Finite Group Databasebeta

← #27179943

Subgroups — #27179943

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×#15692279C211yesmaximal, characteristic
2C4×#13108073C211yesmaximal, characteristic, complemented
3C4×D4×D101C211yesmaximal, characteristic, complemented
4#20265287C211yesmaximal, characteristic, complemented
5#20265279C211yesmaximal, characteristic, complemented
6#20265277C211yesmaximal, characteristic, complemented
7#20265268C211yesmaximal, characteristic, complemented
8#20265264C211yesmaximal, characteristic, complemented
9D8×Dic101C211yesmaximal, characteristic, complemented
10#20265257C211yesmaximal, characteristic, complemented
11#20265216C211yesmaximal, characteristic, complemented
12#20265180C211yesmaximal, characteristic, complemented
13#20265175C211yesmaximal, characteristic, complemented
14#20265129C211yesmaximal, characteristic, complemented
15#442×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×D4×D101C2211yescharacteristic
18C22×C4×D101—21no—
19#15692279C418yescomplemented
20C2×#13108063C2211yescharacteristic
21C2×#13108062C2211yescharacteristic
22C2×#13108059C2211yescharacteristic
23C2×#13108057C2211yescharacteristic
24C2×#13108057C2211yescharacteristic
25C4×Dic202C2211yescharacteristic, complemented
26#15692220C2211yescharacteristic, complemented
27C4×D404C2211yescharacteristic
28D4×Dic101C2211yescharacteristic
29D4×Dic101C2211yescharacteristic, complemented
30#15692217C2211yescharacteristic
31C4×C22⋊D101—21no—
32C4×C22⋊D101—21no—
33#15692215—21no—
34#15692215—21no—
35#15692213C2211yescharacteristic
36#15692211—21no—
37C4⋊C4×D101C2211yescharacteristic
38C22⋊C4×D101—21no—
39C2×C4×Dic101—21no—
40C42×D101C2211yescharacteristic
41#15692189C2211yescharacteristic, complemented
42#15692188C2211yescharacteristic, complemented
43#15692186C2211yescharacteristic, complemented
44#15692180C2211yescharacteristic, complemented
45#15692178C2211yescharacteristic, complemented
46#15692177C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692176C2211yescharacteristic, complemented
49#15692175C2211yescharacteristic
50#15692166C2211yescharacteristic, complemented
51C8×Dic101C2211yescharacteristic, complemented
52C8×Dic101C2211yescharacteristic, complemented
53C2×D8×C101C2211yescharacteristic
54C4×D4×C101C2211yescharacteristic
55C4×D4×C101C2211yescharacteristic, complemented
56#106×C101C2211yescharacteristic, complemented
57#102×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic, complemented
59#97×C101C2211yescharacteristic, complemented
60C23×D101—21no—
61#13108073—22no—
62#13108073C2×C414yescomplemented
63C2×Dic202C2311yescharacteristic
64C2×D404C2311yescharacteristic
65C2×C22⋊D101—21no—
66C2×C22⋊D101—21no—
67D4×D101—22no—
68D4×D101C2×C414yescomplemented
69C22×Dic101—21no—
70C22×Dic101—21no—
71C22×Dic101—21no—
72C2×C4×D101—21no—
73C2×C4×D101—21no—
74C2×C4×D101—21no—
75C2×C4×D101—44no—
76C2×C4×D101D411yescharacteristic
77C2×C4×D101D411yescharacteristic
78C2×C4×D101C2311yescharacteristic
79#13108063C2×C414yescomplemented
80#13108062C2×C414yescomplemented
81#13108059C2×C414yescomplemented
82#13108057—22no—
83#13108057C2×C414yescomplemented
84C2×#11717579C2311yescharacteristic
85C2×#11717579C2311yescharacteristic
86C4⋊Dic101C2311yescharacteristic
87C4⋊Dic101C2311yescharacteristic
88#13108053—21no—
89#13108053—21no—
90#13108053—21no—
91C22⋊Dic101—21no—
92C22⋊Dic101—21no—
93#13108051—21no—
94#13108051—21no—
95#13108051—21no—
96C4×Dic101—21no—
97C4×Dic101—21no—
98C4×Dic101—21no—
99C4×Dic101D411yescharacteristic
100C4×Dic101D411yescharacteristic
101C4×Dic101C2311yescharacteristic
102C2×D4×C101C2311yescharacteristic
103C2×D4×C101C2311yescharacteristic
104C22×C4×C101—21no—
105C22×C4×C101—21no—
106D8×C101C2×C414yescomplemented
107C2×C8×C101C2311yescharacteristic
108C2×C8×C101C2311yescharacteristic
109C4⋊C4×C101C2311yescharacteristic
110C4⋊C4×C101C2311yescharacteristic
111C22⋊C4×C101—21no—
112C22⋊C4×C101—21no—
113C42×C101C2311yescharacteristic
114C23×C101—21no—
115C23×C101—21no—
116C22×D101—21no—
117C22×D101—24no—
118C22×D101—42no—
119C22×D101—42no—
120C22×D101C2×D411yescharacteristic
121D4×C101—21no—
122D4×C101—21no—
123D4×C101C22×C412yes—
124D4×C101C22×C412yescomplemented
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—42no—
128C2×C4×C101—42no—
129C2×C4×C101C2×D411yescharacteristic
130C2×C4×C101C2×D411yescharacteristic
131C2×C4×C101C2411yescharacteristic
132C22⋊D101—24no—
133C22⋊D101—24no—
134C2×Dic101—21no—
135C2×Dic101—24no—
136C2×Dic101—21no—
137C2×Dic101—21no—
138C2×Dic101—42no—
139C2×Dic101—42no—
140C2×Dic101C2×D411yescharacteristic
141C2×Dic101C2×D411yescharacteristic
142C2×Dic101C2×D411yescharacteristic
143D404—21no—
144D404C22×C412yes—
145C4×D101—22no—
146C4×D101—41no—
147C4×D101—42no—
148C4×D101—42no—
149C4×D101—42no—
150C4×D101C22×C412yes—
151C4×D101#4112yes—
152Dic202—21no—
153Dic202C22×C412yescomplemented
154C8×C101—21no—
155C8×C101C22×C412yescomplemented
156#11717579—21no—
157#11717579C22×C412yescomplemented
158C22×C101—22no—
159C22×C101—22no—
160C22×C101—41no—
161C22×C101—41no—
162C22×C101—41no—
163C22×C101—41no—
164C22×C101C22×D411yescharacteristic
165C2×D101—21no—
166C2×D101—21no—
167C2×D101—22no—
168C2×D101—48no—
169C2×D101—41no—
170C2×D101—41no—
171C2×D101C4×D412yes—
172C4×C101—21no—
173C4×C101—21no—
174C4×C101—41no—
175C4×C101—41no—
176C4×C101C23×C411yescharacteristic
177C4×C101C2×#4111yescharacteristic
178Dic101—22no—
179Dic101—21no—
180Dic101—41no—
181Dic101—41no—
182Dic101C4×D412yes—
183Dic101AGL(1,ℤ/8)12yescomplemented
184C2×C101—42no—
185C2×C101—42no—
186C2×C101C2×C4×D411yescharacteristic
187C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C101#57511yescharacteristic
189D101—22no—
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#442D10111yescharacteristic, complemented
198#440—1011no—
199#440—1011no—
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×SD16—1011no—
214C2×D8—1011no—
215C2×D8C2×D10111yescharacteristic
216C2×M16—1011no—
217C2×M16—1011no—
218C4×Q8—1011no—
219C4×Q8—1011no—
220C4×D4—1011no—
221C4×D4—1011no—
222C4×D4—1011no—
223C4×D4—1011no—
224C4×D4—2021no—
225C4×D4—2021no—
226C4×D4—2021no—
227C4×D4—2021no—
228C4×D4C2×D10111yescharacteristic
229C4×D4C2×D10111yescharacteristic, complemented
230#117—1011no—
231#117—2021no—
232C2×C4⋊C4—1011no—
233C2×C22⋊C4—2021no—
234C2×C42—1011no—
235C2×C42—2021no—
236#107—1011no—
237#107—1011no—
238#106—1011no—
239#106C2×D10111yescharacteristic, complemented
240#103—1011no—
241#103—1011no—
242#102—1011no—
243#102—1011no—
244#102—1011no—
245#102—1011no—
246#102C2×D10111yescharacteristic
247#102C2×D10111yescharacteristic, complemented
248#97—1011no—
249#97C2×D10111yescharacteristic, complemented
250C4×C8—1011no—
251C4×C8—1011no—
252C24—2021no—
253#41—1014no—
254#41—2022no—
255C2×Q8—1011no—
256C2×D4—1014no—
257C2×D4—1011no—
258C2×D4—2022no—
259C2×D4—2021no—
260C2×D4—2021no—
261C2×D4C22×D10111yescharacteristic
262C2×D4C22×D10111yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—1011no—
266C22×C4—1011no—
267C22×C4—1011no—
268C22×C4—4044no—
269C22×C4—2021no—
270C22×C4—2021no—
271C22×C4—2021no—
272C22×C4—2021no—
273C22×C4—2021no—
274C22×C4—2021no—
275SD16—1014no—
276SD16—1014no—
277D8—1014no—
278D8C4×D10114yescomplemented
279M16—1014no—
280M16—2022no—
281C2×C8—1011no—
282C2×C8—1011no—
283C2×C8C22×D10111yescharacteristic
284C2×C8C22×D10111yescharacteristic
285C4⋊C4—1011no—
286C4⋊C4—1011no—
287C4⋊C4—2021no—
288C4⋊C4—2021no—
289C4⋊C4—2021no—
290C4⋊C4C22×D10111yescharacteristic
291C4⋊C4C22×D10111yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—2021no—
295C22⋊C4—2021no—
296C22⋊C4—2021no—
297C22⋊C4—2021no—
298C22⋊C4—2021no—
299C42—1011no—
300C42—1011no—
301C42—1011no—
302C42—2021no—
303C42—2021no—
304C42—2021no—
305C42C22×D10111yescharacteristic
306C23—21no—
307C23—21no—
308C23—1011no—
309C23—4042no—
310C23—4042no—
311C23—2024no—
312C23—2021no—
313Q8—1012no—
314Q8—2021no—
315D4—21no—
316D4—21no—
317D4—1012no—
318D4—2024no—
319D4—2024no—
320D4—2021no—
321D4C2×C4×D10112yes—
322D4C2×C4×D10112yescomplemented
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—1011no—
328C2×C4—1011no—
329C2×C4—1012no—
330C2×C4—1012no—
331C2×C4—1011no—
332C2×C4—4042no—
333C2×C4—4041no—
334C2×C4—4042no—
335C2×C4—4042no—
336C2×C4—4042no—
337C2×C4—4042no—
338C2×C4—2022no—
339C2×C4—2024no—
340C2×C4—2021no—
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—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—1012no—
357C22—4048no—
358C22—4041no—
359C22—4041no—
360C22—2022no—
361C22—2021no—
362C22—2021no—
363C22C2×D4×D10111yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—1012no—
369C4—1012no—
370C4—4041no—
371C4—4041no—
372C4—2022no—
373C4—2021no—
374C4C22×C4×D10111yescharacteristic
375C4#41×D10111yescharacteristic
376C2—42no—
377C2—42no—
378C2—4042no—
379C2—2022no—
380C2C4×D4×D10111yescharacteristic
381C2AGL(1,ℤ/8)×D10111yescharacteristic
382C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutatorC4×C101C23×C4
FrattiniC2×C4C23×D101
Fitting#442×C101C2
socleC22×C101C22×D4
prime radicalC2×#15692279C2
cosocleC2×C4×C101C24
hypercenter#442D101
hypocenterC101C4×AGL(1,ℤ/8)
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. C4×#13108073C2semidirect
3. C4×D4×D101C2semidirect
4. #20265287C2semidirect
5. #20265279C2semidirect
6. #20265277C2semidirect
7. #20265268C2semidirect
8. #20265264C2semidirect
9. D8×Dic101C2semidirect
10. #20265257C2semidirect
11. #20265216C2semidirect
12. #20265180C2semidirect
13. #20265175C2semidirect
14. #20265129C2semidirect
15. #442×C101C2semidirect
19. #15692279C4semidirect
25. C4×Dic202C22semidirect
26. #15692220C22semidirect
29. D4×Dic101C22semidirect
41. #15692189C22semidirect
42. #15692188C22semidirect
43. #15692186C22semidirect
44. #15692180C22semidirect
45. #15692178C22semidirect
48. #15692176C22semidirect
50. #15692166C22semidirect
51. C8×Dic101C22semidirect
52. C8×Dic101C22semidirect
55. C4×D4×C101C22semidirect
56. #106×C101C22semidirect
58. #102×C101C22semidirect
59. #97×C101C22semidirect
62. #13108073C2×C4semidirect
68. D4×D101C2×C4semidirect
79. #13108063C2×C4semidirect
80. #13108062C2×C4semidirect
81. #13108059C2×C4semidirect
83. #13108057C2×C4semidirect
106. D8×C101C2×C4semidirect
124. D4×C101C22×C4semidirect
153. Dic202C22×C4semidirect
155. C8×C101C22×C4semidirect
157. #11717579C22×C4semidirect
183. Dic101AGL(1,ℤ/8)semidirect
192. C101C4×AGL(1,ℤ/8)semidirect
197. #442D101semidirect
229. C4×D4C2×D101semidirect
239. #106C2×D101semidirect
247. #102C2×D101semidirect
249. #97C2×D101semidirect
278. D8C4×D101semidirect
322. D4C2×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×#156922792
C4×#131080732
C4×D4×D1012
#202652872
#202652792
#202652772
#202652682
#202652642
D8×Dic1012
#202652572
#202652162
#202651802
#202651752
#202651292
#442×C1012
C4×AGL(1,ℤ/8)101