Finite Group Databasebeta

← #27181616

Subgroups — #27181616

Order 12928.

596 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692282C211yesmaximal, characteristic, complemented
2C2×#15692279C211yesmaximal, characteristic, complemented
3C2×#15692276C211yesmaximal, characteristic, complemented
4C2×D8×D101C211yesmaximal, characteristic, complemented
5#20265745C211yesmaximal, characteristic, complemented
6#20265740C211yesmaximal, characteristic, complemented
7#20265566C211yesmaximal, characteristic, complemented
8#20265565C211yesmaximal, characteristic, complemented
9#20265563C211yesmaximal, characteristic, complemented
10#20265559C211yesmaximal, characteristic, complemented
11#20265558C211yesmaximal, characteristic, complemented
12#20265553C211yesmaximal, characteristic, complemented
13#20265552C211yesmaximal, characteristic, complemented
14#20265134C211yesmaximal, characteristic, complemented
15#496×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×#13108073C2211yescharacteristic
18C2×D4×D101C2211yescharacteristic
19C2×D4×D101C2211yescharacteristic
20#15692282—24no—
21#15692279—24no—
22#15692276—24no—
23C2×#13108064C2211yescharacteristic, complemented
24C2×#13108063C2211yescharacteristic, complemented
25C2×#13108063C2211yescharacteristic, complemented
26C2×#13108062C2211yescharacteristic, complemented
27C2×#13108061C2211yescharacteristic, complemented
28C2×D808C2211yescharacteristic
29C2×D808C2211yescharacteristic, complemented
30C2×#13108059C2211yescharacteristic
31C2×#13108059C2211yescharacteristic
32C2×#13108059C2211yescharacteristic, complemented
33C2×#13108059C2211yescharacteristic, complemented
34C2×#13108059C2211yescharacteristic, complemented
35D8×D101—24no—
36C2×#13108057C2211yescharacteristic
37C2×C8×D101C2211yescharacteristic
38#15692252C2211yescharacteristic, complemented
39#15692251C2211yescharacteristic, complemented
40#15692251C2211yescharacteristic, complemented
41#15692244C2211yescharacteristic, complemented
42#15692243C2211yescharacteristic, complemented
43#15692242C2211yescharacteristic, complemented
44#15692228—22no—
45#15692227—22no—
46#15692225—22no—
47#15692223—22no—
48#15692212C2211yescharacteristic
49#15692165C2211yescharacteristic, complemented
50C4×#11717579C2211yescharacteristic, complemented
51C8×Dic101C2211yescharacteristic, complemented
52C2×SD16×C101C2211yescharacteristic, complemented
53C2×SD16×C101C2211yescharacteristic, complemented
54C2×D8×C101C2211yescharacteristic
55C2×D8×C101C2211yescharacteristic, complemented
56#127×C101C2211yescharacteristic, complemented
57#124×C101C2211yescharacteristic, complemented
58#97×C101C2211yescharacteristic, complemented
59C23×D101—21no—
60C23×D101—21no—
61#13108075—22no—
62#13108075—22no—
63#13108073—22no—
64#13108073—22no—
65C2×Dic202C2311yescharacteristic
66C2×D404—21no—
67C2×D404—22no—
68C2×D404C2311yescharacteristic
69C2×D404C2311yescharacteristic
70C2×D404C2311yescharacteristic
71C2×C22⋊D101—21no—
72C2×C22⋊D101—21no—
73C2×C22⋊D101—21no—
74C2×C22⋊D101—22no—
75C2×C22⋊D101—22no—
76D4×D101—22no—
77D4×D101—22no—
78D4×D101—22no—
79D4×D101—22no—
80C22×Dic101—21no—
81C2×C4×D101—21no—
82C2×C4×D101C2311yescharacteristic
83#13108064—22no—
84#13108063—22no—
85#13108063—22no—
86#13108062—22no—
87#13108061—22no—
88D808—22no—
89D808—22no—
90#13108059—22no—
91#13108059—22no—
92#13108059—22no—
93#13108059—22no—
94#13108059—22no—
95#13108057D414yescomplemented
96C2×#11717579C2311yescharacteristic
97C2×#11717579C2311yescharacteristic
98C8×D101D414yescomplemented
99#13108053D412yes—
100C22⋊Dic101—22no—
101#13108051—22no—
102#13108051—22no—
103#13108051D412yes—
104C4×Dic101C2311yescharacteristic
105C2×Q8×C101C2311yescharacteristic
106C2×D4×C101—22no—
107C2×D4×C101C2311yescharacteristic
108C2×D4×C101C2311yescharacteristic
109C2×D4×C101C2311yescharacteristic
110SD16×C101—22no—
111SD16×C101—22no—
112D8×C101—22no—
113D8×C101—22no—
114C2×C8×C101C2311yescharacteristic
115C2×C8×C101C2311yescharacteristic
116C22⋊C4×C101—22no—
117C42×C101C2311yescharacteristic
118C23×C101—21no—
119C23×C101—21no—
120C23×C101—21no—
121C22×D101—21no—
122C22×D101—21no—
123C22×D101—21no—
124C22×D101—42no—
125C22×D101—42no—
126C22×D101—82no—
127C22×D101—82no—
128C22×D101C2×D411yescharacteristic
129Q8×C101—21no—
130Q8×C101—21no—
131D4×C101—21no—
132D4×C101—21no—
133D4×C101—21no—
134D4×C101—21no—
135D4×C101—21no—
136D4×C101—21no—
137D4×C101—82no—
138C2×C4×C101—21no—
139C2×C4×C101C2×D412yes—
140C2×C4×C101C2411yescharacteristic
141C22⋊D101—42no—
142C22⋊D101—42no—
143C22⋊D101—42no—
144C22⋊D101—82no—
145C22⋊D101—82no—
146C2×Dic101—21no—
147C2×Dic101—42no—
148C2×Dic101C2×D412yes—
149C2×Dic101C2×D411yescharacteristic
150D404—21no—
151D404—21no—
152D404—21no—
153D404—21no—
154D404—21no—
155D404—21no—
156D404—42no—
157D404—82no—
158C4×D101—42no—
159C4×D101C2×D412yes—
160C4×D101C2×D412yes—
161Dic202—21no—
162Dic202—21no—
163C8×C101C2×D412yes—
164C8×C101C2×D412yes—
165#11717579C2×D412yes—
166#11717579C2×D412yes—
167C22×C101—41no—
168C22×C101—41no—
169C22×C101—41no—
170C22×C101—42no—
171C22×C101—42no—
172C22×C101—42no—
173C22×C101C22×D411yescharacteristic
174C2×D101—22no—
175C2×D101—41no—
176C2×D101—41no—
177C2×D101—41no—
178C2×D101—42no—
179C2×D101—42no—
180C2×D101—42no—
181C2×D101—84no—
182C2×D101—84no—
183C2×D101#12712yes—
184C4×C101—22no—
185C4×C101—41no—
186C4×C101C22×D411yescharacteristic
187C4×C101C22×D411yescharacteristic
188Dic101—22no—
189Dic101—41no—
190Dic101#12712yes—
191C2×C101—81no—
192C2×C101—81no—
193C2×C101—81no—
194C2×C101C2×#12711yescharacteristic
195C2×C101#57612yes—
196D101—22no—
197D101—81no—
198D101—81no—
199D101—81no—
200#3155—1011nomaximal, Sylow
201C101#315511yescharacteristic, complemented, Sylow
202C2×AGL(1,ℤ/8)—1011no—
203C2×AGL(1,ℤ/8)—1011no—
204C2×#135—1011no—
205C22×D8—1011no—
206#535—1011no—
207#535—1011no—
208#496—1011no—
209#496—1011no—
210#496—1011no—
211#496D10111yescharacteristic, complemented
212#495—1011no—
213#495—1011no—
214#493—1011no—
215#493—1011no—
216#405—1011no—
217C2×#41—1011no—
218C2×#41—1011no—
219C22×D4—1011no—
220C22×D4—1011no—
221AGL(1,ℤ/8)—2024no—
222AGL(1,ℤ/8)—2024no—
223#135—2024no—
224C2×Q16—1011no—
225C2×SD16—1011no—
226C2×SD16—1011no—
227C2×SD16—1011no—
228C2×SD16—1011no—
229C2×SD16C2×D10111yescharacteristic, complemented
230C2×SD16C2×D10111yescharacteristic, complemented
231C2×D8—1011no—
232C2×D8—1011no—
233C2×D8—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#127—1011no—
245#127—1011no—
246#127C2×D10111yescharacteristic, complemented
247#124—1011no—
248#124—1011no—
249#124—1011no—
250#124C2×D10111yescharacteristic, complemented
251#121—2022no—
252#121—2022no—
253#120—2022no—
254#120—2022no—
255#117—1011no—
256#97—1011no—
257#97C2×D10111yescharacteristic, complemented
258C4×C8—1011no—
259C4×C8—1011no—
260C24—2021no—
261C24—2021no—
262#41—2022no—
263#41—2022no—
264#41—2022no—
265#41—2022no—
266C2×Q8—1011no—
267C2×Q8C22×D10111yescharacteristic
268C2×D4—22no—
269C2×D4—1011no—
270C2×D4—1011no—
271C2×D4—1011no—
272C2×D4—2021no—
273C2×D4—2022no—
274C2×D4—2022no—
275C2×D4—2022no—
276C2×D4—2022no—
277C2×D4—2022no—
278C2×D4—2022no—
279C2×D4—2021no—
280C2×D4—2021no—
281C2×D4—2021no—
282C2×D4—2022no—
283C2×D4C22×D10111yescharacteristic
284C2×D4C22×D10111yescharacteristic
285C2×D4C22×D10111yescharacteristic
286C22×C4—1011no—
287C22×C4—2021no—
288C22×C4—2021no—
289Q16—2022no—
290SD16—22no—
291SD16—22no—
292SD16—2022no—
293SD16—2022no—
294SD16—2022no—
295SD16—2022no—
296D8—22no—
297D8—22no—
298D8—2022no—
299D8—2022no—
300D8—2022no—
301D8—2022no—
302D8—2022no—
303D8—2022no—
304D8—2022no—
305M16—1014no—
306C2×C8—1011no—
307C2×C8—1011no—
308C2×C8—1014no—
309C2×C8C22×D10111yescharacteristic
310C2×C8C22×D10111yescharacteristic
311C4⋊C4—1012no—
312C22⋊C4—22no—
313C22⋊C4—1012no—
314C22⋊C4—2022no—
315C22⋊C4—2022no—
316C22⋊C4—2022no—
317C42—1011no—
318C42C22×D10111yescharacteristic
319C23—21no—
320C23—21no—
321C23—21no—
322C23—8082no—
323C23—8082no—
324C23—1011no—
325C23—4042no—
326C23—4042no—
327C23—2021no—
328C23—2021no—
329C23—2021no—
330Q8—21no—
331Q8—21no—
332Q8—2021no—
333Q8—2021no—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—82no—
341D4—8082no—
342D4—8082no—
343D4—8082no—
344D4—4042no—
345D4—4042no—
346D4—4042no—
347D4—4042no—
348D4—2021no—
349D4—2021no—
350D4—2021no—
351D4—2021no—
352D4—2021no—
353D4—2021no—
354C2×C4—21no—
355C2×C4—1011no—
356C2×C4—1012no—
357C2×C4—1012no—
358C2×C4—1012no—
359C2×C4—4042no—
360C2×C4—4042no—
361C2×C4—2021no—
362C2×C4D4×D10112yes—
363C2×C4C23×D10111yescharacteristic
364C8—1012no—
365C8—1012no—
366C8D4×D10112yes—
367C8D4×D10112yes—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—8084no—
375C22—8084no—
376C22—1012no—
377C22—4041no—
378C22—4041no—
379C22—4041no—
380C22—4042no—
381C22—4042no—
382C22—4042no—
383C22—2022no—
384C22C2×D4×D10111yescharacteristic
385C4—22no—
386C4—41no—
387C4—1012no—
388C4—4041no—
389C4—2022no—
390C4C2×D4×D10111yescharacteristic
391C4C2×D4×D10111yescharacteristic
392C2—81no—
393C2—81no—
394C2—81no—
395C2—8081no—
396C2—8081no—
397C2—8081no—
398C2—2022no—
399C2#127×D10111yescharacteristic
400C2#2026598112yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#496×C101C2
socleC22×C101C22×D4
hypercenter#496D101
hypocenterC101#3155
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#15692282C2semidirect
2. C2×#15692279C2semidirect
3. C2×#15692276C2semidirect
4. C2×D8×D101C2semidirect
5. #20265745C2semidirect
6. #20265740C2semidirect
7. #20265566C2semidirect
8. #20265565C2semidirect
9. #20265563C2semidirect
10. #20265559C2semidirect
11. #20265558C2semidirect
12. #20265553C2semidirect
13. #20265552C2semidirect
14. #20265134C2semidirect
15. #496×C101C2semidirect
23. C2×#13108064C22semidirect
24. C2×#13108063C22semidirect
25. C2×#13108063C22semidirect
26. C2×#13108062C22semidirect
27. C2×#13108061C22semidirect
29. C2×D808C22semidirect
32. C2×#13108059C22semidirect
33. C2×#13108059C22semidirect
34. C2×#13108059C22semidirect
38. #15692252C22semidirect
39. #15692251C22semidirect
40. #15692251C22semidirect
41. #15692244C22semidirect
42. #15692243C22semidirect
43. #15692242C22semidirect
49. #15692165C22semidirect
50. C4×#11717579C22semidirect
51. C8×Dic101C22semidirect
52. C2×SD16×C101C22semidirect
53. C2×SD16×C101C22semidirect
55. C2×D8×C101C22semidirect
56. #127×C101C22semidirect
57. #124×C101C22semidirect
58. #97×C101C22semidirect
95. #13108057D4semidirect
98. C8×D101D4semidirect
201. C101#3155semidirect
211. #496D101semidirect
229. C2×SD16C2×D101semidirect
230. C2×SD16C2×D101semidirect
240. C2×D8C2×D101semidirect
246. #127C2×D101semidirect
250. #124C2×D101semidirect
257. #97C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
C2×#156922792
C2×#156922762
C2×D8×D1012
#202657452
#202657402
#202655662
#202655652
#202655632
#202655592
#202655582
#202655532
#202655522
#202651342
#496×C1012
#3155101