Finite Group Databasebeta

← #30652839

Subgroups — #30652839

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#22550609C213yesmaximal, complemented
2#22550602C211yesmaximal, characteristic, complemented
3#22550589C213yesmaximal, complemented
4#22550576C213yesmaximal, complemented
5#22550565C213yesmaximal, complemented
6#22550545C211yesmaximal, characteristic, complemented
7#22550538C211yesmaximal, characteristic, complemented
8#22550533C211yesmaximal, characteristic, complemented
9#22550491C213yesmaximal, complemented
10#22550477C213yesmaximal, complemented
11#22550410C213yesmaximal, complemented
12#22550410C213yesmaximal, complemented
13#22550358C211yesmaximal, characteristic, complemented
14#22550352C211yesmaximal, characteristic, complemented
15C4×#41×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18#16861208C2213yes—
19#16861205C2213yes—
20#16861203C2211yescharacteristic
21#16861203C2213yes—
22#16861202C2216yes—
23#16861201C2213yes—
24#16861201C2213yes—
25#16861200C2213yes—
26#16861197C2213yes—
27#16861197C2213yes—
28#16861196C2213yes—
29#16861195C2216yes—
30#16861192C2213yes—
31#16861192C2213yes—
32#16861192C2216yes—
33#16861191C2216yes—
34#16861189C2213yes—
35#16861188C2213yes—
36#16861188C2213yes—
37#16861184C2213yes—
38#16861183C2213yes—
39#16861183C2213yes—
40#16861182C2216yes—
41#16861180C2213yes—
42#16861180C2213yes—
43Q8×Dic127C2211yescharacteristic
44Q8×Dic127C2211yescharacteristic
45C4×Dic254C2213yes—
46C4×Dic254C2213yes—
47#16861174C2213yes—
48C4×D508C2213yes—
49C4×D508C2213yes—
50#16861171C2213yes—
51C4×C22⋊D127C2213yes—
52C4×C22⋊D127C2213yes—
53C4×C22⋊D127C2216yes—
54#16861167C2213yes—
55#16861166C2216yes—
56#16861166C2213yes—
57#16861166C2213yes—
58C4⋊C4×D127C2213yes—
59C2×#41×C127C2211yescharacteristic
60C4×Q8×C127C2211yescharacteristic
61C4×Q8×C127C2211yescharacteristic
62C4×D4×C127C2213yes—
63C4×D4×C127C2213yes—
64#117×C127C2213yes—
65C2×C42×C127C2213yes—
66#14523249—24no—
67C2×Dic254C2313yes—
68C2×Dic254C2313yes—
69Q8×D127—24no—
70C2×D508C2313yes—
71C2×D508C2313yes—
72C2×C22⋊D127C2313yes—
73C2×C22⋊D127C2316yes—
74C2×C22⋊D127C2313yes—
75C2×C4×D127C2313yes—
76C2×C4×D127C2313yes—
77C2×C4×D127C2316yes—
78C4⋊Dic127C2313yes—
79C4⋊Dic127C2313yes—
80C4⋊Dic127C2313yes—
81C4⋊Dic127C2313yes—
82Dic127⋊C4C2311yescharacteristic
83Dic127⋊C4C2313yes—
84Dic127⋊C4C2313yes—
85Dic127⋊C4C2313yes—
86Dic127⋊C4C2311yescharacteristic
87Dic127⋊C4C2316yes—
88Dic127⋊C4C2313yes—
89Dic127⋊C4C2313yes—
90Dic127⋊C4C2311yescharacteristic
91C22⋊Dic127C2316yes—
92C22⋊Dic127C2313yes—
93C22⋊Dic127C2313yes—
94(C2×D127)⋊C4C2316yes—
95(C2×D127)⋊C4C2311yescharacteristic
96(C2×D127)⋊C4C2313yes—
97(C2×D127)⋊C4C2313yes—
98(C2×D127)⋊C4C2313yes—
99(C2×D127)⋊C4C2313yes—
100(C2×D127)⋊C4C2311yescharacteristic
101(C2×D127)⋊C4C2313yes—
102(C2×D127)⋊C4C2311yescharacteristic
103C4×Dic127C2316yes—
104C4×Dic127C2313yes—
105C4×Dic127C2313yes—
106#41×C127—24no—
107C2×Q8×C127C2311yescharacteristic
108C2×D4×C127C2313yes—
109C22×C4×C127C2313yes—
110C22×C4×C127C2313yes—
111C22×C4×C127C2313yes—
112C4⋊C4×C127C2313yes—
113C4⋊C4×C127C2313yes—
114C22⋊C4×C127C2316yes—
115C42×C127C2313yes—
116C42×C127C2313yes—
117C42×C127C2311yescharacteristic
118C42×C127C2313yes—
119C22×D127C2411yescharacteristic
120C22×D127C2413yes—
121Dic254—26no—
122Dic254—43no—
123D508—26no—
124D508—43no—
125C22⋊D127—43no—
126C22⋊D127—43no—
127C22⋊D127—46no—
128C2×Dic127C2411yescharacteristic
129C2×Dic127C2413yes—
130C2×Dic127C2411yescharacteristic
131C2×Dic127C2413yes—
132C2×Dic127C2413yes—
133C2×Dic127C2411yescharacteristic
134C4×D127—26no—
135C4×D127—26no—
136C4×D127—46no—
137C23×C127C2413yes—
138Q8×C127#4114yes—
139D4×C127—26no—
140C2×C4×C127—26no—
141C2×C4×C127—43no—
142C2×C4×C127—43no—
143C2×C4×C127C2411yescharacteristic
144C2×C4×C127C2411yescharacteristic
145C2×C4×C127C2411yescharacteristic
146C2×C4×C127C2413yes—
147C2×C4×C127C2413yes—
148C2×C4×C127C2413yes—
149C22×C127—23no—
150C22×C127—23no—
151C22×C127—23no—
152C22×C127C2511yescharacteristic
153C2×D127—21no—
154C2×D127—23no—
155C2×D127—23no—
156C2×D127—21no—
157C2×D127—23no—
158C2×D127—21no—
159C4×C127—21no—
160C4×C127—21no—
161C4×C127—21no—
162C4×C127—23no—
163C4×C127—23no—
164C4×C127C2×#4116yes—
165Dic127—21no—
166Dic127—23no—
167Dic127—23no—
168Dic127—21no—
169Dic127—23no—
170Dic127—21no—
171C2×C127—43no—
172C2×C127C22×#4111yescharacteristic
173C2×C127#58511yescharacteristic
174C2×C127#58511yescharacteristic
175D127—41no—
176D127—43no—
177#3524—1271nomaximal, Sylow
178C127#352411yescharacteristic, complemented, Sylow
179#559—1273no—
180#557—1271no—
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×#41D12711yescharacteristic, complemented
194C2×#41—1271no—
195C2×#41C2×D12711yescharacteristic
196C22×Q8—1271no—
197C4⋊Q8—1273no—
198#127—1273no—
199#126—1276no—
200#126—1271no—
201#126—1273no—
202#125—1273no—
203#125—1273no—
204#125—1273no—
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—1273no—
221#121—1276no—
222C4×Q8—1271no—
223C4×Q8—1271no—
224C4×Q8—1273no—
225C4×Q8—1273no—
226C4×Q8—1273no—
227C4×Q8C2×D12711yescharacteristic
228C4×Q8C2×D12711yescharacteristic
229C4×D4—1273no—
230C4×D4—1273no—
231C4×D4—1276no—
232C4×D4—1273no—
233C4×D4—1273no—
234C4×D4—1273no—
235C4×D4C2×D12713yes—
236C4×D4C2×D12713yes—
237#117—1273no—
238#117—1273no—
239#117—1273no—
240#117—1276no—
241#117C2×D12713yes—
242C2×C4⋊C4—1273no—
243C2×C42C2×D12713yes—
244#41—24no—
245#41—2544no—
246C2×Q8—1273no—
247C2×Q8—1273no—
248C2×Q8—2544no—
249C2×Q8C22×D12711yescharacteristic
250C2×D4—1273no—
251C2×D4—1276no—
252C2×D4—1273no—
253C2×D4—1273no—
254C2×D4—1273no—
255C2×D4C22×D12713yes—
256C22×C4—1276no—
257C22×C4—1273no—
258C22×C4—1273no—
259C22×C4C22×D12713yes—
260C22×C4C22×D12713yes—
261C22×C4C22×D12713yes—
262C4⋊C4—1273no—
263C4⋊C4—1271no—
264C4⋊C4—1273no—
265C4⋊C4—1273no—
266C4⋊C4—1273no—
267C4⋊C4—1276no—
268C4⋊C4—1271no—
269C4⋊C4—1273no—
270C4⋊C4—1271no—
271C4⋊C4—1273no—
272C4⋊C4—1273no—
273C4⋊C4—1273no—
274C4⋊C4—1273no—
275C4⋊C4C22×D12713yes—
276C4⋊C4C22×D12713yes—
277C22⋊C4—1273no—
278C22⋊C4—1276no—
279C22⋊C4—1273no—
280C22⋊C4—1271no—
281C22⋊C4—1273no—
282C22⋊C4—1273no—
283C22⋊C4—1273no—
284C22⋊C4—1273no—
285C22⋊C4—1276no—
286C22⋊C4—1271no—
287C22⋊C4—1273no—
288C22⋊C4—1271no—
289C22⋊C4C22×D12716yes—
290C42—1276no—
291C42—1273no—
292C42—1273no—
293C42C22×D12711yescharacteristic
294C42C22×D12713yes—
295C42C22×D12713yes—
296C42C22×D12713yes—
297C23—1271no—
298C23—1273no—
299C23C23×D12713yes—
300Q8—5083no—
301Q8—2546no—
302Q8#1452324814yes—
303D4—26no—
304D4—5083no—
305D4—5083no—
306D4—5083no—
307D4—5086no—
308D4—2546no—
309C2×C4—26no—
310C2×C4—43no—
311C2×C4—43no—
312C2×C4—1271no—
313C2×C4—1273no—
314C2×C4—1271no—
315C2×C4—1273no—
316C2×C4—1273no—
317C2×C4—1271no—
318C2×C4—5086no—
319C2×C4—2546no—
320C2×C4—2546no—
321C2×C4C23×D12713yes—
322C2×C4C23×D12711yescharacteristic
323C2×C4C23×D12711yescharacteristic
324C2×C4C23×D12711yescharacteristic
325C2×C4C23×D12713yes—
326C2×C4C23×D12713yes—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—2541no—
331C22—2543no—
332C22—2543no—
333C22—2541no—
334C22—2543no—
335C22—2541no—
336C22C24×D12711yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—23no—
341C4—23no—
342C4—2541no—
343C4—2543no—
344C4—2543no—
345C4—2541no—
346C4—2543no—
347C4—2541no—
348C4C2×#1452324816yes—
349C2—43no—
350C2—5081no—
351C2—5083no—
352C2C22×#1452324811yescharacteristic
353C2#2255071211yescharacteristic
354C2#2255071211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D127
commutator, socle, cosocleC22×C127C25
FittingC4×#41×C127C2
hypercenterC4×#41D127
hypocenterC127#3524
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. #22550609C2semidirect
2. #22550602C2semidirect
3. #22550589C2semidirect
4. #22550576C2semidirect
5. #22550565C2semidirect
6. #22550545C2semidirect
7. #22550538C2semidirect
8. #22550533C2semidirect
9. #22550491C2semidirect
10. #22550477C2semidirect
11. #22550410C2semidirect
12. #22550410C2semidirect
13. #22550358C2semidirect
14. #22550352C2semidirect
15. C4×#41×C127C2semidirect
178. C127#3524semidirect
193. C4×#41D127semidirect

Maximal subgroups

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

subgroupindex
#225506092
#225506022
#225505892
#225505762
#225505652
#225505452
#225505382
#225505332
#225504912
#225504772
#225504102
#225503582
#225503522
C4×#41×C1272
#3524127