Finite Group Databasebeta

← #27180621

Subgroups — #27180621

Order 12928.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692284C211yesmaximal, characteristic, complemented
2C2×#15692279C211yesmaximal, characteristic
3#124×D101C211yesmaximal, characteristic, complemented
4C4×#13108073C211yesmaximal, characteristic, complemented
5#20265402C211yesmaximal, characteristic, complemented
6#20265400C211yesmaximal, characteristic, complemented
7#20265398C211yesmaximal, characteristic, complemented
8#20265396C211yesmaximal, characteristic, complemented
9#20265394C211yesmaximal, characteristic, complemented
10#20265392C211yesmaximal, characteristic, complemented
11#20265390C211yesmaximal, characteristic, complemented
12#20265199C211yesmaximal, characteristic, complemented
13#20265184C211yesmaximal, characteristic, complemented
14#20265180C211yesmaximal, characteristic, complemented
15#463×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×Q8×D101C2211yescharacteristic
18C2×D4×D101C2211yescharacteristic
19#15692284—24no—
20#15692279—24no—
21C2×Dic404C2211yescharacteristic, complemented
22C2×#13108064C2211yescharacteristic, complemented
23C2×#13108063C2211yescharacteristic
24C2×#13108062C2211yescharacteristic
25C2×#13108062C2211yescharacteristic, complemented
26C2×#13108059C2211yescharacteristic
27C2×#13108057C2211yescharacteristic
28C2×#13108057C2211yescharacteristic
29#15692244C2211yescharacteristic
30#15692243C2211yescharacteristic
31#15692242C2211yescharacteristic
32#15692241—21no—
33#15692241—21no—
34C4×Dic202C2211yescharacteristic, complemented
35#15692220C2211yescharacteristic, complemented
36D4×Dic101C2211yescharacteristic, complemented
37C4×C22⋊D101—21no—
38#15692215—21no—
39#15692213C2211yescharacteristic
40#15692211—21no—
41C22⋊C4×D101—21no—
42C22⋊C4×D101—21no—
43C2×C4×Dic101—21no—
44C42×D101C2211yescharacteristic
45C4⋊#11717579C2211yescharacteristic, complemented
46#15692184C2211yescharacteristic, complemented
47#15692184C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692179C2211yescharacteristic, complemented
50#15692178C2211yescharacteristic, complemented
51#15692177C2211yescharacteristic
52#15692176C2211yescharacteristic
53#15692175C2211yescharacteristic
54C2×SD16×C101C2211yescharacteristic, complemented
55C2×D8×C101C2211yescharacteristic
56#124×C101C2211yescharacteristic
57C4×D4×C101C2211yescharacteristic, complemented
58C4⋊C8×C101C2211yescharacteristic, complemented
59#103×C101C2211yescharacteristic, complemented
60#102×C101C2211yescharacteristic
61C23×D101—21no—
62#13108073—22no—
63#13108073D414yes—
64C2×Dic202—21no—
65C2×Dic202C2311yescharacteristic
66C2×Dic202C2311yescharacteristic
67Q8×D101—22no—
68Q8×D101—22no—
69C2×D404C2311yescharacteristic
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72D4×D101—22no—
73D4×D101—22no—
74C22×Dic101—21no—
75C22×Dic101—21no—
76C2×C4×D101—21no—
77C2×C4×D101D411yescharacteristic
78C2×C4×D101D411yescharacteristic
79C2×C4×D101C2311yescharacteristic
80Dic404—22no—
81#13108064—22no—
82#13108063—22no—
83#13108062—22no—
84#13108062—22no—
85#13108059—22no—
86#13108057—22no—
87#13108057—22no—
88C2×#11717579C2311yescharacteristic
89C2×#11717579C2311yescharacteristic
90C4⋊Dic101C2311yescharacteristic
91#13108053—21no—
92#13108053—21no—
93C22⋊Dic101—21no—
94C22⋊Dic101—21no—
95C22⋊Dic101—21no—
96#13108051—21no—
97#13108051—21no—
98#13108051—21no—
99#13108051—21no—
100#13108051—21no—
101C4×Dic101—21no—
102C4×Dic101—21no—
103C4×Dic101—21no—
104C4×Dic101D411yescharacteristic
105C4×Dic101D411yescharacteristic
106C4×Dic101C2311yescharacteristic
107C2×Q8×C101C2311yescharacteristic
108C2×D4×C101C2311yescharacteristic
109C2×D4×C101C2311yescharacteristic
110C22×C4×C101—21no—
111SD16×C101—22no—
112D8×C101—22no—
113C2×C8×C101C2311yescharacteristic
114C2×C8×C101C2311yescharacteristic
115C4⋊C4×C101C2311yescharacteristic
116C22⋊C4×C101—21no—
117C22⋊C4×C101—21no—
118C22⋊C4×C101—21no—
119C42×C101C2311yescharacteristic
120C23×C101—21no—
121C23×C101—21no—
122C22×D101—21no—
123C22×D101—42no—
124C22×D101—42no—
125C22×D101—42no—
126C22×D101C2×D411yescharacteristic
127Q8×C101—21no—
128Q8×C101—21no—
129D4×C101—21no—
130D4×C101—21no—
131D4×C101—21no—
132D4×C101C2×D412yes—
133C2×C4×C101—21no—
134C2×C4×C101—21no—
135C2×C4×C101—42no—
136C2×C4×C101C2×D411yescharacteristic
137C2×C4×C101C2×D411yescharacteristic
138C2×C4×C101C2411yescharacteristic
139C22⋊D101—24no—
140C22⋊D101—42no—
141C2×Dic101—21no—
142C2×Dic101—24no—
143C2×Dic101—21no—
144C2×Dic101—21no—
145C2×Dic101—42no—
146C2×Dic101C2×D411yescharacteristic
147C2×Dic101C2×D411yescharacteristic
148C2×Dic101C2×D411yescharacteristic
149D404—21no—
150D404—21no—
151C4×D101—41no—
152C4×D101—41no—
153C4×D101—42no—
154C4×D101C2×D412yes—
155C4×D101#4112yes—
156Dic202—21no—
157Dic202—21no—
158Dic202—21no—
159Dic202—42no—
160Dic202C2×D412yes—
161C8×C101—21no—
162C8×C101—21no—
163#11717579—21no—
164#11717579—21no—
165C22×C101—22no—
166C22×C101—41no—
167C22×C101—41no—
168C22×C101—41no—
169C22×C101—41no—
170C22×C101—41no—
171C22×C101C22×D411yescharacteristic
172C2×D101—21no—
173C2×D101—21no—
174C2×D101—41no—
175C2×D101—41no—
176C2×D101—41no—
177C2×D101—84no—
178C2×D101#12112yes—
179C4×C101—21no—
180C4×C101—21no—
181C4×C101—41no—
182C4×C101—41no—
183C4×C101C22×D411yescharacteristic
184C4×C101C2×#4111yescharacteristic
185Dic101—22no—
186Dic101—21no—
187Dic101—21no—
188Dic101—41no—
189Dic101—41no—
190Dic101#12112yes—
191C2×C101—42no—
192C2×C101—81no—
193C2×C101C2×#12111yescharacteristic
194C2×C101#57511yescharacteristic
195C2×C101#57511yescharacteristic
196D101—22no—
197D101—81no—
198#3040—1011nomaximal, Sylow
199C101#304011yescharacteristic, complemented, Sylow
200C2×#137—1011no—
201C2×AGL(1,ℤ/8)—1011no—
202C2×#124—1011no—
203C4×#41—1011no—
204#464—1011no—
205#464—1011no—
206#464—1011no—
207#464—1011no—
208#463—1011no—
209#463—1011no—
210#463—1011no—
211#463D10111yescharacteristic, complemented
212#423—1011no—
213#419—1011no—
214#418—1011no—
215C2×#41—1011no—
216C22×Q8—1011no—
217C22×D4—1011no—
218#137—2024no—
219AGL(1,ℤ/8)—2024no—
220C2×Q16—1011no—
221C2×Q16—1011no—
222C2×SD16—1011no—
223C2×SD16—1011no—
224C2×SD16—1011no—
225C2×SD16C2×D10111yescharacteristic, complemented
226C2×D8—1011no—
227C2×D8C2×D10111yescharacteristic
228C2×M16—1011no—
229C2×M16—1011no—
230#124—1011no—
231#124—1011no—
232#124—1011no—
233#124—2021no—
234#124—2021no—
235#124C2×D10111yescharacteristic
236C4×Q8—1011no—
237C4×Q8—1011no—
238C4×D4—1011no—
239C4×D4—2021no—
240C4×D4—2021no—
241C4×D4C2×D10111yescharacteristic, complemented
242#117—1011no—
243#117—2021no—
244C2×C22⋊C4—2021no—
245C2×C22⋊C4—2021no—
246C2×C42—1011no—
247C2×C42—2021no—
248C4⋊C8—1011no—
249C4⋊C8—1011no—
250C4⋊C8—1011no—
251C4⋊C8C2×D10111yescharacteristic, complemented
252#103—1011no—
253#103—1011no—
254#103—1011no—
255#103C2×D10111yescharacteristic, complemented
256#102—1011no—
257#102—1011no—
258#102—1011no—
259#102C2×D10111yescharacteristic
260C24—2021no—
261#41—1014no—
262#41—2022no—
263C2×Q8—1011no—
264C2×Q8—1011no—
265C2×Q8—2021no—
266C2×Q8—2022no—
267C2×Q8—2022no—
268C2×Q8C22×D10111yescharacteristic
269C2×D4—1011no—
270C2×D4—2022no—
271C2×D4—2022no—
272C2×D4—2021no—
273C2×D4—2021no—
274C2×D4C22×D10111yescharacteristic
275C2×D4C22×D10111yescharacteristic
276C22×C4—21no—
277C22×C4—1011no—
278C22×C4—1011no—
279C22×C4—1011no—
280C22×C4—2021no—
281C22×C4—2021no—
282C22×C4—2021no—
283Q16—2022no—
284Q16—2022no—
285SD16—22no—
286SD16—2022no—
287SD16—2022no—
288SD16—2022no—
289D8—22no—
290D8—2022no—
291M16—2022no—
292M16—2022no—
293C2×C8—1011no—
294C2×C8—1011no—
295C2×C8C22×D10111yescharacteristic
296C2×C8C22×D10111yescharacteristic
297C4⋊C4—1011no—
298C4⋊C4—2021no—
299C4⋊C4—2021no—
300C4⋊C4C22×D10111yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—2021no—
305C22⋊C4—2021no—
306C22⋊C4—2021no—
307C22⋊C4—2021no—
308C22⋊C4—2021no—
309C22⋊C4—2021no—
310C22⋊C4—2021no—
311C22⋊C4—2021no—
312C42—1011no—
313C42—1011no—
314C42—1011no—
315C42—2021no—
316C42—2021no—
317C42—2021no—
318C42C22×D10111yescharacteristic
319C23—21no—
320C23—21no—
321C23—1011no—
322C23—4042no—
323C23—4042no—
324C23—4042no—
325C23—2021no—
326Q8—21no—
327Q8—21no—
328Q8—1012no—
329Q8—4042no—
330Q8—2021no—
331Q8—2021no—
332Q8—2021no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—4042no—
337D4—2024no—
338D4—2021no—
339D4—2021no—
340D4D4×D10112yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—1011no—
345C2×C4—1012no—
346C2×C4—1012no—
347C2×C4—1011no—
348C2×C4—1011no—
349C2×C4—4041no—
350C2×C4—4041no—
351C2×C4—4042no—
352C2×C4—4042no—
353C2×C4—2024no—
354C2×C4—2021no—
355C2×C4—2021no—
356C2×C4—2021no—
357C2×C4D4×D10111yescharacteristic
358C2×C4D4×D10111yescharacteristic
359C2×C4C23×D10111yescharacteristic
360C8—21no—
361C8—21no—
362C8—2021no—
363C8—2021no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—8084no—
371C22—1012no—
372C22—4041no—
373C22—4041no—
374C22—4041no—
375C22—2021no—
376C22—2021no—
377C22C2×D4×D10111yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—1012no—
383C4—4041no—
384C4—4041no—
385C4—2022no—
386C4—2021no—
387C4—2021no—
388C4C2×D4×D10111yescharacteristic
389C4#41×D10111yescharacteristic
390C2—42no—
391C2—81no—
392C2—8081no—
393C2—2022no—
394C2#121×D10111yescharacteristic
395C2#2026597311yescharacteristic
396C2#2026597411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#463×C101C2
socleC22×C101C22×D4
prime radicalC2×#15692279C2
hypercenter#463D101
hypocenterC101#3040
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×#15692284C2semidirect
3. #124×D101C2semidirect
4. C4×#13108073C2semidirect
5. #20265402C2semidirect
6. #20265400C2semidirect
7. #20265398C2semidirect
8. #20265396C2semidirect
9. #20265394C2semidirect
10. #20265392C2semidirect
11. #20265390C2semidirect
12. #20265199C2semidirect
13. #20265184C2semidirect
14. #20265180C2semidirect
15. #463×C101C2semidirect
21. C2×Dic404C22semidirect
22. C2×#13108064C22semidirect
25. C2×#13108062C22semidirect
34. C4×Dic202C22semidirect
35. #15692220C22semidirect
36. D4×Dic101C22semidirect
45. C4⋊#11717579C22semidirect
46. #15692184C22semidirect
47. #15692184C22semidirect
48. #15692180C22semidirect
49. #15692179C22semidirect
50. #15692178C22semidirect
54. C2×SD16×C101C22semidirect
57. C4×D4×C101C22semidirect
58. C4⋊C8×C101C22semidirect
59. #103×C101C22semidirect
199. C101#3040semidirect
211. #463D101semidirect
225. C2×SD16C2×D101semidirect
241. C4×D4C2×D101semidirect
251. C4⋊C8C2×D101semidirect
255. #103C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922842
C2×#156922792
#124×D1012
C4×#131080732
#202654022
#202654002
#202653982
#202653962
#202653942
#202653922
#202653902
#202651992
#202651842
#202651802
#463×C1012
#3040101