Finite Group Databasebeta

← #27183147

Subgroups — #27183147

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
1D4×D404C211yesmaximal, characteristic
2#20265543C211yesmaximal, characteristic, complemented
3#20265501C211yesmaximal, characteristic, complemented
4#20265490C211yesmaximal, characteristic, complemented
5#20265474C211yesmaximal, characteristic, complemented
6#20265425C211yesmaximal, characteristic, complemented
7#20265408C211yesmaximal, characteristic, complemented
8#20265377C211yesmaximal, characteristic, complemented
9#20265327C211yesmaximal, characteristic, complemented
10#20265327C211yesmaximal, characteristic, complemented
11#20265275C211yesmaximal, characteristic, complemented
12#20265249C211yesmaximal, characteristic, complemented
13#20265181C211yesmaximal, characteristic, complemented
14C2×#15692175C211yesmaximal, characteristic, complemented
15#552×C101C211yesmaximal, characteristic, complemented
16C22×D404C2211yescharacteristic
17C22×D404C2211yescharacteristic
18C2×D4×D101—21no—
19C2×#13108061C2211yescharacteristic
20C2×#13108061C2211yescharacteristic, complemented
21C2×#13108061C2211yescharacteristic, complemented
22C2×#13108058C2211yescharacteristic, complemented
23C22×#11717579C2211yescharacteristic, complemented
24#15692252C2211yescharacteristic
25#15692230—21no—
26#15692229C2211yescharacteristic
27#15692229C2211yescharacteristic
28#15692225—21no—
29#15692225—21no—
30C4×D404C2211yescharacteristic
31#15692188C2211yescharacteristic, complemented
32#15692188C2211yescharacteristic, complemented
33#15692186C2211yescharacteristic, complemented
34C4⋊#11717579C2211yescharacteristic, complemented
35#15692179C2211yescharacteristic, complemented
36#15692175C2211yescharacteristic
37#15692175C2211yescharacteristic
38#15692175C2211yescharacteristic
39#15692175C2211yescharacteristic, complemented
40#15692175C2211yescharacteristic, complemented
41#15692175C2211yescharacteristic, complemented
42#15692175C2211yescharacteristic, complemented
43#15692175C2211yescharacteristic, complemented
44#15692175C2211yescharacteristic, complemented
45C22⋊#11717579C2211yescharacteristic, complemented
46C22⋊#11717579C2211yescharacteristic, complemented
47#15692166C2211yescharacteristic, complemented
48#126×C101—21no—
49#125×C101—21no—
50#125×C101C2211yescharacteristic, complemented
51#123×C101—21no—
52C22⋊Q8×C101C2211yescharacteristic, complemented
53C22⋊Q8×C101C2211yescharacteristic, complemented
54C4×Q8×C101C2211yescharacteristic, complemented
55C4×D4×C101C2211yescharacteristic
56#117×C101—21no—
57#117×C101C2211yescharacteristic, complemented
58C2×C4⋊C4×C101C2211yescharacteristic, complemented
59C23×D101—21no—
60C23×D101—21no—
61C2×D404—22no—
62C2×D404—22no—
63C2×D404—21no—
64C2×D404—21no—
65C2×D404—21no—
66C2×D404—21no—
67C2×D404—41no—
68C2×D404—41no—
69C2×D404C2311yescharacteristic
70C2×D404C2311yescharacteristic
71C2×D404C2311yescharacteristic
72C2×C22⋊D101—21no—
73C2×C22⋊D101—21no—
74D4×D101—44no—
75C2×C4×D101—21no—
76#13108061—22no—
77#13108061—41no—
78#13108061—41no—
79#13108058—41no—
80C2×#11717579—41no—
81C2×#11717579C2311yescharacteristic
82C2×#11717579C2311yescharacteristic
83C2×#11717579C2311yescharacteristic
84C2×#11717579C2311yescharacteristic
85C4⋊Dic101C2311yescharacteristic
86#13108051—21no—
87#13108051—21no—
88#13108051—21no—
89C2×Q8×C101C2311yescharacteristic
90C2×D4×C101D411yescharacteristic
91C22×C4×C101—21no—
92C22×C4×C101C2311yescharacteristic
93C22×C4×C101C2311yescharacteristic
94C4⋊C4×C101—21no—
95C4⋊C4×C101—21no—
96C4⋊C4×C101—21no—
97C4⋊C4×C101—21no—
98C4⋊C4×C101—21no—
99C4⋊C4×C101—21no—
100C4⋊C4×C101D411yescharacteristic
101C4⋊C4×C101C2311yescharacteristic
102C4⋊C4×C101C2311yescharacteristic
103C4⋊C4×C101C2311yescharacteristic
104C22⋊C4×C101—21no—
105C22⋊C4×C101—21no—
106C22⋊C4×C101—21no—
107C22⋊C4×C101D411yescharacteristic
108C22⋊C4×C101D411yescharacteristic
109C42×C101—21no—
110C42×C101—21no—
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—
123Q8×C101—21no—
124Q8×C101—21no—
125D4×C101—22no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—21no—
129C2×C4×C101—21no—
130C2×C4×C101—21no—
131C2×C4×C101—21no—
132C2×C4×C101—21no—
133C2×C4×C101—21no—
134C2×C4×C101—42no—
135C2×C4×C101C2×D411yescharacteristic
136C2×C4×C101C2×D411yescharacteristic
137C2×C4×C101C2×D411yescharacteristic
138C2×C4×C101C2×D411yescharacteristic
139C2×C4×C101C2411yescharacteristic
140C22⋊D101—42no—
141C22⋊D101—42no—
142C2×Dic101—21no—
143D404—21no—
144D404—21no—
145D404—21no—
146D404—21no—
147D404—21no—
148D404—42no—
149D404—42no—
150D404—42no—
151D404—81no—
152D404#4112yes—
153C4×D101—42no—
154#11717579—21no—
155#11717579—21no—
156#11717579—21no—
157#11717579—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×C101—41no—
181C4×C101—41no—
182C4×C101—41no—
183C4×C101—41no—
184C4×C101C2×#4111yescharacteristic
185C4×C101H(5,𝔽2)11yescharacteristic
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#552D10111yescharacteristic, complemented
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#435—1011no—
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×M16—1011no—
219C22×C8—1011no—
220#127—1011no—
221#126—21no—
222#125—21no—
223#125C2×D10111yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C2×D10111yescharacteristic, complemented
226C22⋊Q8C2×D10111yescharacteristic, complemented
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—21no—
236#117C2×D10111yescharacteristic, complemented
237C2×C4⋊C4C2×D10111yescharacteristic, complemented
238#107—1011no—
239#107—1011no—
240#106—1011no—
241C4⋊C8—1011no—
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⋊C8—1011no—
253C22⋊C8—1011no—
254#97—1011no—
255C24—2021no—
256C24—2021no—
257C2×Q8C22×D10111yescharacteristic
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×D4C22⋊D10111yescharacteristic
273C22×C4—21no—
274C22×C4—2021no—
275C22×C4C22×D10111yescharacteristic
276C22×C4C22×D10111yescharacteristic
277SD16—4041no—
278SD16—4041no—
279SD16—2022no—
280M16—4041no—
281C2×C8—1011no—
282C2×C8—1011no—
283C2×C8—1011no—
284C2×C8—1011no—
285C2×C8—4041no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—1011no—
293C4⋊C4C22⋊D10111yescharacteristic
294C4⋊C4C22×D10111yescharacteristic
295C4⋊C4C22×D10111yescharacteristic
296C4⋊C4C22×D10111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—2021no—
301C22⋊C4—2021no—
302C22⋊C4—2021no—
303C22⋊C4C22⋊D10111yescharacteristic
304C22⋊C4C22⋊D10111yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D10111yescharacteristic
308C23—8081no—
309C23—8081no—
310C23—4042no—
311C23—4042no—
312C23—4042no—
313C23—4042no—
314C23—2021no—
315C23—2021no—
316C23—2021no—
317C23C2×C22⋊D10111yescharacteristic
318C23C2×C22⋊D10111yescharacteristic
319Q8—21no—
320Q8—21no—
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—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—4042no—
344C2×C4—2021no—
345C2×C4C2×C22⋊D10111yescharacteristic
346C2×C4C2×C22⋊D10111yescharacteristic
347C2×C4C2×C22⋊D10111yescharacteristic
348C2×C4C2×C22⋊D10111yescharacteristic
349C2×C4C23×D10111yescharacteristic
350C8—2021no—
351C8—2021no—
352C8—2021no—
353C8—2021no—
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#1569228012yes—
371C22C22×C22⋊D10111yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—4041no—
381C4#41×D10111yescharacteristic
382C4#1569230111yescharacteristic
383C2—22no—
384C2—41no—
385C2—8081no—
386C2—8081no—
387C2—4042no—
388C2#2026581511yescharacteristic
389C2C2×#1569228011yescharacteristic
390C2#2026598511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#552×C101C2
socleC22×C101C22×D4
prime radicalD4×D404C2
hypercenter#552D101
hypocenterC101#3310
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #20265543C2semidirect
3. #20265501C2semidirect
4. #20265490C2semidirect
5. #20265474C2semidirect
6. #20265425C2semidirect
7. #20265408C2semidirect
8. #20265377C2semidirect
9. #20265327C2semidirect
10. #20265327C2semidirect
11. #20265275C2semidirect
12. #20265249C2semidirect
13. #20265181C2semidirect
14. C2×#15692175C2semidirect
15. #552×C101C2semidirect
20. C2×#13108061C22semidirect
21. C2×#13108061C22semidirect
22. C2×#13108058C22semidirect
23. C22×#11717579C22semidirect
31. #15692188C22semidirect
32. #15692188C22semidirect
33. #15692186C22semidirect
34. C4⋊#11717579C22semidirect
35. #15692179C22semidirect
39. #15692175C22semidirect
40. #15692175C22semidirect
41. #15692175C22semidirect
42. #15692175C22semidirect
43. #15692175C22semidirect
44. #15692175C22semidirect
45. C22⋊#11717579C22semidirect
46. C22⋊#11717579C22semidirect
47. #15692166C22semidirect
50. #125×C101C22semidirect
52. C22⋊Q8×C101C22semidirect
53. C22⋊Q8×C101C22semidirect
54. C4×Q8×C101C22semidirect
57. #117×C101C22semidirect
58. C2×C4⋊C4×C101C22semidirect
196. C101#3310semidirect
197. #552D101semidirect
223. #125C2×D101semidirect
225. C22⋊Q8C2×D101semidirect
226. C22⋊Q8C2×D101semidirect
232. C4×Q8C2×D101semidirect
236. #117C2×D101semidirect
237. C2×C4⋊C4C2×D101semidirect

Maximal subgroups

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

subgroupindex
D4×D4042
#202655432
#202655012
#202654902
#202654742
#202654252
#202654082
#202653772
#202653272
#202652752
#202652492
#202651812
C2×#156921752
#552×C1012
#3310101