Finite Group Databasebeta

← #27183092

Subgroups — #27183092

Order 12928.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265830C211yesmaximal, characteristic, complemented
2#20265820C211yesmaximal, characteristic, complemented
3#20265506C211yesmaximal, characteristic, complemented
4#20265494C211yesmaximal, characteristic, complemented
5#20265464C211yesmaximal, characteristic, complemented
6#20265429C211yesmaximal, characteristic, complemented
7#20265409C211yesmaximal, characteristic, complemented
8#20265389C211yesmaximal, characteristic, complemented
9#20265325C211yesmaximal, characteristic, complemented
10#20265312C211yesmaximal, characteristic, complemented
11#20265290C211yesmaximal, characteristic, complemented
12#20265250C211yesmaximal, characteristic, complemented
13#20265177C211yesmaximal, characteristic, complemented
14#102×D101C211yesmaximal, characteristic, complemented
15#489×C101C211yesmaximal, characteristic, complemented
16C2×#13108075—21no—
17C2×#13108075C2211yescharacteristic
18C2×D4×D101C2211yescharacteristic
19C2×#13108061C2211yescharacteristic, complemented
20C2×D808C2211yescharacteristic
21C2×#13108059C2211yescharacteristic
22C2×#13108057C2211yescharacteristic, complemented
23C2×C8×D101C2211yescharacteristic
24#15692240—21no—
25#15692240—21no—
26#15692237—21no—
27#15692231—21no—
28#15692231C2211yescharacteristic
29#15692230C2211yescharacteristic
30#15692230C2211yescharacteristic, complemented
31#15692228—21no—
32#15692226C2211yescharacteristic
33#15692225—21no—
34C4×D404C2211yescharacteristic
35C4×D404C2211yescharacteristic
36C4×D404C2211yescharacteristic, complemented
37C4⋊C4×D101C2211yescharacteristic
38C4⋊C4×D101C2211yescharacteristic, complemented
39C22⋊C4×D101—21no—
40#15692187C2211yescharacteristic, complemented
41#15692186C2211yescharacteristic, complemented
42#15692186C2211yescharacteristic, complemented
43C4⋊#11717579C2211yescharacteristic, complemented
44#15692179C2211yescharacteristic, complemented
45#15692177C2211yescharacteristic
46#15692177C2211yescharacteristic, complemented
47#15692176C2211yescharacteristic
48#15692175C2211yescharacteristic
49#15692175C2211yescharacteristic
50#15692167C2211yescharacteristic
51#15692167C2211yescharacteristic, complemented
52C4⋊Q8×C101C2211yescharacteristic, complemented
53#124×C101C2211yescharacteristic
54#103×C101C2211yescharacteristic, complemented
55#103×C101C2211yescharacteristic, complemented
56#102×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58#97×C101C2211yescharacteristic, complemented
59C23×D101—21no—
60#13108075—22no—
61#13108075—41no—
62#13108075—44no—
63C2×D404—21no—
64C2×D404—21no—
65C2×D404—21no—
66C2×D404D411yescharacteristic
67C2×D404C2311yescharacteristic
68C2×D404C2311yescharacteristic
69C2×C22⋊D101—21no—
70C2×C22⋊D101—21no—
71D4×D101—22no—
72D4×D101—41no—
73C2×C4×D101—21no—
74C2×C4×D101—21no—
75C2×C4×D101—21no—
76C2×C4×D101—21no—
77C2×C4×D101—21no—
78C2×C4×D101C2311yescharacteristic
79C2×C4×D101C2311yescharacteristic
80#13108061—41no—
81D808—22no—
82#13108059—41no—
83#13108057—41no—
84C2×#11717579C2311yescharacteristic
85C2×#11717579C2311yescharacteristic
86C8×D101—41no—
87C4⋊Dic101D411yescharacteristic
88C4⋊Dic101C2311yescharacteristic
89C4⋊Dic101C2311yescharacteristic
90#13108053—21no—
91#13108053—21no—
92#13108053—21no—
93C22⋊Dic101—21no—
94#13108051—21no—
95#13108051—21no—
96#13108051—21no—
97#13108051—21no—
98#13108051—21no—
99#13108051D411yescharacteristic
100#13108051D411yescharacteristic
101C2×Q8×C101—21no—
102C2×Q8×C101C2311yescharacteristic
103C2×D4×C101C2311yescharacteristic
104C2×C8×C101C2311yescharacteristic
105C2×C8×C101C2311yescharacteristic
106C4⋊C4×C101—21no—
107C4⋊C4×C101C2311yescharacteristic
108C4⋊C4×C101C2311yescharacteristic
109C22⋊C4×C101—21no—
110C22⋊C4×C101—21no—
111C42×C101C2311yescharacteristic
112C23×C101—21no—
113C22×D101—21no—
114C22×D101—21no—
115C22×D101—42no—
116C22×D101—42no—
117C22×D101—81no—
118C22×D101C2×D411yescharacteristic
119C22×D101C2×D411yescharacteristic
120Q8×C101—21no—
121Q8×C101—21no—
122Q8×C101—42no—
123D4×C101—21no—
124D4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101C2×D411yescharacteristic
129C2×C4×C101C2×D411yescharacteristic
130C2×C4×C101C2411yescharacteristic
131C22⋊D101—42no—
132C22⋊D101—81no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101C2×D411yescharacteristic
136C2×Dic101C2×D411yescharacteristic
137D404—22no—
138D404—21no—
139D404—21no—
140D404—21no—
141D404—42no—
142D404—42no—
143D404—42no—
144D404#4112yescomplemented
145C4×D101—21no—
146C4×D101—21no—
147C4×D101—21no—
148C4×D101—21no—
149C4×D101—42no—
150C4×D101—42no—
151C4×D101—42no—
152C4×D101—42no—
153C4×D101—81no—
154C8×C101—21no—
155C8×C101—21no—
156#11717579—21no—
157#11717579—21no—
158C22×C101—41no—
159C22×C101—41no—
160C22×C101—41no—
161C22×C101C22×D411yescharacteristic
162C2×D101—21no—
163C2×D101—21no—
164C2×D101—21no—
165C2×D101—22no—
166C2×D101—21no—
167C2×D101—21no—
168C2×D101—41no—
169C2×D101—41no—
170C2×D101—41no—
171C2×D101—41no—
172C2×D101—41no—
173C2×D101—84no—
174C2×D101AGL(1,ℤ/8)12yes—
175C4×C101—21no—
176C4×C101—21no—
177C4×C101—41no—
178C4×C101—41no—
179C4×C101—41no—
180C4×C101C2×#4111yescharacteristic
181C4×C101H(5,𝔽2)11yescharacteristic
182Dic101—21no—
183Dic101—21no—
184Dic101—41no—
185Dic101—41no—
186C2×C101—81no—
187C2×C101#54611yescharacteristic
188C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C101#57711yescharacteristic
190D101—22no—
191D101—41no—
192D101—42no—
193D101—81no—
194#3306—1011nomaximal, Sylow
195C101#330611yescharacteristic, complemented, Sylow
196#547—1011no—
197#546—1011no—
198#489D10111yescharacteristic, complemented
199#483—1011no—
200#481—1011no—
201#476—1011no—
202#469—1011no—
203#465—1011no—
204#463—1011no—
205#449—1011no—
206#447—1011no—
207#442—1011no—
208#435—1011no—
209#417—1011no—
210C2×#102—1011no—
211C2×#41—1011no—
212C2×#41—2021no—
213C22×D4—1011no—
214C2×SD16—1011no—
215C2×D8—1011no—
216C2×D8—1011no—
217C2×M16—1011no—
218C22×C8—1011no—
219C4⋊Q8C2×D10111yescharacteristic, complemented
220#124C2×D10111yescharacteristic
221#123—2021no—
222#123—2021no—
223#123—2021no—
224C22⋊Q8—1011no—
225C22⋊Q8—2021no—
226#121—1011no—
227#121—1011no—
228#121—1011no—
229#121—2021no—
230#120—2021no—
231C4×D4—1011no—
232C4×D4—1011no—
233C4×D4—1011no—
234C2×C4⋊C4—1011no—
235C2×C4⋊C4—1011no—
236C2×C22⋊C4—2021no—
237#106—1011no—
238#106—1011no—
239#106—1011no—
240C4⋊C8—1011no—
241#103—1011no—
242#103C2×D10111yescharacteristic, complemented
243#103C2×D10111yescharacteristic, complemented
244#102—1011no—
245#102—1011no—
246#102—1011no—
247#102—1011no—
248#102—1011no—
249#102C2×D10111yescharacteristic
250#102C2×D10111yescharacteristic
251C22⋊C8—1011no—
252C22⋊C8—1011no—
253#97C2×D10111yescharacteristic, complemented
254C24—2021no—
255#41—4044no—
256#41—4041no—
257#41—2022no—
258C2×Q8—21no—
259C2×Q8C22×D10111yescharacteristic
260C2×D4—1011no—
261C2×D4—1011no—
262C2×D4—1011no—
263C2×D4—4041no—
264C2×D4—2021no—
265C2×D4—2021no—
266C2×D4—2021no—
267C2×D4—2021no—
268C2×D4—2022no—
269C2×D4—2021no—
270C2×D4C22×D10111yescharacteristic
271C22×C4—1011no—
272C22×C4—1011no—
273C22×C4—2021no—
274C22×C4—2021no—
275C22×C4—2021no—
276C22×C4—2021no—
277C22×C4—2021no—
278SD16—4041no—
279D8—4041no—
280D8—2022no—
281M16—4041no—
282C2×C8—1011no—
283C2×C8—1011no—
284C2×C8—4041no—
285C2×C8C22×D10111yescharacteristic
286C2×C8C22×D10111yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—1011no—
289C4⋊C4—1011no—
290C4⋊C4—1011no—
291C4⋊C4—2021no—
292C4⋊C4—2021no—
293C4⋊C4—2021no—
294C4⋊C4C22×D10111yescharacteristic
295C4⋊C4C22×D10111yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—1011no—
299C22⋊C4—1011no—
300C22⋊C4—2021no—
301C22⋊C4—2021no—
302C22⋊C4—2021no—
303C22⋊C4—2021no—
304C22⋊C4—2021no—
305C22⋊C4—2021no—
306C42C22×D10111yescharacteristic
307C23—21no—
308C23—8081no—
309C23—1011no—
310C23—1011no—
311C23—4042no—
312C23—4042no—
313C23—2021no—
314C23—2021no—
315Q8—21no—
316Q8—21no—
317Q8—42no—
318D4—21no—
319D4—21no—
320D4—8081no—
321D4—1012no—
322D4—4042no—
323D4—4042no—
324D4—4042no—
325D4—4042no—
326D4—2022no—
327D4—2021no—
328D4—2021no—
329D4—2021no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—8081no—
334C2×C4—1011no—
335C2×C4—1011no—
336C2×C4—4042no—
337C2×C4—4042no—
338C2×C4—4042no—
339C2×C4—4042no—
340C2×C4—2021no—
341C2×C4—2021no—
342C2×C4—2021no—
343C2×C4—2021no—
344C2×C4—2021no—
345C2×C4—2021no—
346C2×C4D4×D10111yescharacteristic
347C2×C4D4×D10111yescharacteristic
348C2×C4C23×D10111yescharacteristic
349C8—21no—
350C8—21no—
351C8—2021no—
352C8—2021no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—8084no—
357C22—1012no—
358C22—4041no—
359C22—4041no—
360C22—4041no—
361C22—4041no—
362C22—4041no—
363C22—2021no—
364C22—2022no—
365C22—2021no—
366C22—2021no—
367C22—2021no—
368C22—2021no—
369C22C2×D4×D10111yescharacteristic
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—4041no—
376C4—4041no—
377C4—2021no—
378C4—2021no—
379C4#41×D10111yescharacteristic
380C4#1569230111yescharacteristic
381C2—81no—
382C2—8081no—
383C2—4042no—
384C2—4041no—
385C2—2022no—
386C2#2026582011yescharacteristic
387C2AGL(1,ℤ/8)×D10111yescharacteristic
388C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#489×C101C2
socleC22×C101C22×D4
hypercenter#489D101
hypocenterC101#3306
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #20265830C2semidirect
2. #20265820C2semidirect
3. #20265506C2semidirect
4. #20265494C2semidirect
5. #20265464C2semidirect
6. #20265429C2semidirect
7. #20265409C2semidirect
8. #20265389C2semidirect
9. #20265325C2semidirect
10. #20265312C2semidirect
11. #20265290C2semidirect
12. #20265250C2semidirect
13. #20265177C2semidirect
14. #102×D101C2semidirect
15. #489×C101C2semidirect
19. C2×#13108061C22semidirect
22. C2×#13108057C22semidirect
30. #15692230C22semidirect
36. C4×D404C22semidirect
38. C4⋊C4×D101C22semidirect
40. #15692187C22semidirect
41. #15692186C22semidirect
42. #15692186C22semidirect
43. C4⋊#11717579C22semidirect
44. #15692179C22semidirect
46. #15692177C22semidirect
51. #15692167C22semidirect
52. C4⋊Q8×C101C22semidirect
54. #103×C101C22semidirect
55. #103×C101C22semidirect
58. #97×C101C22semidirect
144. D404#41semidirect
195. C101#3306semidirect
198. #489D101semidirect
219. C4⋊Q8C2×D101semidirect
242. #103C2×D101semidirect
243. #103C2×D101semidirect
253. #97C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658302
#202658202
#202655062
#202654942
#202654642
#202654292
#202654092
#202653892
#202653252
#202653122
#202652902
#202652502
#202651772
#102×D1012
#489×C1012
#3306101