Finite Group Databasebeta

← #27180744

Subgroups — #27180744

Order 12928.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692282C211yesmaximal, characteristic, complemented
2C22×#13108057C211yesmaximal, characteristic, complemented
3#20265690C211yesmaximal, characteristic, complemented
4#121×D101C211yesmaximal, characteristic, complemented
5#20265431C211yesmaximal, characteristic, complemented
6#20265429C211yesmaximal, characteristic, complemented
7#20265426C211yesmaximal, characteristic, complemented
8#20265420C211yesmaximal, characteristic, complemented
9#20265413C211yesmaximal, characteristic, complemented
10#20265411C211yesmaximal, characteristic, complemented
11#20265408C211yesmaximal, characteristic, complemented
12#20265216C211yesmaximal, characteristic, complemented
13#20265180C211yesmaximal, characteristic, complemented
14#20265177C211yesmaximal, characteristic, complemented
15#465×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×D4×D101—21no—
18C2×D4×D101C2211yescharacteristic
19C22×C4×D101C2211yescharacteristic
20#15692282—24no—
21C2×#13108061C2211yescharacteristic, complemented
22C2×D808C2211yescharacteristic, complemented
23C2×#13108059C2211yescharacteristic, complemented
24C2×#13108057—24no—
25C2×#13108057C2211yescharacteristic
26C2×#13108057C2211yescharacteristic
27C2×#13108057C2211yescharacteristic, complemented
28C2×#13108057C2211yescharacteristic, complemented
29C22×#11717579C2211yescharacteristic, complemented
30#15692240—21no—
31#15692231C2211yescharacteristic, complemented
32#15692230C2211yescharacteristic
33#15692230C2211yescharacteristic, complemented
34#15692229C2211yescharacteristic
35#15692229C2211yescharacteristic, complemented
36#15692228—21no—
37#15692227—21no—
38#15692226C2211yescharacteristic
39#15692225—21no—
40#15692217—21no—
41#15692215—21no—
42#15692213C2211yescharacteristic
43C4⋊C4×D101C2211yescharacteristic
44C22⋊C4×D101—21no—
45#15692189C2211yescharacteristic, complemented
46#15692188C2211yescharacteristic, complemented
47#15692186C2211yescharacteristic, complemented
48#15692179C2211yescharacteristic, complemented
49#15692177C2211yescharacteristic, complemented
50#15692177C2211yescharacteristic, complemented
51#15692176C2211yescharacteristic, complemented
52#15692175C2211yescharacteristic, complemented
53#15692175C2211yescharacteristic, complemented
54C2×SD16×C101C2211yescharacteristic, complemented
55C22×C8×C101C2211yescharacteristic, complemented
56C22⋊Q8×C101C2211yescharacteristic, complemented
57#121×C101C2211yescharacteristic
58#106×C101C2211yescharacteristic, complemented
59#103×C101C2211yescharacteristic, complemented
60#102×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101D411yescharacteristic
63#13108075—22no—
64#13108075—22no—
65C2×D404—21no—
66C2×D404—21no—
67C2×D404—21no—
68C2×D404C2311yescharacteristic
69C2×D404C2311yescharacteristic
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C22⋊D101—21no—
73C2×C22⋊D101—21no—
74D4×D101—22no—
75D4×D101—22no—
76D4×D101—44no—
77C22×Dic101D411yescharacteristic
78C2×C4×D101—22no—
79C2×C4×D101—22no—
80C2×C4×D101—21no—
81C2×C4×D101—21no—
82C2×C4×D101D411yescharacteristic
83C2×C4×D101D411yescharacteristic
84C2×C4×D101C2311yescharacteristic
85C2×C4×D101C2311yescharacteristic
86#13108061—22no—
87D808—22no—
88#13108059—22no—
89#13108057—22no—
90#13108057—22no—
91#13108057—22no—
92#13108057D414yescomplemented
93C2×#11717579—22no—
94C2×#11717579C2311yescharacteristic
95C2×#11717579C2311yescharacteristic
96C4⋊Dic101C2311yescharacteristic
97C4⋊Dic101C2311yescharacteristic
98#13108053—21no—
99#13108053—21no—
100C22⋊Dic101—21no—
101#13108051—21no—
102#13108051—21no—
103#13108051—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051—21no—
107C4×Dic101—21no—
108C2×Q8×C101C2311yescharacteristic
109C2×D4×C101—21no—
110C2×D4×C101C2311yescharacteristic
111C22×C4×C101C2311yescharacteristic
112SD16×C101—22no—
113C2×C8×C101—22no—
114C2×C8×C101C2311yescharacteristic
115C2×C8×C101C2311yescharacteristic
116C4⋊C4×C101—21no—
117C4⋊C4×C101C2311yescharacteristic
118C4⋊C4×C101C2311yescharacteristic
119C22⋊C4×C101—21no—
120C22⋊C4×C101—21no—
121C23×C101—21no—
122C23×C101C2×D411yescharacteristic
123C22×D101—22no—
124C22×D101—22no—
125C22×D101—21no—
126C22×D101—21no—
127C22×D101—41no—
128C22×D101—42no—
129C22×D101—42no—
130C22×D101—42no—
131C22×D101C2×D411yescharacteristic
132C22×D101C2×D411yescharacteristic
133Q8×C101—21no—
134Q8×C101—21no—
135D4×C101—21no—
136D4×C101—21no—
137D4×C101—42no—
138C2×C4×C101—21no—
139C2×C4×C101—21no—
140C2×C4×C101—21no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101C2×D411yescharacteristic
144C2×C4×C101C2411yescharacteristic
145C22⋊D101—42no—
146C22⋊D101—42no—
147C22⋊D101—42no—
148C22⋊D101—42no—
149C2×Dic101—22no—
150C2×Dic101—21no—
151C2×Dic101—21no—
152C2×Dic101C2×D411yescharacteristic
153C2×Dic101C2×D411yescharacteristic
154D404—21no—
155D404—21no—
156D404—21no—
157D404—21no—
158D404—42no—
159D404—42no—
160D404—42no—
161C4×D101—22no—
162C4×D101—21no—
163C4×D101—21no—
164C4×D101—22no—
165C4×D101—42no—
166C4×D101—42no—
167C4×D101C2×D412yes—
168C4×D101#4112yes—
169C8×C101—21no—
170C8×C101C2×D412yescomplemented
171#11717579—21no—
172#11717579C2×D412yescomplemented
173C22×C101—21no—
174C22×C101—21no—
175C22×C101—41no—
176C22×C101—41no—
177C22×C101—41no—
178C22×C101AGL(1,ℤ/8)12yescomplemented
179C22×C101C22×D411yescharacteristic
180C2×D101—21no—
181C2×D101—21no—
182C2×D101—21no—
183C2×D101—21no—
184C2×D101—21no—
185C2×D101—44no—
186C2×D101—41no—
187C2×D101—41no—
188C2×D101—41no—
189C2×D101—41no—
190C2×D101—41no—
191C2×D101—41no—
192C2×D101—84no—
193C2×D101#12112yes—
194C4×C101—21no—
195C4×C101—41no—
196C4×C101—41no—
197C4×C101—41no—
198C4×C101C22×D411yescharacteristic
199C4×C101C2×#4111yescharacteristic
200Dic101—21no—
201Dic101—41no—
202Dic101—41no—
203Dic101#12112yes—
204C2×C101—22no—
205C2×C101—81no—
206C2×C101C2×#12111yescharacteristic
207C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
208C2×C101#57511yescharacteristic
209D101—22no—
210D101—41no—
211D101—81no—
212D101—81no—
213#3055—1011nomaximal, Sylow
214C101#305511yescharacteristic, complemented, Sylow
215C2×AGL(1,ℤ/8)—1011no—
216C22×M16—1011no—
217#525—1011no—
218C2×#121—1011no—
219#469—1011no—
220#469—1011no—
221#468—1011no—
222#468—1011no—
223#466—1011no—
224#466—1011no—
225#465—1011no—
226#465D10111yescharacteristic, complemented
227#428—1011no—
228#418—1011no—
229#417—1011no—
230C2×#41—1011no—
231C22×D4—1011no—
232C22×D4—2021no—
233C23×C4—1011no—
234AGL(1,ℤ/8)—2024no—
235C2×SD16—1011no—
236C2×SD16C2×D10111yescharacteristic, complemented
237C2×D8—1011no—
238C2×D8—1011no—
239C2×M16—1011no—
240C2×M16—1011no—
241C2×M16—1011no—
242C2×M16—1011no—
243C2×M16—2024no—
244C22×C8—1011no—
245C22×C8C2×D10111yescharacteristic, complemented
246#123—2021no—
247C22⋊Q8—1011no—
248C22⋊Q8C2×D10111yescharacteristic, complemented
249#121—1011no—
250#121—1011no—
251#121—1011no—
252#121—1011no—
253#121—1011no—
254#121—2021no—
255#121—2021no—
256#121C2×D10111yescharacteristic
257#120—2021no—
258C4×D4—2021no—
259C4×D4—2021no—
260#117—1011no—
261C2×C4⋊C4—1011no—
262C2×C22⋊C4—2021no—
263#107—1011no—
264#107—1011no—
265#106—1011no—
266#106C2×D10111yescharacteristic, complemented
267#103—1011no—
268#103C2×D10111yescharacteristic, complemented
269#102—1011no—
270#102—1011no—
271#102—1011no—
272#102—1011no—
273#102—1011no—
274#102C2×D10111yescharacteristic, complemented
275C24—1011no—
276C24—2021no—
277#41—2022no—
278#41—2022no—
279C2×Q8C22×D10111yescharacteristic
280C2×D4—21no—
281C2×D4—1011no—
282C2×D4—1011no—
283C2×D4—4044no—
284C2×D4—2021no—
285C2×D4—2021no—
286C2×D4—2021no—
287C2×D4—2022no—
288C2×D4—2022no—
289C2×D4—2021no—
290C2×D4—2021no—
291C2×D4—2021no—
292C2×D4—2021no—
293C2×D4C22×D10111yescharacteristic
294C22×C4—1011no—
295C22×C4—1011no—
296C22×C4—1011no—
297C22×C4—1011no—
298C22×C4—1011no—
299C22×C4—2021no—
300C22×C4—2022no—
301C22×C4—2022no—
302C22×C4—2021no—
303C22×C4C22×D10111yescharacteristic
304SD16—22no—
305SD16—2022no—
306D8—2022no—
307D8—2022no—
308M16—1014no—
309M16—2022no—
310M16—2022no—
311M16—2022no—
312C2×C8—22no—
313C2×C8—1011no—
314C2×C8—1011no—
315C2×C8—2022no—
316C2×C8C22×D10111yescharacteristic
317C2×C8C22×D10111yescharacteristic
318C4⋊C4—21no—
319C4⋊C4—1011no—
320C4⋊C4—1011no—
321C4⋊C4—2021no—
322C4⋊C4—2021no—
323C4⋊C4C22×D10111yescharacteristic
324C4⋊C4C22×D10111yescharacteristic
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—2021no—
328C22⋊C4—2021no—
329C22⋊C4—2021no—
330C22⋊C4—2021no—
331C22⋊C4—2021no—
332C22⋊C4—2021no—
333C22⋊C4—2021no—
334C42—2021no—
335C23—21no—
336C23—1011no—
337C23—1011no—
338C23—4041no—
339C23—4042no—
340C23—4042no—
341C23—4042no—
342C23—2022no—
343C23—2021no—
344C23—2021no—
345C23—2022no—
346C23D4×D10111yescharacteristic
347Q8—21no—
348Q8—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—4042no—
353D4—4042no—
354D4—4042no—
355D4—4042no—
356D4—4042no—
357D4—4042no—
358D4—4042no—
359D4—2021no—
360D4—2021no—
361D4—2021no—
362D4—2021no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—1011no—
369C2×C4—1012no—
370C2×C4—1012no—
371C2×C4—1011no—
372C2×C4—4042no—
373C2×C4—4042no—
374C2×C4—2022no—
375C2×C4—2022no—
376C2×C4—2022no—
377C2×C4—2021no—
378C2×C4—2021no—
379C2×C4—2021no—
380C2×C4—2021no—
381C2×C4D4×D10111yescharacteristic
382C2×C4C23×D10111yescharacteristic
383C8—21no—
384C8—1012no—
385C8—2021no—
386C8D4×D10112yescomplemented
387C22—21no—
388C22—21no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—8084no—
393C22—1012no—
394C22—4044no—
395C22—4041no—
396C22—4041no—
397C22—4041no—
398C22—4041no—
399C22—4041no—
400C22—4041no—
401C22—2021no—
402C22—2021no—
403C22—2021no—
404C22—2021no—
405C22—2021no—
406C22#1569228212yescomplemented
407C22C2×D4×D10111yescharacteristic
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—1012no—
413C4—4041no—
414C4—4041no—
415C4—2021no—
416C4C2×D4×D10111yescharacteristic
417C4#41×D10111yescharacteristic
418C2—22no—
419C2—81no—
420C2—8081no—
421C2—8081no—
422C2—4041no—
423C2—2022no—
424C2#121×D10111yescharacteristic
425C2C2×#1569228211yescharacteristic
426C2#2026597411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#465×C101C2
socleC22×C101C22×D4
hypercenter#465D101
hypocenterC101#3055
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×#15692282C2semidirect
2. C22×#13108057C2semidirect
3. #20265690C2semidirect
4. #121×D101C2semidirect
5. #20265431C2semidirect
6. #20265429C2semidirect
7. #20265426C2semidirect
8. #20265420C2semidirect
9. #20265413C2semidirect
10. #20265411C2semidirect
11. #20265408C2semidirect
12. #20265216C2semidirect
13. #20265180C2semidirect
14. #20265177C2semidirect
15. #465×C101C2semidirect
21. C2×#13108061C22semidirect
22. C2×D808C22semidirect
23. C2×#13108059C22semidirect
27. C2×#13108057C22semidirect
28. C2×#13108057C22semidirect
29. C22×#11717579C22semidirect
31. #15692231C22semidirect
33. #15692230C22semidirect
35. #15692229C22semidirect
45. #15692189C22semidirect
46. #15692188C22semidirect
47. #15692186C22semidirect
48. #15692179C22semidirect
49. #15692177C22semidirect
50. #15692177C22semidirect
51. #15692176C22semidirect
52. #15692175C22semidirect
53. #15692175C22semidirect
54. C2×SD16×C101C22semidirect
55. C22×C8×C101C22semidirect
56. C22⋊Q8×C101C22semidirect
58. #106×C101C22semidirect
59. #103×C101C22semidirect
60. #102×C101C22semidirect
92. #13108057D4semidirect
170. C8×C101C2×D4semidirect
172. #11717579C2×D4semidirect
178. C22×C101AGL(1,ℤ/8)semidirect
214. C101#3055semidirect
226. #465D101semidirect
236. C2×SD16C2×D101semidirect
245. C22×C8C2×D101semidirect
248. C22⋊Q8C2×D101semidirect
266. #106C2×D101semidirect
268. #103C2×D101semidirect
274. #102C2×D101semidirect
386. C8D4×D101semidirect
406. C22#15692282semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
C22×#131080572
#202656902
#121×D1012
#202654312
#202654292
#202654262
#202654202
#202654132
#202654112
#202654082
#202652162
#202651802
#202651772
#465×C1012
#3055101