Finite Group Databasebeta

← #27182954

Subgroups — #27182954

Order 12928.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265882C211yesmaximal, characteristic, complemented
2#20265811C211yesmaximal, characteristic
3#20265522C211yesmaximal, characteristic, complemented
4#20265497C211yesmaximal, characteristic, complemented
5#20265466C211yesmaximal, characteristic, complemented
6#20265407C211yesmaximal, characteristic, complemented
7#20265404C211yesmaximal, characteristic, complemented
8#20265376C211yesmaximal, characteristic, complemented
9#20265330C211yesmaximal, characteristic, complemented
10#20265329C211yesmaximal, characteristic, complemented
11#20265264C211yesmaximal, characteristic, complemented
12#20265240C211yesmaximal, characteristic, complemented
13C2×#15692175C211yesmaximal, characteristic, complemented
14#102×D101C211yesmaximal, characteristic, complemented
15#481×C101C211yesmaximal, characteristic, complemented
16C22×D404C2211yescharacteristic
17C2×D4×D101—21no—
18C2×D4×D101C2211yescharacteristic
19C2×#13108063C2211yescharacteristic
20C2×#13108063C2211yescharacteristic, complemented
21C2×#13108062C2211yescharacteristic, complemented
22C22×#11717579C2211yescharacteristic, complemented
23C2×C8×D101C2211yescharacteristic, complemented
24#15692254C2211yescharacteristic, complemented
25#15692251C2211yescharacteristic
26#15692245—21no—
27#15692235C2211yescharacteristic, complemented
28#15692234—21no—
29#15692234C2211yescharacteristic, complemented
30#15692233—21no—
31#15692230C2211yescharacteristic
32#15692228—21no—
33#15692225—21no—
34#15692223—21no—
35#15692220C2211yescharacteristic, complemented
36#15692217C2211yescharacteristic
37C4×C22⋊D101—21no—
38C4×C22⋊D101C2211yescharacteristic
39C4⋊C4×D101C2211yescharacteristic, complemented
40#15692186C2211yescharacteristic, complemented
41#15692184C2211yescharacteristic, complemented
42#15692178C2211yescharacteristic, complemented
43#15692177C2211yescharacteristic
44#15692177C2211yescharacteristic, complemented
45#15692176C2211yescharacteristic, complemented
46#15692175C2211yescharacteristic
47#15692175C2211yescharacteristic
48#15692175C2211yescharacteristic, complemented
49#15692175C2211yescharacteristic, complemented
50#15692167C2211yescharacteristic, complemented
51C8×Dic101C2211yescharacteristic, complemented
52#121×C101C2211yescharacteristic
53C2×C4⋊C4×C101C2211yescharacteristic, complemented
54#106×C101C2211yescharacteristic, complemented
55#106×C101C2211yescharacteristic, complemented
56#102×C101C2211yescharacteristic, complemented
57#102×C101C2211yescharacteristic, complemented
58C22⋊C8×C101C2211yescharacteristic, complemented
59C23×D101—21no—
60C23×D101—21no—
61C2×Dic202—21no—
62C2×Dic202C2311yescharacteristic
63C2×D404—22no—
64C2×D404—21no—
65C2×D404—21no—
66C2×D404—41no—
67C2×D404C2311yescharacteristic
68C2×D404C2311yescharacteristic
69C2×C22⋊D101—21no—
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C22⋊D101D411yescharacteristic
73D4×D101—22no—
74D4×D101—41no—
75D4×D101—44no—
76C2×C4×D101—21no—
77C2×C4×D101—21no—
78C2×C4×D101C2311yescharacteristic
79#13108063—22no—
80#13108063—41no—
81#13108062—41no—
82C2×#11717579—41no—
83C2×#11717579C2311yescharacteristic
84C2×#11717579C2311yescharacteristic
85C8×D101—41no—
86C4⋊Dic101—21no—
87C4⋊Dic101C2311yescharacteristic
88#13108053—21no—
89#13108053—21no—
90#13108053—21no—
91#13108053—21no—
92#13108053D411yescharacteristic
93C22⋊Dic101—21no—
94C22⋊Dic101D411yescharacteristic
95#13108051—21no—
96#13108051—21no—
97#13108051—21no—
98#13108051D411yescharacteristic
99C4×Dic101—21no—
100C4×Dic101C2311yescharacteristic
101C2×D4×C101—21no—
102C2×D4×C101C2311yescharacteristic
103C22×C4×C101—21no—
104C22×C4×C101C2311yescharacteristic
105C2×C8×C101C2311yescharacteristic
106C2×C8×C101C2311yescharacteristic
107C4⋊C4×C101—21no—
108C4⋊C4×C101C2311yescharacteristic
109C4⋊C4×C101C2311yescharacteristic
110C4⋊C4×C101C2311yescharacteristic
111C22⋊C4×C101—21no—
112C23×C101—21no—
113C23×C101C2×D411yescharacteristic
114C22×D101—21no—
115C22×D101—21no—
116C22×D101—42no—
117C22×D101—42no—
118C22×D101—42no—
119C22×D101—42no—
120C22×D101—81no—
121C22×D101—81no—
122C22×D101C2×D411yescharacteristic
123D4×C101—21no—
124D4×C101—21no—
125D4×C101—42no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—21no—
129C2×C4×C101—21no—
130C2×C4×C101—21no—
131C2×C4×C101—81no—
132C2×C4×C101C2×D411yescharacteristic
133C2×C4×C101C2411yescharacteristic
134C22⋊D101—42no—
135C22⋊D101—42no—
136C22⋊D101—81no—
137C22⋊D101SD1614yescomplemented
138C2×Dic101—21no—
139C2×Dic101—21no—
140C2×Dic101C2×D411yescharacteristic
141C2×Dic101C2×D411yescharacteristic
142C2×Dic101C2×D411yescharacteristic
143D404—21no—
144D404—21no—
145D404—21no—
146D404—42no—
147D404—42no—
148D404#4112yes—
149C4×D101—21no—
150C4×D101—21no—
151C4×D101—42no—
152C4×D101—81no—
153Dic202—21no—
154Dic202—21no—
155Dic202—81no—
156C8×C101—21no—
157C8×C101—21no—
158#11717579—21no—
159#11717579—21no—
160C22×C101—21no—
161C22×C101—21no—
162C22×C101—41no—
163C22×C101—41no—
164C22×C101—41no—
165C22×C101C2×SD1612yes—
166C22×C101C22×D411yescharacteristic
167C2×D101—21no—
168C2×D101—21no—
169C2×D101—22no—
170C2×D101—41no—
171C2×D101—41no—
172C2×D101—41no—
173C2×D101—41no—
174C2×D101—41no—
175C2×D101—84no—
176C2×D101—84no—
177C2×D101C2×SD1612yes—
178C4×C101—21no—
179C4×C101—41no—
180C4×C101—41no—
181C4×C101—41no—
182C4×C101C2×#4111yescharacteristic
183C4×C101H(5,𝔽2)11yescharacteristic
184Dic101—21no—
185Dic101—21no—
186Dic101—41no—
187Dic101—41no—
188Dic101C2×SD1612yes—
189C2×C101—22no—
190C2×C101—81no—
191C2×C101#54611yescharacteristic
192C2×C101C22×SD1611yescharacteristic
193C2×C101#57611yescharacteristic
194D101—22no—
195D101—42no—
196D101—81no—
197#3295—1011nomaximal, Sylow
198C101#329511yescharacteristic, complemented, Sylow
199#554—1011no—
200D42—1011no—
201#486—1011no—
202#481—1011no—
203#481D10111yescharacteristic, complemented
204#476—1011no—
205#465—1011no—
206#465—1011no—
207#460—1011no—
208#450—1011no—
209#450—1011no—
210C4×SD16—1011no—
211C8×D4—1011no—
212C2×#102—1011no—
213C2×#102—1011no—
214C22×D4—1011no—
215C22×D4—1011no—
216C22×D4—2021no—
217C2×SD16—1011no—
218C2×SD16—1011no—
219C2×SD16—1011no—
220C22×C8—1011no—
221C22×C8—1011no—
222C4⋊Q8—1011no—
223#127—1011no—
224#125—2021no—
225C22⋊Q8—1011no—
226C22⋊Q8—1011no—
227C22⋊Q8—2021no—
228C22⋊Q8—2021no—
229#121—1011no—
230#121—2021no—
231#121C2×D10111yescharacteristic
232#120—2021no—
233#120—2021no—
234C4×Q8—1011no—
235C4×D4—1011no—
236C4×D4—1011no—
237C4×D4—2021no—
238C2×C4⋊C4—1011no—
239C2×C4⋊C4C2×D10111yescharacteristic, complemented
240#106—1011no—
241#106C2×D10111yescharacteristic, complemented
242#106C2×D10111yescharacteristic, complemented
243C4⋊C8—1011no—
244#103—1011no—
245#102—1011no—
246#102—1011no—
247#102—1011no—
248#102—1011no—
249#102—1011no—
250#102—1011no—
251#102—1011no—
252#102C2×D10111yescharacteristic, complemented
253#102C2×D10111yescharacteristic, complemented
254C22⋊C8—1011no—
255C22⋊C8C2×D10111yescharacteristic, complemented
256C4×C8—1011no—
257C24—2021no—
258C24—2021no—
259C2×Q8—1011no—
260C2×Q8—2021no—
261C2×D4—21no—
262C2×D4—1011no—
263C2×D4—1011no—
264C2×D4—1011no—
265C2×D4—4041no—
266C2×D4—4044no—
267C2×D4—4041no—
268C2×D4—2022no—
269C2×D4—2021no—
270C2×D4—2021no—
271C2×D4—2021no—
272C2×D4—2021no—
273C2×D4—2022no—
274C2×D4—2021no—
275C2×D4C22×D10111yescharacteristic
276C22×C4—21no—
277C22×C4—1011no—
278C22×C4—2021no—
279C22×C4—2021no—
280C22×C4C22×D10111yescharacteristic
281SD16—4041no—
282SD16—4041no—
283SD16—2022no—
284C2×C8—1011no—
285C2×C8—1011no—
286C2×C8—4041no—
287C2×C8—4041no—
288C2×C8C22×D10111yescharacteristic
289C2×C8C22×D10111yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—1011no—
292C4⋊C4—1011no—
293C4⋊C4—2021no—
294C4⋊C4—2021no—
295C4⋊C4—2021no—
296C4⋊C4—2021no—
297C4⋊C4—2021no—
298C4⋊C4C22×D10111yescharacteristic
299C4⋊C4C22×D10111yescharacteristic
300C4⋊C4C22×D10111yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—1011no—
303C22⋊C4—1011no—
304C22⋊C4—2021no—
305C22⋊C4—2021no—
306C22⋊C4—2021no—
307C22⋊C4—2021no—
308C42—1011no—
309C42—2021no—
310C23—21no—
311C23—8081no—
312C23—8081no—
313C23—1011no—
314C23—4042no—
315C23—4042no—
316C23—4042no—
317C23—4042no—
318C23—2021no—
319C23—2021no—
320C23D4×D10111yescharacteristic
321Q8—8081no—
322Q8—2021no—
323Q8—2021no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—8081no—
328D4—1014no—
329D4—1012no—
330D4—4042no—
331D4—4042no—
332D4—4042no—
333D4—4042no—
334D4—2021no—
335D4—2021no—
336D4—2021no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—81no—
343C2×C4—8081no—
344C2×C4—1011no—
345C2×C4—1011no—
346C2×C4—1011no—
347C2×C4—4042no—
348C2×C4—2021no—
349C2×C4—2021no—
350C2×C4—2021no—
351C2×C4—2021no—
352C2×C4D4×D10111yescharacteristic
353C2×C4C23×D10111yescharacteristic
354C8—21no—
355C8—21no—
356C8—2021no—
357C8—2021no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—8084no—
364C22—8084no—
365C22—1012no—
366C22—4041no—
367C22—4041no—
368C22—4041no—
369C22—4041no—
370C22—4041no—
371C22—2022no—
372C22—2021no—
373C22—2021no—
374C22SD16×D10112yes—
375C22C2×D4×D10111yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—1012no—
381C4—4041no—
382C4—4041no—
383C4—2021no—
384C4—2021no—
385C4#41×D10111yescharacteristic
386C4#1569230111yescharacteristic
387C2—22no—
388C2—81no—
389C2—8081no—
390C2—4042no—
391C2—2022no—
392C2#2026581911yescharacteristic
393C2C2×SD16×D10111yescharacteristic
394C2#2026598111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#481×C101C2
socleC22×C101C22×D4
prime radical#20265811C2
hypercenter#481D101
hypocenterC101#3295
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #20265882C2semidirect
3. #20265522C2semidirect
4. #20265497C2semidirect
5. #20265466C2semidirect
6. #20265407C2semidirect
7. #20265404C2semidirect
8. #20265376C2semidirect
9. #20265330C2semidirect
10. #20265329C2semidirect
11. #20265264C2semidirect
12. #20265240C2semidirect
13. C2×#15692175C2semidirect
14. #102×D101C2semidirect
15. #481×C101C2semidirect
20. C2×#13108063C22semidirect
21. C2×#13108062C22semidirect
22. C22×#11717579C22semidirect
23. C2×C8×D101C22semidirect
24. #15692254C22semidirect
27. #15692235C22semidirect
29. #15692234C22semidirect
35. #15692220C22semidirect
39. C4⋊C4×D101C22semidirect
40. #15692186C22semidirect
41. #15692184C22semidirect
42. #15692178C22semidirect
44. #15692177C22semidirect
45. #15692176C22semidirect
48. #15692175C22semidirect
49. #15692175C22semidirect
50. #15692167C22semidirect
51. C8×Dic101C22semidirect
53. C2×C4⋊C4×C101C22semidirect
54. #106×C101C22semidirect
55. #106×C101C22semidirect
56. #102×C101C22semidirect
57. #102×C101C22semidirect
58. C22⋊C8×C101C22semidirect
137. C22⋊D101SD16semidirect
198. C101#3295semidirect
203. #481D101semidirect
239. C2×C4⋊C4C2×D101semidirect
241. #106C2×D101semidirect
242. #106C2×D101semidirect
252. #102C2×D101semidirect
253. #102C2×D101semidirect
255. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658822
#202658112
#202655222
#202654972
#202654662
#202654072
#202654042
#202653762
#202653302
#202653292
#202652642
#202652402
C2×#156921752
#102×D1012
#481×C1012
#3295101