Finite Group Databasebeta

← #27182738

Subgroups — #27182738

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×#15692282C211yesmaximal, characteristic, complemented
2C2×#15692279C211yesmaximal, characteristic, complemented
3#20265822C211yesmaximal, characteristic, complemented
4#20265820C211yesmaximal, characteristic, complemented
5#20265559C211yesmaximal, characteristic, complemented
6#20265426C211yesmaximal, characteristic, complemented
7#20265422C211yesmaximal, characteristic, complemented
8#20265393C211yesmaximal, characteristic, complemented
9#20265342C211yesmaximal, characteristic, complemented
10#20265336C211yesmaximal, characteristic, complemented
11#20265312C211yesmaximal, characteristic, complemented
12#20265309C211yesmaximal, characteristic, complemented
13C4×#13108059C211yesmaximal, characteristic, complemented
14#20265254C211yesmaximal, characteristic, complemented
15#463×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×#13108073C2211yescharacteristic
18C2×D4×D101C2211yescharacteristic, complemented
19C2×D4×D101C2211yescharacteristic, complemented
20#15692282—24no—
21#15692279—24no—
22C2×#13108064C2211yescharacteristic, complemented
23C2×#13108063C2211yescharacteristic, complemented
24C2×#13108062C2211yescharacteristic, complemented
25C2×#13108061C2211yescharacteristic, complemented
26C2×D808C2211yescharacteristic
27C2×#13108059C2211yescharacteristic
28C2×#13108059C2211yescharacteristic
29C2×#13108057C2211yescharacteristic, complemented
30C2×#13108057C2211yescharacteristic, complemented
31#15692243C2211yescharacteristic, complemented
32#15692239—21no—
33#15692237—21no—
34#15692235C2211yescharacteristic, complemented
35#15692231C2211yescharacteristic, complemented
36#15692230C2211yescharacteristic
37#15692229—21no—
38#15692228—21no—
39#15692226C2211yescharacteristic
40#15692225—21no—
41#15692225—21no—
42C4×D404C2211yescharacteristic
43C4×D404C2211yescharacteristic, complemented
44C2×#13108051—21no—
45C22⋊C4×D101—21no—
46#15692188C2211yescharacteristic, complemented
47#15692180C2211yescharacteristic, complemented
48#15692177C2211yescharacteristic, complemented
49#15692176C2211yescharacteristic, complemented
50#15692175C2211yescharacteristic, complemented
51#15692167C2211yescharacteristic, complemented
52#15692167C2211yescharacteristic, complemented
53C4×#11717579C2211yescharacteristic, complemented
54C2×SD16×C101C2211yescharacteristic, complemented
55C2×D8×C101C2211yescharacteristic
56#124×C101C2211yescharacteristic, complemented
57C4×D4×C101C2211yescharacteristic, complemented
58C4⋊C8×C101C2211yescharacteristic, complemented
59#103×C101C2211yescharacteristic, complemented
60#102×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101—21no—
63#13108075—22no—
64#13108075—41no—
65#13108073—22no—
66#13108073—41no—
67C2×Dic202C2311yescharacteristic
68C2×D404—21no—
69C2×D404—21no—
70C2×D404D411yescharacteristic
71C2×D404C2311yescharacteristic
72C2×D404C2311yescharacteristic
73C2×C22⋊D101—21no—
74C2×C22⋊D101—21no—
75C2×C22⋊D101—21no—
76C2×C22⋊D101—21no—
77D4×D101—22no—
78D4×D101—22no—
79D4×D101—41no—
80D4×D101—41no—
81C22×Dic101—21no—
82C2×C4×D101—21no—
83C2×C4×D101—21no—
84C2×C4×D101C2311yescharacteristic
85C2×C4×D101C2311yescharacteristic
86#13108064—41no—
87#13108063—22no—
88#13108062—22no—
89#13108061—22no—
90D808—22no—
91#13108059—41no—
92#13108059D414yescomplemented
93#13108057—22no—
94#13108057—22no—
95C2×#11717579C2311yescharacteristic
96C2×#11717579C2311yescharacteristic
97C4⋊Dic101—21no—
98C4⋊Dic101D411yescharacteristic
99C4⋊Dic101C2311yescharacteristic
100#13108053—21no—
101C22⋊Dic101—21no—
102#13108051—21no—
103#13108051—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051—21no—
107#13108051—21no—
108#13108051D411yescharacteristic
109#13108051D411yescharacteristic
110C2×Q8×C101C2311yescharacteristic
111C2×D4×C101C2311yescharacteristic
112C2×D4×C101C2311yescharacteristic
113C22×C4×C101—21no—
114SD16×C101—22no—
115D8×C101—22no—
116C2×C8×C101C2311yescharacteristic
117C2×C8×C101C2311yescharacteristic
118C4⋊C4×C101C2311yescharacteristic
119C22⋊C4×C101—21no—
120C22⋊C4×C101—21no—
121C22⋊C4×C101—21no—
122C42×C101C2311yescharacteristic
123C23×C101—21no—
124C23×C101—21no—
125C22×D101—21no—
126C22×D101—21no—
127C22×D101—42no—
128C22×D101—42no—
129C22×D101—42no—
130C22×D101—42no—
131C22×D101—81no—
132C22×D101—81no—
133C22×D101C2×D411yescharacteristic
134C22×D101C2×D411yescharacteristic
135Q8×C101—21no—
136Q8×C101—21no—
137D4×C101—21no—
138D4×C101—21no—
139D4×C101—21no—
140D4×C101C2×D412yes—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—81no—
144C2×C4×C101C2×D411yescharacteristic
145C2×C4×C101C2×D411yescharacteristic
146C2×C4×C101C2411yescharacteristic
147C22⋊D101—42no—
148C22⋊D101—42no—
149C22⋊D101—42no—
150C22⋊D101—81no—
151C2×Dic101—21no—
152C2×Dic101—21no—
153C2×Dic101—42no—
154C2×Dic101C2×D411yescharacteristic
155C2×Dic101C2×D411yescharacteristic
156D404—21no—
157D404—21no—
158D404—21no—
159D404—41no—
160D404—42no—
161D404—81no—
162D404C2×D412yes—
163C4×D101—21no—
164C4×D101—21no—
165C4×D101—21no—
166C4×D101—21no—
167C4×D101—42no—
168C4×D101—81no—
169Dic202—21no—
170Dic202—21no—
171C8×C101—21no—
172C8×C101—21no—
173#11717579—21no—
174#11717579C2×D412yes—
175C22×C101—22no—
176C22×C101—41no—
177C22×C101—41no—
178C22×C101—41no—
179C22×C101—41no—
180C22×C101—41no—
181C22×C101C22×D411yescharacteristic
182C2×D101—21no—
183C2×D101—21no—
184C2×D101—22no—
185C2×D101—21no—
186C2×D101—21no—
187C2×D101—21no—
188C2×D101—21no—
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—21no—
198C4×C101—41no—
199C4×C101—41no—
200C4×C101C22×D411yescharacteristic
201C4×C101H(5,𝔽2)11yescharacteristic
202Dic101—21no—
203Dic101—21no—
204Dic101—41no—
205Dic101—41no—
206C2×C101—42no—
207C2×C101—81no—
208C2×C101D4211yescharacteristic
209C2×C101#57511yescharacteristic
210C2×C101#57611yescharacteristic
211D101—41no—
212D101—41no—
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#468—1011no—
224#463—1011no—
225#463D10111yescharacteristic, complemented
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×D8C2×D10111yescharacteristic
247C2×M16—1011no—
248C2×M16—1011no—
249#124—1011no—
250#124C2×D10111yescharacteristic, complemented
251#123—2021no—
252#123—2021no—
253C22⋊Q8—1011no—
254C22⋊Q8—1011no—
255#121—1011no—
256#121—1011no—
257#121—2021no—
258#121—2021no—
259#120—2021no—
260#120—2021no—
261C4×D4—1011no—
262C4×D4—1011no—
263C4×D4C2×D10111yescharacteristic, complemented
264C2×C22⋊C4—2021no—
265C2×C22⋊C4—2021no—
266#107—1011no—
267C4⋊C8C2×D10111yescharacteristic, complemented
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—1011no—
286C2×D4—1011no—
287C2×D4—1011no—
288C2×D4—4041no—
289C2×D4—4041no—
290C2×D4—2021no—
291C2×D4—2021no—
292C2×D4—2021no—
293C2×D4—2021no—
294C2×D4—2022no—
295C2×D4—2021no—
296C2×D4—2021no—
297C2×D4—2022no—
298C2×D4C22×D10111yescharacteristic
299C2×D4C22×D10111yescharacteristic
300C22×C4—21no—
301C22×C4—1011no—
302C22×C4—1011no—
303C22×C4—2021no—
304C22×C4—2021no—
305C22×C4—2021no—
306Q16—4041no—
307SD16—22no—
308SD16—2022no—
309SD16—2022no—
310SD16—2022no—
311D8—22no—
312D8—1014no—
313D8—4041no—
314D8—2022no—
315M16—2022no—
316M16—2022no—
317C2×C8—1011no—
318C2×C8—1011no—
319C2×C8C22×D10111yescharacteristic
320C2×C8C22×D10111yescharacteristic
321C4⋊C4—1011no—
322C4⋊C4—1011no—
323C4⋊C4—2021no—
324C4⋊C4—2021no—
325C4⋊C4C22×D10111yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—1011no—
330C22⋊C4—1011no—
331C22⋊C4—2021no—
332C22⋊C4—2021no—
333C22⋊C4—2021no—
334C22⋊C4—2021no—
335C22⋊C4—2021no—
336C22⋊C4—2021no—
337C22⋊C4—2021no—
338C42C22×D10111yescharacteristic
339C23—21no—
340C23—21no—
341C23—8081no—
342C23—8081no—
343C23—1011no—
344C23—1011no—
345C23—4042no—
346C23—4042no—
347C23—4042no—
348C23—4042no—
349C23—2021no—
350C23—2021no—
351Q8—21no—
352Q8—21no—
353Q8—2021no—
354Q8—2021no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—8081no—
359D4—8081no—
360D4—1012no—
361D4—4041no—
362D4—4042no—
363D4—4042no—
364D4—4042no—
365D4—4042no—
366D4—2021no—
367D4—2021no—
368D4—2021no—
369D4D4×D10112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—81no—
373C2×C4—8081no—
374C2×C4—1011no—
375C2×C4—1011no—
376C2×C4—4042no—
377C2×C4—4042no—
378C2×C4—2021no—
379C2×C4—2021no—
380C2×C4—2021no—
381C2×C4—2021no—
382C2×C4—2021no—
383C2×C4—2021no—
384C2×C4D4×D10111yescharacteristic
385C2×C4D4×D10111yescharacteristic
386C2×C4C23×D10111yescharacteristic
387C8—21no—
388C8—21no—
389C8—1012no—
390C8—2021no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—8084no—
398C22—8084no—
399C22—4041no—
400C22—4041no—
401C22—4041no—
402C22—4041no—
403C22—4041no—
404C22—2022no—
405C22—2021no—
406C22—2021no—
407C22—2021no—
408C22—2021no—
409C22—2021no—
410C22—2021no—
411C22C2×D4×D10111yescharacteristic
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—4041no—
417C4—4041no—
418C4—2021no—
419C4—2021no—
420C4C2×D4×D10111yescharacteristic
421C4#1569230111yescharacteristic
422C2—42no—
423C2—81no—
424C2—8081no—
425C2—4041no—
426C2—4041no—
427C2—4042no—
428C2#2026581111yescharacteristic
429C2#2026597411yescharacteristic
430C2#2026598111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#463×C101C2
socleC22×C101C22×D4
hypercenter#463D101
hypocenterC101#3273
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#15692282C2semidirect
2. C2×#15692279C2semidirect
3. #20265822C2semidirect
4. #20265820C2semidirect
5. #20265559C2semidirect
6. #20265426C2semidirect
7. #20265422C2semidirect
8. #20265393C2semidirect
9. #20265342C2semidirect
10. #20265336C2semidirect
11. #20265312C2semidirect
12. #20265309C2semidirect
13. C4×#13108059C2semidirect
14. #20265254C2semidirect
15. #463×C101C2semidirect
18. C2×D4×D101C22semidirect
19. C2×D4×D101C22semidirect
22. C2×#13108064C22semidirect
23. C2×#13108063C22semidirect
24. C2×#13108062C22semidirect
25. C2×#13108061C22semidirect
29. C2×#13108057C22semidirect
30. C2×#13108057C22semidirect
31. #15692243C22semidirect
34. #15692235C22semidirect
35. #15692231C22semidirect
43. C4×D404C22semidirect
46. #15692188C22semidirect
47. #15692180C22semidirect
48. #15692177C22semidirect
49. #15692176C22semidirect
50. #15692175C22semidirect
51. #15692167C22semidirect
52. #15692167C22semidirect
53. C4×#11717579C22semidirect
54. C2×SD16×C101C22semidirect
56. #124×C101C22semidirect
57. C4×D4×C101C22semidirect
58. C4⋊C8×C101C22semidirect
59. #103×C101C22semidirect
60. #102×C101C22semidirect
92. #13108059D4semidirect
216. C101#3273semidirect
225. #463D101semidirect
242. C2×SD16C2×D101semidirect
250. #124C2×D101semidirect
263. C4×D4C2×D101semidirect
267. C4⋊C8C2×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×#156922822
C2×#156922792
#202658222
#202658202
#202655592
#202654262
#202654222
#202653932
#202653422
#202653362
#202653122
#202653092
C4×#131080592
#202652542
#463×C1012
#3273101