Finite Group Databasebeta

← #27181043

Subgroups — #27181043

Order 12928.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265684C211yesmaximal, characteristic
2#117×D101C211yesmaximal, characteristic, complemented
3#20265517C211yesmaximal, characteristic, complemented
4#20265516C211yesmaximal, characteristic, complemented
5#20265511C211yesmaximal, characteristic, complemented
6#20265506C211yesmaximal, characteristic, complemented
7#20265500C211yesmaximal, characteristic, complemented
8#20265499C211yesmaximal, characteristic, complemented
9#20265494C211yesmaximal, characteristic, complemented
10#20265489C211yesmaximal, characteristic, complemented
11#20265215C211yesmaximal, characteristic, complemented
12#20265213C211yesmaximal, characteristic, complemented
13#20265173C211yesmaximal, characteristic, complemented
14C2×#15692167C211yesmaximal, characteristic, complemented
15#416×C101C211yesmaximal, characteristic, complemented
16C22×C4×D101C2211yescharacteristic
17C22×#11717579C2211yescharacteristic
18#15692236—21no—
19#15692231C2211yescharacteristic
20#15692231C2211yescharacteristic
21#15692226C2211yescharacteristic
22#15692226C2211yescharacteristic
23#15692223—21no—
24C4×C22⋊D101—21no—
25C4×C22⋊D101—21no—
26#15692214C2211yescharacteristic
27#15692214C2211yescharacteristic
28#15692213C2211yescharacteristic
29#15692213C2211yescharacteristic
30#15692212—21no—
31#15692211—21no—
32C4⋊C4×D101C2211yescharacteristic
33C4⋊C4×D101C2211yescharacteristic
34C22⋊C4×D101—21no—
35C42×D101—21no—
36#15692189C2211yescharacteristic, complemented
37#15692189C2211yescharacteristic, complemented
38#15692188C2211yescharacteristic, complemented
39#15692188C2211yescharacteristic, complemented
40#15692187C2211yescharacteristic, complemented
41#15692187C2211yescharacteristic, complemented
42#15692186C2211yescharacteristic, complemented
43#15692186C2211yescharacteristic, complemented
44#15692179C2211yescharacteristic, complemented
45#15692179C2211yescharacteristic, complemented
46#15692176C2211yescharacteristic, complemented
47#15692176C2211yescharacteristic, complemented
48#15692167C2211yescharacteristic
49#15692167C2211yescharacteristic
50#15692167C2211yescharacteristic
51#15692167C2211yescharacteristic
52C2×#41×C101C2211yescharacteristic
53C22×C8×C101C2211yescharacteristic
54#117×C101C2211yescharacteristic
55#103×C101C2211yescharacteristic, complemented
56#103×C101C2211yescharacteristic, complemented
57#102×C101C2211yescharacteristic, complemented
58#102×C101C2211yescharacteristic, complemented
59C23×D101D411yescharacteristic
60C2×C22⋊D101—21no—
61C2×C22⋊D101—21no—
62C22×Dic101D411yescharacteristic
63C2×C4×D101—22no—
64C2×C4×D101—22no—
65C2×C4×D101—21no—
66C2×C4×D101—21no—
67C2×C4×D101D411yescharacteristic
68C2×C4×D101D411yescharacteristic
69C2×C4×D101C2311yescharacteristic
70C2×C4×D101C2311yescharacteristic
71C2×#11717579—22no—
72C2×#11717579C2311yescharacteristic
73C2×#11717579C2311yescharacteristic
74C4⋊Dic101C2311yescharacteristic
75C4⋊Dic101C2311yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77C4⋊Dic101C2311yescharacteristic
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053—21no—
82C22⋊Dic101—21no—
83C22⋊Dic101—21no—
84C22⋊Dic101—21no—
85C22⋊Dic101—21no—
86#13108051—21no—
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90C4×Dic101—21no—
91C4×Dic101—21no—
92C4×Dic101—21no—
93C4×Dic101—21no—
94#41×C101—22no—
95#41×C101—22no—
96C2×Q8×C101C2311yescharacteristic
97C2×D4×C101—21no—
98C2×D4×C101C2311yescharacteristic
99C22×C4×C101—21no—
100C22×C4×C101C2311yescharacteristic
101C2×C8×C101—22no—
102C2×C8×C101C2311yescharacteristic
103C2×C8×C101C2311yescharacteristic
104C4⋊C4×C101C2311yescharacteristic
105C4⋊C4×C101C2311yescharacteristic
106C22⋊C4×C101—21no—
107C42×C101—21no—
108C23×C101—21no—
109C23×C101C2×D411yescharacteristic
110C22×D101—22no—
111C22×D101—22no—
112C22×D101—41no—
113C22×D101C2×D411yescharacteristic
114C22×D101C2×D411yescharacteristic
115Q8×C101—21no—
116Q8×C101—21no—
117D4×C101—21no—
118D4×C101—21no—
119D4×C101—42no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—42no—
124C2×C4×C101C2×D411yescharacteristic
125C2×C4×C101#4112yes—
126C2×C4×C101#4112yes—
127C2×C4×C101C2411yescharacteristic
128C22⋊D101—42no—
129C22⋊D101—42no—
130C2×Dic101—22no—
131C2×Dic101—21no—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101C2×D411yescharacteristic
136C2×Dic101C2×D411yescharacteristic
137C4×D101—22no—
138C4×D101—22no—
139C4×D101—21no—
140C4×D101—21no—
141C4×D101—21no—
142C4×D101—21no—
143C4×D101—42no—
144C4×D101—42no—
145C8×C101—21no—
146C8×C101—21no—
147#11717579—21no—
148#11717579—21no—
149C22×C101—21no—
150C22×C101—21no—
151C22×C101—41no—
152C22×C101—41no—
153C22×C101—41no—
154C22×C101#12312yes—
155C22×C101C22×D411yescharacteristic
156C2×D101—21no—
157C2×D101—21no—
158C2×D101—21no—
159C2×D101—21no—
160C2×D101—21no—
161C2×D101—44no—
162C2×D101#13512yes—
163C4×C101—41no—
164C4×C101—41no—
165C4×C101—41no—
166C4×C101#12312yes—
167C4×C101C2×#4111yescharacteristic
168C4×C101C2×#4111yescharacteristic
169Dic101—21no—
170Dic101—21no—
171Dic101—41no—
172Dic101—41no—
173Dic101—41no—
174Dic101—41no—
175C2×C101—22no—
176C2×C101—81no—
177C2×C101C2×#12311yescharacteristic
178C2×C101C2×#13511yescharacteristic
179C2×C101#57511yescharacteristic
180D101—22no—
181D101—41no—
182#3091—1011nomaximal, Sylow
183C101#309111yescharacteristic, complemented, Sylow
184#525—1011no—
185C2×#117—1011no—
186#485—1011no—
187#485—1011no—
188#484—1011no—
189#483—1011no—
190#482—1011no—
191#482—1011no—
192#481—1011no—
193#480—1011no—
194#427—1011no—
195#427—1011no—
196#416—1011no—
197#416D10111yescharacteristic, complemented
198C2×C22⋊C8—1011no—
199C2×#41C2×D10111yescharacteristic
200C23×C4—1011no—
201C22×C8—1011no—
202C22×C8C2×D10111yescharacteristic
203#123—2021no—
204C22⋊Q8—1011no—
205C22⋊Q8—1011no—
206#121—1011no—
207#121—1011no—
208#120—2021no—
209C4×D4—2021no—
210C4×D4—2021no—
211#117—1011no—
212#117—1011no—
213#117—1011no—
214#117—1011no—
215#117—2021no—
216#117—2021no—
217#117C2×D10111yescharacteristic
218C2×C4⋊C4—1011no—
219C2×C4⋊C4—1011no—
220C2×C22⋊C4—2021no—
221C2×C42—2021no—
222#107—1011no—
223#107—1011no—
224#107—1011no—
225#107—1011no—
226#106—1011no—
227#106—1011no—
228#106—1011no—
229#106—1011no—
230#103—1011no—
231#103—1011no—
232#103C2×D10111yescharacteristic, complemented
233#103C2×D10111yescharacteristic, complemented
234#102—1011no—
235#102—1011no—
236#102C2×D10111yescharacteristic, complemented
237#102C2×D10111yescharacteristic, complemented
238C22⋊C8—1011no—
239C22⋊C8—1011no—
240C22⋊C8—1011no—
241C22⋊C8—1011no—
242C24—1011no—
243#41—22no—
244#41—22no—
245C2×Q8C22×D10111yescharacteristic
246C2×D4—21no—
247C2×D4—2021no—
248C2×D4—2021no—
249C2×D4C22×D10111yescharacteristic
250C22×C4—21no—
251C22×C4—1011no—
252C22×C4—1011no—
253C22×C4—1011no—
254C22×C4—1011no—
255C22×C4—1011no—
256C22×C4—2021no—
257C22×C4—2021no—
258C22×C4—2022no—
259C22×C4—2022no—
260C22×C4C22×D10111yescharacteristic
261C2×C8—22no—
262C2×C8—1011no—
263C2×C8—1011no—
264C2×C8—2022no—
265C2×C8C22×D10111yescharacteristic
266C2×C8C22×D10111yescharacteristic
267C4⋊C4—1011no—
268C4⋊C4—1011no—
269C4⋊C4—1011no—
270C4⋊C4—1011no—
271C4⋊C4—2021no—
272C4⋊C4—2021no—
273C4⋊C4—2021no—
274C4⋊C4—2021no—
275C4⋊C4C22×D10111yescharacteristic
276C4⋊C4C22×D10111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—2021no—
279C22⋊C4—2021no—
280C22⋊C4—2021no—
281C22⋊C4—2021no—
282C22⋊C4—2021no—
283C22⋊C4—2021no—
284C22⋊C4—2021no—
285C22⋊C4—2021no—
286C42—21no—
287C42—2021no—
288C42—2021no—
289C42—2021no—
290C42—2021no—
291C23—21no—
292C23—1011no—
293C23—1011no—
294C23—4041no—
295C23—2022no—
296C23—2022no—
297C23D4×D10111yescharacteristic
298Q8—21no—
299Q8—21no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—4042no—
304D4—4042no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—1011no—
310C2×C4—1011no—
311C2×C4—4042no—
312C2×C4—4042no—
313C2×C4—2021no—
314C2×C4—2021no—
315C2×C4—2021no—
316C2×C4—2021no—
317C2×C4—2022no—
318C2×C4—2021no—
319C2×C4—2021no—
320C2×C4—2021no—
321C2×C4—2021no—
322C2×C4—2022no—
323C2×C4—2022no—
324C2×C4D4×D10111yescharacteristic
325C2×C4#1310807312yes—
326C2×C4#1310807412yes—
327C2×C4C23×D10111yescharacteristic
328C8—21no—
329C8—21no—
330C8—2021no—
331C8—2021no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—1012no—
338C22—4044no—
339C22—2021no—
340C22—2021no—
341C22—2021no—
342C22—2021no—
343C22—2021no—
344C22#1569223712yes—
345C22C2×D4×D10111yescharacteristic
346C4—41no—
347C4—41no—
348C4—41no—
349C4—4041no—
350C4—4041no—
351C4—4041no—
352C4—4041no—
353C4—2021no—
354C4—2021no—
355C4#1569223712yes—
356C4C2×#1310807311yescharacteristic
357C4C2×#1310807411yescharacteristic
358C2—22no—
359C2—81no—
360C2—4041no—
361C2—2022no—
362C2C2×#1569223711yescharacteristic
363C2#135×D10111yescharacteristic
364C2#2026597411yescharacteristic
subgroupG/H
centerC2×C4D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#416×C101C2
socleC22×C101C22×D4
prime radical#20265684C2
hypercenter#416D101
hypocenterC101#3091
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #117×D101C2semidirect
3. #20265517C2semidirect
4. #20265516C2semidirect
5. #20265511C2semidirect
6. #20265506C2semidirect
7. #20265500C2semidirect
8. #20265499C2semidirect
9. #20265494C2semidirect
10. #20265489C2semidirect
11. #20265215C2semidirect
12. #20265213C2semidirect
13. #20265173C2semidirect
14. C2×#15692167C2semidirect
15. #416×C101C2semidirect
36. #15692189C22semidirect
37. #15692189C22semidirect
38. #15692188C22semidirect
39. #15692188C22semidirect
40. #15692187C22semidirect
41. #15692187C22semidirect
42. #15692186C22semidirect
43. #15692186C22semidirect
44. #15692179C22semidirect
45. #15692179C22semidirect
46. #15692176C22semidirect
47. #15692176C22semidirect
55. #103×C101C22semidirect
56. #103×C101C22semidirect
57. #102×C101C22semidirect
58. #102×C101C22semidirect
183. C101#3091semidirect
197. #416D101semidirect
232. #103C2×D101semidirect
233. #103C2×D101semidirect
236. #102C2×D101semidirect
237. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202656842
#117×D1012
#202655172
#202655162
#202655112
#202655062
#202655002
#202654992
#202654942
#202654892
#202652152
#202652132
#202651732
C2×#156921672
#416×C1012
#3091101