Finite Group Databasebeta

← #27183100

Subgroups — #27183100

Order 12928.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265832C211yesmaximal, characteristic, complemented
2#20265820C211yesmaximal, characteristic, complemented
3#20265536C211yesmaximal, characteristic, complemented
4#20265506C211yesmaximal, characteristic, complemented
5#20265494C211yesmaximal, characteristic, complemented
6#20265468C211yesmaximal, characteristic, complemented
7#20265427C211yesmaximal, characteristic, complemented
8#20265403C211yesmaximal, characteristic, complemented
9#20265321C211yesmaximal, characteristic, complemented
10#20265309C211yesmaximal, characteristic, complemented
11#20265287C211yesmaximal, characteristic, complemented
12#20265247C211yesmaximal, characteristic, complemented
13#20265177C211yesmaximal, characteristic, complemented
14#102×D101C211yesmaximal, characteristic, complemented
15#463×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×#13108074—21no—
18C2×D4×D101C2211yescharacteristic
19C2×#13108059C2211yescharacteristic
20C2×#13108057C2211yescharacteristic, complemented
21C2×C8×D101C2211yescharacteristic
22#15692255C2211yescharacteristic, complemented
23#15692237—21no—
24#15692237—21no—
25#15692237—21no—
26#15692234—21no—
27#15692231C2211yescharacteristic
28#15692228—21no—
29#15692226C2211yescharacteristic
30#15692226C2211yescharacteristic
31#15692226C2211yescharacteristic, complemented
32#15692225—21no—
33C4×D404C2211yescharacteristic
34C4×D404C2211yescharacteristic
35C4⋊C4×D101C2211yescharacteristic
36C4⋊C4×D101C2211yescharacteristic, complemented
37C22⋊C4×D101—21no—
38#15692187C2211yescharacteristic, complemented
39#15692187C2211yescharacteristic, complemented
40#15692186C2211yescharacteristic, complemented
41#15692179C2211yescharacteristic, complemented
42#15692179C2211yescharacteristic, complemented
43#15692177C2211yescharacteristic
44#15692177C2211yescharacteristic
45#15692176C2211yescharacteristic
46#15692176C2211yescharacteristic
47#15692176C2211yescharacteristic, complemented
48#15692175C2211yescharacteristic
49#15692167C2211yescharacteristic
50#15692167C2211yescharacteristic, complemented
51#15692166C2211yescharacteristic, complemented
52C2×SD16×C101C2211yescharacteristic, complemented
53C2×D8×C101C2211yescharacteristic
54#124×C101C2211yescharacteristic
55C4×D4×C101C2211yescharacteristic, complemented
56C4⋊C8×C101C2211yescharacteristic, complemented
57#103×C101C2211yescharacteristic, complemented
58#102×C101C2211yescharacteristic
59C23×D101—21no—
60#13108075—22no—
61#13108075—41no—
62#13108074—44no—
63C2×Dic202—21no—
64C2×D404—21no—
65C2×D404D411yescharacteristic
66C2×D404C2311yescharacteristic
67C2×C22⋊D101—21no—
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C2×C22⋊D101—21no—
71D4×D101—22no—
72D4×D101—41no—
73C2×C4×D101—21no—
74C2×C4×D101—21no—
75C2×C4×D101—21no—
76C2×C4×D101—21no—
77C2×C4×D101C2311yescharacteristic
78C2×C4×D101C2311yescharacteristic
79#13108059—22no—
80#13108057—41no—
81C2×#11717579C2311yescharacteristic
82C2×#11717579C2311yescharacteristic
83C8×D101—41no—
84C4⋊Dic101—21no—
85C4⋊Dic101D411yescharacteristic
86C4⋊Dic101C2311yescharacteristic
87C4⋊Dic101C2311yescharacteristic
88C4⋊Dic101C2311yescharacteristic
89#13108053—21no—
90#13108053—21no—
91#13108053—21no—
92C22⋊Dic101—21no—
93C22⋊Dic101—21no—
94C22⋊Dic101—21no—
95#13108051—21no—
96#13108051—21no—
97#13108051D411yescharacteristic
98#13108051D411yescharacteristic
99C2×Q8×C101C2311yescharacteristic
100C2×D4×C101C2311yescharacteristic
101C2×D4×C101C2311yescharacteristic
102C22×C4×C101—21no—
103SD16×C101—41no—
104D8×C101—41no—
105C2×C8×C101C2311yescharacteristic
106C2×C8×C101C2311yescharacteristic
107C4⋊C4×C101C2311yescharacteristic
108C22⋊C4×C101—21no—
109C22⋊C4×C101—21no—
110C22⋊C4×C101—21no—
111C42×C101C2311yescharacteristic
112C23×C101—21no—
113C23×C101—21no—
114C22×D101—21no—
115C22×D101—42no—
116C22×D101—42no—
117C22×D101—81no—
118C22×D101C2×D411yescharacteristic
119C22×D101C2×D411yescharacteristic
120Q8×C101—21no—
121Q8×C101—21no—
122D4×C101—21no—
123D4×C101—21no—
124D4×C101—21no—
125D4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—42no—
129C2×C4×C101C2×D411yescharacteristic
130C2×C4×C101C2×D411yescharacteristic
131C2×C4×C101C2411yescharacteristic
132C22⋊D101—42no—
133C22⋊D101—42no—
134C22⋊D101—42no—
135C22⋊D101—81no—
136C2×Dic101—21no—
137C2×Dic101—21no—
138C2×Dic101—21no—
139C2×Dic101C2×D411yescharacteristic
140C2×Dic101C2×D411yescharacteristic
141D404—22no—
142D404—21no—
143D404—42no—
144D404#4112yescomplemented
145C4×D101—21no—
146C4×D101—21no—
147C4×D101—21no—
148C4×D101—21no—
149C4×D101—42no—
150C4×D101—42no—
151C4×D101—42no—
152C4×D101—81no—
153Dic202—42no—
154C8×C101—21no—
155C8×C101—21no—
156#11717579—21no—
157#11717579—21no—
158C22×C101—41no—
159C22×C101—41no—
160C22×C101—41no—
161C22×C101—41no—
162C22×C101—41no—
163C22×C101—41no—
164C22×C101C22×D411yescharacteristic
165C2×D101—21no—
166C2×D101—21no—
167C2×D101—21no—
168C2×D101—22no—
169C2×D101—21no—
170C2×D101—21no—
171C2×D101—41no—
172C2×D101—41no—
173C2×D101—84no—
174C2×D101AGL(1,ℤ/8)12yes—
175C4×C101—21no—
176C4×C101—21no—
177C4×C101—41no—
178C4×C101—41no—
179C4×C101C2×#4111yescharacteristic
180C4×C101H(5,𝔽2)11yescharacteristic
181Dic101—21no—
182Dic101—21no—
183Dic101—41no—
184Dic101—41no—
185Dic101—41no—
186C2×C101—81no—
187C2×C101—81no—
188C2×C101#54611yescharacteristic
189C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C101#57711yescharacteristic
191D101—22no—
192D101—41no—
193D101—42no—
194#3306—1011nomaximal, Sylow
195C101#330611yescharacteristic, complemented, Sylow
196#547—1011no—
197#546—1011no—
198#489—1011no—
199#483—1011no—
200#481—1011no—
201#476—1011no—
202#469—1011no—
203#465—1011no—
204#463D10111yescharacteristic, complemented
205#449—1011no—
206#447—1011no—
207#442—1011no—
208#435—1011no—
209#417—1011no—
210C2×#102—1011no—
211C2×#41—1011no—
212C2×#41—2021no—
213C22×D4—1011no—
214C2×SD16C2×D10111yescharacteristic, complemented
215C2×D8—1011no—
216C2×D8C2×D10111yescharacteristic
217C2×M16—1011no—
218C22×C8—1011no—
219C4⋊Q8—1011no—
220#124C2×D10111yescharacteristic
221#123—2021no—
222#123—2021no—
223#123—2021no—
224C22⋊Q8—1011no—
225C22⋊Q8—2021no—
226#121—1011no—
227#121—1011no—
228#121—1011no—
229#121—2021no—
230#120—2021no—
231C4×D4—1011no—
232C4×D4—1011no—
233C4×D4C2×D10111yescharacteristic, complemented
234C2×C4⋊C4—1011no—
235C2×C4⋊C4—1011no—
236C2×C22⋊C4—2021no—
237#106—1011no—
238#106—1011no—
239#106—1011no—
240C4⋊C8C2×D10111yescharacteristic, complemented
241#103—1011no—
242#103—1011no—
243#103C2×D10111yescharacteristic, complemented
244#102—1011no—
245#102—1011no—
246#102—1011no—
247#102—1011no—
248#102—1011no—
249#102—1011no—
250#102C2×D10111yescharacteristic
251C22⋊C8—1011no—
252C22⋊C8—1011no—
253#97—1011no—
254C24—2021no—
255#41—4044no—
256#41—4041no—
257#41—2022no—
258C2×Q8—2021no—
259C2×Q8C22×D10111yescharacteristic
260C2×D4—1011no—
261C2×D4—1011no—
262C2×D4—4041no—
263C2×D4—2021no—
264C2×D4—2021no—
265C2×D4—2022no—
266C2×D4—2021no—
267C2×D4—2021no—
268C2×D4—2021no—
269C2×D4C22×D10111yescharacteristic
270C2×D4C22×D10111yescharacteristic
271C22×C4—21no—
272C22×C4—1011no—
273C22×C4—1011no—
274C22×C4—2021no—
275C22×C4—2021no—
276C22×C4—2021no—
277C22×C4—2021no—
278SD16—41no—
279D8—41no—
280D8—2022no—
281M16—4041no—
282C2×C8—1011no—
283C2×C8—1011no—
284C2×C8—4041no—
285C2×C8C22×D10111yescharacteristic
286C2×C8C22×D10111yescharacteristic
287C4⋊C4—1011no—
288C4⋊C4—1011no—
289C4⋊C4—1011no—
290C4⋊C4—1011no—
291C4⋊C4—2021no—
292C4⋊C4—2021no—
293C4⋊C4—2021no—
294C4⋊C4—2021no—
295C4⋊C4C22×D10111yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—1011no—
300C22⋊C4—1011no—
301C22⋊C4—2021no—
302C22⋊C4—2021no—
303C22⋊C4—2021no—
304C22⋊C4—2021no—
305C22⋊C4—2021no—
306C42C22×D10111yescharacteristic
307C23—21no—
308C23—21no—
309C23—8081no—
310C23—1011no—
311C23—1011no—
312C23—4042no—
313C23—4042no—
314C23—2021no—
315Q8—21no—
316Q8—21no—
317Q8—4042no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—8081no—
323D4—1012no—
324D4—4042no—
325D4—4042no—
326D4—4042no—
327D4—4042no—
328D4—2022no—
329D4—2021no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—8081no—
334C2×C4—1011no—
335C2×C4—1011no—
336C2×C4—4042no—
337C2×C4—4042no—
338C2×C4—4042no—
339C2×C4—2021no—
340C2×C4—2021no—
341C2×C4—2021no—
342C2×C4—2021no—
343C2×C4—2021no—
344C2×C4—2021no—
345C2×C4—2021no—
346C2×C4D4×D10111yescharacteristic
347C2×C4D4×D10111yescharacteristic
348C2×C4C23×D10111yescharacteristic
349C8—21no—
350C8—21no—
351C8—2021no—
352C8—2021no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—8084no—
360C22—1012no—
361C22—4041no—
362C22—4041no—
363C22—2021no—
364C22—2022no—
365C22—2021no—
366C22—2021no—
367C22—2021no—
368C22—2021no—
369C22C2×D4×D10111yescharacteristic
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—4041no—
375C4—4041no—
376C4—4041no—
377C4—2021no—
378C4—2021no—
379C4#41×D10111yescharacteristic
380C4#1569230011yescharacteristic
381C2—81no—
382C2—81no—
383C2—4042no—
384C2—4041no—
385C2—2022no—
386C2#2026582411yescharacteristic
387C2AGL(1,ℤ/8)×D10111yescharacteristic
388C2#2026598411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#463×C101C2
socleC22×C101C22×D4
hypercenter#463D101
hypocenterC101#3306
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #20265832C2semidirect
2. #20265820C2semidirect
3. #20265536C2semidirect
4. #20265506C2semidirect
5. #20265494C2semidirect
6. #20265468C2semidirect
7. #20265427C2semidirect
8. #20265403C2semidirect
9. #20265321C2semidirect
10. #20265309C2semidirect
11. #20265287C2semidirect
12. #20265247C2semidirect
13. #20265177C2semidirect
14. #102×D101C2semidirect
15. #463×C101C2semidirect
20. C2×#13108057C22semidirect
22. #15692255C22semidirect
31. #15692226C22semidirect
36. C4⋊C4×D101C22semidirect
38. #15692187C22semidirect
39. #15692187C22semidirect
40. #15692186C22semidirect
41. #15692179C22semidirect
42. #15692179C22semidirect
47. #15692176C22semidirect
50. #15692167C22semidirect
51. #15692166C22semidirect
52. C2×SD16×C101C22semidirect
55. C4×D4×C101C22semidirect
56. C4⋊C8×C101C22semidirect
57. #103×C101C22semidirect
144. D404#41semidirect
195. C101#3306semidirect
204. #463D101semidirect
214. C2×SD16C2×D101semidirect
233. C4×D4C2×D101semidirect
240. C4⋊C8C2×D101semidirect
243. #103C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658322
#202658202
#202655362
#202655062
#202654942
#202654682
#202654272
#202654032
#202653212
#202653092
#202652872
#202652472
#202651772
#102×D1012
#463×C1012
#3306101