Finite Group Databasebeta

← #27181310

Subgroups — #27181310

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#20265651C211yesmaximal, characteristic, complemented
2#20265515C211yesmaximal, characteristic, complemented
3#20265498C211yesmaximal, characteristic, complemented
4#20265464C211yesmaximal, characteristic, complemented
5#20265454C211yesmaximal, characteristic, complemented
6#20265389C211yesmaximal, characteristic, complemented
7#20265389C211yesmaximal, characteristic, complemented
8#20265327C211yesmaximal, characteristic, complemented
9#20265310C211yesmaximal, characteristic
10#20265287C211yesmaximal, characteristic, complemented
11#20265275C211yesmaximal, characteristic, complemented
12C4×#13108061C211yesmaximal, characteristic, complemented
13C4×#13108059C211yesmaximal, characteristic, complemented
14#20265238C211yesmaximal, characteristic, complemented
15#542×C101C211yesmaximal, characteristic, complemented
16C22×D404C2211yescharacteristic
17C2×#13108061C2211yescharacteristic
18C2×#13108061C2211yescharacteristic
19C2×#13108059C2211yescharacteristic
20C2×#13108059C2211yescharacteristic
21C4×D404C2211yescharacteristic
22C4×D404C2211yescharacteristic
23C4×D404C2211yescharacteristic
24C4×D404C2211yescharacteristic
25#15692217—21no—
26#15692217—21no—
27#15692214C2211yescharacteristic
28C22⋊C4×D101—21no—
29C22⋊C4×D101—21no—
30#15692188C2211yescharacteristic, complemented
31#15692188C2211yescharacteristic, complemented
32#15692186C2211yescharacteristic, complemented
33#15692186C2211yescharacteristic, complemented
34C4⋊#11717579C2211yescharacteristic, complemented
35C4⋊#11717579C2211yescharacteristic, complemented
36#15692179C2211yescharacteristic, complemented
37#15692179C2211yescharacteristic, complemented
38#15692176C2211yescharacteristic, complemented
39#15692176C2211yescharacteristic, complemented
40#15692175C2211yescharacteristic
41#15692175C2211yescharacteristic
42#15692175C2211yescharacteristic
43#15692175C2211yescharacteristic
44C22⋊#11717579C2211yescharacteristic
45C22⋊#11717579C2211yescharacteristic
46#15692166C2211yescharacteristic, complemented
47C4×#11717579C2211yescharacteristic, complemented
48C4⋊Q8×C101C2211yescharacteristic, complemented
49#126×C101—21no—
50#124×C101—21no—
51#124×C101C2211yescharacteristic, complemented
52#123×C101—21no—
53C22⋊Q8×C101—21no—
54C22⋊Q8×C101C2211yescharacteristic
55#121×C101C2211yescharacteristic
56C4×Q8×C101C2211yescharacteristic, complemented
57C4×D4×C101C2211yescharacteristic, complemented
58#117×C101C2211yescharacteristic
59C23×D101—21no—
60C2×D404—21no—
61C2×D404—22no—
62C2×D404—22no—
63C2×D404C2311yescharacteristic
64C2×D404C2311yescharacteristic
65C2×C4×D101—21no—
66C2×C4×D101—21no—
67C2×C4×D101—21no—
68C2×C4×D101—21no—
69#13108061—22no—
70#13108061—22no—
71#13108059—22no—
72#13108059—22no—
73C2×#11717579C2311yescharacteristic
74C2×#11717579C2311yescharacteristic
75C2×#11717579C2311yescharacteristic
76C2×#11717579C2311yescharacteristic
77C4⋊Dic101C2311yescharacteristic
78C4⋊Dic101C2311yescharacteristic
79C22⋊Dic101—21no—
80#13108051—21no—
81#13108051—21no—
82#13108051—21no—
83#13108051—21no—
84C4×Dic101—21no—
85C2×Q8×C101—21no—
86C2×Q8×C101C2311yescharacteristic
87C2×D4×C101—21no—
88C2×D4×C101C2311yescharacteristic
89C22×C4×C101—21no—
90C22×C4×C101C2311yescharacteristic
91C4⋊C4×C101—21no—
92C4⋊C4×C101—21no—
93C4⋊C4×C101—21no—
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—21no—
109C23×C101C2×D411yescharacteristic
110C22×D101—21no—
111C22×D101—21no—
112C22×D101—24no—
113C22×D101—81no—
114C22×D101—81no—
115Q8×C101—21no—
116Q8×C101—21no—
117Q8×C101—81no—
118D4×C101—21no—
119D4×C101—21no—
120D4×C101—81no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—81no—
127C2×C4×C101C2×D411yescharacteristic
128C2×C4×C101C2×D411yescharacteristic
129C2×C4×C101C2×D411yescharacteristic
130C2×C4×C101C2×D411yescharacteristic
131C2×C4×C101C2×D411yescharacteristic
132C2×C4×C101C2411yescharacteristic
133C2×Dic101—21no—
134C2×Dic101—21no—
135D404—24no—
136D404—21no—
137D404—21no—
138D404#4112yes—
139D404#4112yes—
140C4×D101—81no—
141C4×D101—81no—
142C4×D101—81no—
143C4×D101—81no—
144#11717579—21no—
145#11717579—21no—
146#11717579—21no—
147#11717579—21no—
148C22×C101—21no—
149C22×C101—21no—
150C22×C101—21no—
151C22×C101—41no—
152C22×C101—41no—
153C22×C101—41no—
154C22×C101C22×D411yescharacteristic
155C2×D101—22no—
156C2×D101—22no—
157C2×D101—41no—
158C2×D101—41no—
159C2×D101—41no—
160C2×D101—41no—
161C2×D101—84no—
162C4×C101—21no—
163C4×C101—21no—
164C4×C101—21no—
165C4×C101—21no—
166C4×C101—21no—
167C4×C101—41no—
168C4×C101—41no—
169C4×C101—41no—
170C4×C101C2×#4111yescharacteristic
171C4×C101C2×#4111yescharacteristic
172Dic101—41no—
173Dic101—41no—
174C2×C101—41no—
175C2×C101—81no—
176C2×C101#52511yescharacteristic
177C2×C101#57611yescharacteristic
178C2×C101#57711yescharacteristic
179D101—42no—
180D101—42no—
181#3118—1011nomaximal, Sylow
182C101#311811yescharacteristic, complemented, Sylow
183#542D10111yescharacteristic, complemented
184#518—1011no—
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×D4—1011no—
199C2×SD16—1011no—
200C2×SD16—1011no—
201C2×D8—1011no—
202C2×D8—1011no—
203C4⋊Q8C2×D10111yescharacteristic, complemented
204#126—21no—
205#124—21no—
206#124C2×D10111yescharacteristic, complemented
207#123—21no—
208C22⋊Q8—21no—
209C22⋊Q8C2×D10111yescharacteristic
210#121C2×D10111yescharacteristic
211C4×Q8C2×D10111yescharacteristic, complemented
212C4×D4—1011no—
213C4×D4—1011no—
214C4×D4—1011no—
215C4×D4—1011no—
216C4×D4—2021no—
217C4×D4—2021no—
218C4×D4C2×D10111yescharacteristic, complemented
219#117—1011no—
220#117C2×D10111yescharacteristic
221C2×C22⋊C4—2021no—
222C2×C22⋊C4—2021no—
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—2021no—
242C2×Q8—21no—
243C2×Q8C22×D10111yescharacteristic
244C2×D4—21no—
245C2×D4—1011no—
246C2×D4—1011no—
247C2×D4—2022no—
248C2×D4—2022no—
249C2×D4—2021no—
250C2×D4C22×D10111yescharacteristic
251C22×C4—21no—
252C22×C4—2021no—
253C22×C4—2021no—
254C22×C4—2021no—
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—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—1011no—
269C4⋊C4—1011no—
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—21no—
291C23—8081no—
292C23—8081no—
293C23—2024no—
294C23—2021no—
295C23—2021no—
296C23C2×C22⋊D10111yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—81no—
300D4—21no—
301D4—21no—
302D4—81no—
303D4—1012no—
304D4—1012no—
305D4—2024no—
306D4—2021no—
307D4—2021no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—81no—
314C2×C4—8081no—
315C2×C4—8081no—
316C2×C4—8081no—
317C2×C4—8081no—
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—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—8084no—
337C22—4041no—
338C22—4041no—
339C22—4041no—
340C22—4041no—
341C22—2022no—
342C22—2022no—
343C22C22×C22⋊D10111yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—4041no—
353C4—4041no—
354C4#41×D10111yescharacteristic
355C4#41×D10111yescharacteristic
356C2—41no—
357C2—81no—
358C2—4042no—
359C2—4042no—
360C2#2026568411yescharacteristic
361C2#2026598011yescharacteristic
362C2#2026598511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#542×C101C2
socleC22×C101C22×D4
prime radical#20265310C2
hypercenter#542D101
hypocenterC101#3118
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #20265651C2semidirect
2. #20265515C2semidirect
3. #20265498C2semidirect
4. #20265464C2semidirect
5. #20265454C2semidirect
6. #20265389C2semidirect
7. #20265389C2semidirect
8. #20265327C2semidirect
10. #20265287C2semidirect
11. #20265275C2semidirect
12. C4×#13108061C2semidirect
13. C4×#13108059C2semidirect
14. #20265238C2semidirect
15. #542×C101C2semidirect
30. #15692188C22semidirect
31. #15692188C22semidirect
32. #15692186C22semidirect
33. #15692186C22semidirect
34. C4⋊#11717579C22semidirect
35. C4⋊#11717579C22semidirect
36. #15692179C22semidirect
37. #15692179C22semidirect
38. #15692176C22semidirect
39. #15692176C22semidirect
46. #15692166C22semidirect
47. C4×#11717579C22semidirect
48. C4⋊Q8×C101C22semidirect
51. #124×C101C22semidirect
56. C4×Q8×C101C22semidirect
57. C4×D4×C101C22semidirect
182. C101#3118semidirect
183. #542D101semidirect
203. C4⋊Q8C2×D101semidirect
206. #124C2×D101semidirect
211. C4×Q8C2×D101semidirect
218. C4×D4C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202656512
#202655152
#202654982
#202654642
#202654542
#202653892
#202653272
#202653102
#202652872
#202652752
C4×#131080612
C4×#131080592
#202652382
#542×C1012
#3118101