Finite Group Databasebeta

← #27179937

Subgroups — #27179937

Order 12928.

548 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692282C211yesmaximal, characteristic
2C4×#13108075C211yesmaximal, characteristic, complemented
3C4×D4×D101C211yesmaximal, characteristic, complemented
4#20265290C211yesmaximal, characteristic, complemented
5#20265287C211yesmaximal, characteristic, complemented
6#20265275C211yesmaximal, characteristic, complemented
7SD16×Dic101C211yesmaximal, characteristic, complemented
8#20265262C211yesmaximal, characteristic, complemented
9#20265259C211yesmaximal, characteristic, complemented
10#20265257C211yesmaximal, characteristic, complemented
11#20265218C211yesmaximal, characteristic, complemented
12#20265180C211yesmaximal, characteristic, complemented
13#20265177C211yesmaximal, characteristic, complemented
14#20265129C211yesmaximal, characteristic, complemented
15#440×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×D4×D101C2211yescharacteristic
18C22×C4×D101—21no—
19#15692282C418yescomplemented
20C2×#13108061C2211yescharacteristic
21C2×D808C2211yescharacteristic
22C2×#13108059C2211yescharacteristic
23C2×#13108057C2211yescharacteristic
24C2×#13108057C2211yescharacteristic
25Q8×Dic101C2211yescharacteristic, complemented
26C4×D404—21no—
27C4×D404C2211yescharacteristic
28C4×D404C2211yescharacteristic, complemented
29D4×Dic101C2211yescharacteristic
30#15692217—21no—
31#15692217C2211yescharacteristic
32#15692217C2211yescharacteristic, complemented
33C4×C22⋊D101—21no—
34#15692215—21no—
35#15692213—21no—
36#15692213C2211yescharacteristic
37C4⋊C4×D101C2211yescharacteristic
38C22⋊C4×D101—21no—
39C42×D101—21no—
40C42×D101C2211yescharacteristic
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#15692166C2211yescharacteristic, complemented
51C8×Dic101C2211yescharacteristic, 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×D101—44no—
77C2×C4×D101D411yescharacteristic
78C2×C4×D101D411yescharacteristic
79C2×C4×D101C2311yescharacteristic
80#13108061C2×C414yescomplemented
81D808C2×C414yescomplemented
82#13108059C2×C414yescomplemented
83#13108057—22no—
84#13108057C2×C414yescomplemented
85C2×#11717579C2311yescharacteristic
86C2×#11717579C2311yescharacteristic
87C4⋊Dic101—21no—
88C4⋊Dic101C2311yescharacteristic
89C4⋊Dic101C2311yescharacteristic
90#13108053—21no—
91C22⋊Dic101—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051—21no—
95#13108051—21no—
96#13108051—21no—
97C4×Dic101—21no—
98C4×Dic101—21no—
99C4×Dic101D411yescharacteristic
100C4×Dic101D411yescharacteristic
101C4×Dic101C2311yescharacteristic
102C2×Q8×C101C2311yescharacteristic
103C2×D4×C101C2311yescharacteristic
104C22×C4×C101—21no—
105SD16×C101C2×C414yescomplemented
106C2×C8×C101C2311yescharacteristic
107C2×C8×C101C2311yescharacteristic
108C4⋊C4×C101—21no—
109C4⋊C4×C101C2311yescharacteristic
110C4⋊C4×C101C2311yescharacteristic
111C22⋊C4×C101—21no—
112C42×C101—21no—
113C42×C101C2311yescharacteristic
114C23×C101—21no—
115C22×D101—21no—
116C22×D101—21no—
117C22×D101—24no—
118C22×D101—42no—
119C22×D101—42no—
120C22×D101C2×D411yescharacteristic
121Q8×C101—21no—
122Q8×C101C22×C412yescomplemented
123D4×C101—21no—
124D4×C101C22×C412yes—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—42no—
129C2×C4×C101C2×D411yescharacteristic
130C2×C4×C101C2×D411yescharacteristic
131C2×C4×C101C2411yescharacteristic
132C22⋊D101—24no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101—42no—
136C2×Dic101C2×D411yescharacteristic
137C2×Dic101C2×D411yescharacteristic
138C2×Dic101C2×D411yescharacteristic
139D404—24no—
140D404—21no—
141D404—21no—
142D404C22×C412yes—
143D404C22×C412yescomplemented
144C4×D101—24no—
145C4×D101—22no—
146C4×D101—41no—
147C4×D101—42no—
148C4×D101—42no—
149C4×D101—42no—
150C4×D101—42no—
151C4×D101—42no—
152C4×D101C22×C412yes—
153C4×D101#4112yes—
154C8×C101—21no—
155C8×C101C22×C412yescomplemented
156#11717579—21no—
157#11717579C22×C412yescomplemented
158C22×C101—22no—
159C22×C101—41no—
160C22×C101—41no—
161C22×C101C22×D411yescharacteristic
162C2×D101—21no—
163C2×D101—21no—
164C2×D101—22no—
165C2×D101—22no—
166C2×D101—48no—
167C2×D101—41no—
168C2×D101—41no—
169C2×D101—41no—
170C2×D101—41no—
171C2×D101C4×D412yes—
172C4×C101—22no—
173C4×C101—21no—
174C4×C101—21no—
175C4×C101—41no—
176C4×C101—41no—
177C4×C101C23×C411yescharacteristic
178C4×C101C2×#4111yescharacteristic
179Dic101—21no—
180Dic101—41no—
181Dic101—41no—
182Dic101C4×D412yes—
183Dic101AGL(1,ℤ/8)12yescomplemented
184C2×C101—42no—
185C2×C101C2×C4×D411yescharacteristic
186C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
187C2×C101#57511yescharacteristic
188D101—22no—
189D101—42no—
190D101—42no—
191C4×AGL(1,ℤ/8)—1011nomaximal, Sylow
192C101C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
193C2×AGL(1,ℤ/8)—1011no—
194C4×#41—1011no—
195C2×C4×D4—1011no—
196#442—1011no—
197#442—1011no—
198#440—1011no—
199#440D10111yescharacteristic, complemented
200C4×SD16—1011no—
201C4×SD16—1011no—
202C4×D8—1011no—
203C4×D8—1011no—
204#428—1011no—
205#418—1011no—
206#417—1011no—
207C4×M16—1011no—
208C2×#41—1011no—
209C22×D4—1011no—
210C23×C4—2021no—
211AGL(1,ℤ/8)—1018no—
212C2×SD16—1011no—
213C2×SD16C2×D10111yescharacteristic
214C2×D8—1011no—
215C2×D8—1011no—
216C2×M16—1011no—
217C2×M16—1011no—
218C4×Q8—1011no—
219C4×Q8C2×D10111yescharacteristic, complemented
220C4×D4—1011no—
221C4×D4—1011no—
222C4×D4—1011no—
223C4×D4—1011no—
224C4×D4—1011no—
225C4×D4—2021no—
226C4×D4—2021no—
227C4×D4—2021no—
228C4×D4—2021no—
229C4×D4C2×D10111yescharacteristic
230#117—1011no—
231#117—2021no—
232C2×C4⋊C4—1011no—
233C2×C22⋊C4—2021no—
234C2×C42—1011no—
235C2×C42—2021no—
236#107—1011no—
237#107C2×D10111yescharacteristic, complemented
238#106—1011no—
239#106—1011no—
240#103—1011no—
241#103C2×D10111yescharacteristic, complemented
242#102—1011no—
243#102—1011no—
244#102—1011no—
245#102—1011no—
246#102—1011no—
247#102C2×D10111yescharacteristic
248#97—1011no—
249#97C2×D10111yescharacteristic, complemented
250C4×C8—1011no—
251C4×C8—1011no—
252C24—2021no—
253#41—1014no—
254#41—2022no—
255C2×Q8C22×D10111yescharacteristic
256C2×D4—1014no—
257C2×D4—1011no—
258C2×D4—1011no—
259C2×D4—2021no—
260C2×D4—2022no—
261C2×D4—2021no—
262C2×D4C22×D10111yescharacteristic
263C22×C4—21no—
264C22×C4—1011no—
265C22×C4—1011no—
266C22×C4—1011no—
267C22×C4—4044no—
268C22×C4—2021no—
269C22×C4—2021no—
270C22×C4—2021no—
271C22×C4—2021no—
272C22×C4—2021no—
273C22×C4—2021no—
274C22×C4—2021no—
275SD16—1014no—
276SD16C4×D10114yescomplemented
277D8—1014no—
278D8—1014no—
279M16—1014no—
280M16—2022no—
281C2×C8—1011no—
282C2×C8—1011no—
283C2×C8C22×D10111yescharacteristic
284C2×C8C22×D10111yescharacteristic
285C4⋊C4—21no—
286C4⋊C4—1011no—
287C4⋊C4—1011no—
288C4⋊C4—2021no—
289C4⋊C4—2021no—
290C4⋊C4C22×D10111yescharacteristic
291C4⋊C4C22×D10111yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—2021no—
294C22⋊C4—2021no—
295C22⋊C4—2021no—
296C22⋊C4—2021no—
297C22⋊C4—2021no—
298C22⋊C4—2021no—
299C42—21no—
300C42—1011no—
301C42—1011no—
302C42—1011no—
303C42—2021no—
304C42—2021no—
305C42C22×D10111yescharacteristic
306C23—21no—
307C23—1011no—
308C23—4042no—
309C23—4042no—
310C23—2024no—
311C23—2021no—
312C23—2021no—
313Q8—21no—
314Q8C2×C4×D10112yescomplemented
315D4—21no—
316D4—1012no—
317D4—1012no—
318D4—2024no—
319D4—2024no—
320D4—2021no—
321D4—2021no—
322D4C2×C4×D10112yes—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—1011no—
328C2×C4—1011no—
329C2×C4—1012no—
330C2×C4—1012no—
331C2×C4—1011no—
332C2×C4—4042no—
333C2×C4—4042no—
334C2×C4—4041no—
335C2×C4—4042no—
336C2×C4—4042no—
337C2×C4—4042no—
338C2×C4—4042no—
339C2×C4—2022no—
340C2×C4—2024no—
341C2×C4—2021no—
342C2×C4—2021no—
343C2×C4D4×D10111yescharacteristic
344C2×C4D4×D10111yescharacteristic
345C2×C4C23×D10111yescharacteristic
346C8—21no—
347C8—1012no—
348C8—2021no—
349C8C2×C4×D10112yescomplemented
350C22—22no—
351C22—41no—
352C22—41no—
353C22—1012no—
354C22—4048no—
355C22—4041no—
356C22—4041no—
357C22—4041no—
358C22—4041no—
359C22—2022no—
360C22—2022no—
361C22—2021no—
362C22—2021no—
363C22C2×D4×D10111yescharacteristic
364C4—22no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—1012no—
370C4—1012no—
371C4—4041no—
372C4—4041no—
373C4—2021no—
374C4C22×C4×D10111yescharacteristic
375C4#41×D10111yescharacteristic
376C2—42no—
377C2—4042no—
378C2—4042no—
379C2—2022no—
380C2C4×D4×D10111yescharacteristic
381C2AGL(1,ℤ/8)×D10111yescharacteristic
382C2#2026597611yescharacteristic
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
hypocenterC101C4×AGL(1,ℤ/8)
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. C4×#13108075C2semidirect
3. C4×D4×D101C2semidirect
4. #20265290C2semidirect
5. #20265287C2semidirect
6. #20265275C2semidirect
7. SD16×Dic101C2semidirect
8. #20265262C2semidirect
9. #20265259C2semidirect
10. #20265257C2semidirect
11. #20265218C2semidirect
12. #20265180C2semidirect
13. #20265177C2semidirect
14. #20265129C2semidirect
15. #440×C101C2semidirect
19. #15692282C4semidirect
25. Q8×Dic101C22semidirect
28. C4×D404C22semidirect
32. #15692217C22semidirect
41. #15692188C22semidirect
42. #15692187C22semidirect
43. #15692186C22semidirect
44. #15692179C22semidirect
46. #15692177C22semidirect
49. #15692175C22semidirect
50. #15692166C22semidirect
51. C8×Dic101C22semidirect
52. C8×Dic101C22semidirect
54. C4×Q8×C101C22semidirect
56. #107×C101C22semidirect
57. #103×C101C22semidirect
59. #97×C101C22semidirect
62. #13108075C2×C4semidirect
68. D4×D101C2×C4semidirect
80. #13108061C2×C4semidirect
81. D808C2×C4semidirect
82. #13108059C2×C4semidirect
84. #13108057C2×C4semidirect
105. SD16×C101C2×C4semidirect
122. Q8×C101C22×C4semidirect
143. D404C22×C4semidirect
155. C8×C101C22×C4semidirect
157. #11717579C22×C4semidirect
183. Dic101AGL(1,ℤ/8)semidirect
192. C101C4×AGL(1,ℤ/8)semidirect
199. #440D101semidirect
219. C4×Q8C2×D101semidirect
237. #107C2×D101semidirect
241. #103C2×D101semidirect
249. #97C2×D101semidirect
276. SD16C4×D101semidirect
314. Q8C2×C4×D101semidirect
349. C8C2×C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
C4×#131080752
C4×D4×D1012
#202652902
#202652872
#202652752
SD16×Dic1012
#202652622
#202652592
#202652572
#202652182
#202651802
#202651772
#202651292
#440×C1012
C4×AGL(1,ℤ/8)101