Finite Group Databasebeta

← #27180708

Subgroups — #27180708

Order 12928.

520 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692276C211yesmaximal, characteristic, complemented
2C22×C8×D101C211yesmaximal, characteristic, complemented
3#20265690C211yesmaximal, characteristic, complemented
4#20265687C211yesmaximal, characteristic, complemented
5#20265417C211yesmaximal, characteristic, complemented
6#20265416C211yesmaximal, characteristic, complemented
7#20265412C211yesmaximal, characteristic, complemented
8#20265411C211yesmaximal, characteristic, complemented
9#20265407C211yesmaximal, characteristic, complemented
10#20265406C211yesmaximal, characteristic, complemented
11#20265403C211yesmaximal, characteristic, complemented
12#20265212C211yesmaximal, characteristic, complemented
13#20265172C211yesmaximal, characteristic, complemented
14#20265170C211yesmaximal, characteristic, complemented
15#465×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×#13108073C2211yescharacteristic
18C22×C4×D101C2211yescharacteristic
19#15692276—24no—
20C2×#13108064C2211yescharacteristic, complemented
21C2×#13108063C2211yescharacteristic, complemented
22C2×#13108059C2211yescharacteristic, complemented
23C22×#11717579C2211yescharacteristic
24C2×C8×D101—24no—
25C2×C8×D101C2211yescharacteristic
26C2×C8×D101C2211yescharacteristic
27C2×C8×D101C2211yescharacteristic
28C2×C8×D101C2211yescharacteristic
29#15692240—21no—
30#15692237—21no—
31#15692235C2211yescharacteristic
32#15692234C2211yescharacteristic
33#15692231C2211yescharacteristic
34#15692230C2211yescharacteristic
35#15692229C2211yescharacteristic
36#15692226C2211yescharacteristic
37#15692225—21no—
38#15692223—21no—
39D4×Dic101—21no—
40#15692217—21no—
41#15692215—21no—
42#15692215—21no—
43#15692213C2211yescharacteristic
44#15692213C2211yescharacteristic
45#15692188C2211yescharacteristic, complemented
46#15692188C2211yescharacteristic, complemented
47#15692187C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692179C2211yescharacteristic, complemented
50#15692178C2211yescharacteristic, complemented
51#15692177C2211yescharacteristic, complemented
52#15692176C2211yescharacteristic, complemented
53#15692175C2211yescharacteristic, complemented
54C2×SD16×C101C2211yescharacteristic, complemented
55C22×C8×C101C2211yescharacteristic
56C22⋊Q8×C101C2211yescharacteristic
57#121×C101C2211yescharacteristic
58#106×C101C2211yescharacteristic, complemented
59#103×C101C2211yescharacteristic, complemented
60#102×C101C2211yescharacteristic, complemented
61C23×D101D411yescharacteristic
62#13108075—22no—
63#13108075—22no—
64#13108073—22no—
65#13108073—22no—
66C2×Dic202C2311yescharacteristic
67C2×D404—21no—
68C2×D404—21no—
69C2×D404C2311yescharacteristic
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C22⋊D101—21no—
73C22×Dic101—21no—
74C22×Dic101D411yescharacteristic
75C2×C4×D101—22no—
76C2×C4×D101—22no—
77C2×C4×D101—21no—
78C2×C4×D101D411yescharacteristic
79C2×C4×D101D411yescharacteristic
80C2×C4×D101C2311yescharacteristic
81C2×C4×D101C2311yescharacteristic
82#13108064—22no—
83#13108063—22no—
84#13108059—22no—
85C2×#11717579—22no—
86C2×#11717579C2311yescharacteristic
87C2×#11717579C2311yescharacteristic
88C8×D101—22no—
89C8×D101—22no—
90C8×D101—22no—
91C8×D101D414yescomplemented
92C4⋊Dic101—21no—
93C4⋊Dic101C2311yescharacteristic
94C4⋊Dic101C2311yescharacteristic
95#13108053—21no—
96#13108053—21no—
97C22⋊Dic101—21no—
98C22⋊Dic101—21no—
99#13108051—21no—
100#13108051—21no—
101#13108051—21no—
102#13108051—21no—
103#13108051—21no—
104#13108051—21no—
105C4×Dic101—21no—
106C4×Dic101—21no—
107C2×Q8×C101C2311yescharacteristic
108C2×D4×C101—21no—
109C2×D4×C101C2311yescharacteristic
110C22×C4×C101C2311yescharacteristic
111SD16×C101—22no—
112C2×C8×C101—22no—
113C2×C8×C101C2311yescharacteristic
114C2×C8×C101C2311yescharacteristic
115C4⋊C4×C101—21no—
116C4⋊C4×C101C2311yescharacteristic
117C4⋊C4×C101C2311yescharacteristic
118C22⋊C4×C101—21no—
119C22⋊C4×C101—21no—
120C23×C101—21no—
121C23×C101C2×D411yescharacteristic
122C22×D101—22no—
123C22×D101—22no—
124C22×D101—22no—
125C22×D101—21no—
126C22×D101C2×D411yescharacteristic
127C22×D101C2×D411yescharacteristic
128Q8×C101—21no—
129Q8×C101—21no—
130D4×C101—21no—
131D4×C101—21no—
132D4×C101—42no—
133C2×C4×C101—21no—
134C2×C4×C101—21no—
135C2×C4×C101—21no—
136C2×C4×C101—21no—
137C2×C4×C101—21no—
138C2×C4×C101C2×D411yescharacteristic
139C2×C4×C101C2411yescharacteristic
140C22⋊D101—42no—
141C22⋊D101—42no—
142C22⋊D101—42no—
143C2×Dic101—22no—
144C2×Dic101—21no—
145C2×Dic101—21no—
146C2×Dic101—21no—
147C2×Dic101—42no—
148C2×Dic101C2×D411yescharacteristic
149C2×Dic101C2×D411yescharacteristic
150D404—21no—
151D404—21no—
152D404—42no—
153D404—42no—
154C4×D101—22no—
155C4×D101—21no—
156C4×D101—21no—
157C4×D101—22no—
158C4×D101—42no—
159C4×D101C2×D412yes—
160C4×D101#4112yes—
161Dic202—21no—
162Dic202—21no—
163C8×C101—21no—
164C8×C101C2×D412yes—
165#11717579—21no—
166#11717579C2×D412yes—
167C22×C101—21no—
168C22×C101—21no—
169C22×C101—41no—
170C22×C101—41no—
171C22×C101—41no—
172C22×C101#13512yescomplemented
173C22×C101C22×D411yescharacteristic
174C2×D101—28no—
175C2×D101—21no—
176C2×D101—21no—
177C2×D101—21no—
178C2×D101—21no—
179C2×D101—41no—
180C2×D101—41no—
181C2×D101—41no—
182C2×D101#12112yes—
183C2×D101#13512yescomplemented
184C4×C101—21no—
185C4×C101—41no—
186C4×C101—41no—
187C4×C101—41no—
188C4×C101C22×D411yescharacteristic
189C4×C101C2×#4111yescharacteristic
190Dic101—21no—
191Dic101—41no—
192Dic101—41no—
193Dic101—41no—
194Dic101#12112yes—
195C2×C101—22no—
196C2×C101—81no—
197C2×C101C2×#12111yescharacteristic
198C2×C101C2×#13511yescharacteristic
199C2×C101C2×#13511yescharacteristic
200D101—22no—
201D101—22no—
202D101—81no—
203#3050—1011nomaximal, Sylow
204C101#305011yescharacteristic, complemented, Sylow
205C2×#135—1011no—
206C23×C8—1011no—
207#525—1011no—
208#525—1011no—
209#467—1011no—
210#467—1011no—
211#466—1011no—
212#466—1011no—
213#465—1011no—
214#465—1011no—
215#465—1011no—
216#465D10111yescharacteristic, complemented
217#427—1011no—
218#416—1011no—
219#416—1011no—
220C2×#41—1011no—
221C2×#41—1011no—
222C23×C4—1011no—
223#135—2024no—
224C2×Q16—1011no—
225C2×SD16—1011no—
226C2×SD16C2×D10111yescharacteristic, complemented
227C2×D8—1011no—
228C22×C8—1011no—
229C22×C8—1011no—
230C22×C8—1011no—
231C22×C8—1011no—
232C22×C8—1011no—
233C22×C8—2024no—
234C22×C8C2×D10111yescharacteristic
235#123—2021no—
236#123—2021no—
237C22⋊Q8—1011no—
238C22⋊Q8—1011no—
239C22⋊Q8—1011no—
240C22⋊Q8C2×D10111yescharacteristic
241#121—1011no—
242#121—1011no—
243#121—1011no—
244#121C2×D10111yescharacteristic
245#120—2021no—
246#120—2021no—
247C4×D4—2021no—
248C4×D4—2021no—
249C4×D4—2021no—
250C4×D4—2021no—
251#117—1011no—
252#117—1011no—
253#107—1011no—
254#107—1011no—
255#106—1011no—
256#106C2×D10111yescharacteristic, complemented
257#103—1011no—
258#103—1011no—
259#103—1011no—
260#103C2×D10111yescharacteristic, complemented
261#102—1011no—
262#102—1011no—
263#102—1011no—
264#102C2×D10111yescharacteristic, complemented
265C24—1011no—
266#41—2022no—
267#41—2022no—
268#41—2022no—
269#41—2022no—
270C2×Q8—1011no—
271C2×Q8C22×D10111yescharacteristic
272C2×D4—21no—
273C2×D4—1011no—
274C2×D4—2021no—
275C2×D4—2021no—
276C2×D4—2021no—
277C2×D4—2021no—
278C2×D4—2021no—
279C2×D4C22×D10111yescharacteristic
280C22×C4—1011no—
281C22×C4—1011no—
282C22×C4—1011no—
283C22×C4—1011no—
284C22×C4—1011no—
285C22×C4—2021no—
286C22×C4—2022no—
287C22×C4—2022no—
288C22×C4—2021no—
289C22×C4C22×D10111yescharacteristic
290Q16—2022no—
291SD16—22no—
292SD16—2022no—
293D8—2022no—
294C2×C8—22no—
295C2×C8—1011no—
296C2×C8—1014no—
297C2×C8—1011no—
298C2×C8—2022no—
299C2×C8—2022no—
300C2×C8—2022no—
301C2×C8—2022no—
302C2×C8C22×D10111yescharacteristic
303C2×C8C22×D10111yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—1011no—
306C4⋊C4—1011no—
307C4⋊C4—2021no—
308C4⋊C4—2021no—
309C4⋊C4—2021no—
310C4⋊C4C22×D10111yescharacteristic
311C4⋊C4C22×D10111yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—2021no—
315C22⋊C4—2021no—
316C22⋊C4—2021no—
317C22⋊C4—2021no—
318C22⋊C4—2021no—
319C22⋊C4—2021no—
320C22⋊C4—2021no—
321C22⋊C4—2021no—
322C42—2021no—
323C42—2021no—
324C23—21no—
325C23—1011no—
326C23—1011no—
327C23—2022no—
328C23—2022no—
329C23—2021no—
330C23—2022no—
331C23D4×D10111yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—2021no—
335Q8—2021no—
336D4—21no—
337D4—21no—
338D4—42no—
339D4—4042no—
340D4—4042no—
341D4—4042no—
342D4—4042no—
343D4—4042no—
344D4—2021no—
345D4—2021no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—1011no—
352C2×C4—1012no—
353C2×C4—1012no—
354C2×C4—1011no—
355C2×C4—4042no—
356C2×C4—4042no—
357C2×C4—2022no—
358C2×C4—2022no—
359C2×C4—2022no—
360C2×C4—2021no—
361C2×C4—2021no—
362C2×C4—2021no—
363C2×C4—2021no—
364C2×C4—2021no—
365C2×C4D4×D10111yescharacteristic
366C2×C4C23×D10111yescharacteristic
367C8—21no—
368C8—1012no—
369C8—2021no—
370C8D4×D10112yes—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—1012no—
377C22—1012no—
378C22—4041no—
379C22—4041no—
380C22—4041no—
381C22—2028no—
382C22—2021no—
383C22—2021no—
384C22—2021no—
385C22—2021no—
386C22#1569227612yescomplemented
387C22C2×D4×D10111yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—1012no—
393C4—4041no—
394C4—4041no—
395C4—4041no—
396C4—2021no—
397C4C2×D4×D10111yescharacteristic
398C4#41×D10111yescharacteristic
399C2—22no—
400C2—81no—
401C2—8081no—
402C2—2022no—
403C2—2022no—
404C2#121×D10111yescharacteristic
405C2#135×D10111yescharacteristic
406C2C2×#1569227611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#465×C101C2
socleC22×C101C22×D4
hypercenter#465D101
hypocenterC101#3050
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#15692276C2semidirect
2. C22×C8×D101C2semidirect
3. #20265690C2semidirect
4. #20265687C2semidirect
5. #20265417C2semidirect
6. #20265416C2semidirect
7. #20265412C2semidirect
8. #20265411C2semidirect
9. #20265407C2semidirect
10. #20265406C2semidirect
11. #20265403C2semidirect
12. #20265212C2semidirect
13. #20265172C2semidirect
14. #20265170C2semidirect
15. #465×C101C2semidirect
20. C2×#13108064C22semidirect
21. C2×#13108063C22semidirect
22. C2×#13108059C22semidirect
45. #15692188C22semidirect
46. #15692188C22semidirect
47. #15692187C22semidirect
48. #15692180C22semidirect
49. #15692179C22semidirect
50. #15692178C22semidirect
51. #15692177C22semidirect
52. #15692176C22semidirect
53. #15692175C22semidirect
54. C2×SD16×C101C22semidirect
58. #106×C101C22semidirect
59. #103×C101C22semidirect
60. #102×C101C22semidirect
91. C8×D101D4semidirect
172. C22×C101#135semidirect
183. C2×D101#135semidirect
204. C101#3050semidirect
216. #465D101semidirect
226. C2×SD16C2×D101semidirect
256. #106C2×D101semidirect
260. #103C2×D101semidirect
264. #102C2×D101semidirect
386. C22#15692276semidirect

Maximal subgroups

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

subgroupindex
C2×#156922762
C22×C8×D1012
#202656902
#202656872
#202654172
#202654162
#202654122
#202654112
#202654072
#202654062
#202654032
#202652122
#202651722
#202651702
#465×C1012
#3050101