Finite Group Databasebeta

← #27182740

Subgroups — #27182740

Order 12928.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692283C211yesmaximal, characteristic, complemented
2C2×#15692282C211yesmaximal, characteristic, complemented
3#20265824C211yesmaximal, characteristic, complemented
4#20265818C211yesmaximal, characteristic, complemented
5#20265558C211yesmaximal, characteristic, complemented
6#20265424C211yesmaximal, characteristic, complemented
7#20265395C211yesmaximal, characteristic, complemented
8#20265391C211yesmaximal, characteristic, complemented
9#20265342C211yesmaximal, characteristic, complemented
10#20265337C211yesmaximal, characteristic, complemented
11#20265312C211yesmaximal, characteristic, complemented
12#20265310C211yesmaximal, characteristic, complemented
13#20265259C211yesmaximal, characteristic, complemented
14#20265253C211yesmaximal, characteristic, complemented
15#468×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×#13108074C2211yescharacteristic
18C22×D404C2211yescharacteristic, complemented
19C2×D4×D101C2211yescharacteristic, complemented
20#15692283—24no—
21#15692282—24no—
22C2×#13108064C2211yescharacteristic, complemented
23C2×#13108063C2211yescharacteristic, complemented
24C2×#13108063C2211yescharacteristic, complemented
25C2×#13108061C2211yescharacteristic, complemented
26C2×D808C2211yescharacteristic
27C2×D808C2211yescharacteristic
28C2×#13108059C2211yescharacteristic
29C2×#13108059C2211yescharacteristic
30C2×#13108057C2211yescharacteristic, complemented
31#15692244C2211yescharacteristic, complemented
32#15692242C2211yescharacteristic, complemented
33#15692240—21no—
34#15692237—21no—
35#15692235C2211yescharacteristic, complemented
36#15692230—21no—
37#15692230C2211yescharacteristic
38#15692226—21no—
39#15692225—21no—
40#15692223—21no—
41#15692217C2211yescharacteristic, complemented
42#15692217C2211yescharacteristic, complemented
43C4×C22⋊D101C2211yescharacteristic
44C22⋊C4×D101—21no—
45C22⋊C4×D101—21no—
46#15692184C2211yescharacteristic, complemented
47#15692178C2211yescharacteristic, complemented
48#15692175C2211yescharacteristic, complemented
49#15692175C2211yescharacteristic, complemented
50#15692175C2211yescharacteristic, complemented
51C22⋊#11717579C2211yescharacteristic, complemented
52#15692167C2211yescharacteristic, complemented
53C8×Dic101C2211yescharacteristic, complemented
54C2×SD16×C101C2211yescharacteristic, complemented
55C2×M16×C101C2211yescharacteristic, complemented
56C22⋊Q8×C101C2211yescharacteristic, complemented
57#121×C101C2211yescharacteristic
58#107×C101C2211yescharacteristic, complemented
59#103×C101C2211yescharacteristic, complemented
60#102×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101—21no—
63#13108075—22no—
64#13108075—41no—
65#13108074—22no—
66#13108074—41no—
67C2×Dic202C2311yescharacteristic
68C2×D404—22no—
69C2×D404—21no—
70C2×D404—21no—
71C2×D404—21no—
72C2×D404—41no—
73C2×D404C2311yescharacteristic
74C2×D404C2311yescharacteristic
75C2×D404C2311yescharacteristic
76C2×C22⋊D101—21no—
77C2×C22⋊D101—21no—
78C2×C22⋊D101D411yescharacteristic
79D4×D101—22no—
80D4×D101—41no—
81C2×C4×D101—21no—
82C2×C4×D101—21no—
83C2×C4×D101—21no—
84C2×C4×D101—21no—
85C2×C4×D101C2311yescharacteristic
86#13108064—41no—
87#13108063—22no—
88#13108063—22no—
89#13108061—22no—
90D808—22no—
91D808D414yescomplemented
92#13108059—22no—
93#13108059—41no—
94#13108057—22no—
95C2×#11717579C2311yescharacteristic
96C2×#11717579C2311yescharacteristic
97C4⋊Dic101—21no—
98#13108053D411yescharacteristic
99C22⋊Dic101—21no—
100C22⋊Dic101—21no—
101C22⋊Dic101D411yescharacteristic
102#13108051—21no—
103#13108051—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051—21no—
107#13108051—21no—
108#13108051D411yescharacteristic
109C4×Dic101C2311yescharacteristic
110C2×Q8×C101C2311yescharacteristic
111C2×D4×C101—21no—
112C2×D4×C101C2311yescharacteristic
113C22×C4×C101C2311yescharacteristic
114SD16×C101—22no—
115M16×C101—22no—
116C2×C8×C101C2311yescharacteristic
117C2×C8×C101C2311yescharacteristic
118C4⋊C4×C101—21no—
119C4⋊C4×C101C2311yescharacteristic
120C4⋊C4×C101C2311yescharacteristic
121C22⋊C4×C101—21no—
122C22⋊C4×C101—21no—
123C23×C101—21no—
124C23×C101C2×D411yescharacteristic
125C22×D101—21no—
126C22×D101—21no—
127C22×D101—21no—
128C22×D101—42no—
129C22×D101—42no—
130C22×D101—42no—
131C22×D101—42no—
132C22×D101—81no—
133C22×D101—81no—
134C22×D101C2×D411yescharacteristic
135Q8×C101—21no—
136Q8×C101—21no—
137D4×C101—21no—
138D4×C101—21no—
139D4×C101—81no—
140C2×C4×C101—21no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—21no—
145C2×C4×C101C2×D411yescharacteristic
146C2×C4×C101C2411yescharacteristic
147C22⋊D101—41no—
148C22⋊D101—42no—
149C22⋊D101—42no—
150C2×Dic101—21no—
151C2×Dic101C2×D411yescharacteristic
152C2×Dic101C2×D411yescharacteristic
153C2×Dic101C2×D411yescharacteristic
154D404—21no—
155D404—21no—
156D404—21no—
157D404—21no—
158D404—42no—
159D404—42no—
160D404—81no—
161D404C2×D412yes—
162D404C2×D412yes—
163C4×D101—21no—
164C4×D101—21no—
165C4×D101—42no—
166C4×D101—42no—
167C4×D101—81no—
168C4×D101—81no—
169Dic202—21no—
170Dic202—21no—
171C8×C101—21no—
172C8×C101C2×D412yes—
173#11717579—21no—
174#11717579—21no—
175C22×C101—21no—
176C22×C101—21no—
177C22×C101—21no—
178C22×C101—41no—
179C22×C101—41no—
180C22×C101—41no—
181C22×C101C22×D411yescharacteristic
182C2×D101—21no—
183C2×D101—22no—
184C2×D101—22no—
185C2×D101—21no—
186C2×D101—21no—
187C2×D101—41no—
188C2×D101—41no—
189C2×D101—41no—
190C2×D101—41no—
191C2×D101—41no—
192C2×D101—41no—
193C2×D101—41no—
194C2×D101—84no—
195C2×D101—84no—
196C4×C101—21no—
197C4×C101—41no—
198C4×C101—41no—
199C4×C101—41no—
200C4×C101C22×D411yescharacteristic
201C4×C101H(5,𝔽2)11yescharacteristic
202Dic101—21no—
203Dic101—21no—
204Dic101—21no—
205Dic101—41no—
206C2×C101—41no—
207C2×C101—81no—
208C2×C101D4211yescharacteristic
209C2×C101#57511yescharacteristic
210C2×C101#57611yescharacteristic
211D101—41no—
212D101—42no—
213D101—42no—
214D101—81no—
215#3273—1011nomaximal, Sylow
216C101#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—1011no—
218C2×AGL(1,ℤ/8)—1011no—
219#546—1011no—
220#546—1011no—
221#495—1011no—
222#468—1011no—
223#468D10111yescharacteristic, complemented
224#463—1011no—
225#463—1011no—
226#452—1011no—
227#452—1011no—
228#447—1011no—
229#447—1011no—
230C4×D8—1011no—
231#436—1011no—
232C2×#41—1011no—
233C2×#41—1011no—
234C22×D4—1011no—
235C22×D4—1011no—
236AGL(1,ℤ/8)—2024no—
237AGL(1,ℤ/8)—2024no—
238C2×Q16—1011no—
239C2×SD16—1011no—
240C2×SD16—1011no—
241C2×SD16—1011no—
242C2×SD16C2×D10111yescharacteristic, complemented
243C2×D8—1011no—
244C2×D8—1011no—
245C2×D8—1011no—
246C2×D8—1011no—
247C2×M16—1011no—
248C2×M16C2×D10111yescharacteristic, complemented
249#124—1011no—
250#124—1011no—
251#123—2021no—
252#123—2021no—
253C22⋊Q8—1011no—
254C22⋊Q8C2×D10111yescharacteristic, complemented
255#121—1011no—
256#121—2021no—
257#121—2021no—
258#121C2×D10111yescharacteristic
259#120—2021no—
260#120—2021no—
261C4×D4—1011no—
262C4×D4—1011no—
263C4×D4—1011no—
264C2×C22⋊C4—2021no—
265C2×C22⋊C4—2021no—
266#107C2×D10111yescharacteristic, complemented
267C4⋊C8—1011no—
268#103—1011no—
269#103C2×D10111yescharacteristic, complemented
270#102—1011no—
271#102—1011no—
272#102—1011no—
273#102C2×D10111yescharacteristic, complemented
274C22⋊C8—1011no—
275C22⋊C8—1011no—
276C4×C8—1011no—
277C24—2021no—
278C24—2021no—
279#41—4041no—
280#41—4041no—
281#41—2022no—
282#41—2022no—
283C2×Q8—1011no—
284C2×Q8C22×D10111yescharacteristic
285C2×D4—21no—
286C2×D4—1011no—
287C2×D4—1011no—
288C2×D4—1011no—
289C2×D4—1011no—
290C2×D4—4041no—
291C2×D4—4041no—
292C2×D4—2021no—
293C2×D4—2022no—
294C2×D4—2021no—
295C2×D4—2021no—
296C2×D4—2022no—
297C2×D4—2021no—
298C2×D4—2021no—
299C2×D4C22×D10111yescharacteristic
300C22×C4—1011no—
301C22×C4—2021no—
302C22×C4—2021no—
303C22×C4—2021no—
304C22×C4—2021no—
305C22×C4C22×D10111yescharacteristic
306Q16—4041no—
307SD16—22no—
308SD16—2022no—
309SD16—2022no—
310SD16—2022no—
311D8—1014no—
312D8—4041no—
313D8—2022no—
314D8—2022no—
315M16—22no—
316M16—2022no—
317C2×C8—1011no—
318C2×C8—1011no—
319C2×C8C22×D10111yescharacteristic
320C2×C8C22×D10111yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—1011no—
323C4⋊C4—2021no—
324C4⋊C4C22×D10111yescharacteristic
325C4⋊C4C22×D10111yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—1011no—
329C22⋊C4—1011no—
330C22⋊C4—2021no—
331C22⋊C4—2021no—
332C22⋊C4—2021no—
333C22⋊C4—2021no—
334C22⋊C4—2021no—
335C22⋊C4—2021no—
336C22⋊C4—2021no—
337C22⋊C4—2021no—
338C42—1011no—
339C23—21no—
340C23—8081no—
341C23—8081no—
342C23—1011no—
343C23—4042no—
344C23—4042no—
345C23—4042no—
346C23—4042no—
347C23—2021no—
348C23—2021no—
349C23—2021no—
350C23D4×D10111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—2021no—
354Q8—2021no—
355D4—21no—
356D4—21no—
357D4—81no—
358D4—8081no—
359D4—1012no—
360D4—1012no—
361D4—4042no—
362D4—4042no—
363D4—4042no—
364D4—4041no—
365D4—4042no—
366D4—2021no—
367D4—2021no—
368D4—2021no—
369D4—2021no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—8081no—
376C2×C4—8081no—
377C2×C4—1011no—
378C2×C4—1011no—
379C2×C4—1011no—
380C2×C4—4042no—
381C2×C4—4042no—
382C2×C4—2021no—
383C2×C4—2021no—
384C2×C4—2021no—
385C2×C4D4×D10111yescharacteristic
386C2×C4C23×D10111yescharacteristic
387C8—21no—
388C8—2021no—
389C8—2021no—
390C8D4×D10112yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—8084no—
398C22—8084no—
399C22—4041no—
400C22—4041no—
401C22—4041no—
402C22—4041no—
403C22—4041no—
404C22—4041no—
405C22—4041no—
406C22—2022no—
407C22—2022no—
408C22—2021no—
409C22—2021no—
410C22—2021no—
411C22C2×D4×D10111yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—4041no—
417C4—2021no—
418C4—2021no—
419C4—2021no—
420C4C2×D4×D10111yescharacteristic
421C4#1569230111yescharacteristic
422C2—41no—
423C2—81no—
424C2—8081no—
425C2—4041no—
426C2—4042no—
427C2—4042no—
428C2#2026581111yescharacteristic
429C2#2026597611yescharacteristic
430C2#2026598111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#468×C101C2
socleC22×C101C22×D4
hypercenter#468D101
hypocenterC101#3273
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#15692283C2semidirect
2. C2×#15692282C2semidirect
3. #20265824C2semidirect
4. #20265818C2semidirect
5. #20265558C2semidirect
6. #20265424C2semidirect
7. #20265395C2semidirect
8. #20265391C2semidirect
9. #20265342C2semidirect
10. #20265337C2semidirect
11. #20265312C2semidirect
12. #20265310C2semidirect
13. #20265259C2semidirect
14. #20265253C2semidirect
15. #468×C101C2semidirect
18. C22×D404C22semidirect
19. C2×D4×D101C22semidirect
22. C2×#13108064C22semidirect
23. C2×#13108063C22semidirect
24. C2×#13108063C22semidirect
25. C2×#13108061C22semidirect
30. C2×#13108057C22semidirect
31. #15692244C22semidirect
32. #15692242C22semidirect
35. #15692235C22semidirect
41. #15692217C22semidirect
42. #15692217C22semidirect
46. #15692184C22semidirect
47. #15692178C22semidirect
48. #15692175C22semidirect
49. #15692175C22semidirect
50. #15692175C22semidirect
51. C22⋊#11717579C22semidirect
52. #15692167C22semidirect
53. C8×Dic101C22semidirect
54. C2×SD16×C101C22semidirect
55. C2×M16×C101C22semidirect
56. C22⋊Q8×C101C22semidirect
58. #107×C101C22semidirect
59. #103×C101C22semidirect
60. #102×C101C22semidirect
91. D808D4semidirect
216. C101#3273semidirect
223. #468D101semidirect
242. C2×SD16C2×D101semidirect
248. C2×M16C2×D101semidirect
254. C22⋊Q8C2×D101semidirect
266. #107C2×D101semidirect
269. #103C2×D101semidirect
273. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922832
C2×#156922822
#202658242
#202658182
#202655582
#202654242
#202653952
#202653912
#202653422
#202653372
#202653122
#202653102
#202652592
#202652532
#468×C1012
#3273101