Finite Group Databasebeta

← #27183158

Subgroups — #27183158

Order 12928.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265868C211yesmaximal, characteristic, complemented
2D4×D404C211yesmaximal, characteristic
3#20265542C211yesmaximal, characteristic, complemented
4#20265504C211yesmaximal, characteristic, complemented
5#20265492C211yesmaximal, characteristic, complemented
6#20265477C211yesmaximal, characteristic, complemented
7#20265426C211yesmaximal, characteristic, complemented
8#20265409C211yesmaximal, characteristic, complemented
9#20265377C211yesmaximal, characteristic, complemented
10#20265326C211yesmaximal, characteristic, complemented
11#20265326C211yesmaximal, characteristic, complemented
12#20265250C211yesmaximal, characteristic, complemented
13#20265180C211yesmaximal, characteristic, complemented
14#102×D101C211yesmaximal, characteristic, complemented
15#440×C101C211yesmaximal, characteristic, complemented
16C22×D404—21no—
17C2×D4×D101C2211yescharacteristic
18C2×D4×D101C2211yescharacteristic
19C2×#13108061C2211yescharacteristic, complemented
20C2×#13108061C2211yescharacteristic, complemented
21C2×#13108057C2211yescharacteristic, complemented
22C2×C8×D101C2211yescharacteristic, complemented
23#15692252C2211yescharacteristic
24#15692249—21no—
25#15692247C2211yescharacteristic, complemented
26#15692245—21no—
27#15692240—21no—
28#15692231C2211yescharacteristic, complemented
29#15692231C2211yescharacteristic, complemented
30#15692230C2211yescharacteristic
31#15692230C2211yescharacteristic
32#15692229—21no—
33#15692225—21no—
34#15692225—21no—
35C4×D404C2211yescharacteristic
36#15692213C2211yescharacteristic, complemented
37#15692212—21no—
38C4⋊C4×D101C2211yescharacteristic, complemented
39#15692188C2211yescharacteristic, complemented
40#15692186C2211yescharacteristic, complemented
41C4⋊#11717579C2211yescharacteristic, complemented
42#15692177C2211yescharacteristic, complemented
43#15692177C2211yescharacteristic, complemented
44#15692177C2211yescharacteristic, complemented
45#15692177C2211yescharacteristic, complemented
46#15692176C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692175C2211yescharacteristic, complemented
49#15692175C2211yescharacteristic, complemented
50#15692167C2211yescharacteristic, complemented
51#15692167C2211yescharacteristic, complemented
52C2×SD16×C101C2211yescharacteristic
53C4×Q8×C101C2211yescharacteristic, complemented
54C4×D4×C101C2211yescharacteristic
55#107×C101C2211yescharacteristic, complemented
56#103×C101C2211yescharacteristic, complemented
57#102×C101C2211yescharacteristic
58#97×C101C2211yescharacteristic, complemented
59C23×D101—21no—
60C23×D101—21no—
61C2×D404—21no—
62C2×D404—21no—
63C2×D404—21no—
64C2×D404—21no—
65C2×D404—44no—
66C2×D404D411yescharacteristic
67C2×D404C2311yescharacteristic
68C2×D404C2311yescharacteristic
69C2×C22⋊D101—21no—
70C2×C22⋊D101—21no—
71D4×D101—22no—
72D4×D101—22no—
73D4×D101—41no—
74D4×D101—41no—
75C2×C4×D101—21no—
76C2×C4×D101C2311yescharacteristic
77C2×C4×D101C2311yescharacteristic
78#13108061—41no—
79#13108061—41no—
80#13108057—41no—
81C2×#11717579C2311yescharacteristic
82C2×#11717579C2311yescharacteristic
83C8×D101—41no—
84C4⋊Dic101D411yescharacteristic
85C4⋊Dic101C2311yescharacteristic
86C4⋊Dic101C2311yescharacteristic
87#13108053—21no—
88#13108053—21no—
89#13108053—21no—
90#13108053—21no—
91#13108053—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051—21no—
95#13108051—21no—
96#13108051—21no—
97#13108051D411yescharacteristic
98#13108051D411yescharacteristic
99C4×Dic101—21no—
100C2×Q8×C101C2311yescharacteristic
101C2×D4×C101C2311yescharacteristic
102C22×C4×C101—21no—
103SD16×C101—22no—
104C2×C8×C101C2311yescharacteristic
105C2×C8×C101C2311yescharacteristic
106C4⋊C4×C101—21no—
107C4⋊C4×C101C2311yescharacteristic
108C4⋊C4×C101C2311yescharacteristic
109C22⋊C4×C101—21no—
110C42×C101—21no—
111C42×C101C2311yescharacteristic
112C23×C101—21no—
113C22×D101—21no—
114C22×D101—21no—
115C22×D101—42no—
116C22×D101—42no—
117C22×D101—42no—
118C22×D101—42no—
119C22×D101—81no—
120C22×D101—81no—
121C22×D101C2×D411yescharacteristic
122C22×D101C2×D411yescharacteristic
123Q8×C101—21no—
124Q8×C101—21no—
125D4×C101—21no—
126D4×C101#4112yes—
127C2×C4×C101—21no—
128C2×C4×C101—21no—
129C2×C4×C101—21no—
130C2×C4×C101—42no—
131C2×C4×C101C2×D411yescharacteristic
132C2×C4×C101C2×D411yescharacteristic
133C2×C4×C101C2411yescharacteristic
134C22⋊D101—42no—
135C22⋊D101—42no—
136C2×Dic101—21no—
137C2×Dic101—21no—
138C2×Dic101C2×D411yescharacteristic
139C2×Dic101C2×D411yescharacteristic
140D404—22no—
141D404—21no—
142D404—21no—
143D404—21no—
144D404—21no—
145D404—42no—
146D404—42no—
147D404—42no—
148D404—81no—
149C4×D101—21no—
150C4×D101—21no—
151C4×D101—21no—
152C4×D101—21no—
153C4×D101—42no—
154C8×C101—21no—
155C8×C101—21no—
156#11717579—21no—
157#11717579—21no—
158C22×C101—22no—
159C22×C101—41no—
160C22×C101—41no—
161C22×C101C22×D411yescharacteristic
162C2×D101—21no—
163C2×D101—21no—
164C2×D101—21no—
165C2×D101—21no—
166C2×D101—21no—
167C2×D101—41no—
168C2×D101—41no—
169C2×D101—41no—
170C2×D101—41no—
171C2×D101—41no—
172C2×D101—41no—
173C2×D101—84no—
174C2×D101—84no—
175C2×D101AGL(1,ℤ/8)12yes—
176C4×C101—21no—
177C4×C101—21no—
178C4×C101—41no—
179C4×C101—41no—
180C4×C101—41no—
181C4×C101C2×#4111yescharacteristic
182C4×C101H(5,𝔽2)11yescharacteristic
183Dic101—21no—
184Dic101—21no—
185Dic101—41no—
186Dic101—41no—
187C2×C101—42no—
188C2×C101#54611yescharacteristic
189C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C101#57711yescharacteristic
191D101—22no—
192D101—41no—
193D101—81no—
194D101—81no—
195#3310—1011nomaximal, Sylow
196C101#331011yescharacteristic, complemented, Sylow
197#552—1011no—
198D42—1011no—
199#490—1011no—
200#482—1011no—
201#480—1011no—
202#478—1011no—
203#468—1011no—
204#465—1011no—
205#460—1011no—
206#450—1011no—
207#450—1011no—
208#440D10111yescharacteristic, complemented
209#435—1011no—
210#418—1011no—
211C2×#102—1011no—
212C22×D4—1011no—
213C22×D4—1011no—
214C22×D4—2021no—
215C2×SD16—1011no—
216C2×SD16—1011no—
217C2×SD16C2×D10111yescharacteristic
218C2×M16—1011no—
219C22×C8—1011no—
220#127—1011no—
221#126—2021no—
222#125—1011no—
223#125—2021no—
224#123—2021no—
225C22⋊Q8—1011no—
226C22⋊Q8—1011no—
227#121—1011no—
228#121—1011no—
229#121—2021no—
230#120—2021no—
231#120—2021no—
232C4×Q8C2×D10111yescharacteristic, complemented
233C4×D4—1011no—
234C4×D4C2×D10111yescharacteristic
235#117—1011no—
236#117—2021no—
237C2×C4⋊C4—1011no—
238#107—1011no—
239#107C2×D10111yescharacteristic, complemented
240#106—1011no—
241C4⋊C8—1011no—
242#103C2×D10111yescharacteristic, complemented
243#102—1011no—
244#102—1011no—
245#102—1011no—
246#102—1011no—
247#102—1011no—
248#102—1011no—
249#102—1011no—
250#102—1011no—
251#102C2×D10111yescharacteristic
252C22⋊C8—1011no—
253C22⋊C8—1011no—
254#97C2×D10111yescharacteristic, complemented
255C24—2021no—
256C24—2021no—
257C2×Q8C22×D10111yescharacteristic
258C2×D4—1011no—
259C2×D4—1011no—
260C2×D4—1011no—
261C2×D4—4044no—
262C2×D4—4041no—
263C2×D4—4041no—
264C2×D4—2021no—
265C2×D4—2021no—
266C2×D4—2021no—
267C2×D4—2021no—
268C2×D4—2022no—
269C2×D4—2022no—
270C2×D4—2021no—
271C2×D4—2021no—
272C2×D4C22×D10111yescharacteristic
273C22×C4—21no—
274C22×C4—1011no—
275C22×C4—1011no—
276C22×C4—2021no—
277SD16—22no—
278SD16—4041no—
279SD16—4041no—
280M16—4041no—
281C2×C8—1011no—
282C2×C8—1011no—
283C2×C8—4041no—
284C2×C8C22×D10111yescharacteristic
285C2×C8C22×D10111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—1011no—
288C4⋊C4—1011no—
289C4⋊C4—1011no—
290C4⋊C4—2021no—
291C4⋊C4—2021no—
292C4⋊C4—2021no—
293C4⋊C4—2021no—
294C4⋊C4—2021no—
295C4⋊C4C22×D10111yescharacteristic
296C4⋊C4C22×D10111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—1011no—
299C22⋊C4—1011no—
300C22⋊C4—2021no—
301C22⋊C4—2021no—
302C22⋊C4—2021no—
303C22⋊C4—2021no—
304C22⋊C4—2021no—
305C42—21no—
306C42—2021no—
307C42C22×D10111yescharacteristic
308C23—21no—
309C23—8081no—
310C23—8081no—
311C23—1011no—
312C23—1011no—
313C23—4042no—
314C23—4042no—
315C23—4042no—
316C23—4042no—
317C23—2021no—
318C23—2021no—
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—8081no—
323D4—4042no—
324D4—4042no—
325D4—4042no—
326D4—4042no—
327D4—4042no—
328D4—2022no—
329D4—2021no—
330D4—2021no—
331D4—2021no—
332D4—2021no—
333D4#1310807412yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—1011no—
339C2×C4—1011no—
340C2×C4—4042no—
341C2×C4—2021no—
342C2×C4—2021no—
343C2×C4—2021no—
344C2×C4—2021no—
345C2×C4—2021no—
346C2×C4—2021no—
347C2×C4D4×D10111yescharacteristic
348C2×C4D4×D10111yescharacteristic
349C2×C4C23×D10111yescharacteristic
350C8—21no—
351C8—21no—
352C8—2021no—
353C8—2021no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—8084no—
358C22—8084no—
359C22—1012no—
360C22—4041no—
361C22—4041no—
362C22—4041no—
363C22—4041no—
364C22—4041no—
365C22—4041no—
366C22—2021no—
367C22—2021no—
368C22—2021no—
369C22—2021no—
370C22—2021no—
371C22C2×D4×D10111yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—4041no—
378C4—4041no—
379C4—2021no—
380C4—2021no—
381C4C2×#1310807411yescharacteristic
382C4#1569230111yescharacteristic
383C2—42no—
384C2—8081no—
385C2—8081no—
386C2—4041no—
387C2—2022no—
388C2#2026582311yescharacteristic
389C2AGL(1,ℤ/8)×D10111yescharacteristic
390C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#440×C101C2
socleC22×C101C22×D4
prime radicalD4×D404C2
hypercenter#440D101
hypocenterC101#3310
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #20265868C2semidirect
3. #20265542C2semidirect
4. #20265504C2semidirect
5. #20265492C2semidirect
6. #20265477C2semidirect
7. #20265426C2semidirect
8. #20265409C2semidirect
9. #20265377C2semidirect
10. #20265326C2semidirect
11. #20265326C2semidirect
12. #20265250C2semidirect
13. #20265180C2semidirect
14. #102×D101C2semidirect
15. #440×C101C2semidirect
19. C2×#13108061C22semidirect
20. C2×#13108061C22semidirect
21. C2×#13108057C22semidirect
22. C2×C8×D101C22semidirect
25. #15692247C22semidirect
28. #15692231C22semidirect
29. #15692231C22semidirect
36. #15692213C22semidirect
38. C4⋊C4×D101C22semidirect
39. #15692188C22semidirect
40. #15692186C22semidirect
41. C4⋊#11717579C22semidirect
42. #15692177C22semidirect
43. #15692177C22semidirect
44. #15692177C22semidirect
45. #15692177C22semidirect
48. #15692175C22semidirect
49. #15692175C22semidirect
50. #15692167C22semidirect
51. #15692167C22semidirect
53. C4×Q8×C101C22semidirect
55. #107×C101C22semidirect
56. #103×C101C22semidirect
58. #97×C101C22semidirect
196. C101#3310semidirect
208. #440D101semidirect
232. C4×Q8C2×D101semidirect
239. #107C2×D101semidirect
242. #103C2×D101semidirect
254. #97C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658682
D4×D4042
#202655422
#202655042
#202654922
#202654772
#202654262
#202654092
#202653772
#202653262
#202652502
#202651802
#102×D1012
#440×C1012
#3310101