Finite Group Databasebeta

← #27180480

Subgroups — #27180480

Order 12928.

724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20266013C211yesmaximal, characteristic, complemented
2#20265975C212yesmaximal, complemented
3#20265359C212yesmaximal, complemented
4#20265354C212yesmaximal, complemented
5#20265349C212yesmaximal, complemented
6#20265344C212yesmaximal, complemented
7C2×#15692183C212yesmaximal, complemented
8#20265161C211yesmaximal, characteristic, complemented
9#532×C101C211yesmaximal, characteristic, complemented
10#15692303C2212yes—
11#15692302—22no—
12#15692301—22no—
13#15692300—22no—
14#15692300C2212yes—
15C2×#13108074—22no—
16C2×#13108074C2212yes—
17#41×D101—24no—
18#15692286C2214yescomplemented
19#15692285—22no—
20#15692281C2214yescomplemented
21#15692280—22no—
22#15692275—24no—
23C2×#13108058C2212yescomplemented
24#15692183C2214yescomplemented
25#15692183C2214yescomplemented
26#15692174C2212yescomplemented
27#15692172C2212yescomplemented
28C2×#41×C101C2211yescharacteristic
29C2×#41×C101C2211yescharacteristic, complemented
30C4⋊Q8×C101C2211yescharacteristic, complemented
31#127×C101C2211yescharacteristic, complemented
32#124×C101C2211yescharacteristic, complemented
33#124×C101C2211yescharacteristic, complemented
34#121×C101—22no—
35C4×D4×C101—22no—
36C2×C42×C101C2211yescharacteristic, complemented
37#13108075—24no—
38#13108074—24no—
39#13108074—24no—
40#13108074—22no—
41#13108074—24no—
42#13108074—22no—
43#13108074D414yescomplemented
44#13108074C2314yes—
45#13108073—28no—
46#13108073—24no—
47C2×Dic202—22no—
48C2×Dic202D412yescomplemented
49Q8×D101—24no—
50C2×D404—22no—
51C2×D404D412yescomplemented
52C2×C22⋊D101—24no—
53C2×C22⋊D101—22no—
54D4×D101—28no—
55D4×D101—24no—
56C2×C4×D101—24no—
57C2×C4×D101—22no—
58#13108064—24no—
59#13108062—24no—
60#13108061—24no—
61#13108059—24no—
62#13108058—22no—
63#13108058C2314yes—
64C2×#11717579—22no—
65#41×C101—22no—
66#41×C101—22no—
67#41×C101—22no—
68#41×C101—22no—
69C2×Q8×C101C2311yescharacteristic
70C2×Q8×C101C2311yescharacteristic
71C2×D4×C101—22no—
72C2×D4×C101—21no—
73C2×D4×C101D412yes—
74C2×D4×C101C2311yescharacteristic
75C2×D4×C101C2311yescharacteristic
76C22×C4×C101—21no—
77C22×C4×C101—21no—
78C22×C4×C101D412yes—
79C22×C4×C101C2311yescharacteristic
80C4⋊C4×C101—22no—
81C22⋊C4×C101—22no—
82C22⋊C4×C101—22no—
83C42×C101—21no—
84C42×C101C2311yescharacteristic
85C42×C101C2311yescharacteristic
86C23×C101—21no—
87C23×C101C2×D412yes—
88C23×C101C2×D411yescharacteristic
89C22×D101—24no—
90C22×D101—22no—
91Q8×C101—22no—
92Q8×C101—22no—
93D4×C101—24no—
94D4×C101—22no—
95D4×C101—22no—
96D4×C101—44no—
97D4×C101—42no—
98C2×C4×C101—21no—
99C2×C4×C101—22no—
100C2×C4×C101—22no—
101C2×C4×C101—21no—
102C2×C4×C101—21no—
103C2×C4×C101—42no—
104C2×C4×C101—41no—
105C2×C4×C101—41no—
106C2×C4×C101C2×D412yes—
107C2×C4×C101C2×D412yes—
108C2×C4×C101C2×D412yes—
109C2×C4×C101C2×D411yescharacteristic
110C2×C4×C101C2411yescharacteristic
111C22⋊D101—28no—
112C22⋊D101—24no—
113C22⋊D101—24no—
114C2×Dic101—24no—
115C2×Dic101—22no—
116D404—24no—
117D404—22no—
118D404C2×D414yes—
119C4×D101—28no—
120C4×D101—24no—
121C4×D101—24no—
122Dic202—24no—
123Dic202—22no—
124Dic202C2×D414yes—
125#11717579—24no—
126C22×C101—22no—
127C22×C101—41no—
128C22×C101—42no—
129C22×C101—42no—
130C22×C101—41no—
131C22×C101#12012yes—
132C22×C101C22×D411yescharacteristic
133C2×D101—24no—
134C2×D101—82no—
135C2×D101—84no—
136C4×C101—22no—
137C4×C101—22no—
138C4×C101—41no—
139C4×C101—41no—
140C4×C101#12011yescharacteristic
141C4×C101#12011yescharacteristic
142C4×C101C22×D412yes—
143Dic101—24no—
144C2×C101—21no—
145C2×C101—22no—
146C2×C101—42no—
147C2×C101—81no—
148C2×C101C2×#12011yescharacteristic
149D101—44no—
150#3018—1011nomaximal, Sylow
151C101#301811yescharacteristic, complemented, Sylow
152#585—1011no—
153#575—1012no—
154#532D10111yescharacteristic, complemented
155#456—1012no—
156#455—1012no—
157#454—1012no—
158#453—1012no—
159C2×#104—1012no—
160#413—1011no—
1612−1+4—1012no—
1622−1+4—2022no—
163H(5,𝔽2)—1012no—
164H(5,𝔽2)—2022no—
165H(5,𝔽2)—2022no—
166C2×#41—1012no—
167C2×#41—2024no—
168C2×#41—2022no—
169C2×#41C2×D10111yescharacteristic
170C2×#41C2×D10111yescharacteristic, complemented
171#137—1014no—
172#137—2022no—
173AGL(1,ℤ/8)—1014no—
174AGL(1,ℤ/8)—2022no—
175#135—2024no—
176C2×M16—1012no—
177C4⋊Q8C2×D10111yescharacteristic, complemented
178#127C2×D10111yescharacteristic, complemented
179#124C2×D10111yescharacteristic, complemented
180#124C2×D10111yescharacteristic, complemented
181#121—22no—
182C4×D4—22no—
183C2×C42C2×D10111yescharacteristic, complemented
184#104—1014no—
185#104—1014no—
186#101—1012no—
187#100—1012no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41—1014no—
193#41—1014no—
194#41—2024no—
195#41—2024no—
196#41—2024no—
197#41—2028no—
198#41—2024no—
199#41—2022no—
200#41—2022no—
201#41—2024no—
202C2×Q8—1012no—
203C2×Q8—2022no—
204C2×Q8—2024no—
205C2×Q8C22×D10111yescharacteristic
206C2×Q8C22×D10111yescharacteristic
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—1012no—
210C2×D4—2022no—
211C2×D4—2024no—
212C2×D4—2028no—
213C2×D4—2022no—
214C2×D4—2024no—
215C2×D4C22⋊D10112yes—
216C2×D4C22×D10111yescharacteristic
217C2×D4C22×D10111yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—2024no—
221C22×C4—2022no—
222C22×C4C22⋊D10112yes—
223C22×C4C22×D10111yescharacteristic
224Q16—2024no—
225SD16—2024no—
226SD16—2024no—
227D8—2024no—
228M16—1014no—
229M16—2022no—
230C2×C8—2022no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—21no—
235C42C22×D10111yescharacteristic
236C42C22×D10111yescharacteristic
237C23—21no—
238C23—2024no—
239C23—2022no—
240C23C2×C22⋊D10112yes—
241C23C2×C22⋊D10111yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—1014no—
245Q8—2024no—
246Q8—2022no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—44no—
251D4—42no—
252D4—1014no—
253D4—2024no—
254D4—2028no—
255D4—2024no—
256D4—2024no—
257D4—2022no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—2028no—
267C2×C4—2024no—
268C2×C4—2022no—
269C2×C4—2024no—
270C2×C4—2024no—
271C2×C4D4×D10112yes—
272C2×C4D4×D10112yes—
273C2×C4C2×C22⋊D10112yes—
274C2×C4C2×C22⋊D10111yescharacteristic
275C2×C4C23×D10111yescharacteristic
276C8—2024no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—8082no—
283C22—8084no—
284C22—2024no—
285C22#1569222312yes—
286C22C22×C22⋊D10111yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—2024no—
292C4#1569222311yescharacteristic
293C4#1569222311yescharacteristic
294C4C2×D4×D10112yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—81no—
299C2—4044no—
300C2C2×#1569222311yescharacteristic
subgroupG/H
centerC4#15692223
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#532×C101C2
socleC2×C101C2×#120
hypercenter#532D101
hypocenterC101#3018
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #20266013C2semidirect
2. #20265975C2semidirect
3. #20265359C2semidirect
4. #20265354C2semidirect
5. #20265349C2semidirect
6. #20265344C2semidirect
7. C2×#15692183C2semidirect
8. #20265161C2semidirect
9. #532×C101C2semidirect
18. #15692286C22semidirect
20. #15692281C22semidirect
23. C2×#13108058C22semidirect
24. #15692183C22semidirect
25. #15692183C22semidirect
26. #15692174C22semidirect
27. #15692172C22semidirect
29. C2×#41×C101C22semidirect
30. C4⋊Q8×C101C22semidirect
31. #127×C101C22semidirect
32. #124×C101C22semidirect
33. #124×C101C22semidirect
36. C2×C42×C101C22semidirect
43. #13108074D4semidirect
48. C2×Dic202D4semidirect
51. C2×D404D4semidirect
151. C101#3018semidirect
154. #532D101semidirect
170. C2×#41C2×D101semidirect
177. C4⋊Q8C2×D101semidirect
178. #127C2×D101semidirect
179. #124C2×D101semidirect
180. #124C2×D101semidirect
183. C2×C42C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202660132
#202659752
#202653592
#202653542
#202653492
#202653442
C2×#156921832
#202651612
#532×C1012
#3018101