Finite Group Databasebeta

← #27180630

Subgroups — #27180630

Order 12928.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692279C211yesmaximal, characteristic, complemented
2C2×D8×D101C211yesmaximal, characteristic, complemented
3#20265740C211yesmaximal, characteristic, complemented
4#20265649C211yesmaximal, characteristic, complemented
5#20265393C211yesmaximal, characteristic, complemented
6#20265392C211yesmaximal, characteristic, complemented
7#20265391C211yesmaximal, characteristic, complemented
8#20265390C211yesmaximal, characteristic, complemented
9#20265371C211yesmaximal, characteristic, complemented
10#20265370C211yesmaximal, characteristic, complemented
11#20265369C211yesmaximal, characteristic, complemented
12#20265202C211yesmaximal, characteristic, complemented
13#20265180C211yesmaximal, characteristic, complemented
14#20265170C211yesmaximal, characteristic, complemented
15#459×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×D4×D101C2211yescharacteristic
18C2×D4×D101C2211yescharacteristic
19#15692279—24no—
20C2×#13108063C2211yescharacteristic, complemented
21C2×#13108062C2211yescharacteristic, complemented
22C2×D808C2211yescharacteristic
23C2×#13108059C2211yescharacteristic
24C2×#13108059C2211yescharacteristic
25C2×#13108059C2211yescharacteristic, complemented
26D8×D101—24no—
27C2×#13108057C2211yescharacteristic
28C2×C8×D101C2211yescharacteristic
29#15692251C2211yescharacteristic, complemented
30#15692243C2211yescharacteristic, complemented
31#15692242C2211yescharacteristic, complemented
32#15692227—21no—
33#15692227—21no—
34#15692223—21no—
35#15692223—21no—
36C4×D404C2211yescharacteristic, complemented
37D4×Dic101C2211yescharacteristic, complemented
38#15692217C2211yescharacteristic, complemented
39C4×C22⋊D101—21no—
40#15692215—21no—
41#15692213C2211yescharacteristic
42#15692212C2211yescharacteristic
43C2×#13108051—21no—
44C22⋊C4×D101—21no—
45C4⋊#11717579C2211yescharacteristic, complemented
46#15692184C2211yescharacteristic, complemented
47#15692184C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692178C2211yescharacteristic, complemented
50#15692177C2211yescharacteristic, complemented
51#15692176C2211yescharacteristic, complemented
52#15692176C2211yescharacteristic, complemented
53#15692175C2211yescharacteristic, complemented
54C2×D8×C101C2211yescharacteristic
55C2×D8×C101C2211yescharacteristic, complemented
56#127×C101C2211yescharacteristic, complemented
57C4×D4×C101C2211yescharacteristic, complemented
58C4⋊C8×C101C2211yescharacteristic, complemented
59#102×C101C2211yescharacteristic, complemented
60#102×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101—21no—
63#13108073—22no—
64#13108073—22no—
65C2×Dic202C2311yescharacteristic
66C2×D404C2311yescharacteristic
67C2×D404C2311yescharacteristic
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C22⋊D101—21no—
73D4×D101—22no—
74D4×D101—22no—
75D4×D101—22no—
76D4×D101D414yescomplemented
77C22×Dic101—21no—
78C22×Dic101—21no—
79C2×C4×D101—21no—
80C2×C4×D101—21no—
81C2×C4×D101C2311yescharacteristic
82#13108063—22no—
83#13108062—22no—
84D808—22no—
85#13108059—22no—
86#13108059—22no—
87#13108059—22no—
88#13108057—22no—
89C2×#11717579C2311yescharacteristic
90C2×#11717579C2311yescharacteristic
91C8×D101—22no—
92C4⋊Dic101C2311yescharacteristic
93#13108053D411yescharacteristic
94#13108053D411yescharacteristic
95C22⋊Dic101—21no—
96C22⋊Dic101—21no—
97C22⋊Dic101—21no—
98#13108051—21no—
99#13108051—21no—
100#13108051—21no—
101#13108051—21no—
102#13108051—21no—
103#13108051D411yescharacteristic
104#13108051D411yescharacteristic
105C4×Dic101—21no—
106C4×Dic101C2311yescharacteristic
107C2×D4×C101—21no—
108C2×D4×C101—21no—
109C2×D4×C101C2311yescharacteristic
110C2×D4×C101C2311yescharacteristic
111C2×D4×C101C2311yescharacteristic
112C22×C4×C101—21no—
113D8×C101—22no—
114D8×C101—22no—
115C2×C8×C101C2311yescharacteristic
116C2×C8×C101C2311yescharacteristic
117C4⋊C4×C101C2311yescharacteristic
118C22⋊C4×C101—21no—
119C42×C101C2311yescharacteristic
120C23×C101—21no—
121C23×C101—21no—
122C23×C101—21no—
123C22×D101—21no—
124C22×D101—24no—
125C22×D101—21no—
126C22×D101—42no—
127C22×D101—81no—
128C22×D101—81no—
129C22×D101—81no—
130C22×D101—81no—
131C22×D101C2×D411yescharacteristic
132D4×C101—21no—
133D4×C101—21no—
134D4×C101—21no—
135D4×C101—21no—
136D4×C101—21no—
137D4×C101—81no—
138D4×C101—81no—
139D4×C101C2×D412yes—
140C2×C4×C101—21no—
141C2×C4×C101—81no—
142C2×C4×C101C2×D411yescharacteristic
143C2×C4×C101C2×D411yescharacteristic
144C2×C4×C101C2411yescharacteristic
145C22⋊D101—24no—
146C22⋊D101—42no—
147C22⋊D101—42no—
148C22⋊D101—81no—
149C22⋊D101—81no—
150C2×Dic101—21no—
151C2×Dic101—21no—
152C2×Dic101—42no—
153C2×Dic101—81no—
154C2×Dic101C2×D411yescharacteristic
155C2×Dic101C2×D411yescharacteristic
156C2×Dic101C2×D411yescharacteristic
157D404—21no—
158D404—21no—
159D404—21no—
160D404C2×D412yes—
161C4×D101—81no—
162C4×D101—81no—
163C4×D101C2×D412yes—
164C4×D101#4112yes—
165Dic202—21no—
166Dic202—21no—
167C8×C101—21no—
168C8×C101—21no—
169#11717579—21no—
170#11717579—21no—
171C22×C101—22no—
172C22×C101—41no—
173C22×C101—41no—
174C22×C101—41no—
175C22×C101—41no—
176C22×C101—41no—
177C22×C101—41no—
178C22×C101—41no—
179C22×C101—41no—
180C22×C101C22×D411yescharacteristic
181C2×D101—21no—
182C2×D101—21no—
183C2×D101—22no—
184C2×D101—41no—
185C2×D101—41no—
186C2×D101—41no—
187C2×D101—41no—
188C2×D101—41no—
189C2×D101—84no—
190C2×D101—84no—
191C2×D101#12112yes—
192C4×C101—21no—
193C4×C101—21no—
194C4×C101—41no—
195C4×C101C22×D411yescharacteristic
196C4×C101C2×#4111yescharacteristic
197Dic101—21no—
198Dic101—21no—
199Dic101—41no—
200Dic101—41no—
201Dic101#12112yes—
202C2×C101—42no—
203C2×C101—81no—
204C2×C101—81no—
205C2×C101C2×#12111yescharacteristic
206C2×C101#57611yescharacteristic
207C2×C101#57611yescharacteristic
208D101—22no—
209D101—42no—
210D101—81no—
211#3041—1011nomaximal, Sylow
212C101#304111yescharacteristic, complemented, Sylow
213C2×AGL(1,ℤ/8)—1011no—
214C22×D8—1011no—
215#535—1011no—
216#518—1011no—
217#463—1011no—
218#463—1011no—
219#463—1011no—
220#463—1011no—
221#459—1011no—
222#459—1011no—
223#459—1011no—
224#459D10111yescharacteristic, complemented
225#424—1011no—
226#418—1011no—
227#416—1011no—
228C2×#41—1011no—
229C22×D4—1011no—
230C22×D4—1011no—
231AGL(1,ℤ/8)—2024no—
232C2×SD16—1011no—
233C2×SD16—1011no—
234C2×D8—1011no—
235C2×D8—1011no—
236C2×D8—1011no—
237C2×D8—1011no—
238C2×D8—2024no—
239C2×D8C2×D10111yescharacteristic
240C2×D8C2×D10111yescharacteristic, complemented
241C2×M16—1011no—
242C22×C8—1011no—
243#127—1011no—
244#127C2×D10111yescharacteristic, complemented
245#124—1011no—
246#124—1011no—
247#121—2021no—
248#121—2021no—
249#120—2021no—
250#120—2021no—
251C4×D4—1011no—
252C4×D4—1011no—
253C4×D4—1011no—
254C4×D4—2021no—
255C4×D4—2021no—
256C4×D4C2×D10111yescharacteristic, complemented
257#117—1011no—
258#117—1011no—
259C2×C22⋊C4—2021no—
260C2×C22⋊C4—2021no—
261C4⋊C8—1011no—
262C4⋊C8—1011no—
263C4⋊C8—1011no—
264C4⋊C8C2×D10111yescharacteristic, complemented
265#103—1011no—
266#103—1011no—
267#102—1011no—
268#102—1011no—
269#102—1011no—
270#102—1011no—
271#102C2×D10111yescharacteristic, complemented
272#102C2×D10111yescharacteristic, complemented
273C24—2021no—
274C24—2021no—
275#41—2022no—
276#41—2022no—
277C2×Q8—1011no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—1014no—
281C2×D4—1011no—
282C2×D4—1011no—
283C2×D4—2022no—
284C2×D4—2022no—
285C2×D4—2022no—
286C2×D4—2021no—
287C2×D4—2021no—
288C2×D4—2021no—
289C2×D4—2021no—
290C2×D4—2021no—
291C2×D4C22×D10111yescharacteristic
292C2×D4C22×D10111yescharacteristic
293C2×D4C22×D10111yescharacteristic
294C22×C4—21no—
295C22×C4—1011no—
296C22×C4—2021no—
297C22×C4—2021no—
298C22×C4—2021no—
299C22×C4—2021no—
300SD16—2022no—
301SD16—2022no—
302D8—22no—
303D8—22no—
304D8—2022no—
305D8—2022no—
306D8—2022no—
307D8—2022no—
308M16—2022no—
309C2×C8—1011no—
310C2×C8—1011no—
311C2×C8—2022no—
312C2×C8C22×D10111yescharacteristic
313C2×C8C22×D10111yescharacteristic
314C4⋊C4—1011no—
315C4⋊C4—1011no—
316C4⋊C4—1011no—
317C4⋊C4C22×D10111yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—1011no—
320C22⋊C4—1011no—
321C22⋊C4—2021no—
322C22⋊C4—2021no—
323C22⋊C4—2021no—
324C22⋊C4—2021no—
325C22⋊C4—2021no—
326C22⋊C4—2021no—
327C22⋊C4—2021no—
328C22⋊C4—2021no—
329C42—1011no—
330C42—2021no—
331C42C22×D10111yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—8081no—
336C23—8081no—
337C23—8081no—
338C23—8081no—
339C23—1011no—
340C23—4042no—
341C23—2024no—
342C23—2021no—
343C23—2021no—
344Q8—2021no—
345Q8—2021no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—81no—
352D4—81no—
353D4—8081no—
354D4—8081no—
355D4—1012no—
356D4—4042no—
357D4—4042no—
358D4—2024no—
359D4—2021no—
360D4—2021no—
361D4—2021no—
362D4D4×D10112yes—
363C2×C4—21no—
364C2×C4—81no—
365C2×C4—8081no—
366C2×C4—8081no—
367C2×C4—8081no—
368C2×C4—1011no—
369C2×C4—1012no—
370C2×C4—1012no—
371C2×C4—1011no—
372C2×C4—1011no—
373C2×C4—4042no—
374C2×C4—2021no—
375C2×C4—2021no—
376C2×C4D4×D10111yescharacteristic
377C2×C4D4×D10111yescharacteristic
378C2×C4C23×D10111yescharacteristic
379C8—21no—
380C8—21no—
381C8—2021no—
382C8—2021no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—8084no—
393C22—8084no—
394C22—1012no—
395C22—4041no—
396C22—4041no—
397C22—4041no—
398C22—4041no—
399C22—4041no—
400C22—2022no—
401C22—2021no—
402C22—2021no—
403C22C2×D4×D10111yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—1012no—
408C4—4041no—
409C4—4041no—
410C4—2021no—
411C4—2021no—
412C4C2×D4×D10111yescharacteristic
413C4#41×D10111yescharacteristic
414C2—42no—
415C2—81no—
416C2—81no—
417C2—8081no—
418C2—4042no—
419C2—2022no—
420C2#121×D10111yescharacteristic
421C2#2026597811yescharacteristic
422C2#2026598111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#459×C101C2
socleC22×C101C22×D4
hypercenter#459D101
hypocenterC101#3041
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#15692279C2semidirect
2. C2×D8×D101C2semidirect
3. #20265740C2semidirect
4. #20265649C2semidirect
5. #20265393C2semidirect
6. #20265392C2semidirect
7. #20265391C2semidirect
8. #20265390C2semidirect
9. #20265371C2semidirect
10. #20265370C2semidirect
11. #20265369C2semidirect
12. #20265202C2semidirect
13. #20265180C2semidirect
14. #20265170C2semidirect
15. #459×C101C2semidirect
20. C2×#13108063C22semidirect
21. C2×#13108062C22semidirect
25. C2×#13108059C22semidirect
29. #15692251C22semidirect
30. #15692243C22semidirect
31. #15692242C22semidirect
36. C4×D404C22semidirect
37. D4×Dic101C22semidirect
38. #15692217C22semidirect
45. C4⋊#11717579C22semidirect
46. #15692184C22semidirect
47. #15692184C22semidirect
48. #15692180C22semidirect
49. #15692178C22semidirect
50. #15692177C22semidirect
51. #15692176C22semidirect
52. #15692176C22semidirect
53. #15692175C22semidirect
55. C2×D8×C101C22semidirect
56. #127×C101C22semidirect
57. C4×D4×C101C22semidirect
58. C4⋊C8×C101C22semidirect
59. #102×C101C22semidirect
60. #102×C101C22semidirect
76. D4×D101D4semidirect
212. C101#3041semidirect
224. #459D101semidirect
240. C2×D8C2×D101semidirect
244. #127C2×D101semidirect
256. C4×D4C2×D101semidirect
264. C4⋊C8C2×D101semidirect
271. #102C2×D101semidirect
272. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922792
C2×D8×D1012
#202657402
#202656492
#202653932
#202653922
#202653912
#202653902
#202653712
#202653702
#202653692
#202652022
#202651802
#202651702
#459×C1012
#3041101