Finite Group Databasebeta

← #27181313

Subgroups — #27181313

Order 12928.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265796C211yesmaximal, characteristic, complemented
2#20265649C211yesmaximal, characteristic, complemented
3#20265517C211yesmaximal, characteristic, complemented
4#20265499C211yesmaximal, characteristic, complemented
5#20265467C211yesmaximal, characteristic, complemented
6#20265458C211yesmaximal, characteristic, complemented
7#20265389C211yesmaximal, characteristic, complemented
8#20265326C211yesmaximal, characteristic, complemented
9#20265309C211yesmaximal, characteristic
10#20265286C211yesmaximal, characteristic, complemented
11#20265278C211yesmaximal, characteristic, complemented
12#20265262C211yesmaximal, characteristic, complemented
13D8×Dic101C211yesmaximal, characteristic, complemented
14#20265237C211yesmaximal, characteristic, complemented
15#463×C101C211yesmaximal, characteristic, complemented
16C2×D4×D101C2211yescharacteristic
17C2×#13108061C2211yescharacteristic
18C2×#13108059C2211yescharacteristic
19#15692255C2211yescharacteristic, complemented
20#15692248—21no—
21#15692241—21no—
22#15692239—21no—
23#15692232—21no—
24#15692231C2211yescharacteristic
25#15692226C2211yescharacteristic
26Q8×Dic101C2211yescharacteristic, complemented
27C4×D404C2211yescharacteristic
28D4×Dic101C2211yescharacteristic
29D4×Dic101C2211yescharacteristic, complemented
30#15692217C2211yescharacteristic
31C4×C22⋊D101—21no—
32#15692215—21no—
33#15692213C2211yescharacteristic
34#15692212C2211yescharacteristic
35C2×#13108051—21no—
36C22⋊C4×D101—21no—
37#15692189C2211yescharacteristic, complemented
38#15692189C2211yescharacteristic, complemented
39#15692186C2211yescharacteristic, complemented
40#15692186C2211yescharacteristic, complemented
41C4⋊#11717579C2211yescharacteristic, complemented
42#15692179C2211yescharacteristic, complemented
43#15692177C2211yescharacteristic
44#15692177C2211yescharacteristic
45#15692176C2211yescharacteristic
46#15692176C2211yescharacteristic
47#15692176C2211yescharacteristic, complemented
48#15692167C2211yescharacteristic
49#15692167C2211yescharacteristic
50#15692165C2211yescharacteristic, complemented
51C8×Dic101C2211yescharacteristic, complemented
52C2×SD16×C101C2211yescharacteristic
53C2×D8×C101C2211yescharacteristic
54#124×C101C2211yescharacteristic, complemented
55C4×D4×C101C2211yescharacteristic
56C4⋊C8×C101C2211yescharacteristic, complemented
57#103×C101C2211yescharacteristic, complemented
58#102×C101C2211yescharacteristic, complemented
59C23×D101—21no—
60C2×Dic202—21no—
61C2×D404C2311yescharacteristic
62C2×C22⋊D101—21no—
63C2×C22⋊D101—21no—
64D4×D101—22no—
65D4×D101—22no—
66C22×Dic101—21no—
67C22×Dic101—21no—
68C2×C4×D101—21no—
69C2×C4×D101—21no—
70C2×C4×D101C2311yescharacteristic
71#13108061—22no—
72#13108059—22no—
73C2×#11717579C2311yescharacteristic
74C2×#11717579C2311yescharacteristic
75C4⋊Dic101—21no—
76C4⋊Dic101—21no—
77C4⋊Dic101C2311yescharacteristic
78C4⋊Dic101C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80#13108053—21no—
81#13108053D411yescharacteristic
82#13108053D411yescharacteristic
83C22⋊Dic101—21no—
84C22⋊Dic101—21no—
85C22⋊Dic101—21no—
86#13108051—21no—
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90#13108051D411yescharacteristic
91#13108051D411yescharacteristic
92C4×Dic101—21no—
93C4×Dic101—21no—
94C4×Dic101C2311yescharacteristic
95C2×Q8×C101C2311yescharacteristic
96C2×D4×C101C2311yescharacteristic
97C2×D4×C101C2311yescharacteristic
98C22×C4×C101—21no—
99SD16×C101—22no—
100D8×C101—22no—
101C2×C8×C101C2311yescharacteristic
102C2×C8×C101C2311yescharacteristic
103C4⋊C4×C101C2311yescharacteristic
104C22⋊C4×C101—21no—
105C22⋊C4×C101—21no—
106C22⋊C4×C101—21no—
107C42×C101C2311yescharacteristic
108C23×C101—21no—
109C23×C101—21no—
110C22×D101—21no—
111C22×D101—24no—
112C22×D101—81no—
113C22×D101—81no—
114C22×D101C2×D411yescharacteristic
115Q8×C101—21no—
116Q8×C101—21no—
117D4×C101—21no—
118D4×C101—21no—
119D4×C101—21no—
120D4×C101#4112yes—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—81no—
124C2×C4×C101C2×D411yescharacteristic
125C2×C4×C101C2×D411yescharacteristic
126C2×C4×C101C2411yescharacteristic
127C22⋊D101—24no—
128C22⋊D101—81no—
129C2×Dic101—21no—
130C2×Dic101—21no—
131C2×Dic101—21no—
132C2×Dic101—81no—
133C2×Dic101—81no—
134C2×Dic101C2×D411yescharacteristic
135C2×Dic101C2×D411yescharacteristic
136C2×Dic101C2×D411yescharacteristic
137D404—21no—
138D404#4112yes—
139C4×D101—21no—
140C4×D101—21no—
141C4×D101—81no—
142C4×D101—81no—
143Dic202—81no—
144C8×C101—21no—
145C8×C101—21no—
146#11717579—21no—
147#11717579—21no—
148C22×C101—22no—
149C22×C101—41no—
150C22×C101—41no—
151C22×C101—41no—
152C22×C101—41no—
153C22×C101—41no—
154C22×C101C22×D411yescharacteristic
155C2×D101—21no—
156C2×D101—22no—
157C2×D101—21no—
158C2×D101—21no—
159C2×D101—41no—
160C2×D101—41no—
161C2×D101—84no—
162C4×C101—21no—
163C4×C101—21no—
164C4×C101—41no—
165C4×C101—41no—
166C4×C101C2×#4111yescharacteristic
167C4×C101C2×#4111yescharacteristic
168Dic101—21no—
169Dic101—21no—
170Dic101—21no—
171Dic101—41no—
172Dic101—41no—
173Dic101—41no—
174C2×C101—42no—
175C2×C101—81no—
176C2×C101#52511yescharacteristic
177C2×C101#57611yescharacteristic
178C2×C101#57711yescharacteristic
179D101—41no—
180D101—42no—
181#3118—1011nomaximal, Sylow
182C101#311811yescharacteristic, complemented, Sylow
183#542—1011no—
184#518—1011no—
185#485—1011no—
186#482—1011no—
187#476—1011no—
188#474—1011no—
189#463—1011no—
190#463D10111yescharacteristic, complemented
191#450—1011no—
192#447—1011no—
193#442—1011no—
194#440—1011no—
195C4×SD16—1011no—
196C4×D8—1011no—
197#433—1011no—
198C22×D4—1011no—
199C2×SD16—1011no—
200C2×SD16C2×D10111yescharacteristic
201C2×D8—1011no—
202C2×D8C2×D10111yescharacteristic
203C4⋊Q8—1011no—
204#126—2021no—
205#124—2021no—
206#124C2×D10111yescharacteristic, complemented
207#123—2021no—
208C22⋊Q8—1011no—
209C22⋊Q8—2021no—
210#121—1011no—
211C4×Q8—1011no—
212C4×D4—1011no—
213C4×D4—1011no—
214C4×D4—1011no—
215C4×D4—1011no—
216C4×D4—2021no—
217C4×D4—2021no—
218C4×D4C2×D10111yescharacteristic
219#117—1011no—
220#117—1011no—
221C2×C22⋊C4—2021no—
222C2×C22⋊C4—2021no—
223#107—1011no—
224#107—1011no—
225#106—1011no—
226#106—1011no—
227C4⋊C8—1011no—
228C4⋊C8C2×D10111yescharacteristic, complemented
229#103—1011no—
230#103C2×D10111yescharacteristic, complemented
231#102—1011no—
232#102—1011no—
233#102—1011no—
234#102—1011no—
235#102—1011no—
236#102C2×D10111yescharacteristic, complemented
237C22⋊C8—1011no—
238C22⋊C8—1011no—
239#97—1011no—
240C4×C8—1011no—
241C24—2021no—
242C2×Q8—2021no—
243C2×Q8C22×D10111yescharacteristic
244C2×D4—1011no—
245C2×D4—2021no—
246C2×D4—2022no—
247C2×D4—2021no—
248C2×D4—2022no—
249C2×D4C22×D10111yescharacteristic
250C2×D4C22×D10111yescharacteristic
251C22×C4—21no—
252C22×C4—1011no—
253C22×C4—2021no—
254C22×C4—2021no—
255C22×C4—2021no—
256C22×C4—2021no—
257SD16—22no—
258SD16—2022no—
259D8—22no—
260D8—2022no—
261C2×C8—1011no—
262C2×C8—1011no—
263C2×C8C22×D10111yescharacteristic
264C2×C8C22×D10111yescharacteristic
265C4⋊C4—1011no—
266C4⋊C4—1011no—
267C4⋊C4—1011no—
268C4⋊C4—1011no—
269C4⋊C4—1011no—
270C4⋊C4—2021no—
271C4⋊C4—2021no—
272C4⋊C4—2021no—
273C4⋊C4C22×D10111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—1011no—
278C22⋊C4—1011no—
279C22⋊C4—2021no—
280C22⋊C4—2021no—
281C22⋊C4—2021no—
282C22⋊C4—2021no—
283C22⋊C4—2021no—
284C22⋊C4—2021no—
285C22⋊C4—2021no—
286C42—1011no—
287C42—2021no—
288C42—2021no—
289C42C22×D10111yescharacteristic
290C23—21no—
291C23—21no—
292C23—8081no—
293C23—8081no—
294C23—1011no—
295C23—2024no—
296C23—2021no—
297Q8—21no—
298Q8—21no—
299Q8—8081no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—8081no—
304D4—1012no—
305D4—2024no—
306D4—2021no—
307D4#1310807312yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—8081no—
312C2×C4—8081no—
313C2×C4—8081no—
314C2×C4—8081no—
315C2×C4—1011no—
316C2×C4—1011no—
317C2×C4—1011no—
318C2×C4—2021no—
319C2×C4—2021no—
320C2×C4—2021no—
321C2×C4—2021no—
322C2×C4—2021no—
323C2×C4D4×D10111yescharacteristic
324C2×C4D4×D10111yescharacteristic
325C2×C4C23×D10111yescharacteristic
326C8—21no—
327C8—21no—
328C8—2021no—
329C8—2021no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—8084no—
337C22—4041no—
338C22—4041no—
339C22—2021no—
340C22—2021no—
341C22—2022no—
342C22—2021no—
343C22C2×D4×D10111yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—4041no—
349C4—4041no—
350C4—4041no—
351C4—2021no—
352C4—2021no—
353C4—2021no—
354C4#41×D10111yescharacteristic
355C4C2×#1310807311yescharacteristic
356C2—42no—
357C2—81no—
358C2—4041no—
359C2—4042no—
360C2#2026568711yescharacteristic
361C2#2026597911yescharacteristic
362C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#463×C101C2
socleC22×C101C22×D4
prime radical#20265309C2
hypercenter#463D101
hypocenterC101#3118
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. #20265796C2semidirect
2. #20265649C2semidirect
3. #20265517C2semidirect
4. #20265499C2semidirect
5. #20265467C2semidirect
6. #20265458C2semidirect
7. #20265389C2semidirect
8. #20265326C2semidirect
10. #20265286C2semidirect
11. #20265278C2semidirect
12. #20265262C2semidirect
13. D8×Dic101C2semidirect
14. #20265237C2semidirect
15. #463×C101C2semidirect
19. #15692255C22semidirect
26. Q8×Dic101C22semidirect
29. D4×Dic101C22semidirect
37. #15692189C22semidirect
38. #15692189C22semidirect
39. #15692186C22semidirect
40. #15692186C22semidirect
41. C4⋊#11717579C22semidirect
42. #15692179C22semidirect
47. #15692176C22semidirect
50. #15692165C22semidirect
51. C8×Dic101C22semidirect
54. #124×C101C22semidirect
56. C4⋊C8×C101C22semidirect
57. #103×C101C22semidirect
58. #102×C101C22semidirect
182. C101#3118semidirect
190. #463D101semidirect
206. #124C2×D101semidirect
228. C4⋊C8C2×D101semidirect
230. #103C2×D101semidirect
236. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657962
#202656492
#202655172
#202654992
#202654672
#202654582
#202653892
#202653262
#202653092
#202652862
#202652782
#202652622
D8×Dic1012
#202652372
#463×C1012
#3118101