Finite Group Databasebeta

← #27183155

Subgroups — #27183155

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#20265867C211yesmaximal, characteristic, complemented
2D4×D404C211yesmaximal, characteristic
3#20265542C211yesmaximal, characteristic, complemented
4#20265502C211yesmaximal, characteristic, complemented
5#20265491C211yesmaximal, characteristic, complemented
6#20265478C211yesmaximal, characteristic, complemented
7#20265423C211yesmaximal, characteristic, complemented
8#20265406C211yesmaximal, characteristic, complemented
9#20265378C211yesmaximal, characteristic, complemented
10#20265330C211yesmaximal, characteristic, complemented
11#20265330C211yesmaximal, characteristic, complemented
12#20265279C211yesmaximal, characteristic, complemented
13#20265183C211yesmaximal, characteristic, complemented
14C2×#15692177C211yesmaximal, characteristic, complemented
15#435×C101C211yesmaximal, characteristic, complemented
16C22×D404C2211yescharacteristic
17C22×D404C2211yescharacteristic
18C2×D4×D101—21no—
19C2×#13108063C2211yescharacteristic
20C2×#13108063C2211yescharacteristic, complemented
21C2×#13108063C2211yescharacteristic, complemented
22#15692252C2211yescharacteristic
23#15692248—21no—
24#15692247C2211yescharacteristic, complemented
25#15692245—21no—
26#15692236—21no—
27#15692234C2211yescharacteristic, complemented
28#15692234C2211yescharacteristic, complemented
29#15692230—21no—
30#15692229C2211yescharacteristic
31#15692229C2211yescharacteristic
32#15692225—21no—
33#15692225—21no—
34C4×Dic202C2211yescharacteristic, complemented
35C4×D404C2211yescharacteristic
36#15692214C2211yescharacteristic, complemented
37#15692211—21no—
38C2×C4⋊Dic101C2211yescharacteristic, complemented
39#15692189C2211yescharacteristic, complemented
40#15692189C2211yescharacteristic, complemented
41#15692187C2211yescharacteristic, complemented
42#15692180C2211yescharacteristic, complemented
43#15692177C2211yescharacteristic
44#15692177C2211yescharacteristic
45#15692177C2211yescharacteristic
46#15692177C2211yescharacteristic, complemented
47#15692177C2211yescharacteristic, complemented
48#15692177C2211yescharacteristic, complemented
49#15692177C2211yescharacteristic, complemented
50#15692177C2211yescharacteristic, complemented
51#15692177C2211yescharacteristic, complemented
52C2×M16×C101C2211yescharacteristic, complemented
53C22×C8×C101C2211yescharacteristic, complemented
54C4×D4×C101C2211yescharacteristic
55C4⋊C8×C101C2211yescharacteristic, complemented
56C22⋊C8×C101C2211yescharacteristic, complemented
57C22⋊C8×C101C2211yescharacteristic, complemented
58#97×C101C2211yescharacteristic, complemented
59C23×D101—21no—
60C23×D101—21no—
61C2×Dic202C2311yescharacteristic
62C2×D404—22no—
63C2×D404—22no—
64C2×D404—21no—
65C2×D404—21no—
66C2×D404—21no—
67C2×D404—21no—
68C2×D404—41no—
69C2×D404—41no—
70C2×D404C2311yescharacteristic
71C2×D404C2311yescharacteristic
72C2×D404C2311yescharacteristic
73C2×C22⋊D101—21no—
74C2×C22⋊D101—21no—
75D4×D101—44no—
76C22×Dic101—21no—
77C2×C4×D101—21no—
78#13108063—22no—
79#13108063—41no—
80#13108063—41no—
81C4⋊Dic101—21no—
82C4⋊Dic101C2311yescharacteristic
83C4⋊Dic101C2311yescharacteristic
84C4⋊Dic101C2311yescharacteristic
85C4⋊Dic101C2311yescharacteristic
86#13108053—21no—
87#13108053—21no—
88#13108053—21no—
89#13108053—21no—
90#13108053—21no—
91C22⋊Dic101—21no—
92C22⋊Dic101—21no—
93C22⋊Dic101—21no—
94#13108051—21no—
95#13108051—21no—
96#13108051—21no—
97C4×Dic101—21no—
98C4×Dic101—21no—
99C2×D4×C101D411yescharacteristic
100C22×C4×C101C2311yescharacteristic
101C22×C4×C101C2311yescharacteristic
102M16×C101—41no—
103C2×C8×C101—41no—
104C2×C8×C101C2311yescharacteristic
105C2×C8×C101C2311yescharacteristic
106C2×C8×C101C2311yescharacteristic
107C2×C8×C101C2311yescharacteristic
108C4⋊C4×C101D411yescharacteristic
109C22⋊C4×C101D411yescharacteristic
110C22⋊C4×C101D411yescharacteristic
111C42×C101C2311yescharacteristic
112C23×C101C2×D411yescharacteristic
113C23×C101C2×D411yescharacteristic
114C22×D101—21no—
115C22×D101—21no—
116C22×D101—21no—
117C22×D101—42no—
118C22×D101—42no—
119C22×D101—42no—
120C22×D101—42no—
121C22×D101—81no—
122C22×D101—81no—
123D4×C101—22no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101C2×D411yescharacteristic
129C2×C4×C101C2×D411yescharacteristic
130C2×C4×C101C2×D411yescharacteristic
131C2×C4×C101C2×D411yescharacteristic
132C2×C4×C101C2411yescharacteristic
133C22⋊D101—42no—
134C22⋊D101—42no—
135C2×Dic101—21no—
136C2×Dic101—21no—
137C2×Dic101—21no—
138C2×Dic101—21no—
139C2×Dic101—21no—
140C2×Dic101—42no—
141D404—21no—
142D404—21no—
143D404—21no—
144D404—21no—
145D404—21no—
146D404—42no—
147D404—42no—
148D404—42no—
149D404—81no—
150D404#4112yes—
151C4×D101—42no—
152Dic202—21no—
153Dic202—21no—
154C8×C101—21no—
155C8×C101—21no—
156C8×C101—21no—
157C8×C101—21no—
158C22×C101—21no—
159C22×C101—21no—
160C22×C101—21no—
161C22×C101—21no—
162C22×C101—21no—
163C22×C101AGL(1,ℤ/8)12yes—
164C22×C101C22×D411yescharacteristic
165C2×D101—22no—
166C2×D101—41no—
167C2×D101—41no—
168C2×D101—41no—
169C2×D101—41no—
170C2×D101—41no—
171C2×D101—41no—
172C2×D101—41no—
173C2×D101—41no—
174C2×D101—84no—
175C2×D101—84no—
176C4×C101—21no—
177C4×C101—21no—
178C4×C101—21no—
179C4×C101—21no—
180C4×C101C2×#4111yescharacteristic
181C4×C101H(5,𝔽2)11yescharacteristic
182Dic101—41no—
183Dic101—41no—
184Dic101—41no—
185Dic101—41no—
186Dic101—41no—
187C2×C101—22no—
188C2×C101—41no—
189C2×C101#54611yescharacteristic
190C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C101#57711yescharacteristic
192D101—42no—
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#440—1011no—
209#435D10111yescharacteristic, complemented
210#418—1011no—
211C2×#102—1011no—
212C22×D4—1011no—
213C22×D4—1011no—
214C22×D4—2021no—
215C2×SD16—1011no—
216C2×SD16—1011no—
217C2×SD16—1011no—
218C2×M16C2×D10111yescharacteristic, complemented
219C22×C8C2×D10111yescharacteristic, complemented
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×Q8—1011no—
233C4×D4—1011no—
234C4×D4C2×D10111yescharacteristic
235#117—1011no—
236#117—2021no—
237C2×C4⋊C4—1011no—
238#107—1011no—
239#107—1011no—
240#106—1011no—
241C4⋊C8C2×D10111yescharacteristic, complemented
242#103—1011no—
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#102—1011no—
252C22⋊C8C2×D10111yescharacteristic, complemented
253C22⋊C8C2×D10111yescharacteristic, complemented
254#97C2×D10111yescharacteristic, complemented
255C24—2021no—
256C24—2021no—
257C2×Q8—1011no—
258C2×D4—1011no—
259C2×D4—1011no—
260C2×D4—1011no—
261C2×D4—4041no—
262C2×D4—4041no—
263C2×D4—4044no—
264C2×D4—2022no—
265C2×D4—2022no—
266C2×D4—2021no—
267C2×D4—2021no—
268C2×D4—2021no—
269C2×D4—2021no—
270C2×D4—2021no—
271C2×D4—2021no—
272C2×D4D40411yescharacteristic
273C22×C4—2021no—
274C22×C4—2021no—
275C22×C4C22×D10111yescharacteristic
276C22×C4C22×D10111yescharacteristic
277SD16—4041no—
278SD16—4041no—
279SD16—2022no—
280M16—41no—
281C2×C8—41no—
282C2×C8C22×D10111yescharacteristic
283C2×C8C22×D10111yescharacteristic
284C2×C8C22×D10111yescharacteristic
285C2×C8C22×D10111yescharacteristic
286C4⋊C4—1011no—
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⋊C4—2021no—
296C4⋊C4D40411yescharacteristic
297C22⋊C4—2021no—
298C22⋊C4—2021no—
299C22⋊C4—2021no—
300C22⋊C4—2021no—
301C22⋊C4—2021no—
302C22⋊C4—2021no—
303C22⋊C4D40411yescharacteristic
304C22⋊C4D40411yescharacteristic
305C42—2021no—
306C42—2021no—
307C42C22×D10111yescharacteristic
308C23—8081no—
309C23—8081no—
310C23—4042no—
311C23—4042no—
312C23—4042no—
313C23—4042no—
314C23—2021no—
315C23—2021no—
316C23—2021no—
317C23C2×D40411yescharacteristic
318C23C2×D40411yescharacteristic
319Q8—2021no—
320Q8—2021no—
321D4—22no—
322D4—8081no—
323D4—1012no—
324D4—4042no—
325D4—4042no—
326D4—4042no—
327D4—4042no—
328D4—4042no—
329D4—2021no—
330D4—2021no—
331D4—2021no—
332D4—2021no—
333D4—2021no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—4042no—
339C2×C4—4042no—
340C2×C4—2021no—
341C2×C4—2021no—
342C2×C4—2021no—
343C2×C4—2021no—
344C2×C4—2021no—
345C2×C4C2×D40411yescharacteristic
346C2×C4C2×D40411yescharacteristic
347C2×C4C2×D40411yescharacteristic
348C2×C4C2×D40411yescharacteristic
349C2×C4C23×D10111yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—8084no—
360C22—8084no—
361C22—4041no—
362C22—4041no—
363C22—4041no—
364C22—4041no—
365C22—4041no—
366C22—4041no—
367C22—4041no—
368C22—4041no—
369C22—2022no—
370C22#1569228312yes—
371C22C22×D40411yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—4041no—
377C4—4041no—
378C4—4041no—
379C4—4041no—
380C4—4041no—
381C4#41×D10111yescharacteristic
382C4#1569230011yescharacteristic
383C2—22no—
384C2—41no—
385C2—8081no—
386C2—8081no—
387C2—4042no—
388C2#2026582211yescharacteristic
389C2C2×#1569228311yescharacteristic
390C2#2026598911yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#435×C101C2
socleC22×C101C22×D4
prime radicalD4×D404C2
hypercenter#435D101
hypocenterC101#3310
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #20265867C2semidirect
3. #20265542C2semidirect
4. #20265502C2semidirect
5. #20265491C2semidirect
6. #20265478C2semidirect
7. #20265423C2semidirect
8. #20265406C2semidirect
9. #20265378C2semidirect
10. #20265330C2semidirect
11. #20265330C2semidirect
12. #20265279C2semidirect
13. #20265183C2semidirect
14. C2×#15692177C2semidirect
15. #435×C101C2semidirect
20. C2×#13108063C22semidirect
21. C2×#13108063C22semidirect
24. #15692247C22semidirect
27. #15692234C22semidirect
28. #15692234C22semidirect
34. C4×Dic202C22semidirect
36. #15692214C22semidirect
38. C2×C4⋊Dic101C22semidirect
39. #15692189C22semidirect
40. #15692189C22semidirect
41. #15692187C22semidirect
42. #15692180C22semidirect
46. #15692177C22semidirect
47. #15692177C22semidirect
48. #15692177C22semidirect
49. #15692177C22semidirect
50. #15692177C22semidirect
51. #15692177C22semidirect
52. C2×M16×C101C22semidirect
53. C22×C8×C101C22semidirect
55. C4⋊C8×C101C22semidirect
56. C22⋊C8×C101C22semidirect
57. C22⋊C8×C101C22semidirect
58. #97×C101C22semidirect
196. C101#3310semidirect
209. #435D101semidirect
218. C2×M16C2×D101semidirect
219. C22×C8C2×D101semidirect
241. C4⋊C8C2×D101semidirect
252. C22⋊C8C2×D101semidirect
253. C22⋊C8C2×D101semidirect
254. #97C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658672
D4×D4042
#202655422
#202655022
#202654912
#202654782
#202654232
#202654062
#202653782
#202653302
#202652792
#202651832
C2×#156921772
#435×C1012
#3310101