Finite Group Databasebeta

← #27179986

Subgroups — #27179986

Order 12928.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D101C211yesmaximal, characteristic
2#20265653C211yesmaximal, characteristic, complemented
3#20265649C211yesmaximal, characteristic
4#20265280C211yesmaximal, characteristic, complemented
5#20265277C211yesmaximal, characteristic, complemented
6#20265275C211yesmaximal, characteristic, complemented
7#20265274C211yesmaximal, characteristic, complemented
8C4×#13108063C211yesmaximal, characteristic, complemented
9SD16×Dic101C211yesmaximal, characteristic, complemented
10#20265264C211yesmaximal, characteristic, complemented
11#20265212C211yesmaximal, characteristic, complemented
12#20265184C211yesmaximal, characteristic, complemented
13#20265180C211yesmaximal, characteristic
14#20265131C211yesmaximal, characteristic, complemented
15C4×SD16×C101C211yesmaximal, characteristic, complemented
16C2×Q8×D101C2211yescharacteristic
17C2×D4×D101C2211yescharacteristic
18C2×#13108063C2211yescharacteristic
19C2×#13108062C2211yescharacteristic
20C2×#13108061C2211yescharacteristic
21SD16×D101C418yescomplemented
22C2×#13108057C2211yescharacteristic
23C2×C8×D101C2211yescharacteristic
24Q8×Dic101C2211yescharacteristic, complemented
25C4×Dic202—21no—
26C4×Dic202C2211yescharacteristic, complemented
27#15692220—21no—
28#15692220C2211yescharacteristic, complemented
29C4×D404C2211yescharacteristic
30D4×Dic101C2211yescharacteristic
31#15692217C2211yescharacteristic
32C4×C22⋊D101—21no—
33#15692215—21no—
34#15692213—21no—
35#15692213C2211yescharacteristic
36#15692213C2211yescharacteristic
37#15692212—21no—
38#15692212C2211yescharacteristic
39C2×#13108051—21no—
40C22⋊C4×D101—21no—
41#15692188C2211yescharacteristic, complemented
42#15692188C2211yescharacteristic, complemented
43#15692187C2211yescharacteristic, complemented
44#15692180C2211yescharacteristic, complemented
45#15692179C2211yescharacteristic, complemented
46#15692178C2211yescharacteristic, complemented
47#15692177C2211yescharacteristic
48#15692176C2211yescharacteristic
49#15692175C2211yescharacteristic
50#15692166C2211yescharacteristic, complemented
51#15692165C2211yescharacteristic, complemented
52C8×Dic101C2211yescharacteristic, complemented
53C2×SD16×C101C2211yescharacteristic
54C4×Q8×C101C2211yescharacteristic, complemented
55C4×D4×C101C2211yescharacteristic
56#106×C101C2211yescharacteristic, complemented
57#103×C101C2211yescharacteristic, complemented
58#102×C101C2211yescharacteristic
59C4×C8×C101C2211yescharacteristic, complemented
60C23×D101—21no—
61C2×Dic202—21no—
62C2×Dic202C2311yescharacteristic
63Q8×D101—22no—
64Q8×D101C2×C414yescomplemented
65C2×D404C2311yescharacteristic
66C2×C22⋊D101—21no—
67D4×D101—22no—
68D4×D101C2×C414yes—
69C22×Dic101—21no—
70C2×C4×D101—21no—
71C2×C4×D101—21no—
72C2×C4×D101—21no—
73C2×C4×D101C2311yescharacteristic
74#13108063C2×C414yescomplemented
75#13108062C2×C414yescomplemented
76#13108061C2×C414yescomplemented
77#13108057—22no—
78C2×#11717579C2311yescharacteristic
79C2×#11717579C2311yescharacteristic
80C8×D101C2×C414yescomplemented
81C4⋊Dic101—21no—
82C4⋊Dic101C2311yescharacteristic
83C4⋊Dic101C2311yescharacteristic
84#13108053—21no—
85#13108053—21no—
86#13108053D411yescharacteristic
87#13108053D411yescharacteristic
88C22⋊Dic101—21no—
89#13108051—21no—
90#13108051—21no—
91#13108051—21no—
92#13108051—21no—
93#13108051D411yescharacteristic
94#13108051D411yescharacteristic
95C4×Dic101—21no—
96C4×Dic101—21no—
97C4×Dic101—21no—
98C4×Dic101—21no—
99C4×Dic101—21no—
100C4×Dic101C2311yescharacteristic
101C2×Q8×C101C2311yescharacteristic
102C2×D4×C101C2311yescharacteristic
103C22×C4×C101—21no—
104SD16×C101C2×C414yescomplemented
105C2×C8×C101C2311yescharacteristic
106C2×C8×C101C2311yescharacteristic
107C4⋊C4×C101—21no—
108C4⋊C4×C101C2311yescharacteristic
109C4⋊C4×C101C2311yescharacteristic
110C22⋊C4×C101—21no—
111C42×C101—21no—
112C42×C101C2311yescharacteristic
113C23×C101—21no—
114C22×D101—21no—
115C22×D101—24no—
116C22×D101—81no—
117C22×D101—81no—
118C22×D101C2×D411yescharacteristic
119Q8×C101—21no—
120Q8×C101C22×C412yes—
121D4×C101—21no—
122D4×C101C22×C412yes—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—81no—
127C2×C4×C101C2×D411yescharacteristic
128C2×C4×C101C2×D411yescharacteristic
129C2×C4×C101C2411yescharacteristic
130C22⋊D101—24no—
131C2×Dic101—21no—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101—81no—
135C2×Dic101C2×D411yescharacteristic
136C2×Dic101C2×D411yescharacteristic
137C2×Dic101C2×D411yescharacteristic
138D404—21no—
139D404C22×C412yes—
140C4×D101—24no—
141C4×D101—81no—
142C4×D101—81no—
143C4×D101C22×C412yes—
144C4×D101#4112yes—
145Dic202—21no—
146Dic202—24no—
147Dic202C22×C412yes—
148C8×C101—21no—
149C8×C101C22×C412yes—
150#11717579—21no—
151#11717579C22×C412yes—
152C22×C101—22no—
153C22×C101—41no—
154C22×C101—41no—
155C22×C101C22×D411yescharacteristic
156C2×D101—21no—
157C2×D101—21no—
158C2×D101—22no—
159C2×D101—41no—
160C2×D101—41no—
161C2×D101—84no—
162C2×D101C4×D412yes—
163C4×C101—22no—
164C4×C101—21no—
165C4×C101—21no—
166C4×C101—41no—
167C4×C101—41no—
168C4×C101C23×C411yescharacteristic
169C4×C101C2×#4111yescharacteristic
170Dic101—22no—
171Dic101—21no—
172Dic101—21no—
173Dic101—41no—
174Dic101—41no—
175Dic101C4×D412yes—
176C2×C101—42no—
177C2×C101C2×C4×D411yescharacteristic
178C2×C101#57711yescharacteristic
179C2×C101#57711yescharacteristic
180D101—22no—
181D101—42no—
182#2949—1011nomaximal, Sylow
183C101#294911yescharacteristic, complemented, Sylow
184C22×SD16—1011no—
185#519—1011no—
186#518—1011no—
187#440—1011no—
188#440—1011no—
189#440—1011no—
190#440—1011no—
191C4×SD16—1011no—
192C4×SD16—1011no—
193C4×SD16—1011no—
194C4×SD16D10111yescharacteristic, complemented
195#427—1011no—
196#419—1011no—
197#418—1011no—
198#405—1011no—
199C22×Q8—1011no—
200C22×D4—1011no—
201C2×SD16—1011no—
202C2×SD16—1011no—
203C2×SD16—1018no—
204C2×SD16—1011no—
205C2×SD16C2×D10111yescharacteristic
206C2×M16—1011no—
207C22×C8—1011no—
208C4×Q8—1011no—
209C4×Q8—1011no—
210C4×Q8—1011no—
211C4×Q8—2021no—
212C4×Q8—2021no—
213C4×Q8C2×D10111yescharacteristic, complemented
214C4×D4—1011no—
215C4×D4—1011no—
216C4×D4—1011no—
217C4×D4—2021no—
218C4×D4—2021no—
219C4×D4C2×D10111yescharacteristic
220#117—1011no—
221#117—1011no—
222#117—1011no—
223#117—2021no—
224#117—2021no—
225C2×C22⋊C4—2021no—
226C2×C22⋊C4—2021no—
227#107—1011no—
228#107—1011no—
229#106—1011no—
230#106C2×D10111yescharacteristic, complemented
231#103—1011no—
232#103—1011no—
233#103—1011no—
234#103C2×D10111yescharacteristic, complemented
235#102—1011no—
236#102—1011no—
237#102—1011no—
238#102C2×D10111yescharacteristic
239#97—1011no—
240#97—1011no—
241C4×C8—1011no—
242C4×C8C2×D10111yescharacteristic, complemented
243C24—2021no—
244C2×Q8—1014no—
245C2×Q8—1011no—
246C2×Q8—2021no—
247C2×Q8—2022no—
248C2×Q8C22×D10111yescharacteristic
249C2×D4—1014no—
250C2×D4—1011no—
251C2×D4—2022no—
252C2×D4—2021no—
253C2×D4C22×D10111yescharacteristic
254C22×C4—21no—
255C22×C4—1011no—
256C22×C4—2021no—
257C22×C4—2021no—
258C22×C4—2021no—
259C22×C4—2021no—
260SD16—1014no—
261SD16—1014no—
262SD16—1014no—
263SD16C4×D10114yescomplemented
264M16—2022no—
265C2×C8—1011no—
266C2×C8—1014no—
267C2×C8—1011no—
268C2×C8C22×D10111yescharacteristic
269C2×C8C22×D10111yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—1011no—
272C4⋊C4—1011no—
273C4⋊C4—1011no—
274C4⋊C4—1011no—
275C4⋊C4—2021no—
276C4⋊C4—2021no—
277C4⋊C4—2021no—
278C4⋊C4C22×D10111yescharacteristic
279C4⋊C4C22×D10111yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—1011no—
282C22⋊C4—1011no—
283C22⋊C4—2021no—
284C22⋊C4—2021no—
285C22⋊C4—2021no—
286C22⋊C4—2021no—
287C22⋊C4—2021no—
288C42—21no—
289C42—1011no—
290C42—2021no—
291C42—2021no—
292C42—2021no—
293C42—2021no—
294C42—2021no—
295C42C22×D10111yescharacteristic
296C23—21no—
297C23—8081no—
298C23—8081no—
299C23—1011no—
300C23—2024no—
301C23—2021no—
302Q8—21no—
303Q8—1012no—
304Q8—2024no—
305Q8—2021no—
306Q8C2×C4×D10112yes—
307D4—21no—
308D4—1012no—
309D4—2024no—
310D4—2021no—
311D4C2×C4×D10112yes—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—8081no—
317C2×C4—8081no—
318C2×C4—8081no—
319C2×C4—1011no—
320C2×C4—1012no—
321C2×C4—1012no—
322C2×C4—1011no—
323C2×C4—1011no—
324C2×C4—2024no—
325C2×C4—2021no—
326C2×C4—2021no—
327C2×C4—2021no—
328C2×C4D4×D10111yescharacteristic
329C2×C4D4×D10111yescharacteristic
330C2×C4C23×D10111yescharacteristic
331C8—21no—
332C8—1012no—
333C8—2021no—
334C8C2×C4×D10112yes—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—8084no—
339C22—1012no—
340C22—4041no—
341C22—4041no—
342C22—2022no—
343C22—2021no—
344C22—2021no—
345C22C2×D4×D10111yescharacteristic
346C4—21no—
347C4—22no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—1012no—
352C4—4041no—
353C4—4041no—
354C4—2021no—
355C4—2022no—
356C4—2021no—
357C4C22×C4×D10111yescharacteristic
358C4#41×D10111yescharacteristic
359C2—42no—
360C2—4042no—
361C2—2022no—
362C2C4×D4×D10111yescharacteristic
363C2#2026598311yescharacteristic
364C2#2026598411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutatorC4×C101C23×C4
FrattiniC2×C4C23×D101
FittingC4×SD16×C101C2
socleC22×C101C22×D4
prime radicalC2×D4×D101C22
cosocleC2×C4×C101C24
hypercenterC4×SD16D101
hypocenterC101#2949
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #20265653C2semidirect
4. #20265280C2semidirect
5. #20265277C2semidirect
6. #20265275C2semidirect
7. #20265274C2semidirect
8. C4×#13108063C2semidirect
9. SD16×Dic101C2semidirect
10. #20265264C2semidirect
11. #20265212C2semidirect
12. #20265184C2semidirect
14. #20265131C2semidirect
15. C4×SD16×C101C2semidirect
21. SD16×D101C4semidirect
24. Q8×Dic101C22semidirect
26. C4×Dic202C22semidirect
28. #15692220C22semidirect
41. #15692188C22semidirect
42. #15692188C22semidirect
43. #15692187C22semidirect
44. #15692180C22semidirect
45. #15692179C22semidirect
46. #15692178C22semidirect
50. #15692166C22semidirect
51. #15692165C22semidirect
52. C8×Dic101C22semidirect
54. C4×Q8×C101C22semidirect
56. #106×C101C22semidirect
57. #103×C101C22semidirect
59. C4×C8×C101C22semidirect
64. Q8×D101C2×C4semidirect
74. #13108063C2×C4semidirect
75. #13108062C2×C4semidirect
76. #13108061C2×C4semidirect
80. C8×D101C2×C4semidirect
104. SD16×C101C2×C4semidirect
183. C101#2949semidirect
194. C4×SD16D101semidirect
213. C4×Q8C2×D101semidirect
230. #106C2×D101semidirect
234. #103C2×D101semidirect
242. C4×C8C2×D101semidirect
263. SD16C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1012
#202656532
#202656492
#202652802
#202652772
#202652752
#202652742
C4×#131080632
SD16×Dic1012
#202652642
#202652122
#202651842
#202651802
#202651312
C4×SD16×C1012
#2949101