Finite Group Databasebeta

← #27182961

Subgroups — #27182961

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#20265823C211yesmaximal, characteristic
2#20265534C211yesmaximal, characteristic, complemented
3#20265507C211yesmaximal, characteristic, complemented
4#20265501C211yesmaximal, characteristic, complemented
5#20265474C211yesmaximal, characteristic, complemented
6#20265416C211yesmaximal, characteristic, complemented
7#20265408C211yesmaximal, characteristic, complemented
8#20265399C211yesmaximal, characteristic, complemented
9#20265337C211yesmaximal, characteristic, complemented
10#20265327C211yesmaximal, characteristic, complemented
11C4×#13108061C211yesmaximal, characteristic, complemented
12D4×#11717579C211yesmaximal, characteristic, complemented
13#20265171C211yesmaximal, characteristic, complemented
14C2×#15692175C211yesmaximal, characteristic, complemented
15#552×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C22×D404C2211yescharacteristic
18C2×#13108064C2211yescharacteristic, complemented
19C2×#13108061C2211yescharacteristic
20C2×#13108061C2211yescharacteristic
21C22×#11717579C2211yescharacteristic
22C22×#11717579C2211yescharacteristic, complemented
23#15692243C2211yescharacteristic
24#15692240—21no—
25#15692234C2211yescharacteristic
26#15692229C2211yescharacteristic
27#15692228—21no—
28#15692223—21no—
29C4×D404C2211yescharacteristic
30C22⋊C4×D101—21no—
31#15692188C2211yescharacteristic, complemented
32#15692186C2211yescharacteristic, complemented
33#15692186C2211yescharacteristic, complemented
34C4⋊#11717579C2211yescharacteristic, complemented
35#15692179C2211yescharacteristic, complemented
36#15692178C2211yescharacteristic, complemented
37#15692178C2211yescharacteristic, complemented
38#15692178C2211yescharacteristic, complemented
39#15692175C2211yescharacteristic
40#15692175C2211yescharacteristic
41#15692175C2211yescharacteristic
42#15692175C2211yescharacteristic
43#15692175C2211yescharacteristic
44#15692175C2211yescharacteristic
45C22⋊#11717579C2211yescharacteristic
46C22⋊#11717579C2211yescharacteristic, complemented
47C4×#11717579C2211yescharacteristic, complemented
48#126×C101—21no—
49#125×C101—21no—
50#125×C101C2211yescharacteristic, complemented
51#123×C101—21no—
52C22⋊Q8×C101C2211yescharacteristic
53C22⋊Q8×C101C2211yescharacteristic, complemented
54C4×Q8×C101C2211yescharacteristic, complemented
55C4×D4×C101C2211yescharacteristic
56#117×C101—21no—
57#117×C101C2211yescharacteristic, complemented
58C2×C4⋊C4×C101C2211yescharacteristic
59C23×D101—21no—
60#13108074—22no—
61#13108074—41no—
62C2×Dic202C2311yescharacteristic
63C2×D404—22no—
64C2×D404—21no—
65C2×D404—41no—
66C2×D404C2311yescharacteristic
67C2×D404C2311yescharacteristic
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C2×C4×D101—21no—
71C2×C4×D101—21no—
72#13108064—41no—
73#13108061—22no—
74#13108061—41no—
75C2×#11717579—41no—
76C2×#11717579—41no—
77C2×#11717579C2311yescharacteristic
78C2×#11717579C2311yescharacteristic
79C2×#11717579C2311yescharacteristic
80C2×#11717579C2311yescharacteristic
81C4⋊Dic101C2311yescharacteristic
82#13108053—21no—
83C22⋊Dic101—21no—
84#13108051—21no—
85#13108051—21no—
86#13108051—21no—
87#13108051—21no—
88C2×Q8×C101C2311yescharacteristic
89C2×D4×C101D411yescharacteristic
90C22×C4×C101—21no—
91C22×C4×C101C2311yescharacteristic
92C22×C4×C101C2311yescharacteristic
93C4⋊C4×C101—21no—
94C4⋊C4×C101—21no—
95C4⋊C4×C101—21no—
96C4⋊C4×C101—21no—
97C4⋊C4×C101—21no—
98C4⋊C4×C101—21no—
99C4⋊C4×C101D411yescharacteristic
100C4⋊C4×C101C2311yescharacteristic
101C4⋊C4×C101C2311yescharacteristic
102C4⋊C4×C101C2311yescharacteristic
103C22⋊C4×C101—21no—
104C22⋊C4×C101—21no—
105C22⋊C4×C101—21no—
106C22⋊C4×C101D411yescharacteristic
107C22⋊C4×C101D411yescharacteristic
108C42×C101—21no—
109C42×C101—21no—
110C42×C101C2311yescharacteristic
111C23×C101C2×D411yescharacteristic
112C23×C101C2×D411yescharacteristic
113C22×D101—21no—
114C22×D101—21no—
115C22×D101—42no—
116C22×D101—42no—
117C22×D101—81no—
118Q8×C101—21no—
119Q8×C101—21no—
120D4×C101—22no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—21no—
129C2×C4×C101—42no—
130C2×C4×C101C2×D411yescharacteristic
131C2×C4×C101C2×D411yescharacteristic
132C2×C4×C101C2×D411yescharacteristic
133C2×C4×C101C2×D411yescharacteristic
134C2×C4×C101C2411yescharacteristic
135C22⋊D101—42no—
136C22⋊D101—81no—
137C2×Dic101—21no—
138C2×Dic101—21no—
139D404—21no—
140D404—21no—
141D404—21no—
142D404—42no—
143D404#4112yes—
144C4×D101—42no—
145C4×D101—81no—
146Dic202—21no—
147Dic202—21no—
148#11717579—21no—
149#11717579—21no—
150#11717579—21no—
151#11717579—21no—
152C22×C101—21no—
153C22×C101—21no—
154C22×C101—21no—
155C22×C101—21no—
156C22×C101—21no—
157C22×C101#13512yes—
158C22×C101C22×D411yescharacteristic
159C2×D101—22no—
160C2×D101—41no—
161C2×D101—41no—
162C2×D101—41no—
163C2×D101—41no—
164C2×D101—41no—
165C2×D101—84no—
166C4×C101—21no—
167C4×C101—21no—
168C4×C101—21no—
169C4×C101—21no—
170C4×C101—41no—
171C4×C101—41no—
172C4×C101—41no—
173C4×C101—41no—
174C4×C101C2×#4111yescharacteristic
175C4×C101H(5,𝔽2)11yescharacteristic
176Dic101—41no—
177Dic101—41no—
178C2×C101—22no—
179C2×C101—41no—
180C2×C101#54611yescharacteristic
181C2×C101C2×#13511yescharacteristic
182C2×C101#57711yescharacteristic
183D101—42no—
184D101—81no—
185#3296—1011nomaximal, Sylow
186C101#329611yescharacteristic, complemented, Sylow
187#552D10111yescharacteristic, complemented
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#416—1011no—
201C2×#102—1011no—
202C2×#41—1011no—
203C22×D4—1011no—
204C2×Q16—1011no—
205C2×SD16—1011no—
206C2×SD16—1011no—
207C22×C8—1011no—
208C22×C8—1011no—
209#126—21no—
210#125—21no—
211#125C2×D10111yescharacteristic, complemented
212#124—1011no—
213#123—21no—
214#123—2021no—
215C22⋊Q8—1011no—
216C22⋊Q8C2×D10111yescharacteristic
217C22⋊Q8C2×D10111yescharacteristic, complemented
218#121—1011no—
219#121—2021no—
220#120—2021no—
221C4×Q8C2×D10111yescharacteristic, complemented
222C4×D4—1011no—
223C4×D4C2×D10111yescharacteristic
224#117—21no—
225#117C2×D10111yescharacteristic, complemented
226C2×C4⋊C4C2×D10111yescharacteristic
227C2×C22⋊C4—2021no—
228#107—1011no—
229#106—1011no—
230#106—1011no—
231C4⋊C8—1011no—
232#103—1011no—
233#103—1011no—
234#103—1011no—
235#103—1011no—
236#102—1011no—
237#102—1011no—
238#102—1011no—
239#102—1011no—
240#102—1011no—
241#102—1011no—
242C22⋊C8—1011no—
243C22⋊C8—1011no—
244C4×C8—1011no—
245C24—2021no—
246#41—4041no—
247#41—2022no—
248C2×Q8—1011no—
249C2×Q8C22×D10111yescharacteristic
250C2×D4—1011no—
251C2×D4—1011no—
252C2×D4—4041no—
253C2×D4—2022no—
254C2×D4—2021no—
255C2×D4—2021no—
256C2×D4—2021no—
257C2×D4C22⋊D10111yescharacteristic
258C22×C4—21no—
259C22×C4—2021no—
260C22×C4—2021no—
261C22×C4C22×D10111yescharacteristic
262C22×C4C22×D10111yescharacteristic
263Q16—4041no—
264SD16—4041no—
265SD16—2022no—
266C2×C8—1011no—
267C2×C8—1011no—
268C2×C8—1011no—
269C2×C8—1011no—
270C2×C8—4041no—
271C2×C8—4041no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—1011no—
279C4⋊C4—2021no—
280C4⋊C4C22⋊D10111yescharacteristic
281C4⋊C4C22×D10111yescharacteristic
282C4⋊C4C22×D10111yescharacteristic
283C4⋊C4C22×D10111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—2021no—
288C22⋊C4—2021no—
289C22⋊C4—2021no—
290C22⋊C4—2021no—
291C22⋊C4—2021no—
292C22⋊C4C22⋊D10111yescharacteristic
293C22⋊C4C22⋊D10111yescharacteristic
294C42—21no—
295C42—21no—
296C42C22×D10111yescharacteristic
297C23—8081no—
298C23—4042no—
299C23—4042no—
300C23—2021no—
301C23—2021no—
302C23C2×C22⋊D10111yescharacteristic
303C23C2×C22⋊D10111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2021no—
307Q8—2021no—
308D4—22no—
309D4—8081no—
310D4—1012no—
311D4—4042no—
312D4—4042no—
313D4—2021no—
314D4—2021no—
315D4—2021no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—8081no—
326C2×C4—4042no—
327C2×C4—2021no—
328C2×C4—2021no—
329C2×C4C2×C22⋊D10111yescharacteristic
330C2×C4C2×C22⋊D10111yescharacteristic
331C2×C4C2×C22⋊D10111yescharacteristic
332C2×C4C2×C22⋊D10111yescharacteristic
333C2×C4C23×D10111yescharacteristic
334C8—2021no—
335C8—2021no—
336C8—2021no—
337C8—2021no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—8084no—
344C22—4041no—
345C22—4041no—
346C22—4041no—
347C22—4041no—
348C22—4041no—
349C22—2022no—
350C22#1569227512yes—
351C22C22×C22⋊D10111yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—4041no—
361C4—4041no—
362C4#41×D10111yescharacteristic
363C4#1569230111yescharacteristic
364C2—22no—
365C2—41no—
366C2—8081no—
367C2—4042no—
368C2#2026581511yescharacteristic
369C2C2×#1569227511yescharacteristic
370C2#2026598511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#552×C101C2
socleC22×C101C22×D4
prime radical#20265823C2
hypercenter#552D101
hypocenterC101#3296
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #20265534C2semidirect
3. #20265507C2semidirect
4. #20265501C2semidirect
5. #20265474C2semidirect
6. #20265416C2semidirect
7. #20265408C2semidirect
8. #20265399C2semidirect
9. #20265337C2semidirect
10. #20265327C2semidirect
11. C4×#13108061C2semidirect
12. D4×#11717579C2semidirect
13. #20265171C2semidirect
14. C2×#15692175C2semidirect
15. #552×C101C2semidirect
18. C2×#13108064C22semidirect
22. C22×#11717579C22semidirect
31. #15692188C22semidirect
32. #15692186C22semidirect
33. #15692186C22semidirect
34. C4⋊#11717579C22semidirect
35. #15692179C22semidirect
36. #15692178C22semidirect
37. #15692178C22semidirect
38. #15692178C22semidirect
46. C22⋊#11717579C22semidirect
47. C4×#11717579C22semidirect
50. #125×C101C22semidirect
53. C22⋊Q8×C101C22semidirect
54. C4×Q8×C101C22semidirect
57. #117×C101C22semidirect
186. C101#3296semidirect
187. #552D101semidirect
211. #125C2×D101semidirect
217. C22⋊Q8C2×D101semidirect
221. C4×Q8C2×D101semidirect
225. #117C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658232
#202655342
#202655072
#202655012
#202654742
#202654162
#202654082
#202653992
#202653372
#202653272
C4×#131080612
D4×#117175792
#202651712
C2×#156921752
#552×C1012
#3296101