Finite Group Databasebeta

← #27180648

Subgroups — #27180648

Order 12928.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692284C211yesmaximal, characteristic, complemented
2C2×SD16×D101C211yesmaximal, characteristic, complemented
3#20265740C211yesmaximal, characteristic
4#20265653C211yesmaximal, characteristic, complemented
5#20265400C211yesmaximal, characteristic, complemented
6#20265399C211yesmaximal, characteristic, complemented
7#20265398C211yesmaximal, characteristic, complemented
8#20265397C211yesmaximal, characteristic, complemented
9#20265376C211yesmaximal, characteristic, complemented
10#20265375C211yesmaximal, characteristic, complemented
11#20265374C211yesmaximal, characteristic, complemented
12#20265202C211yesmaximal, characteristic, complemented
13#20265180C211yesmaximal, characteristic, complemented
14#20265170C211yesmaximal, characteristic, complemented
15#460×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×Q8×D101C2211yescharacteristic
18C2×D4×D101C2211yescharacteristic
19#15692284—24no—
20C2×Dic404C2211yescharacteristic, complemented
21C2×#13108064C2211yescharacteristic, complemented
22C2×#13108063C2211yescharacteristic
23C2×#13108062C2211yescharacteristic
24C2×#13108062C2211yescharacteristic, complemented
25C2×#13108061C2211yescharacteristic
26SD16×D101—24no—
27C2×#13108057C2211yescharacteristic
28C2×C8×D101C2211yescharacteristic
29#15692251C2211yescharacteristic
30#15692243C2211yescharacteristic
31#15692242C2211yescharacteristic
32#15692227—21no—
33#15692227—21no—
34#15692223—21no—
35#15692223—21no—
36Q8×Dic101C2211yescharacteristic, complemented
37C4×Dic202—21no—
38C4×Dic202C2211yescharacteristic, complemented
39#15692220—21no—
40#15692220C2211yescharacteristic, complemented
41#15692213—21no—
42#15692213C2211yescharacteristic
43#15692212—21no—
44#15692212C2211yescharacteristic
45C4⋊#11717579C2211yescharacteristic, complemented
46#15692184C2211yescharacteristic, complemented
47#15692184C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692178C2211yescharacteristic, complemented
50#15692177C2211yescharacteristic
51#15692176C2211yescharacteristic
52#15692176C2211yescharacteristic, complemented
53#15692175C2211yescharacteristic
54C2×SD16×C101C2211yescharacteristic
55C2×SD16×C101C2211yescharacteristic, complemented
56#127×C101C2211yescharacteristic
57C4×Q8×C101C2211yescharacteristic, complemented
58C4⋊C8×C101C2211yescharacteristic, complemented
59#102×C101C2211yescharacteristic
60#102×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108073—22no—
63#13108073—22no—
64C2×Dic202—21no—
65C2×Dic202C2311yescharacteristic
66C2×Dic202C2311yescharacteristic
67Q8×D101—22no—
68Q8×D101D414yescomplemented
69C2×D404C2311yescharacteristic
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C22⋊D101—21no—
73C2×C22⋊D101—21no—
74D4×D101—22no—
75D4×D101—22no—
76C22×Dic101—21no—
77C2×C4×D101—21no—
78C2×C4×D101C2311yescharacteristic
79Dic404—22no—
80#13108064—22no—
81#13108063—22no—
82#13108062—22no—
83#13108062—22no—
84#13108061—22no—
85#13108057—22no—
86C2×#11717579C2311yescharacteristic
87C2×#11717579C2311yescharacteristic
88C8×D101—22no—
89C4⋊Dic101—21no—
90C4⋊Dic101C2311yescharacteristic
91#13108053—21no—
92#13108053—21no—
93#13108053D411yescharacteristic
94#13108053D411yescharacteristic
95C22⋊Dic101—21no—
96C22⋊Dic101—21no—
97#13108051—21no—
98#13108051—21no—
99#13108051—21no—
100#13108051D411yescharacteristic
101#13108051D411yescharacteristic
102C4×Dic101—21no—
103C4×Dic101—21no—
104C4×Dic101—21no—
105C4×Dic101—21no—
106C4×Dic101C2311yescharacteristic
107C2×Q8×C101C2311yescharacteristic
108C2×D4×C101—21no—
109C2×D4×C101—21no—
110C2×D4×C101C2311yescharacteristic
111C2×D4×C101C2311yescharacteristic
112SD16×C101—22no—
113SD16×C101—22no—
114C2×C8×C101C2311yescharacteristic
115C2×C8×C101C2311yescharacteristic
116C4⋊C4×C101—21no—
117C4⋊C4×C101C2311yescharacteristic
118C42×C101—21no—
119C42×C101C2311yescharacteristic
120C23×C101—21no—
121C23×C101—21no—
122C22×D101—21no—
123C22×D101—42no—
124C22×D101—81no—
125C22×D101—81no—
126C22×D101C2×D411yescharacteristic
127Q8×C101—21no—
128Q8×C101C2×D412yes—
129D4×C101—21no—
130D4×C101—21no—
131D4×C101—21no—
132D4×C101—21no—
133D4×C101—81no—
134D4×C101—81no—
135C2×C4×C101—21no—
136C2×C4×C101—21no—
137C2×C4×C101C2×D411yescharacteristic
138C2×C4×C101C2×D411yescharacteristic
139C2×C4×C101C2411yescharacteristic
140C22⋊D101—42no—
141C22⋊D101—42no—
142C22⋊D101—81no—
143C22⋊D101—81no—
144C2×Dic101—21no—
145C2×Dic101—21no—
146C2×Dic101—21no—
147C2×Dic101—42no—
148C2×Dic101C2×D411yescharacteristic
149C2×Dic101C2×D411yescharacteristic
150C2×Dic101C2×D411yescharacteristic
151D404—21no—
152D404—21no—
153C4×D101—24no—
154C4×D101C2×D412yes—
155C4×D101#4112yes—
156Dic202—21no—
157Dic202—24no—
158Dic202—21no—
159Dic202—21no—
160Dic202C2×D412yes—
161C8×C101—21no—
162C8×C101—21no—
163#11717579—21no—
164#11717579—21no—
165C22×C101—41no—
166C22×C101—41no—
167C22×C101—41no—
168C22×C101—41no—
169C22×C101—41no—
170C22×C101—41no—
171C22×C101C22×D411yescharacteristic
172C2×D101—21no—
173C2×D101—21no—
174C2×D101—41no—
175C2×D101—41no—
176C2×D101—41no—
177C2×D101—84no—
178C2×D101#12112yes—
179C4×C101—22no—
180C4×C101—21no—
181C4×C101—21no—
182C4×C101—41no—
183C4×C101C22×D411yescharacteristic
184C4×C101C2×#4111yescharacteristic
185Dic101—22no—
186Dic101—21no—
187Dic101—21no—
188Dic101—41no—
189Dic101—41no—
190Dic101#12112yes—
191C2×C101—81no—
192C2×C101—81no—
193C2×C101C2×#12111yescharacteristic
194C2×C101#57711yescharacteristic
195C2×C101#57711yescharacteristic
196D101—22no—
197D101—81no—
198#3043—1011nomaximal, Sylow
199C101#304311yescharacteristic, complemented, Sylow
200C2×#137—1011no—
201C22×SD16—1011no—
202#535—1011no—
203#519—1011no—
204#464—1011no—
205#464—1011no—
206#464—1011no—
207#464—1011no—
208#460—1011no—
209#460—1011no—
210#460—1011no—
211#460D10111yescharacteristic, complemented
212#424—1011no—
213#418—1011no—
214#416—1011no—
215C2×#41—1011no—
216C22×Q8—1011no—
217C22×D4—1011no—
218#137—2024no—
219C2×Q16—1011no—
220C2×Q16—1011no—
221C2×SD16—1011no—
222C2×SD16—1011no—
223C2×SD16—1011no—
224C2×SD16—1011no—
225C2×SD16—2024no—
226C2×SD16C2×D10111yescharacteristic
227C2×SD16C2×D10111yescharacteristic, complemented
228C2×M16—1011no—
229C22×C8—1011no—
230#127—1011no—
231#127C2×D10111yescharacteristic
232#124—1011no—
233#124—1011no—
234#121—2021no—
235#121—2021no—
236#120—2021no—
237#120—2021no—
238C4×Q8—1011no—
239C4×Q8—1011no—
240C4×Q8—1011no—
241C4×Q8—2021no—
242C4×Q8—2021no—
243C4×Q8C2×D10111yescharacteristic, complemented
244#117—1011no—
245#117—1011no—
246#117—2021no—
247#117—2021no—
248C4⋊C8—1011no—
249C4⋊C8—1011no—
250C4⋊C8—1011no—
251C4⋊C8C2×D10111yescharacteristic, complemented
252#103—1011no—
253#103—1011no—
254#102—1011no—
255#102—1011no—
256#102—1011no—
257#102—1011no—
258#102C2×D10111yescharacteristic
259#102C2×D10111yescharacteristic, complemented
260C24—2021no—
261#41—2022no—
262#41—2022no—
263C2×Q8—1014no—
264C2×Q8—1011no—
265C2×Q8—1011no—
266C2×Q8—2021no—
267C2×Q8—2022no—
268C2×Q8C22×D10111yescharacteristic
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—1011no—
272C2×D4—2022no—
273C2×D4—2022no—
274C2×D4—2021no—
275C2×D4—2021no—
276C2×D4—2021no—
277C2×D4—2021no—
278C2×D4C22×D10111yescharacteristic
279C2×D4C22×D10111yescharacteristic
280C22×C4—1011no—
281C22×C4—2021no—
282C22×C4—2021no—
283Q16—2022no—
284Q16—2022no—
285SD16—22no—
286SD16—22no—
287SD16—2022no—
288SD16—2022no—
289SD16—2022no—
290SD16—2022no—
291M16—2022no—
292C2×C8—1011no—
293C2×C8—1011no—
294C2×C8—2022no—
295C2×C8C22×D10111yescharacteristic
296C2×C8C22×D10111yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—1011no—
299C4⋊C4—1011no—
300C4⋊C4—1011no—
301C4⋊C4—2021no—
302C4⋊C4—2021no—
303C4⋊C4—2021no—
304C4⋊C4C22×D10111yescharacteristic
305C22⋊C4—1011no—
306C22⋊C4—1011no—
307C22⋊C4—2021no—
308C22⋊C4—2021no—
309C22⋊C4—2021no—
310C22⋊C4—2021no—
311C22⋊C4—2021no—
312C42—21no—
313C42—1011no—
314C42—2021no—
315C42—2021no—
316C42—2021no—
317C42—2021no—
318C42C22×D10111yescharacteristic
319C23—21no—
320C23—21no—
321C23—8081no—
322C23—8081no—
323C23—1011no—
324C23—4042no—
325C23—2021no—
326Q8—21no—
327Q8—1012no—
328Q8—2024no—
329Q8—2021no—
330Q8—2021no—
331Q8—2021no—
332Q8D4×D10112yes—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—81no—
338D4—81no—
339D4—8081no—
340D4—8081no—
341D4—4042no—
342D4—4042no—
343D4—2021no—
344D4—2021no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—1011no—
348C2×C4—1012no—
349C2×C4—1012no—
350C2×C4—1011no—
351C2×C4—1011no—
352C2×C4—4042no—
353C2×C4—2024no—
354C2×C4—2021no—
355C2×C4—2021no—
356C2×C4—2021no—
357C2×C4D4×D10111yescharacteristic
358C2×C4D4×D10111yescharacteristic
359C2×C4C23×D10111yescharacteristic
360C8—21no—
361C8—21no—
362C8—2021no—
363C8—2021no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—8084no—
371C22—1012no—
372C22—4041no—
373C22—4041no—
374C22—4041no—
375C22—2021no—
376C22—2021no—
377C22C2×D4×D10111yescharacteristic
378C4—21no—
379C4—22no—
380C4—21no—
381C4—41no—
382C4—1012no—
383C4—4041no—
384C4—4041no—
385C4—2021no—
386C4—2022no—
387C4—2021no—
388C4C2×D4×D10111yescharacteristic
389C4#41×D10111yescharacteristic
390C2—81no—
391C2—81no—
392C2—8081no—
393C2—2022no—
394C2#121×D10111yescharacteristic
395C2#2026598311yescharacteristic
396C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#460×C101C2
socleC22×C101C22×D4
prime radical#20265740C2
hypercenter#460D101
hypocenterC101#3043
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#15692284C2semidirect
2. C2×SD16×D101C2semidirect
4. #20265653C2semidirect
5. #20265400C2semidirect
6. #20265399C2semidirect
7. #20265398C2semidirect
8. #20265397C2semidirect
9. #20265376C2semidirect
10. #20265375C2semidirect
11. #20265374C2semidirect
12. #20265202C2semidirect
13. #20265180C2semidirect
14. #20265170C2semidirect
15. #460×C101C2semidirect
20. C2×Dic404C22semidirect
21. C2×#13108064C22semidirect
24. C2×#13108062C22semidirect
36. Q8×Dic101C22semidirect
38. C4×Dic202C22semidirect
40. #15692220C22semidirect
45. C4⋊#11717579C22semidirect
46. #15692184C22semidirect
47. #15692184C22semidirect
48. #15692180C22semidirect
49. #15692178C22semidirect
52. #15692176C22semidirect
55. C2×SD16×C101C22semidirect
57. C4×Q8×C101C22semidirect
58. C4⋊C8×C101C22semidirect
60. #102×C101C22semidirect
68. Q8×D101D4semidirect
199. C101#3043semidirect
211. #460D101semidirect
227. C2×SD16C2×D101semidirect
243. C4×Q8C2×D101semidirect
251. C4⋊C8C2×D101semidirect
259. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922842
C2×SD16×D1012
#202657402
#202656532
#202654002
#202653992
#202653982
#202653972
#202653762
#202653752
#202653742
#202652022
#202651802
#202651702
#460×C1012
#3043101