C4×D127C2313yes—
78C2×C4×D127C2313yes—
79C2×C4×D127C2313yes—
80C2×C4×D127C2316yes—
81C2×C4×D127C2313yes—
82C4⋊Dic127C2313yes—
83C4⋊Dic127C2313yes—
84C4⋊Dic127C2311yescharacteristic
85C4⋊Dic127C2313yes—
86Dic127⋊C4C2311yescharacteristic
87Dic127⋊C4C2313yes—
88Dic127⋊C4C2313yes—
89Dic127⋊C4C2313yes—
90(C2×D127)⋊C4C2311yescharacteristic
91(C2×D127)⋊C4C2313yes—
92(C2×D127)⋊C4C2313yes—
93(C2×D127)⋊C4C2313yes—
94(C2×D127)⋊C4C2316yes—
95(C2×D127)⋊C4C2316yes—
96(C2×D127)⋊C4C2313yes—
97(C2×D127)⋊C4C2316yes—
98(C2×D127)⋊C4C2313yes—
99(C2×D127)⋊C4C2311yescharacteristic
100(C2×D127)⋊C4C2313yes—
101(C2×D127)⋊C4C2313yes—
102(C2×D127)⋊C4C2311yescharacteristic
103C4×Dic127C2313yes—
104C4×Dic127C2316yes—
105C4×Dic127C2311yescharacteristic
106C4×Dic127C2313yes—
107C2×Q8×C127C2313yes—
108C2×Q8×C127C2311yescharacteristic
109C4⋊C4×C127C2313yes—
110C4⋊C4×C127C2313yes—
111C4⋊C4×C127C2313yes—
112C4⋊C4×C127C2313yes—
113C4⋊C4×C127C2316yes—
114C4⋊C4×C127C2311yescharacteristic
115C4⋊C4×C127C2313yes—
116C42×C127C2313yes—
117C42×C127C2313yes—
118C42×C127C2313yes—
119C22×D127C2411yescharacteristic
120C22×D127C2413yes—
121C22×D127C2413yes—
122Dic254—26no—
123D508—26no—
124D508—26no—
125D508—43no—
126D508—46no—
127D508—43no—
128D508—43no—
129C2×Dic127C2411yescharacteristic
130C2×Dic127C2413yes—
131C2×Dic127C2411yescharacteristic
132C2×Dic127C2413yes—
133C2×Dic127C2411yescharacteristic
134C4×D127—26no—
135C4×D127—26no—
136C4×D127—26no—
137C4×D127—43no—
138C4×D127—43no—
139C4×D127—46no—
140Q8×C127—43no—
141Q8×C127#4114yes—
142C2×C4×C127C2411yescharacteristic
143C2×C4×C127C2411yescharacteristic
144C2×C4×C127C2411yescharacteristic
145C2×C4×C127C2413yes—
146C2×C4×C127C2413yes—
147C2×C4×C127C2413yes—
148C2×C4×C127C2413yes—
149C22×C127C2511yescharacteristic
150C2×D127—21no—
151C2×D127—23no—
152C2×D127—23no—
153C2×D127—23no—
154C2×D127—23no—
155C2×D127—23no—
156C2×D127—21no—
157C2×D127—23no—
158C2×D127—21no—
159C4×C127—21no—
160C4×C127—23no—
161C4×C127—23no—
162C4×C127—21no—
163C4×C127—23no—
164C4×C127—21no—
165C4×C127C2×#4116yes—
166Dic127—21no—
167Dic127—21no—
168Dic127—23no—
169Dic127—21no—
170Dic127—23no—
171C2×C127C22×#4111yescharacteristic
172C2×C127#58511yescharacteristic
173C2×C127#58511yescharacteristic
174D127—41no—
175D127—43no—
176D127—43no—
177#3524—1271nomaximal, Sylow
178C127#352411yescharacteristic, complemented, Sylow
179#559—1273no—
180#557D12711yescharacteristic, complemented
181#555—1273no—
182#553—1273no—
183#552—1273no—
184#550—1271no—
185#548—1271no—
186#548—1271no—
187#542—1273no—
188#540—1273no—
189#529—1273no—
190#529—1273no—
191#520—1271no—
192#519—1271no—
193C4×#41—1271no—
194C2×#41—1271no—
195C2×#41—1271no—
196C22×Q8—1271no—
197C4⋊Q8C2×D12713yes—
198#127—1273no—
199#126—1271no—
200#126—1273no—
201#126—1276no—
202#125—1273no—
203#125C2×D12713yes—
204#125C2×D12713yes—
205#124—1273no—
206#124—1276no—
207#124—1273no—
208#124—1273no—
209#123—1276no—
210#123—1273no—
211#123—1273no—
212#123—1276no—
213C22⋊Q8—1273no—
214C22⋊Q8—1273no—
215C22⋊Q8—1273no—
216#121—1273no—
217#121—1273no—
218#121—1273no—
219#121—1273no—
220#121—1276no—
221#121—1273no—
222C4×Q8—1271no—
223C4×Q8—1273no—
224C4×Q8—1273no—
225C4×Q8C2×D12711yescharacteristic
226C4×Q8C2×D12711yescharacteristic
227C4×Q8C2×D12711yescharacteristic
228C4×Q8C2×D12713yes—
229C4×D4—1273no—
230C4×D4—1273no—
231C4×D4—1276no—
232C4×D4—1273no—
233C4×D4—1273no—
234C4×D4—1273no—
235C4×D4—1273no—
236C4×D4—1273no—
237#117—1273no—
238#117—1273no—
239#117—1276no—
240#117—1273no—
241#117—1273no—
242C2×C4⋊C4—1273no—
243C2×C42—1273no—
244#41—2544no—
245#41—2544no—
246C2×Q8—1273no—
247C2×Q8—2544no—
248C2×Q8C22×D12713yes—
249C2×Q8C22×D12711yescharacteristic
250C2×D4—1273no—
251C2×D4—1273no—
252C2×D4—1276no—
253C2×D4—1273no—
254C2×D4—1273no—
255C2×D4—1273no—
256C22×C4—1273no—
257C22×C4—1273no—
258C22×C4—1276no—
259C22×C4—1273no—
260C22×C4—1273no—
261C22×C4—1273no—
262C4⋊C4—1273no—
263C4⋊C4—1271no—
264C4⋊C4—1273no—
265C4⋊C4—1273no—
266C4⋊C4—1273no—
267C4⋊C4—1273no—
268C4⋊C4—1271no—
269C4⋊C4—1273no—
270C4⋊C4C22×D12713yes—
271C4⋊C4C22×D12713yes—
272C4⋊C4C22×D12713yes—
273C4⋊C4C22×D12716yes—
274C4⋊C4C22×D12713yes—
275C4⋊C4C22×D12711yescharacteristic
276C4⋊C4C22×D12713yes—
277C22⋊C4—1271no—
278C22⋊C4—1273no—
279C22⋊C4—1273no—
280C22⋊C4—1273no—
281C22⋊C4—1276no—
282C22⋊C4—1273no—
283C22⋊C4—1276no—
284C22⋊C4—1271no—
285C22⋊C4—1273no—
286C22⋊C4—1271no—
287C22⋊C4—1273no—
288C22⋊C4—1276no—
289C22⋊C4—1273no—
290C42—1271no—
291C42—1273no—
292C42—1273no—
293C42—1276no—
294C42C22×D12713yes—
295C42C22×D12713yes—
296C42C22×D12713yes—
297C23—1271no—
298C23—1273no—
299C23—1273no—
300Q8—43no—
301Q8—2546no—
302Q8#1452324914yes—
303D4—5083no—
304D4—5086no—
305D4—5083no—
306D4—5083no—
307D4—2546no—
308D4—2546no—
309C2×C4—1271no—
310C2×C4—1271no—
311C2×C4—1273no—
312C2×C4—1273no—
313C2×C4—1271no—
314C2×C4—5083no—
315C2×C4—5083no—
316C2×C4—5086no—
317C2×C4—2546no—
318C2×C4—2546no—
319C2×C4—2546no—
320C2×C4C23×D12713yes—
321C2×C4C23×D12711yescharacteristic
322C2×C4C23×D12713yes—
323C2×C4C23×D12711yescharacteristic
324C2×C4C23×D12711yescharacteristic
325C2×C4C23×D12713yes—
326C2×C4C23×D12713yes—
327C22—2541no—
328C22—2543no—
329C22—2543no—
330C22—2543no—
331C22—2541no—
332C22—2543no—
333C22—2543no—
334C22—2543no—
335C22—2541no—
336C22C24×D12711yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—2541no—
344C4—2541no—
345C4—2543no—
346C4—2541no—
347C4—2543no—
348C4C2×#1452324916yes—
349C2—5081no—
350C2—5083no—
351C2—5083no—
352C2C22×#1452324911yescharacteristic
353C2#2255071211yescharacteristic
354C2#2255071311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D127
commutator, socle, cosocleC22×C127C25
Fitting#557×C127C2
hypercenter#557D127
hypocenterC127#3524
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #22550611C2semidirect
2. #22550592C2semidirect
3. #22550579C2semidirect
4. #22550564C2semidirect
5. #22550548C2semidirect
6. #22550538C2semidirect
7. #22550538C2semidirect
8. #22550500C2semidirect
9. #22550482C2semidirect
10. #22550413C2semidirect
11. #22550411C2semidirect
12. #22550358C2semidirect
13. #22550352C2semidirect
14. C4×#14523249C2semidirect
15. #557×C127C2semidirect
178. C127#3524semidirect
180. #557D127semidirect

Maximal subgroups

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

subgroupindex
#225506112
#225505922
#225505792
#225505642
#225505482
#225505382
#225505002
#225504822
#225504132
#225504112
#225503582
#225503522
C4×#145232492
#557×C1272
#3524127