Finite Group Databasebeta

← #27180004

Subgroups — #27180004

Order 12928.

484 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692276C211yesmaximal, characteristic
2#20265659C211yesmaximal, characteristic, complemented
3#20265656C211yesmaximal, characteristic, complemented
4#20265279C211yesmaximal, characteristic, complemented
5#20265278C211yesmaximal, characteristic, complemented
6#20265276C211yesmaximal, characteristic, complemented
7C4×#13108064C211yesmaximal, characteristic, complemented
8#20265269C211yesmaximal, characteristic, complemented
9C4×#13108059C211yesmaximal, characteristic, complemented
10#20265257C211yesmaximal, characteristic, complemented
11#107×D101C211yesmaximal, characteristic, complemented
12#20265177C211yesmaximal, characteristic, complemented
13#20265175C211yesmaximal, characteristic, complemented
14#20265134C211yesmaximal, characteristic, complemented
15#440×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×#13108073C2211yescharacteristic
18#15692276C418yescomplemented
19C2×#13108064C2211yescharacteristic
20C2×#13108063C2211yescharacteristic
21C2×#13108059C2211yescharacteristic
22C2×#13108057C2211yescharacteristic
23C2×C8×D101C2211yescharacteristic
24Q8×Dic101C2211yescharacteristic
25C4×Dic202C2211yescharacteristic
26#15692220C2211yescharacteristic
27C4×D404—21no—
28C4×D404C2211yescharacteristic
29D4×Dic101C2211yescharacteristic
30#15692217—21no—
31#15692217C2211yescharacteristic
32C4×C22⋊D101—21no—
33#15692215—21no—
34#15692212—21no—
35#15692212C2211yescharacteristic
36#15692211—21no—
37C2×#13108053—21no—
38C4⋊C4×D101—21no—
39C4⋊C4×D101C2211yescharacteristic
40C4⋊C4×D101C2211yescharacteristic
41#15692189C2211yescharacteristic
42#15692188C2211yescharacteristic
43#15692188C2211yescharacteristic
44#15692180C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692178C2211yescharacteristic
47#15692177C2211yescharacteristic
48#15692176C2211yescharacteristic
49#15692175C2211yescharacteristic
50#15692165C2211yescharacteristic
51C4×#11717579C2211yescharacteristic
52C8×Dic101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic
54C4×Q8×C101C2211yescharacteristic
55C4×D4×C101C2211yescharacteristic
56#107×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59#97×C101C2211yescharacteristic
60#13108075—22no—
61#13108075C2×C414yescomplemented
62#13108073—22no—
63#13108073C2×C414yescomplemented
64C2×Dic202C2311yescharacteristic
65C2×D404—21no—
66C2×D404C2311yescharacteristic
67C2×C22⋊D101—21no—
68C22×Dic101—21no—
69C22×Dic101—21no—
70C2×C4×D101—21no—
71C2×C4×D101—21no—
72C2×C4×D101—21no—
73C2×C4×D101—21no—
74C2×C4×D101—21no—
75C2×C4×D101C2311yescharacteristic
76#13108064C2×C414yescomplemented
77#13108063C2×C414yescomplemented
78#13108059C2×C414yescomplemented
79#13108057—22no—
80C2×#11717579C2311yescharacteristic
81C2×#11717579C2311yescharacteristic
82C8×D101C2×C414yescomplemented
83C4⋊Dic101—21no—
84C4⋊Dic101C2311yescharacteristic
85C4⋊Dic101C2311yescharacteristic
86#13108053—21no—
87#13108053—21no—
88#13108053—21no—
89#13108053—21no—
90#13108053D411yescharacteristic
91#13108053D411yescharacteristic
92C22⋊Dic101—21no—
93#13108051—21no—
94#13108051—21no—
95#13108051D411yescharacteristic
96#13108051D411yescharacteristic
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×D101C2×D411yescharacteristic
116Q8×C101—21no—
117Q8×C101C22×C412yes—
118D4×C101—21no—
119D4×C101C22×C412yes—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—81no—
124C2×C4×C101C2×D411yescharacteristic
125C2×C4×C101C2×D411yescharacteristic
126C2×C4×C101C2411yescharacteristic
127C22⋊D101—24no—
128C2×Dic101—21no—
129C2×Dic101—24no—
130C2×Dic101—21no—
131C2×Dic101—21no—
132C2×Dic101—81no—
133C2×Dic101C2×D411yescharacteristic
134C2×Dic101C2×D411yescharacteristic
135C2×Dic101C2×D411yescharacteristic
136D404—24no—
137D404—21no—
138D404C22×C412yes—
139C4×D101—24no—
140C4×D101—81no—
141C4×D101—81no—
142C4×D101—81no—
143C4×D101—81no—
144C4×D101C22×C412yes—
145C4×D101#4112yes—
146Dic202—21no—
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×D101C4×D412yes—
162C4×C101—22no—
163C4×C101—21no—
164C4×C101—21no—
165C4×C101—41no—
166C4×C101—41no—
167C4×C101C23×C411yescharacteristic
168C4×C101C2×#4111yescharacteristic
169Dic101—22no—
170Dic101—21no—
171Dic101—21no—
172Dic101—41no—
173Dic101—41no—
174Dic101C4×D412yes—
175C2×C101—42no—
176C2×C101C2×C4×D411yescharacteristic
177C2×C101#57611yescharacteristic
178C2×C101#57811yescharacteristic
179D101—22no—
180D101—42no—
181#2951—1011nomaximal, Sylow
182C101#295111yescharacteristic, complemented, Sylow
183C2×#135—1011no—
184#520—1011no—
185#520—1011no—
186#440—1011no—
187#440—1011no—
188#440—1011no—
189#440D10111yescharacteristic, complemented
190C4×Q16—1011no—
191C4×Q16—1011no—
192C4×D8—1011no—
193C4×D8—1011no—
194C2×#107—1011no—
195#417—1011no—
196#417—1011no—
197#405—1011no—
198C2×#41—1011no—
199C2×#41—1011no—
200#135—1018no—
201C2×Q16—1011no—
202C2×SD16—1011no—
203C2×SD16C2×D10111yescharacteristic
204C2×D8—1011no—
205C2×M16—1011no—
206C22×C8—1011no—
207C4×Q8—1011no—
208C4×Q8—1011no—
209C4×Q8—1011no—
210C4×Q8C2×D10111yescharacteristic
211C4×D4—1011no—
212C4×D4—1011no—
213C4×D4—1011no—
214C4×D4—2021no—
215C4×D4—2021no—
216C4×D4—2021no—
217C4×D4—2021no—
218C4×D4C2×D10111yescharacteristic
219#117—1011no—
220#117—2021no—
221#117—2021no—
222C2×C4⋊C4—1011no—
223C2×C4⋊C4—1011no—
224C2×C4⋊C4—2021no—
225C2×C4⋊C4—2021no—
226#107—1011no—
227#107—1011no—
228#107—1011no—
229#107C2×D10111yescharacteristic
230#103—1011no—
231#103—1011no—
232#103—1011no—
233#103C2×D10111yescharacteristic
234#102—1011no—
235#102—1011no—
236#102—1011no—
237#102C2×D10111yescharacteristic
238#97—1011no—
239#97C2×D10111yescharacteristic
240C4×C8—1011no—
241C4×C8—1011no—
242#41—1014no—
243#41—1014no—
244#41—2022no—
245#41—2022no—
246C2×Q8—1011no—
247C2×Q8C22×D10111yescharacteristic
248C2×D4—1011no—
249C2×D4—2021no—
250C2×D4—2021no—
251C2×D4C22×D10111yescharacteristic
252C22×C4—21no—
253C22×C4—1011no—
254C22×C4—2021no—
255C22×C4—2021no—
256C22×C4—2021no—
257C22×C4—2021no—
258C22×C4—2021no—
259C22×C4—2021no—
260C22×C4—2021no—
261Q16—1014no—
262SD16—1014no—
263SD16C4×D10114yescomplemented
264D8—1014no—
265M16—2022no—
266C2×C8—1011no—
267C2×C8—1014no—
268C2×C8—1011no—
269C2×C8C22×D10111yescharacteristic
270C2×C8C22×D10111yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—1011no—
273C4⋊C4—1011no—
274C4⋊C4—1011no—
275C4⋊C4—1011no—
276C4⋊C4—2021no—
277C4⋊C4—2021no—
278C4⋊C4—2021no—
279C4⋊C4—2021no—
280C4⋊C4—2021no—
281C4⋊C4C22×D10111yescharacteristic
282C4⋊C4C22×D10111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—1011no—
285C22⋊C4—1011no—
286C22⋊C4—2021no—
287C22⋊C4—2021no—
288C22⋊C4—2021no—
289C42—21no—
290C42—1011no—
291C42—2021no—
292C42—2021no—
293C42—2021no—
294C42C22×D10111yescharacteristic
295C23—21no—
296C23—1011no—
297C23—2021no—
298Q8—21no—
299Q8—1012no—
300Q8—2021no—
301Q8C2×C4×D10112yes—
302D4—21no—
303D4—1012no—
304D4—2024no—
305D4—2024no—
306D4—2021no—
307D4C2×C4×D10112yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—8081no—
313C2×C4—8081no—
314C2×C4—8081no—
315C2×C4—8081no—
316C2×C4—8081no—
317C2×C4—1011no—
318C2×C4—1012no—
319C2×C4—1012no—
320C2×C4—1011no—
321C2×C4—1011no—
322C2×C4—2024no—
323C2×C4—2024no—
324C2×C4—2021no—
325C2×C4—2021no—
326C2×C4—2021no—
327C2×C4D4×D10111yescharacteristic
328C2×C4D4×D10111yescharacteristic
329C2×C4C23×D10111yescharacteristic
330C8—21no—
331C8—1012no—
332C8—2021no—
333C8C2×C4×D10112yes—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—1012no—
338C22—4041no—
339C22—4041no—
340C22—2022no—
341C22—2021no—
342C22—2021no—
343C22C2×D4×D10111yescharacteristic
344C4—21no—
345C4—22no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—1012no—
350C4—4041no—
351C4—4041no—
352C4—2021no—
353C4—2022no—
354C4—2021no—
355C4C22×C4×D10111yescharacteristic
356C4#41×D10111yescharacteristic
357C2—42no—
358C2—4042no—
359C2—2022no—
360C2C4×D4×D10111yescharacteristic
361C2#2026598111yescharacteristic
362C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutatorC4×C101C23×C4
FrattiniC2×C4C23×D101
Fitting#440×C101C2
socleC22×C101C22×D4
prime radicalC2×#15692276C2
cosocleC2×C4×C101C24
hypercenter#440D101
hypocenterC101#2951
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #20265659C2semidirect
3. #20265656C2semidirect
4. #20265279C2semidirect
5. #20265278C2semidirect
6. #20265276C2semidirect
7. C4×#13108064C2semidirect
8. #20265269C2semidirect
9. C4×#13108059C2semidirect
10. #20265257C2semidirect
11. #107×D101C2semidirect
12. #20265177C2semidirect
13. #20265175C2semidirect
14. #20265134C2semidirect
15. #440×C101C2semidirect
18. #15692276C4semidirect
61. #13108075C2×C4semidirect
63. #13108073C2×C4semidirect
76. #13108064C2×C4semidirect
77. #13108063C2×C4semidirect
78. #13108059C2×C4semidirect
82. C8×D101C2×C4semidirect
104. SD16×C101C2×C4semidirect
182. C101#2951semidirect
189. #440D101semidirect
263. SD16C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922762
#202656592
#202656562
#202652792
#202652782
#202652762
C4×#131080642
#202652692
C4×#131080592
#202652572
#107×D1012
#202651772
#202651752
#202651342
#440×C1012
#2951101