Finite Group Databasebeta

← #27179963

Subgroups — #27179963

Order 12928.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692282C211yesmaximal, characteristic
2#20265659C211yesmaximal, characteristic, complemented
3#20265649C211yesmaximal, characteristic, complemented
4#20265290C211yesmaximal, characteristic, complemented
5#20265286C211yesmaximal, characteristic, complemented
6#20265280C211yesmaximal, characteristic, complemented
7SD16×Dic101C211yesmaximal, characteristic, complemented
8C4×#13108061C211yesmaximal, characteristic, complemented
9#20265259C211yesmaximal, characteristic, complemented
10C4×#13108059C211yesmaximal, characteristic, complemented
11#20265218C211yesmaximal, characteristic, complemented
12#20265172C211yesmaximal, characteristic, complemented
13#102×D101C211yesmaximal, characteristic, complemented
14#20265134C211yesmaximal, characteristic, complemented
15#440×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×D4×D101C2211yescharacteristic
18#15692282C418yescomplemented
19C2×#13108061C2211yescharacteristic
20C2×D808C2211yescharacteristic
21C2×#13108059C2211yescharacteristic
22C2×#13108057C2211yescharacteristic
23C2×C8×D101C2211yescharacteristic
24Q8×Dic101C2211yescharacteristic, complemented
25C4×D404—21no—
26C4×D404C2211yescharacteristic
27C4×D404C2211yescharacteristic, complemented
28D4×Dic101C2211yescharacteristic
29#15692217—21no—
30#15692217C2211yescharacteristic
31#15692217C2211yescharacteristic, complemented
32C4×C22⋊D101—21no—
33#15692215—21no—
34#15692213C2211yescharacteristic
35#15692212—21no—
36#15692212C2211yescharacteristic
37C4⋊C4×D101—21no—
38C4⋊C4×D101C2211yescharacteristic
39C2×#13108051—21no—
40C22⋊C4×D101—21no—
41#15692188C2211yescharacteristic, complemented
42#15692187C2211yescharacteristic, complemented
43#15692186C2211yescharacteristic, complemented
44#15692179C2211yescharacteristic, complemented
45#15692177C2211yescharacteristic
46#15692177C2211yescharacteristic, complemented
47#15692176C2211yescharacteristic
48#15692175C2211yescharacteristic
49#15692175C2211yescharacteristic, complemented
50#15692165C2211yescharacteristic, complemented
51C4×#11717579C2211yescharacteristic, complemented
52C8×Dic101C2211yescharacteristic, complemented
53C2×SD16×C101C2211yescharacteristic
54C4×Q8×C101C2211yescharacteristic, complemented
55C4×D4×C101C2211yescharacteristic
56#107×C101C2211yescharacteristic, complemented
57#103×C101C2211yescharacteristic, complemented
58#102×C101C2211yescharacteristic
59#97×C101C2211yescharacteristic, complemented
60C23×D101—21no—
61#13108075—22no—
62#13108075C2×C414yescomplemented
63C2×D404—21no—
64C2×D404C2311yescharacteristic
65C2×D404C2311yescharacteristic
66C2×C22⋊D101—21no—
67D4×D101—22no—
68D4×D101C2×C414yescomplemented
69C22×Dic101—21no—
70C2×C4×D101—21no—
71C2×C4×D101—21no—
72C2×C4×D101—21no—
73C2×C4×D101—21no—
74C2×C4×D101—21no—
75C2×C4×D101—21no—
76C2×C4×D101C2311yescharacteristic
77#13108061C2×C414yescomplemented
78D808C2×C414yescomplemented
79#13108059C2×C414yescomplemented
80#13108057C2×C414yescomplemented
81C2×#11717579C2311yescharacteristic
82C2×#11717579C2311yescharacteristic
83C8×D101—22no—
84C4⋊Dic101—21no—
85C4⋊Dic101C2311yescharacteristic
86C4⋊Dic101C2311yescharacteristic
87#13108053—21no—
88#13108053D411yescharacteristic
89#13108053D411yescharacteristic
90C22⋊Dic101—21no—
91#13108051—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051—21no—
95#13108051—21no—
96#13108051D411yescharacteristic
97#13108051D411yescharacteristic
98C4×Dic101—21no—
99C4×Dic101—21no—
100C4×Dic101C2311yescharacteristic
101C2×Q8×C101C2311yescharacteristic
102C2×D4×C101C2311yescharacteristic
103C22×C4×C101—21no—
104SD16×C101C2×C414yescomplemented
105C2×C8×C101C2311yescharacteristic
106C2×C8×C101C2311yescharacteristic
107C4⋊C4×C101—21no—
108C4⋊C4×C101C2311yescharacteristic
109C4⋊C4×C101C2311yescharacteristic
110C22⋊C4×C101—21no—
111C42×C101—21no—
112C42×C101C2311yescharacteristic
113C23×C101—21no—
114C22×D101—21no—
115C22×D101—21no—
116C22×D101—24no—
117C22×D101—81no—
118C22×D101—81no—
119C22×D101C2×D411yescharacteristic
120Q8×C101—21no—
121Q8×C101C22×C412yes—
122D4×C101—21no—
123D4×C101C22×C412yes—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—81no—
128C2×C4×C101C2×D411yescharacteristic
129C2×C4×C101C2×D411yescharacteristic
130C2×C4×C101C2411yescharacteristic
131C22⋊D101—24no—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101—81no—
135C2×Dic101C2×D411yescharacteristic
136C2×Dic101C2×D411yescharacteristic
137C2×Dic101C2×D411yescharacteristic
138D404—24no—
139D404—21no—
140D404—21no—
141D404C22×C412yes—
142D404C22×C412yes—
143C4×D101—24no—
144C4×D101—81no—
145C4×D101—81no—
146C4×D101—81no—
147C4×D101—81no—
148C4×D101—81no—
149C4×D101C22×C412yes—
150C4×D101#4112yes—
151C8×C101—21no—
152C8×C101C22×C412yes—
153#11717579—21no—
154#11717579C22×C412yes—
155C22×C101—22no—
156C22×C101—41no—
157C22×C101—41no—
158C22×C101C22×D411yescharacteristic
159C2×D101—21no—
160C2×D101—21no—
161C2×D101—22no—
162C2×D101—22no—
163C2×D101—41no—
164C2×D101—41no—
165C2×D101—41no—
166C2×D101—41no—
167C2×D101—84no—
168C2×D101C4×D412yes—
169C4×C101—22no—
170C4×C101—21no—
171C4×C101—21no—
172C4×C101—41no—
173C4×C101—41no—
174C4×C101C23×C411yescharacteristic
175C4×C101C2×#4111yescharacteristic
176Dic101—21no—
177Dic101—21no—
178Dic101—41no—
179Dic101—41no—
180Dic101C4×D412yes—
181C2×C101—42no—
182C2×C101C2×C4×D411yescharacteristic
183C2×C101#57611yescharacteristic
184C2×C101#57711yescharacteristic
185D101—22no—
186D101—42no—
187D101—42no—
188#2946—1011nomaximal, Sylow
189C101#294611yescharacteristic, complemented, Sylow
190C2×AGL(1,ℤ/8)—1011no—
191#520—1011no—
192#518—1011no—
193#442—1011no—
194#442—1011no—
195#440—1011no—
196#440D10111yescharacteristic, complemented
197C4×SD16—1011no—
198C4×SD16—1011no—
199C4×D8—1011no—
200C4×D8—1011no—
201#428—1011no—
202#416—1011no—
203C2×#102—1011no—
204#405—1011no—
205C2×#41—1011no—
206C22×D4—1011no—
207AGL(1,ℤ/8)—1018no—
208C2×SD16—1011no—
209C2×SD16C2×D10111yescharacteristic
210C2×D8—1011no—
211C2×D8—1011no—
212C2×M16—1011no—
213C22×C8—1011no—
214C4×Q8—1011no—
215C4×Q8C2×D10111yescharacteristic, complemented
216C4×D4—1011no—
217C4×D4—1011no—
218C4×D4—1011no—
219C4×D4—1011no—
220C4×D4—1011no—
221C4×D4—2021no—
222C4×D4—2021no—
223C4×D4—2021no—
224C4×D4—2021no—
225C4×D4C2×D10111yescharacteristic
226#117—1011no—
227#117—1011no—
228#117—2021no—
229C2×C4⋊C4—1011no—
230C2×C4⋊C4—2021no—
231C2×C22⋊C4—2021no—
232C2×C22⋊C4—2021no—
233#107—1011no—
234#107C2×D10111yescharacteristic, complemented
235#106—1011no—
236#106—1011no—
237#103—1011no—
238#103C2×D10111yescharacteristic, complemented
239#102—1011no—
240#102—1011no—
241#102—1011no—
242#102—1011no—
243#102—1011no—
244#102C2×D10111yescharacteristic
245#97—1011no—
246#97C2×D10111yescharacteristic, complemented
247C4×C8—1011no—
248C4×C8—1011no—
249C24—2021no—
250#41—1014no—
251#41—2022no—
252C2×Q8C22×D10111yescharacteristic
253C2×D4—1014no—
254C2×D4—1011no—
255C2×D4—1011no—
256C2×D4—2021no—
257C2×D4—2022no—
258C2×D4—2021no—
259C2×D4C22×D10111yescharacteristic
260C22×C4—21no—
261C22×C4—1011no—
262C22×C4—2021no—
263C22×C4—2021no—
264C22×C4—2021no—
265C22×C4—2021no—
266C22×C4—2021no—
267C22×C4—2021no—
268C22×C4—2021no—
269SD16—1014no—
270SD16C4×D10114yescomplemented
271D8—1014no—
272D8—1014no—
273M16—1014no—
274C2×C8—1011no—
275C2×C8—1011no—
276C2×C8—2022no—
277C2×C8C22×D10111yescharacteristic
278C2×C8C22×D10111yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—1011no—
281C4⋊C4—1011no—
282C4⋊C4—1011no—
283C4⋊C4—1011no—
284C4⋊C4—2021no—
285C4⋊C4—2021no—
286C4⋊C4C22×D10111yescharacteristic
287C4⋊C4C22×D10111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—1011no—
290C22⋊C4—1011no—
291C22⋊C4—2021no—
292C22⋊C4—2021no—
293C22⋊C4—2021no—
294C22⋊C4—2021no—
295C22⋊C4—2021no—
296C22⋊C4—2021no—
297C42—21no—
298C42—1011no—
299C42—2021no—
300C42—2021no—
301C42C22×D10111yescharacteristic
302C23—21no—
303C23—8081no—
304C23—8081no—
305C23—1011no—
306C23—2024no—
307C23—2021no—
308C23—2021no—
309Q8—21no—
310Q8C2×C4×D10112yes—
311D4—21no—
312D4—1012no—
313D4—1012no—
314D4—2024no—
315D4—2024no—
316D4—2021no—
317D4—2021no—
318D4C2×C4×D10112yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—81no—
323C2×C4—8081no—
324C2×C4—8081no—
325C2×C4—8081no—
326C2×C4—8081no—
327C2×C4—8081no—
328C2×C4—8081no—
329C2×C4—1011no—
330C2×C4—1012no—
331C2×C4—1012no—
332C2×C4—1011no—
333C2×C4—1011no—
334C2×C4—2024no—
335C2×C4—2021no—
336C2×C4—2021no—
337C2×C4D4×D10111yescharacteristic
338C2×C4D4×D10111yescharacteristic
339C2×C4C23×D10111yescharacteristic
340C8—21no—
341C8—1012no—
342C8—2021no—
343C8C2×C4×D10112yes—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—8084no—
348C22—1012no—
349C22—4041no—
350C22—4041no—
351C22—4041no—
352C22—4041no—
353C22—2022no—
354C22—2022no—
355C22—2021no—
356C22—2021no—
357C22C2×D4×D10111yescharacteristic
358C4—21no—
359C4—22no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—1012no—
364C4—4041no—
365C4—4041no—
366C4—2021no—
367C4—2021no—
368C4C22×C4×D10111yescharacteristic
369C4#41×D10111yescharacteristic
370C2—42no—
371C2—4042no—
372C2—4042no—
373C2—2022no—
374C2C4×D4×D10111yescharacteristic
375C2#2026598111yescharacteristic
376C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutatorC4×C101C23×C4
FrattiniC2×C4C23×D101
Fitting#440×C101C2
socleC22×C101C22×D4
prime radicalC2×#15692282C2
cosocleC2×C4×C101C24
hypercenter#440D101
hypocenterC101#2946
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #20265659C2semidirect
3. #20265649C2semidirect
4. #20265290C2semidirect
5. #20265286C2semidirect
6. #20265280C2semidirect
7. SD16×Dic101C2semidirect
8. C4×#13108061C2semidirect
9. #20265259C2semidirect
10. C4×#13108059C2semidirect
11. #20265218C2semidirect
12. #20265172C2semidirect
13. #102×D101C2semidirect
14. #20265134C2semidirect
15. #440×C101C2semidirect
18. #15692282C4semidirect
24. Q8×Dic101C22semidirect
27. C4×D404C22semidirect
31. #15692217C22semidirect
41. #15692188C22semidirect
42. #15692187C22semidirect
43. #15692186C22semidirect
44. #15692179C22semidirect
46. #15692177C22semidirect
49. #15692175C22semidirect
50. #15692165C22semidirect
51. C4×#11717579C22semidirect
52. C8×Dic101C22semidirect
54. C4×Q8×C101C22semidirect
56. #107×C101C22semidirect
57. #103×C101C22semidirect
59. #97×C101C22semidirect
62. #13108075C2×C4semidirect
68. D4×D101C2×C4semidirect
77. #13108061C2×C4semidirect
78. D808C2×C4semidirect
79. #13108059C2×C4semidirect
80. #13108057C2×C4semidirect
104. SD16×C101C2×C4semidirect
189. C101#2946semidirect
196. #440D101semidirect
215. C4×Q8C2×D101semidirect
234. #107C2×D101semidirect
238. #103C2×D101semidirect
246. #97C2×D101semidirect
270. SD16C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
#202656592
#202656492
#202652902
#202652862
#202652802
SD16×Dic1012
C4×#131080612
#202652592
C4×#131080592
#202652182
#202651722
#102×D1012
#202651342
#440×C1012
#2946101