Finite Group Databasebeta

← #27183163

Subgroups — #27183163

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#20265818C211yesmaximal, characteristic
3#20265538C211yesmaximal, characteristic, complemented
4#20265515C211yesmaximal, characteristic, complemented
5#20265503C211yesmaximal, characteristic, complemented
6#20265466C211yesmaximal, characteristic, complemented
7#20265409C211yesmaximal, characteristic, complemented
8#20265401C211yesmaximal, characteristic, complemented
9#20265338C211yesmaximal, characteristic, complemented
10#20265327C211yesmaximal, characteristic, complemented
11#20265280C211yesmaximal, characteristic, complemented
12#20265246C211yesmaximal, characteristic, complemented
13#20265183C211yesmaximal, characteristic, complemented
14#20265172C211yesmaximal, characteristic, complemented
15#470×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C22×D404C2211yescharacteristic
18C2×#13108061C2211yescharacteristic
19C2×#13108061C2211yescharacteristic
20C2×C8×D101C2211yescharacteristic, complemented
21#15692254C2211yescharacteristic, complemented
22#15692250—21no—
23#15692248—21no—
24#15692244C2211yescharacteristic
25#15692241—21no—
26#15692240—21no—
27#15692231C2211yescharacteristic, complemented
28#15692230—21no—
29#15692230C2211yescharacteristic
30#15692225—21no—
31Q8×Dic101C2211yescharacteristic, complemented
32#15692220—21no—
33#15692217C2211yescharacteristic
34C4×C22⋊D101C2211yescharacteristic
35#15692214C2211yescharacteristic
36#15692213C2211yescharacteristic, complemented
37C22⋊C4×D101—21no—
38#15692188C2211yescharacteristic, complemented
39#15692186C2211yescharacteristic, complemented
40#15692184C2211yescharacteristic, complemented
41#15692179C2211yescharacteristic, complemented
42#15692179C2211yescharacteristic, complemented
43#15692177C2211yescharacteristic
44#15692177C2211yescharacteristic
45#15692177C2211yescharacteristic
46#15692177C2211yescharacteristic
47#15692175C2211yescharacteristic
48#15692175C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic
50#15692167C2211yescharacteristic, complemented
51#15692165C2211yescharacteristic, complemented
52C2×Q16×C101C2211yescharacteristic, complemented
53C2×M16×C101C2211yescharacteristic
54C22⋊Q8×C101C2211yescharacteristic
55C22⋊Q8×C101C2211yescharacteristic
56#106×C101C2211yescharacteristic, complemented
57#103×C101C2211yescharacteristic, complemented
58#103×C101C2211yescharacteristic, complemented
59C23×D101—21no—
60#13108075—22no—
61#13108075—41no—
62C2×Dic202—21no—
63C2×D404—22no—
64C2×D404—21no—
65C2×D404—21no—
66C2×D404—21no—
67C2×D404—41no—
68C2×D404C2311yescharacteristic
69C2×D404C2311yescharacteristic
70C2×C22⋊D101D411yescharacteristic
71C2×C4×D101—21no—
72C2×C4×D101—21no—
73C2×C4×D101C2311yescharacteristic
74#13108061—22no—
75#13108061—41no—
76C2×#11717579C2311yescharacteristic
77C2×#11717579C2311yescharacteristic
78C8×D101—41no—
79C4⋊Dic101—21no—
80C4⋊Dic101C2311yescharacteristic
81C4⋊Dic101C2311yescharacteristic
82#13108053—21no—
83#13108053—21no—
84#13108053D411yescharacteristic
85C22⋊Dic101—21no—
86C22⋊Dic101D411yescharacteristic
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90#13108051—21no—
91#13108051—21no—
92#13108051—21no—
93#13108051D411yescharacteristic
94C4×Dic101—21no—
95C4×Dic101—21no—
96C4×Dic101—21no—
97C4×Dic101C2311yescharacteristic
98C2×Q8×C101C2311yescharacteristic
99C2×Q8×C101C2311yescharacteristic
100C22×C4×C101C2311yescharacteristic
101Q16×C101—41no—
102M16×C101—41no—
103C2×C8×C101C2311yescharacteristic
104C2×C8×C101C2311yescharacteristic
105C4⋊C4×C101—21no—
106C4⋊C4×C101—21no—
107C4⋊C4×C101C2311yescharacteristic
108C4⋊C4×C101C2311yescharacteristic
109C22⋊C4×C101—21no—
110C22⋊C4×C101—21no—
111C23×C101C2×D411yescharacteristic
112C22×D101—21no—
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—
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×C101C2×D411yescharacteristic
129C2×C4×C101C2411yescharacteristic
130C22⋊D101—41no—
131C2×Dic101—21no—
132C2×Dic101—21no—
133C2×Dic101C2×D411yescharacteristic
134C2×Dic101C2×D411yescharacteristic
135C2×Dic101C2×D411yescharacteristic
136D404—21no—
137D404—21no—
138D404—21no—
139D404—42no—
140D404—42no—
141D404—81no—
142D404#4112yes—
143C4×D101—21no—
144C4×D101—21no—
145C4×D101—42no—
146C4×D101—81no—
147Dic202—42no—
148C8×C101—21no—
149C8×C101—21no—
150#11717579—21no—
151#11717579—21no—
152C22×C101—21no—
153C22×C101—21no—
154C22×C101—21no—
155C22×C101C22×D411yescharacteristic
156C2×D101—21no—
157C2×D101—22no—
158C2×D101—21no—
159C2×D101—21no—
160C2×D101—41no—
161C2×D101—41no—
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×C101—41no—
171C4×C101C2×#4111yescharacteristic
172C4×C101H(5,𝔽2)11yescharacteristic
173Dic101—21no—
174Dic101—21no—
175Dic101—21no—
176Dic101—41no—
177Dic101—41no—
178C2×C101—41no—
179C2×C101#54611yescharacteristic
180C2×C101#57511yescharacteristic
181C2×C101#57711yescharacteristic
182D101—41no—
183D101—42no—
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#470D10111yescharacteristic, complemented
194#465—1011no—
195#464—1011no—
196#452—1011no—
197#450—1011no—
198#440—1011no—
199#435—1011no—
200#418—1011no—
201#416—1011no—
202C2×#41—1011no—
203C22×D4—1011no—
204C2×Q16C2×D10111yescharacteristic, complemented
205C2×SD16—1011no—
206C2×SD16—1011no—
207C2×M16C2×D10111yescharacteristic
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⋊Q8C2×D10111yescharacteristic
217C22⋊Q8C2×D10111yescharacteristic
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#106C2×D10111yescharacteristic, complemented
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#102—1011no—
241#102—1011no—
242C22⋊C8—1011no—
243C22⋊C8—1011no—
244#97—1011no—
245C24—2021no—
246#41—4041no—
247#41—2022no—
248C2×Q8—2021no—
249C2×Q8C22×D10111yescharacteristic
250C2×Q8C22×D10111yescharacteristic
251C2×D4—1011no—
252C2×D4—1011no—
253C2×D4—1011no—
254C2×D4—4041no—
255C2×D4—2021no—
256C2×D4—2022no—
257C2×D4—2021no—
258C2×D4—2021no—
259C22×C4—1011no—
260C22×C4—2021no—
261C22×C4—2021no—
262C22×C4C22×D10111yescharacteristic
263Q16—41no—
264SD16—4041no—
265SD16—2022no—
266M16—41no—
267C2×C8—1011no—
268C2×C8—1011no—
269C2×C8—4041no—
270C2×C8C22×D10111yescharacteristic
271C2×C8C22×D10111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—1011no—
275C4⋊C4—1011no—
276C4⋊C4—1011no—
277C4⋊C4—2021no—
278C4⋊C4—2021no—
279C4⋊C4—2021no—
280C4⋊C4C22×D10111yescharacteristic
281C4⋊C4C22×D10111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—1011no—
285C22⋊C4—1011no—
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—8081no—
298C23—1011no—
299C23—4042no—
300C23—4042no—
301C23—2021no—
302C23—2021no—
303C23D4×D10111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308Q8—4042no—
309D4—8081no—
310D4—1012no—
311D4—4042no—
312D4—4042no—
313D4—4041no—
314D4—2021no—
315D4—2021no—
316D4—2021no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—8081no—
324C2×C4—1011no—
325C2×C4—1011no—
326C2×C4—1011no—
327C2×C4—4042no—
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—21no—
341C22—8084no—
342C22—4041no—
343C22—4041no—
344C22—4041no—
345C22—4041no—
346C22—4041no—
347C22—2021no—
348C22—2022no—
349C22—2021no—
350C22—2021no—
351C22C2×D4×D10111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—4041no—
358C4—4041no—
359C4—2021no—
360C4—2021no—
361C4—2021no—
362C4#41×D10111yescharacteristic
363C4#1569230011yescharacteristic
364C2—41no—
365C2—8081no—
366C2—4042no—
367C2—4041no—
368C2#2026581611yescharacteristic
369C2#2026597611yescharacteristic
370C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#470×C101C2
socleC22×C101C22×D4
prime radical#20265818C2
hypercenter#470D101
hypocenterC101#3311
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #20265898C2semidirect
3. #20265538C2semidirect
4. #20265515C2semidirect
5. #20265503C2semidirect
6. #20265466C2semidirect
7. #20265409C2semidirect
8. #20265401C2semidirect
9. #20265338C2semidirect
10. #20265327C2semidirect
11. #20265280C2semidirect
12. #20265246C2semidirect
13. #20265183C2semidirect
14. #20265172C2semidirect
15. #470×C101C2semidirect
20. C2×C8×D101C22semidirect
21. #15692254C22semidirect
27. #15692231C22semidirect
31. Q8×Dic101C22semidirect
36. #15692213C22semidirect
38. #15692188C22semidirect
39. #15692186C22semidirect
40. #15692184C22semidirect
41. #15692179C22semidirect
42. #15692179C22semidirect
50. #15692167C22semidirect
51. #15692165C22semidirect
52. C2×Q16×C101C22semidirect
56. #106×C101C22semidirect
57. #103×C101C22semidirect
58. #103×C101C22semidirect
186. C101#3311semidirect
193. #470D101semidirect
204. C2×Q16C2×D101semidirect
230. #106C2×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
#202658982
#202658182
#202655382
#202655152
#202655032
#202654662
#202654092
#202654012
#202653382
#202653272
#202652802
#202652462
#202651832
#202651722
#470×C1012
#3311101