Finite Group Databasebeta

← #27181316

Subgroups — #27181316

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#20265798C211yesmaximal, characteristic, complemented
2#20265518C211yesmaximal, characteristic, complemented
3#20265501C211yesmaximal, characteristic, complemented
4#20265467C211yesmaximal, characteristic, complemented
5#20265457C211yesmaximal, characteristic, complemented
6#20265392C211yesmaximal, characteristic, complemented
7#20265392C211yesmaximal, characteristic, complemented
8#20265328C211yesmaximal, characteristic, complemented
9#20265311C211yesmaximal, characteristic
10#20265287C211yesmaximal, characteristic, complemented
11#20265277C211yesmaximal, characteristic, complemented
12C4×#13108062C211yesmaximal, characteristic, complemented
13C4×#13108059C211yesmaximal, characteristic, complemented
14#20265238C211yesmaximal, characteristic, complemented
15#518×C101C211yesmaximal, characteristic, complemented
16C2×#13108062C2211yescharacteristic
17C2×#13108062C2211yescharacteristic
18C2×#13108059C2211yescharacteristic
19C2×#13108059C2211yescharacteristic
20#15692255C2211yescharacteristic, complemented
21#15692250—21no—
22#15692243C2211yescharacteristic, complemented
23#15692241—21no—
24#15692237—21no—
25#15692234C2211yescharacteristic
26#15692233—21no—
27#15692229C2211yescharacteristic
28C4×Dic202C2211yescharacteristic, complemented
29C4×D404C2211yescharacteristic, complemented
30#15692188C2211yescharacteristic, complemented
31#15692188C2211yescharacteristic, complemented
32#15692186C2211yescharacteristic, complemented
33#15692186C2211yescharacteristic, complemented
34C4⋊#11717579C2211yescharacteristic, complemented
35C4⋊#11717579C2211yescharacteristic, complemented
36#15692178C2211yescharacteristic, complemented
37#15692178C2211yescharacteristic, complemented
38#15692176C2211yescharacteristic
39#15692176C2211yescharacteristic
40#15692176C2211yescharacteristic
41#15692176C2211yescharacteristic
42#15692175C2211yescharacteristic, complemented
43#15692175C2211yescharacteristic, complemented
44C22⋊#11717579C2211yescharacteristic
45C22⋊#11717579C2211yescharacteristic
46#15692166C2211yescharacteristic, complemented
47C4×#11717579C2211yescharacteristic, complemented
48C22×D4×C101C2211yescharacteristic
49C4×D4×C101—21no—
50C4×D4×C101—21no—
51C4×D4×C101C2211yescharacteristic
52C4×D4×C101C2211yescharacteristic
53C4×D4×C101C2211yescharacteristic
54C4×D4×C101C2211yescharacteristic
55#117×C101C2211yescharacteristic
56#117×C101C2211yescharacteristic
57C2×C22⋊C4×C101—21no—
58C2×C22⋊C4×C101—21no—
59C2×Dic202—21no—
60C2×Dic202C2311yescharacteristic
61C2×D404C2311yescharacteristic
62C2×C22⋊D101—21no—
63C2×C4×D101—21no—
64#13108062—22no—
65#13108062—22no—
66#13108059—22no—
67#13108059—22no—
68C2×#11717579C2311yescharacteristic
69C2×#11717579C2311yescharacteristic
70C2×#11717579C2311yescharacteristic
71C2×#11717579C2311yescharacteristic
72C4⋊Dic101—21no—
73C4⋊Dic101C2311yescharacteristic
74C4⋊Dic101C2311yescharacteristic
75#13108053—21no—
76#13108053—21no—
77C22⋊Dic101—21no—
78#13108051—21no—
79#13108051—21no—
80#13108051—21no—
81#13108051—21no—
82C4×Dic101—21no—
83C24×C101—21no—
84C2×D4×C101—22no—
85C2×D4×C101—22no—
86C2×D4×C101—21no—
87C2×D4×C101C2311yescharacteristic
88C2×D4×C101C2311yescharacteristic
89C22×C4×C101—21no—
90C22×C4×C101—21no—
91C22×C4×C101—21no—
92C22×C4×C101—21no—
93C22×C4×C101C2311yescharacteristic
94C4⋊C4×C101D411yescharacteristic
95C4⋊C4×C101D411yescharacteristic
96C4⋊C4×C101C2311yescharacteristic
97C4⋊C4×C101C2311yescharacteristic
98C22⋊C4×C101—21no—
99C22⋊C4×C101—21no—
100C22⋊C4×C101—21no—
101C22⋊C4×C101—21no—
102C22⋊C4×C101—21no—
103C22⋊C4×C101D411yescharacteristic
104C22⋊C4×C101D411yescharacteristic
105C42×C101—21no—
106C42×C101C2311yescharacteristic
107C42×C101C2311yescharacteristic
108C23×C101—24no—
109C23×C101—21no—
110C23×C101—21no—
111C23×C101—81no—
112C23×C101—81no—
113C23×C101C2×D411yescharacteristic
114C22×D101—21no—
115D4×C101—24no—
116D4×C101—21no—
117D4×C101—21no—
118D4×C101#4112yes—
119D4×C101#4112yes—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—81no—
125C2×C4×C101—81no—
126C2×C4×C101—81no—
127C2×C4×C101—81no—
128C2×C4×C101C2×D411yescharacteristic
129C2×C4×C101C2×D411yescharacteristic
130C2×C4×C101C2×D411yescharacteristic
131C2×C4×C101C2×D411yescharacteristic
132C2×C4×C101C2×D411yescharacteristic
133C2×C4×C101C2411yescharacteristic
134C22⋊D101—81no—
135C2×Dic101—21no—
136C2×Dic101—21no—
137C2×Dic101—21no—
138D404—21no—
139D404—21no—
140C4×D101—81no—
141Dic202—21no—
142Dic202—21no—
143Dic202—81no—
144#11717579—21no—
145#11717579—21no—
146#11717579—21no—
147#11717579—21no—
148C22×C101—21no—
149C22×C101—22no—
150C22×C101—22no—
151C22×C101—21no—
152C22×C101—21no—
153C22×C101—41no—
154C22×C101—41no—
155C22×C101—41no—
156C22×C101—41no—
157C22×C101—84no—
158C22×C101C22×D411yescharacteristic
159C2×D101—41no—
160C2×D101—41no—
161C2×D101—41no—
162C4×C101—21no—
163C4×C101—21no—
164C4×C101—21no—
165C4×C101—21no—
166C4×C101—21no—
167C4×C101—41no—
168C4×C101—41no—
169C4×C101C2×#4111yescharacteristic
170C4×C101C2×#4111yescharacteristic
171Dic101—41no—
172Dic101—41no—
173Dic101—41no—
174C2×C101—41no—
175C2×C101—42no—
176C2×C101—42no—
177C2×C101#52511yescharacteristic
178C2×C101#57611yescharacteristic
179C2×C101#57711yescharacteristic
180D101—81no—
181#3118—1011nomaximal, Sylow
182C101#311811yescharacteristic, complemented, Sylow
183#542—1011no—
184#518D10111yescharacteristic, complemented
185#485—1011no—
186#482—1011no—
187#476—1011no—
188#474—1011no—
189#463—1011no—
190#463—1011no—
191#450—1011no—
192#447—1011no—
193#442—1011no—
194#440—1011no—
195C4×SD16—1011no—
196C4×D8—1011no—
197#433—1011no—
198C22×D4C2×D10111yescharacteristic
199C2×SD16—1011no—
200C2×SD16—1011no—
201C2×D8—1011no—
202C2×D8—1011no—
203C4⋊Q8—1011no—
204#126—2021no—
205#124—1011no—
206#124—2021no—
207#123—2021no—
208C22⋊Q8—1011no—
209C22⋊Q8—2021no—
210#121—1011no—
211C4×Q8—1011no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—1011no—
215C4×D4C2×D10111yescharacteristic
216C4×D4C2×D10111yescharacteristic
217C4×D4C2×D10111yescharacteristic
218C4×D4C2×D10111yescharacteristic
219#117C2×D10111yescharacteristic
220#117C2×D10111yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223#107—1011no—
224#107—1011no—
225#106—1011no—
226#106—1011no—
227C4⋊C8—1011no—
228C4⋊C8—1011no—
229#103—1011no—
230#103—1011no—
231#102—1011no—
232#102—1011no—
233#102—1011no—
234#102—1011no—
235#102—1011no—
236#102—1011no—
237C22⋊C8—1011no—
238C22⋊C8—1011no—
239#97—1011no—
240C4×C8—1011no—
241C24—21no—
242C2×Q8—1011no—
243C2×Q8—2021no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—1011no—
248C2×D4—2021no—
249C2×D4C22×D10111yescharacteristic
250C2×D4C22×D10111yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—2021no—
256C22×C4C22×D10111yescharacteristic
257SD16—2022no—
258SD16—2022no—
259D8—2022no—
260D8—2022no—
261C2×C8—1011no—
262C2×C8—1011no—
263C2×C8—1011no—
264C2×C8—1011no—
265C4⋊C4—1011no—
266C4⋊C4—1011no—
267C4⋊C4—2021no—
268C4⋊C4—2021no—
269C4⋊C4—2021no—
270C4⋊C4C22⋊D10111yescharacteristic
271C4⋊C4C22⋊D10111yescharacteristic
272C4⋊C4C22×D10111yescharacteristic
273C4⋊C4C22×D10111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2021no—
280C22⋊C4—2021no—
281C22⋊C4—2021no—
282C22⋊C4—2021no—
283C22⋊C4—2021no—
284C22⋊C4C22⋊D10111yescharacteristic
285C22⋊C4C22⋊D10111yescharacteristic
286C42—21no—
287C42—2021no—
288C42C22×D10111yescharacteristic
289C42C22×D10111yescharacteristic
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—2021no—
296C23C2×C22⋊D10111yescharacteristic
297Q8—8081no—
298Q8—2021no—
299Q8—2021no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—8081no—
304D4—2021no—
305D4—2021no—
306D4#1310807412yes—
307D4#1310807412yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—8081no—
317C2×C4—2021no—
318C2×C4—2021no—
319C2×C4—2021no—
320C2×C4C2×C22⋊D10111yescharacteristic
321C2×C4C2×C22⋊D10111yescharacteristic
322C2×C4C2×C22⋊D10111yescharacteristic
323C2×C4C2×C22⋊D10111yescharacteristic
324C2×C4C2×C22⋊D10111yescharacteristic
325C2×C4C23×D10111yescharacteristic
326C8—2021no—
327C8—2021no—
328C8—2021no—
329C8—2021no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—22no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—4041no—
341C22—4041no—
342C22—4041no—
343C22C22×C22⋊D10111yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—4041no—
352C4—4041no—
353C4—4041no—
354C4C2×#1310807411yescharacteristic
355C4C2×#1310807411yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—8081no—
360C2#2026568811yescharacteristic
361C2#2026598011yescharacteristic
362C2#2026598811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#518×C101C2
socleC22×C101C22×D4
prime radical#20265311C2
hypercenter#518D101
hypocenterC101#3118
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #20265798C2semidirect
2. #20265518C2semidirect
3. #20265501C2semidirect
4. #20265467C2semidirect
5. #20265457C2semidirect
6. #20265392C2semidirect
7. #20265392C2semidirect
8. #20265328C2semidirect
10. #20265287C2semidirect
11. #20265277C2semidirect
12. C4×#13108062C2semidirect
13. C4×#13108059C2semidirect
14. #20265238C2semidirect
15. #518×C101C2semidirect
20. #15692255C22semidirect
22. #15692243C22semidirect
28. C4×Dic202C22semidirect
29. C4×D404C22semidirect
30. #15692188C22semidirect
31. #15692188C22semidirect
32. #15692186C22semidirect
33. #15692186C22semidirect
34. C4⋊#11717579C22semidirect
35. C4⋊#11717579C22semidirect
36. #15692178C22semidirect
37. #15692178C22semidirect
42. #15692175C22semidirect
43. #15692175C22semidirect
46. #15692166C22semidirect
47. C4×#11717579C22semidirect
182. C101#3118semidirect
184. #518D101semidirect

Maximal subgroups

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

subgroupindex
#202657982
#202655182
#202655012
#202654672
#202654572
#202653922
#202653282
#202653112
#202652872
#202652772
C4×#131080622
C4×#131080592
#202652382
#518×C1012
#3118101