Finite Group Databasebeta

← #27182974

Subgroups — #27182974

Order 12928.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265868C211yesmaximal, characteristic, complemented
2#20265820C211yesmaximal, characteristic
3#20265531C211yesmaximal, characteristic, complemented
4#20265506C211yesmaximal, characteristic, complemented
5#20265499C211yesmaximal, characteristic, complemented
6#20265478C211yesmaximal, characteristic, complemented
7#20265415C211yesmaximal, characteristic, complemented
8#20265403C211yesmaximal, characteristic, complemented
9#20265338C211yesmaximal, characteristic, complemented
10#20265326C211yesmaximal, characteristic, complemented
11C4×#13108061C211yesmaximal, characteristic, complemented
12#20265242C211yesmaximal, characteristic, complemented
13#20265172C211yesmaximal, characteristic, complemented
14#102×D101C211yesmaximal, characteristic, complemented
15#464×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×D4×D101C2211yescharacteristic
18C2×#13108061C2211yescharacteristic
19C2×C8×D101C2211yescharacteristic
20C2×C8×D101C2211yescharacteristic, complemented
21#15692249—21no—
22#15692247C2211yescharacteristic, complemented
23#15692245—21no—
24#15692240—21no—
25#15692237—21no—
26#15692231C2211yescharacteristic
27#15692231C2211yescharacteristic
28#15692231C2211yescharacteristic, complemented
29#15692228—21no—
30#15692226C2211yescharacteristic
31#15692225—21no—
32C4×D404C2211yescharacteristic
33C4×D404C2211yescharacteristic
34#15692213C2211yescharacteristic, complemented
35#15692212—21no—
36C4⋊C4×D101C2211yescharacteristic
37C22⋊C4×D101—21no—
38#15692189C2211yescharacteristic, complemented
39#15692187C2211yescharacteristic, complemented
40#15692186C2211yescharacteristic, complemented
41#15692179C2211yescharacteristic, complemented
42#15692179C2211yescharacteristic, complemented
43#15692179C2211yescharacteristic, complemented
44#15692177C2211yescharacteristic
45#15692177C2211yescharacteristic
46#15692176C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692175C2211yescharacteristic
49#15692167C2211yescharacteristic
50#15692167C2211yescharacteristic, complemented
51C4×#11717579C2211yescharacteristic, complemented
52C2×Q16×C101C2211yescharacteristic, complemented
53C2×SD16×C101C2211yescharacteristic
54#124×C101C2211yescharacteristic
55C4×Q8×C101C2211yescharacteristic, complemented
56C4⋊C8×C101C2211yescharacteristic, complemented
57#103×C101C2211yescharacteristic, complemented
58#102×C101C2211yescharacteristic
59C23×D101—21no—
60#13108075—22no—
61#13108075—41no—
62C2×D404—21no—
63C2×D404D411yescharacteristic
64C2×D404C2311yescharacteristic
65C2×C22⋊D101—21no—
66C2×C22⋊D101—21no—
67D4×D101—22no—
68D4×D101—41no—
69C2×C4×D101—21no—
70C2×C4×D101—21no—
71C2×C4×D101—21no—
72C2×C4×D101C2311yescharacteristic
73C2×C4×D101C2311yescharacteristic
74#13108061—22no—
75C2×#11717579C2311yescharacteristic
76C2×#11717579C2311yescharacteristic
77C8×D101—41no—
78C8×D101—41no—
79C4⋊Dic101D411yescharacteristic
80C4⋊Dic101C2311yescharacteristic
81C4⋊Dic101C2311yescharacteristic
82C4⋊Dic101C2311yescharacteristic
83#13108053—21no—
84#13108053—21no—
85#13108053—21no—
86#13108053—21no—
87#13108053—21no—
88#13108053—21no—
89C22⋊Dic101—21no—
90#13108051—21no—
91#13108051—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051—21no—
95#13108051D411yescharacteristic
96#13108051D411yescharacteristic
97C4×Dic101—21no—
98C2×Q8×C101C2311yescharacteristic
99C2×Q8×C101C2311yescharacteristic
100C2×D4×C101C2311yescharacteristic
101Q16×C101—41no—
102SD16×C101—41no—
103C2×C8×C101C2311yescharacteristic
104C2×C8×C101C2311yescharacteristic
105C4⋊C4×C101—21no—
106C4⋊C4×C101C2311yescharacteristic
107C22⋊C4×C101—21no—
108C22⋊C4×C101—21no—
109C42×C101—21no—
110C42×C101C2311yescharacteristic
111C23×C101—21no—
112C22×D101—21no—
113C22×D101—42no—
114C22×D101—42no—
115C22×D101—81no—
116C22×D101C2×D411yescharacteristic
117C22×D101C2×D411yescharacteristic
118Q8×C101—21no—
119Q8×C101—21no—
120Q8×C101—21no—
121Q8×C101—21no—
122D4×C101—21no—
123D4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101C2×D411yescharacteristic
128C2×C4×C101C2×D411yescharacteristic
129C2×C4×C101C2411yescharacteristic
130C22⋊D101—42no—
131C22⋊D101—81no—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101C2×D411yescharacteristic
136C2×Dic101C2×D411yescharacteristic
137D404—22no—
138D404—21no—
139D404—42no—
140D404#4112yes—
141C4×D101—21no—
142C4×D101—21no—
143C4×D101—21no—
144C4×D101—21no—
145C4×D101—42no—
146C4×D101—42no—
147C4×D101—81no—
148C8×C101—21no—
149C8×C101—21no—
150#11717579—21no—
151#11717579—21no—
152C22×C101—41no—
153C22×C101—41no—
154C22×C101—41no—
155C22×C101C22×D411yescharacteristic
156C2×D101—21no—
157C2×D101—21no—
158C2×D101—21no—
159C2×D101—22no—
160C2×D101—21no—
161C2×D101—21no—
162C2×D101—41no—
163C2×D101—41no—
164C2×D101—84no—
165C2×D101#13512yes—
166C4×C101—21no—
167C4×C101—21no—
168C4×C101—41no—
169C4×C101—41no—
170C4×C101—41no—
171C4×C101C2×#4111yescharacteristic
172C4×C101H(5,𝔽2)11yescharacteristic
173Dic101—21no—
174Dic101—21no—
175Dic101—41no—
176Dic101—41no—
177Dic101—41no—
178C2×C101—81no—
179C2×C101#54611yescharacteristic
180C2×C101C2×#13511yescharacteristic
181C2×C101#57711yescharacteristic
182D101—22no—
183D101—41no—
184D101—42no—
185#3296—1011nomaximal, Sylow
186C101#329611yescharacteristic, complemented, Sylow
187#552—1011no—
188#546—1011no—
189#488—1011no—
190#483—1011no—
191#482—1011no—
192#478—1011no—
193#467—1011no—
194#465—1011no—
195#464D10111yescharacteristic, complemented
196#452—1011no—
197#450—1011no—
198C4×SD16—1011no—
199C8×D4—1011no—
200#416—1011no—
201C2×#102—1011no—
202C2×#41—1011no—
203C22×D4—1011no—
204C2×Q16C2×D10111yescharacteristic, complemented
205C2×SD16—1011no—
206C2×SD16C2×D10111yescharacteristic
207C22×C8—1011no—
208C22×C8—1011no—
209#126—2021no—
210#125—1011no—
211#125—2021no—
212#124C2×D10111yescharacteristic
213#123—2021no—
214#123—2021no—
215C22⋊Q8—1011no—
216C22⋊Q8—1011no—
217C22⋊Q8—1011no—
218#121—1011no—
219#121—2021no—
220#120—2021no—
221C4×Q8C2×D10111yescharacteristic, complemented
222C4×D4—1011no—
223C4×D4—1011no—
224#117—1011no—
225#117—2021no—
226C2×C4⋊C4—1011no—
227C2×C22⋊C4—2021no—
228#107—1011no—
229#106—1011no—
230#106—1011no—
231C4⋊C8C2×D10111yescharacteristic, complemented
232#103—1011no—
233#103—1011no—
234#103—1011no—
235#103C2×D10111yescharacteristic, complemented
236#102—1011no—
237#102—1011no—
238#102—1011no—
239#102—1011no—
240#102—1011no—
241#102C2×D10111yescharacteristic
242C22⋊C8—1011no—
243C22⋊C8—1011no—
244C4×C8—1011no—
245C24—2021no—
246#41—4041no—
247#41—2022no—
248C2×Q8C22×D10111yescharacteristic
249C2×Q8C22×D10111yescharacteristic
250C2×D4—1011no—
251C2×D4—1011no—
252C2×D4—4041no—
253C2×D4—2021no—
254C2×D4—2021no—
255C2×D4—2022no—
256C2×D4—2021no—
257C2×D4C22×D10111yescharacteristic
258C22×C4—1011no—
259C22×C4—1011no—
260C22×C4—2021no—
261C22×C4—2021no—
262C22×C4—2021no—
263Q16—41no—
264SD16—41no—
265SD16—2022no—
266C2×C8—1011no—
267C2×C8—1011no—
268C2×C8—4041no—
269C2×C8—4041no—
270C2×C8C22×D10111yescharacteristic
271C2×C8C22×D10111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—1011no—
274C4⋊C4—1011no—
275C4⋊C4—1011no—
276C4⋊C4—1011no—
277C4⋊C4—2021no—
278C4⋊C4—2021no—
279C4⋊C4—2021no—
280C4⋊C4—2021no—
281C4⋊C4—2021no—
282C4⋊C4—2021no—
283C4⋊C4C22×D10111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—1011no—
287C22⋊C4—1011no—
288C22⋊C4—2021no—
289C22⋊C4—2021no—
290C22⋊C4—2021no—
291C22⋊C4—2021no—
292C22⋊C4—2021no—
293C22⋊C4—2021no—
294C42—21no—
295C42—2021no—
296C42C22×D10111yescharacteristic
297C23—21no—
298C23—8081no—
299C23—1011no—
300C23—1011no—
301C23—4042no—
302C23—4042no—
303C23—2021no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—21no—
309D4—21no—
310D4—8081no—
311D4—1012no—
312D4—4042no—
313D4—4042no—
314D4—2022no—
315D4—2021no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—8081no—
320C2×C4—1011no—
321C2×C4—1011no—
322C2×C4—4042no—
323C2×C4—4042no—
324C2×C4—2021no—
325C2×C4—2021no—
326C2×C4—2021no—
327C2×C4—2021no—
328C2×C4—2021no—
329C2×C4—2021no—
330C2×C4—2021no—
331C2×C4D4×D10111yescharacteristic
332C2×C4D4×D10111yescharacteristic
333C2×C4C23×D10111yescharacteristic
334C8—21no—
335C8—21no—
336C8—2021no—
337C8—2021no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—8084no—
342C22—1012no—
343C22—4041no—
344C22—4041no—
345C22—2021no—
346C22—2022no—
347C22—2021no—
348C22—2021no—
349C22—2021no—
350C22—2021no—
351C22C2×D4×D10111yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—4041no—
358C4—4041no—
359C4—4041no—
360C4—2021no—
361C4—2021no—
362C4#41×D10111yescharacteristic
363C4#1569230011yescharacteristic
364C2—81no—
365C2—4042no—
366C2—4041no—
367C2—2022no—
368C2#2026582411yescharacteristic
369C2#135×D10111yescharacteristic
370C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#464×C101C2
socleC22×C101C22×D4
prime radical#20265820C2
hypercenter#464D101
hypocenterC101#3296
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #20265868C2semidirect
3. #20265531C2semidirect
4. #20265506C2semidirect
5. #20265499C2semidirect
6. #20265478C2semidirect
7. #20265415C2semidirect
8. #20265403C2semidirect
9. #20265338C2semidirect
10. #20265326C2semidirect
11. C4×#13108061C2semidirect
12. #20265242C2semidirect
13. #20265172C2semidirect
14. #102×D101C2semidirect
15. #464×C101C2semidirect
20. C2×C8×D101C22semidirect
22. #15692247C22semidirect
28. #15692231C22semidirect
34. #15692213C22semidirect
38. #15692189C22semidirect
39. #15692187C22semidirect
40. #15692186C22semidirect
41. #15692179C22semidirect
42. #15692179C22semidirect
43. #15692179C22semidirect
50. #15692167C22semidirect
51. C4×#11717579C22semidirect
52. C2×Q16×C101C22semidirect
55. C4×Q8×C101C22semidirect
56. C4⋊C8×C101C22semidirect
57. #103×C101C22semidirect
186. C101#3296semidirect
195. #464D101semidirect
204. C2×Q16C2×D101semidirect
221. C4×Q8C2×D101semidirect
231. C4⋊C8C2×D101semidirect
235. #103C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658682
#202658202
#202655312
#202655062
#202654992
#202654782
#202654152
#202654032
#202653382
#202653262
C4×#131080612
#202652422
#202651722
#102×D1012
#464×C1012
#3296101