Finite Group Databasebeta

← #27181605

Subgroups — #27181605

Order 12928.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692284C211yesmaximal, characteristic, complemented
2C2×#15692279C211yesmaximal, characteristic, complemented
3C2×#15692274C211yesmaximal, characteristic, complemented
4C2×SD16×D101C211yesmaximal, characteristic, complemented
5#20265747C211yesmaximal, characteristic, complemented
6#20265740C211yesmaximal, characteristic, complemented
7#20265571C211yesmaximal, characteristic, complemented
8#20265568C211yesmaximal, characteristic, complemented
9#20265565C211yesmaximal, characteristic, complemented
10#20265560C211yesmaximal, characteristic, complemented
11#20265559C211yesmaximal, characteristic, complemented
12#20265549C211yesmaximal, characteristic, complemented
13#20265548C211yesmaximal, characteristic, complemented
14#20265134C211yesmaximal, characteristic, complemented
15#496×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×#13108073C2211yescharacteristic
18C2×Q8×D101C2211yescharacteristic
19C2×D4×D101C2211yescharacteristic
20#15692284—24no—
21#15692279—24no—
22#15692274—24no—
23C2×Dic404C2211yescharacteristic, complemented
24C2×Dic404C2211yescharacteristic, complemented
25C2×#13108064C2211yescharacteristic, complemented
26C2×#13108063C2211yescharacteristic
27C2×#13108063C2211yescharacteristic
28C2×#13108062C2211yescharacteristic
29C2×#13108062C2211yescharacteristic
30C2×#13108062C2211yescharacteristic, complemented
31C2×#13108062C2211yescharacteristic, complemented
32C2×#13108062C2211yescharacteristic, complemented
33C2×#13108061C2211yescharacteristic
34SD16×D101—24no—
35C2×#13108059C2211yescharacteristic
36C2×#13108057C2211yescharacteristic
37C2×C8×D101C2211yescharacteristic
38#15692255C2211yescharacteristic, complemented
39#15692253C2211yescharacteristic, complemented
40#15692251C2211yescharacteristic
41#15692243C2211yescharacteristic
42#15692242C2211yescharacteristic
43#15692242C2211yescharacteristic, complemented
44#15692237—22no—
45#15692232—22no—
46#15692227—22no—
47#15692223—22no—
48#15692212C2211yescharacteristic
49#15692165C2211yescharacteristic, complemented
50C4×#11717579C2211yescharacteristic, complemented
51C8×Dic101C2211yescharacteristic, complemented
52C2×SD16×C101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic, complemented
54C2×D8×C101C2211yescharacteristic
55C2×D8×C101C2211yescharacteristic, complemented
56#127×C101C2211yescharacteristic
57#124×C101C2211yescharacteristic, complemented
58#97×C101C2211yescharacteristic, complemented
59C23×D101—21no—
60#13108073—22no—
61#13108073—22no—
62#13108073—22no—
63#13108073—22no—
64C2×Dic202—21no—
65C2×Dic202C2311yescharacteristic
66C2×Dic202C2311yescharacteristic
67C2×Dic202C2311yescharacteristic
68Q8×D101—22no—
69Q8×D101—22no—
70C2×D404C2311yescharacteristic
71C2×C22⋊D101—21no—
72C2×C22⋊D101—21no—
73C2×C22⋊D101—21no—
74C2×C22⋊D101—22no—
75D4×D101—22no—
76D4×D101—22no—
77C22×Dic101—21no—
78C22×Dic101—21no—
79C2×C4×D101—21no—
80C2×C4×D101C2311yescharacteristic
81Dic404—22no—
82Dic404—22no—
83#13108064—22no—
84#13108063—22no—
85#13108063—22no—
86#13108062—22no—
87#13108062—22no—
88#13108062—22no—
89#13108062—22no—
90#13108062—22no—
91#13108061—22no—
92#13108059—22no—
93#13108057D414yescomplemented
94C2×#11717579C2311yescharacteristic
95C2×#11717579C2311yescharacteristic
96C8×D101D414yescomplemented
97C4⋊Dic101—22no—
98#13108053—22no—
99#13108053D412yes—
100C22⋊Dic101—22no—
101C22⋊Dic101—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—42no—
123C22×D101—82no—
124C22×D101C2×D411yescharacteristic
125Q8×C101—21no—
126Q8×C101—21no—
127D4×C101—21no—
128D4×C101—21no—
129D4×C101—21no—
130D4×C101—21no—
131D4×C101—21no—
132D4×C101—21no—
133D4×C101—82no—
134C2×C4×C101—21no—
135C2×C4×C101C2×D412yes—
136C2×C4×C101C2411yescharacteristic
137C22⋊D101—42no—
138C22⋊D101—42no—
139C22⋊D101—42no—
140C22⋊D101—82no—
141C2×Dic101—21no—
142C2×Dic101—21no—
143C2×Dic101—21no—
144C2×Dic101—42no—
145C2×Dic101—42no—
146C2×Dic101C2×D412yes—
147C2×Dic101C2×D411yescharacteristic
148D404—21no—
149D404—21no—
150C4×D101—42no—
151C4×D101C2×D412yes—
152C4×D101C2×D412yes—
153Dic202—21no—
154Dic202—21no—
155Dic202—21no—
156Dic202—21no—
157Dic202—21no—
158Dic202—21no—
159Dic202—42no—
160C8×C101C2×D412yes—
161C8×C101C2×D412yes—
162#11717579C2×D412yes—
163#11717579C2×D412yes—
164C22×C101—41no—
165C22×C101—41no—
166C22×C101—41no—
167C22×C101—42no—
168C22×C101—42no—
169C22×C101—42no—
170C22×C101C22×D411yescharacteristic
171C2×D101—22no—
172C2×D101—41no—
173C2×D101—42no—
174C2×D101—84no—
175C2×D101#12712yes—
176C4×C101—22no—
177C4×C101—41no—
178C4×C101C22×D411yescharacteristic
179C4×C101C22×D411yescharacteristic
180Dic101—22no—
181Dic101—41no—
182Dic101—41no—
183Dic101—41no—
184Dic101#12712yes—
185C2×C101—81no—
186C2×C101—81no—
187C2×C101—81no—
188C2×C101C2×#12711yescharacteristic
189C2×C101#57712yes—
190D101—22no—
191D101—81no—
192#3154—1011nomaximal, Sylow
193C101#315411yescharacteristic, complemented, Sylow
194C2×#137—1011no—
195C2×AGL(1,ℤ/8)—1011no—
196C2×#135—1011no—
197C22×SD16—1011no—
198#536—1011no—
199#535—1011no—
200#497—1011no—
201#497—1011no—
202#496—1011no—
203#496D10111yescharacteristic, complemented
204#495—1011no—
205#495—1011no—
206#492—1011no—
207#492—1011no—
208#405—1011no—
209C2×#41—1011no—
210C2×#41—1011no—
211C22×Q8—1011no—
212C22×D4—1011no—
213#137—2024no—
214AGL(1,ℤ/8)—2024no—
215#135—2024no—
216C2×Q16—1011no—
217C2×Q16—1011no—
218C2×Q16—1011no—
219C2×SD16—1011no—
220C2×SD16—1011no—
221C2×SD16—1011no—
222C2×SD16—1011no—
223C2×SD16—1011no—
224C2×SD16—1011no—
225C2×SD16—1011no—
226C2×SD16—1011no—
227C2×SD16—2024no—
228C2×SD16C2×D10111yescharacteristic
229C2×SD16C2×D10111yescharacteristic, complemented
230C2×D8—1011no—
231C2×D8C2×D10111yescharacteristic
232C2×D8C2×D10111yescharacteristic, complemented
233C2×M16—1011no—
234C22×C8—1011no—
235C4⋊Q8—1011no—
236C4⋊Q8—1011no—
237#127—1011no—
238#127C2×D10111yescharacteristic
239#124—1011no—
240#124—1011no—
241#124—1011no—
242#124C2×D10111yescharacteristic, complemented
243#123—2022no—
244C22⋊Q8—2022no—
245#121—2022no—
246#120—2022no—
247#117—1011no—
248#97—1011no—
249#97C2×D10111yescharacteristic, complemented
250C4×C8—1011no—
251C4×C8—1011no—
252C24—2021no—
253#41—2022no—
254#41—2022no—
255#41—2022no—
256#41—2022no—
257C2×Q8—1011no—
258C2×Q8—1011no—
259C2×Q8—1011no—
260C2×Q8—2021no—
261C2×Q8—2022no—
262C2×Q8—2022no—
263C2×Q8C22×D10111yescharacteristic
264C2×D4—22no—
265C2×D4—1011no—
266C2×D4—2022no—
267C2×D4—2022no—
268C2×D4—2022no—
269C2×D4—2021no—
270C2×D4—2021no—
271C2×D4—2021no—
272C2×D4C22×D10111yescharacteristic
273C2×D4C22×D10111yescharacteristic
274C2×D4C22×D10111yescharacteristic
275C22×C4—1011no—
276C22×C4—2021no—
277C22×C4—2021no—
278C22×C4—2021no—
279Q16—2022no—
280Q16—2022no—
281Q16—2022no—
282SD16—22no—
283SD16—22no—
284SD16—2022no—
285SD16—2022no—
286SD16—2022no—
287SD16—2022no—
288SD16—2022no—
289SD16—2022no—
290SD16—2022no—
291SD16—2022no—
292D8—22no—
293D8—22no—
294D8—2022no—
295M16—1014no—
296C2×C8—1011no—
297C2×C8—1011no—
298C2×C8—1014no—
299C2×C8C22×D10111yescharacteristic
300C2×C8C22×D10111yescharacteristic
301C4⋊C4—1012no—
302C4⋊C4—2022no—
303C4⋊C4—2022no—
304C22⋊C4—22no—
305C22⋊C4—1012no—
306C22⋊C4—2022no—
307C22⋊C4—2022no—
308C22⋊C4—2022no—
309C42—1011no—
310C42C22×D10111yescharacteristic
311C23—21no—
312C23—21no—
313C23—21no—
314C23—8082no—
315C23—1011no—
316C23—4042no—
317C23—2021no—
318Q8—21no—
319Q8—21no—
320Q8—4042no—
321Q8—2021no—
322Q8—2021no—
323Q8—2021no—
324Q8—2021no—
325Q8—2021no—
326Q8—2021no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—82no—
334D4—8082no—
335D4—4042no—
336D4—4042no—
337D4—4042no—
338D4—2021no—
339D4—2021no—
340C2×C4—21no—
341C2×C4—1011no—
342C2×C4—1012no—
343C2×C4—1012no—
344C2×C4—1012no—
345C2×C4—4042no—
346C2×C4—4042no—
347C2×C4—4042no—
348C2×C4—2021no—
349C2×C4—2021no—
350C2×C4—2021no—
351C2×C4D4×D10112yes—
352C2×C4C23×D10111yescharacteristic
353C8—1012no—
354C8—1012no—
355C8D4×D10112yes—
356C8D4×D10112yes—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—8084no—
364C22—1012no—
365C22—4042no—
366C22—4041no—
367C22—2022no—
368C22C2×D4×D10111yescharacteristic
369C4—22no—
370C4—41no—
371C4—1012no—
372C4—4041no—
373C4—4041no—
374C4—4041no—
375C4—2022no—
376C4C2×D4×D10111yescharacteristic
377C4C2×D4×D10111yescharacteristic
378C2—81no—
379C2—81no—
380C2—81no—
381C2—8081no—
382C2—2022no—
383C2#127×D10111yescharacteristic
384C2#2026598712yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#496×C101C2
socleC22×C101C22×D4
hypercenter#496D101
hypocenterC101#3154
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×#15692284C2semidirect
2. C2×#15692279C2semidirect
3. C2×#15692274C2semidirect
4. C2×SD16×D101C2semidirect
5. #20265747C2semidirect
6. #20265740C2semidirect
7. #20265571C2semidirect
8. #20265568C2semidirect
9. #20265565C2semidirect
10. #20265560C2semidirect
11. #20265559C2semidirect
12. #20265549C2semidirect
13. #20265548C2semidirect
14. #20265134C2semidirect
15. #496×C101C2semidirect
23. C2×Dic404C22semidirect
24. C2×Dic404C22semidirect
25. C2×#13108064C22semidirect
30. C2×#13108062C22semidirect
31. C2×#13108062C22semidirect
32. C2×#13108062C22semidirect
38. #15692255C22semidirect
39. #15692253C22semidirect
43. #15692242C22semidirect
49. #15692165C22semidirect
50. C4×#11717579C22semidirect
51. C8×Dic101C22semidirect
53. C2×SD16×C101C22semidirect
55. C2×D8×C101C22semidirect
57. #124×C101C22semidirect
58. #97×C101C22semidirect
93. #13108057D4semidirect
96. C8×D101D4semidirect
193. C101#3154semidirect
203. #496D101semidirect
229. C2×SD16C2×D101semidirect
232. C2×D8C2×D101semidirect
242. #124C2×D101semidirect
249. #97C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922842
C2×#156922792
C2×#156922742
C2×SD16×D1012
#202657472
#202657402
#202655712
#202655682
#202655652
#202655602
#202655592
#202655492
#202655482
#202651342
#496×C1012
#3154101