Finite Group Databasebeta

← #27182968

Subgroups — #27182968

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#20265866C211yesmaximal, characteristic, complemented
2#20265815C211yesmaximal, characteristic
3#20265533C211yesmaximal, characteristic, complemented
4#20265506C211yesmaximal, characteristic, complemented
5#20265501C211yesmaximal, characteristic, complemented
6#20265479C211yesmaximal, characteristic, complemented
7#20265417C211yesmaximal, characteristic, complemented
8#20265406C211yesmaximal, characteristic, complemented
9#20265402C211yesmaximal, characteristic, complemented
10#20265339C211yesmaximal, characteristic, complemented
11#20265329C211yesmaximal, characteristic, complemented
12#20265268C211yesmaximal, characteristic, complemented
13C8×C22⋊D101C211yesmaximal, characteristic, complemented
14#102×D101C211yesmaximal, characteristic, complemented
15#416×C101C211yesmaximal, characteristic, complemented
16C2×D4×D101C2211yescharacteristic
17C2×#13108064C2211yescharacteristic, complemented
18C2×#13108063C2211yescharacteristic
19C2×#13108062C2211yescharacteristic
20C2×C8×D101C2211yescharacteristic
21#15692248—21no—
22#15692247—21no—
23#15692246C2211yescharacteristic, complemented
24#15692244C2211yescharacteristic
25#15692238—21no—
26#15692237—21no—
27#15692235C2211yescharacteristic
28#15692234C2211yescharacteristic, complemented
29#15692231C2211yescharacteristic
30#15692229C2211yescharacteristic
31#15692227—21no—
32#15692223—21no—
33#15692220C2211yescharacteristic, complemented
34D4×Dic101C2211yescharacteristic
35C4×C22⋊D101C2211yescharacteristic
36#15692212—21no—
37C4⋊C4×D101C2211yescharacteristic
38C2×#13108051—21no—
39#15692188C2211yescharacteristic, complemented
40#15692187C2211yescharacteristic, complemented
41#15692186C2211yescharacteristic, complemented
42#15692184C2211yescharacteristic, complemented
43#15692180C2211yescharacteristic, complemented
44#15692179C2211yescharacteristic, complemented
45#15692177C2211yescharacteristic
46#15692176C2211yescharacteristic
47#15692175C2211yescharacteristic
48#15692175C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic, complemented
50#15692167C2211yescharacteristic
51C8×Dic101C2211yescharacteristic, complemented
52C2×#41×C101C2211yescharacteristic
53C22×C8×C101C2211yescharacteristic, complemented
54#117×C101C2211yescharacteristic, complemented
55#103×C101C2211yescharacteristic, complemented
56#103×C101C2211yescharacteristic, complemented
57#102×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59C23×D101—21no—
60C2×Dic202C2311yescharacteristic
61C2×D404C2311yescharacteristic
62C2×C22⋊D101—21no—
63C2×C22⋊D101—21no—
64C2×C22⋊D101D411yescharacteristic
65D4×D101—22no—
66D4×D101—41no—
67C22×Dic101—21no—
68C2×C4×D101—21no—
69C2×C4×D101C2311yescharacteristic
70#13108064—41no—
71#13108063—41no—
72#13108062—22no—
73C2×#11717579C2311yescharacteristic
74C2×#11717579C2311yescharacteristic
75C8×D101—41no—
76C4⋊Dic101—21no—
77C4⋊Dic101—21no—
78C4⋊Dic101C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108053D411yescharacteristic
86C22⋊Dic101—21no—
87C22⋊Dic101—21no—
88C22⋊Dic101D411yescharacteristic
89#13108051—21no—
90#13108051—21no—
91#13108051—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051D411yescharacteristic
95C4×Dic101—21no—
96C4×Dic101C2311yescharacteristic
97#41×C101—22no—
98#41×C101—41no—
99C2×Q8×C101C2311yescharacteristic
100C2×D4×C101—21no—
101C2×D4×C101C2311yescharacteristic
102C22×C4×C101—21no—
103C22×C4×C101C2311yescharacteristic
104C2×C8×C101—41no—
105C2×C8×C101C2311yescharacteristic
106C2×C8×C101C2311yescharacteristic
107C4⋊C4×C101C2311yescharacteristic
108C4⋊C4×C101C2311yescharacteristic
109C22⋊C4×C101—21no—
110C42×C101—21no—
111C23×C101—21no—
112C23×C101C2×D411yescharacteristic
113C22×D101—21no—
114C22×D101—42no—
115C22×D101—42no—
116C22×D101—81no—
117C22×D101C2×D411yescharacteristic
118Q8×C101—21no—
119Q8×C101—21no—
120D4×C101—21no—
121D4×C101—42no—
122D4×C101#4112yes—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—42no—
129C2×C4×C101C2×D411yescharacteristic
130C2×C4×C101C2411yescharacteristic
131C22⋊D101—22no—
132C22⋊D101—42no—
133C22⋊D101—81no—
134C2×Dic101—21no—
135C2×Dic101—21no—
136C2×Dic101—21no—
137C2×Dic101—81no—
138C2×Dic101C2×D411yescharacteristic
139C2×Dic101C2×D411yescharacteristic
140C2×Dic101C2×D411yescharacteristic
141D404—21no—
142D404—21no—
143C4×D101—21no—
144C4×D101—21no—
145C4×D101—42no—
146Dic202—21no—
147Dic202—21no—
148C8×C101—21no—
149C8×C101—21no—
150#11717579—21no—
151#11717579—21no—
152C22×C101—22no—
153C22×C101—21no—
154C22×C101—21no—
155C22×C101—21no—
156C22×C101—41no—
157C22×C101—41no—
158C22×C101C22×D411yescharacteristic
159C2×D101—21no—
160C2×D101—21no—
161C2×D101—41no—
162C2×D101—41no—
163C2×D101—41no—
164C2×D101—84no—
165C2×D101#13512yes—
166C4×C101—21no—
167C4×C101—41no—
168C4×C101—41no—
169C4×C101—41no—
170C4×C101C2×#4111yescharacteristic
171C4×C101H(5,𝔽2)11yescharacteristic
172Dic101—21no—
173Dic101—21no—
174Dic101—21no—
175Dic101—41no—
176Dic101—41no—
177Dic101—41no—
178C2×C101—42no—
179C2×C101—41no—
180C2×C101#54611yescharacteristic
181C2×C101C2×#13511yescharacteristic
182C2×C101#57711yescharacteristic
183D101—22no—
184D101—81no—
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#464—1011no—
196#452—1011no—
197#450—1011no—
198C4×SD16—1011no—
199C8×D4—1011no—
200#416D10111yescharacteristic, complemented
201C2×#102—1011no—
202C2×#41C2×D10111yescharacteristic
203C22×D4—1011no—
204C2×Q16—1011no—
205C2×SD16—1011no—
206C2×SD16—1011no—
207C22×C8—1011no—
208C22×C8C2×D10111yescharacteristic, complemented
209#126—2021no—
210#125—1011no—
211#125—2021no—
212#124—1011no—
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×Q8—1011no—
222C4×D4—1011no—
223C4×D4—1011no—
224#117—2021no—
225#117C2×D10111yescharacteristic, complemented
226C2×C4⋊C4—1011no—
227C2×C22⋊C4—2021no—
228#107—1011no—
229#106—1011no—
230#106—1011no—
231C4⋊C8—1011no—
232#103—1011no—
233#103—1011no—
234#103C2×D10111yescharacteristic, complemented
235#103C2×D10111yescharacteristic, complemented
236#102—1011no—
237#102—1011no—
238#102—1011no—
239#102—1011no—
240#102C2×D10111yescharacteristic
241#102C2×D10111yescharacteristic
242C22⋊C8—1011no—
243C22⋊C8—1011no—
244C4×C8—1011no—
245C24—2021no—
246#41—22no—
247#41—41no—
248C2×Q8—1011no—
249C2×Q8C22×D10111yescharacteristic
250C2×D4—21no—
251C2×D4—1011no—
252C2×D4—1011no—
253C2×D4—4041no—
254C2×D4—2021no—
255C2×D4—2021no—
256C2×D4—2022no—
257C2×D4C22×D10111yescharacteristic
258C22×C4—21no—
259C22×C4—1011no—
260C22×C4—2021no—
261C22×C4—2021no—
262C22×C4C22×D10111yescharacteristic
263Q16—4041no—
264SD16—4041no—
265SD16—2022no—
266C2×C8—41no—
267C2×C8—1011no—
268C2×C8—1011no—
269C2×C8—4041no—
270C2×C8C22×D10111yescharacteristic
271C2×C8C22×D10111yescharacteristic
272C4⋊C4—1011no—
273C4⋊C4—1011no—
274C4⋊C4—1011no—
275C4⋊C4—2021no—
276C4⋊C4—2021no—
277C4⋊C4—2021no—
278C4⋊C4—2021no—
279C4⋊C4—2021no—
280C4⋊C4—2021no—
281C4⋊C4—2021no—
282C4⋊C4C22×D10111yescharacteristic
283C4⋊C4C22×D10111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—1011no—
286C22⋊C4—1011no—
287C22⋊C4—2021no—
288C22⋊C4—2021no—
289C22⋊C4—2021no—
290C22⋊C4—2021no—
291C22⋊C4—2021no—
292C22⋊C4—2021no—
293C22⋊C4—2021no—
294C42—21no—
295C42—1011no—
296C42—2021no—
297C23—21no—
298C23—8081no—
299C23—1011no—
300C23—4042no—
301C23—4042no—
302C23—2021no—
303C23D4×D10111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2021no—
307Q8—2021no—
308D4—21no—
309D4—42no—
310D4—8081no—
311D4—4042no—
312D4—2022no—
313D4—2021no—
314D4—2021no—
315D4#1310807412yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—8081no—
323C2×C4—1011no—
324C2×C4—1011no—
325C2×C4—1011no—
326C2×C4—4042no—
327C2×C4—2021no—
328C2×C4—2021no—
329C2×C4—2021no—
330C2×C4—2021no—
331C2×C4—2021no—
332C2×C4D4×D10111yescharacteristic
333C2×C4C23×D10111yescharacteristic
334C8—21no—
335C8—21no—
336C8—2021no—
337C8—2021no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—8084no—
345C22—1012no—
346C22—4041no—
347C22—4041no—
348C22—4041no—
349C22—2021no—
350C22—2021no—
351C22C2×D4×D10111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—4041no—
357C4—4041no—
358C4—4041no—
359C4—2021no—
360C4—2021no—
361C4—2021no—
362C4C2×#1310807411yescharacteristic
363C4#1569230011yescharacteristic
364C2—41no—
365C2—42no—
366C2—8081no—
367C2—2022no—
368C2#2026581711yescharacteristic
369C2#135×D10111yescharacteristic
370C2#2026598411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#416×C101C2
socleC22×C101C22×D4
prime radical#20265815C2
hypercenter#416D101
hypocenterC101#3296
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #20265866C2semidirect
3. #20265533C2semidirect
4. #20265506C2semidirect
5. #20265501C2semidirect
6. #20265479C2semidirect
7. #20265417C2semidirect
8. #20265406C2semidirect
9. #20265402C2semidirect
10. #20265339C2semidirect
11. #20265329C2semidirect
12. #20265268C2semidirect
13. C8×C22⋊D101C2semidirect
14. #102×D101C2semidirect
15. #416×C101C2semidirect
17. C2×#13108064C22semidirect
23. #15692246C22semidirect
28. #15692234C22semidirect
33. #15692220C22semidirect
39. #15692188C22semidirect
40. #15692187C22semidirect
41. #15692186C22semidirect
42. #15692184C22semidirect
43. #15692180C22semidirect
44. #15692179C22semidirect
49. C22⋊#11717579C22semidirect
51. C8×Dic101C22semidirect
53. C22×C8×C101C22semidirect
54. #117×C101C22semidirect
55. #103×C101C22semidirect
56. #103×C101C22semidirect
186. C101#3296semidirect
200. #416D101semidirect
208. C22×C8C2×D101semidirect
225. #117C2×D101semidirect
234. #103C2×D101semidirect
235. #103C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658662
#202658152
#202655332
#202655062
#202655012
#202654792
#202654172
#202654062
#202654022
#202653392
#202653292
#202652682
C8×C22⋊D1012
#102×D1012
#416×C1012
#3296101