Finite Group Databasebeta

← #27183172

Subgroups — #27183172

Order 12928.

402 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#20265898C211yesmaximal, characteristic, complemented
2#20265815C211yesmaximal, characteristic
3#20265538C211yesmaximal, characteristic, complemented
4#20265517C211yesmaximal, characteristic, complemented
5#20265502C211yesmaximal, characteristic, complemented
6#20265465C211yesmaximal, characteristic, complemented
7#20265432C211yesmaximal, characteristic, complemented
8#20265408C211yesmaximal, characteristic, complemented
9#20265401C211yesmaximal, characteristic, complemented
10#20265326C211yesmaximal, characteristic, complemented
11#20265278C211yesmaximal, characteristic, complemented
12#20265245C211yesmaximal, characteristic, complemented
13#20265180C211yesmaximal, characteristic, complemented
14#20265173C211yesmaximal, characteristic, complemented
15#452×C101C211yesmaximal, characteristic, complemented
16C2×D4×D101C2211yescharacteristic
17C2×#13108061C2211yescharacteristic
18C2×#13108057C2211yescharacteristic
19C22×#11717579C2211yescharacteristic, complemented
20#15692254C2211yescharacteristic, complemented
21#15692250—21no—
22#15692248—21no—
23#15692244C2211yescharacteristic
24#15692241—21no—
25#15692238—21no—
26#15692231C2211yescharacteristic
27#15692231C2211yescharacteristic
28#15692229C2211yescharacteristic
29#15692227—21no—
30#15692223—21no—
31Q8×Dic101C2211yescharacteristic, complemented
32#15692220—21no—
33D4×Dic101C2211yescharacteristic
34C4×C22⋊D101C2211yescharacteristic
35#15692214C2211yescharacteristic, complemented
36#15692213C2211yescharacteristic
37C2×#13108051—21no—
38#15692189C2211yescharacteristic, complemented
39#15692187C2211yescharacteristic, complemented
40#15692186C2211yescharacteristic, complemented
41#15692184C2211yescharacteristic, complemented
42#15692179C2211yescharacteristic, complemented
43#15692179C2211yescharacteristic, complemented
44#15692179C2211yescharacteristic, complemented
45#15692177C2211yescharacteristic
46#15692177C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692176C2211yescharacteristic
49#15692175C2211yescharacteristic
50#15692167C2211yescharacteristic
51#15692165C2211yescharacteristic, complemented
52C2×#41×C101C2211yescharacteristic
53C2×Q16×C101C2211yescharacteristic, complemented
54C2×SD16×C101C2211yescharacteristic
55C22⋊Q8×C101C2211yescharacteristic, complemented
56#103×C101C2211yescharacteristic, complemented
57#102×C101C2211yescharacteristic
58C22⋊C8×C101C2211yescharacteristic, complemented
59C23×D101—21no—
60C2×Dic202—21no—
61C2×D404C2311yescharacteristic
62C2×C22⋊D101—21no—
63C2×C22⋊D101—21no—
64C2×C22⋊D101D411yescharacteristic
65D4×D101—22no—
66D4×D101—41no—
67C22×Dic101—21no—
68C2×C4×D101C2311yescharacteristic
69#13108061—41no—
70#13108057—41no—
71C2×#11717579—41no—
72C2×#11717579C2311yescharacteristic
73C2×#11717579C2311yescharacteristic
74C4⋊Dic101—21no—
75C4⋊Dic101C2311yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77C4⋊Dic101C2311yescharacteristic
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053D411yescharacteristic
82C22⋊Dic101—21no—
83C22⋊Dic101—21no—
84C22⋊Dic101D411yescharacteristic
85#13108051—21no—
86#13108051—21no—
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90#13108051—21no—
91#13108051D411yescharacteristic
92C4×Dic101—21no—
93C4×Dic101—21no—
94C4×Dic101—21no—
95C4×Dic101C2311yescharacteristic
96#41×C101—22no—
97#41×C101—41no—
98C2×Q8×C101C2311yescharacteristic
99C2×Q8×C101C2311yescharacteristic
100C2×D4×C101—21no—
101C2×D4×C101C2311yescharacteristic
102C22×C4×C101—21no—
103C22×C4×C101C2311yescharacteristic
104Q16×C101—41no—
105SD16×C101—22no—
106C2×C8×C101C2311yescharacteristic
107C2×C8×C101C2311yescharacteristic
108C4⋊C4×C101—21no—
109C4⋊C4×C101C2311yescharacteristic
110C22⋊C4×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—
120Q8×C101—21no—
121Q8×C101—21no—
122D4×C101—21no—
123D4×C101—42no—
124D4×C101#4112yes—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—21no—
129C2×C4×C101—21no—
130C2×C4×C101—42no—
131C2×C4×C101C2×D411yescharacteristic
132C2×C4×C101C2411yescharacteristic
133C22⋊D101—41no—
134C22⋊D101—42no—
135C22⋊D101—81no—
136C2×Dic101—21no—
137C2×Dic101—21no—
138C2×Dic101—21no—
139C2×Dic101—81no—
140C2×Dic101C2×D411yescharacteristic
141C2×Dic101C2×D411yescharacteristic
142C2×Dic101C2×D411yescharacteristic
143D404—21no—
144D404—21no—
145C4×D101—21no—
146C4×D101—21no—
147Dic202—42no—
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—21no—
162C2×D101—41no—
163C2×D101—41no—
164C2×D101—41no—
165C2×D101—84no—
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×C101#57511yescharacteristic
182C2×C101#57711yescharacteristic
183D101—41no—
184D101—81no—
185#3311—1011nomaximal, Sylow
186C101#331111yescharacteristic, complemented, Sylow
187#556—1011no—
188#546—1011no—
189#489—1011no—
190#485—1011no—
191#482—1011no—
192#476—1011no—
193#470—1011no—
194#465—1011no—
195#464—1011no—
196#452D10111yescharacteristic, complemented
197#450—1011no—
198#440—1011no—
199#435—1011no—
200#418—1011no—
201#416—1011no—
202C2×#41C2×D10111yescharacteristic
203C22×D4—1011no—
204C2×Q16C2×D10111yescharacteristic, complemented
205C2×SD16—1011no—
206C2×SD16C2×D10111yescharacteristic
207C2×M16—1011no—
208C22×C8—1011no—
209C4⋊Q8—1011no—
210#126—2021no—
211#126—2021no—
212#124—1011no—
213#124—2021no—
214#123—2021no—
215C22⋊Q8—1011no—
216C22⋊Q8—1011no—
217C22⋊Q8C2×D10111yescharacteristic, complemented
218#121—1011no—
219#121—2021no—
220#120—2021no—
221C4×Q8—1011no—
222C4×Q8—2021no—
223C4×D4—1011no—
224C4×D4—1011no—
225#117—1011no—
226#117—1011no—
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#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⋊C8C2×D10111yescharacteristic, complemented
244#97—1011no—
245C24—2021no—
246#41—22no—
247#41—41no—
248C2×Q8—2021no—
249C2×Q8C22×D10111yescharacteristic
250C2×Q8C22×D10111yescharacteristic
251C2×D4—21no—
252C2×D4—1011no—
253C2×D4—1011no—
254C2×D4—4041no—
255C2×D4—2021no—
256C2×D4—2021no—
257C2×D4—2022no—
258C2×D4C22×D10111yescharacteristic
259C22×C4—21no—
260C22×C4—1011no—
261C22×C4—2021no—
262C22×C4C22×D10111yescharacteristic
263Q16—41no—
264SD16—22no—
265SD16—4041no—
266M16—4041no—
267C2×C8—1011no—
268C2×C8—1011no—
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⋊C4C22×D10111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—1011no—
284C22⋊C4—1011no—
285C22⋊C4—2021no—
286C22⋊C4—2021no—
287C22⋊C4—2021no—
288C22⋊C4—2021no—
289C22⋊C4—2021no—
290C22⋊C4—2021no—
291C22⋊C4—2021no—
292C22⋊C4—2021no—
293C42—1011no—
294C42—2021no—
295C42—2021no—
296C42—2021no—
297C23—21no—
298C23—8081no—
299C23—1011no—
300C23—4042no—
301C23—4042no—
302C23—2021no—
303C23D4×D10111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308Q8—4042no—
309D4—21no—
310D4—42no—
311D4—8081no—
312D4—4042no—
313D4—4041no—
314D4—2021no—
315D4—2021no—
316D4#1310807312yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—8081no—
324C2×C4—1011no—
325C2×C4—1011no—
326C2×C4—1011no—
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—4041no—
346C22—4041no—
347C22—4041no—
348C22—2021no—
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×#1310807311yescharacteristic
363C4#1569230011yescharacteristic
364C2—41no—
365C2—42no—
366C2—8081no—
367C2—4041no—
368C2#2026581411yescharacteristic
369C2#2026597411yescharacteristic
370C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#452×C101C2
socleC22×C101C22×D4
prime radical#20265815C2
hypercenter#452D101
hypocenterC101#3311
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #20265898C2semidirect
3. #20265538C2semidirect
4. #20265517C2semidirect
5. #20265502C2semidirect
6. #20265465C2semidirect
7. #20265432C2semidirect
8. #20265408C2semidirect
9. #20265401C2semidirect
10. #20265326C2semidirect
11. #20265278C2semidirect
12. #20265245C2semidirect
13. #20265180C2semidirect
14. #20265173C2semidirect
15. #452×C101C2semidirect
19. C22×#11717579C22semidirect
20. #15692254C22semidirect
31. Q8×Dic101C22semidirect
35. #15692214C22semidirect
38. #15692189C22semidirect
39. #15692187C22semidirect
40. #15692186C22semidirect
41. #15692184C22semidirect
42. #15692179C22semidirect
43. #15692179C22semidirect
44. #15692179C22semidirect
51. #15692165C22semidirect
53. C2×Q16×C101C22semidirect
55. C22⋊Q8×C101C22semidirect
56. #103×C101C22semidirect
58. C22⋊C8×C101C22semidirect
186. C101#3311semidirect
196. #452D101semidirect
204. C2×Q16C2×D101semidirect
217. C22⋊Q8C2×D101semidirect
235. #103C2×D101semidirect
243. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658982
#202658152
#202655382
#202655172
#202655022
#202654652
#202654322
#202654082
#202654012
#202653262
#202652782
#202652452
#202651802
#202651732
#452×C1012
#3311101