Finite Group Databasebeta

← #27181610

Subgroups — #27181610

Order 12928.

564 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#15692286C211yesmaximal, characteristic, complemented
2C2×#15692280C211yesmaximal, characteristic, complemented
3C2×#15692275C211yesmaximal, characteristic, complemented
4C22×#13108061C211yesmaximal, characteristic, complemented
5#20265746C211yesmaximal, characteristic, complemented
6#20265569C212yesmaximal, complemented
7#20265567C212yesmaximal, complemented
8#20265559C212yesmaximal, complemented
9#20265551C212yesmaximal, complemented
10#20265133C211yesmaximal, characteristic, complemented
11#536×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C22×D404C2211yescharacteristic
14#15692286—24no—
15#15692280—24no—
16#15692275—24no—
17C2×#13108064C2212yescomplemented
18C2×#13108064C2211yescharacteristic, complemented
19C2×#13108062C2211yescharacteristic, complemented
20C2×#13108061—24no—
21C2×#13108061C2212yes—
22C2×#13108061C2212yes—
23C2×#13108061C2212yes—
24C2×#13108061C2211yescharacteristic, complemented
25C2×#13108061C2212yescomplemented
26C2×#13108059C2212yes—
27C2×#13108059C2211yescharacteristic, complemented
28C2×#13108058C2211yescharacteristic
29C22×#11717579C2211yescharacteristic
30#15692252C2212yes—
31#15692243C2212yes—
32#15692230—22no—
33#15692225—22no—
34#15692166C2212yescomplemented
35C4×#11717579C2212yescomplemented
36C2×#41×C101C2211yescharacteristic
37C22×Q8×C101C2211yescharacteristic
38C4⋊Q8×C101C2212yescomplemented
39#124×C101C2212yescomplemented
40#123×C101—22no—
41C22⋊Q8×C101—22no—
42#117×C101C2211yescharacteristic
43C23×D101—21no—
44#13108074—22no—
45#13108074—22no—
46C2×Dic202C2311yescharacteristic
47C2×D404—22no—
48C2×D404—22no—
49C2×D404—21no—
50C2×D404—24no—
51C2×D404C2312yes—
52C2×D404C2311yescharacteristic
53C2×C22⋊D101—21no—
54C2×C4×D101—21no—
55#13108064—22no—
56#13108064—24no—
57#13108062—22no—
58#13108061—24no—
59#13108061—24no—
60#13108061—24no—
61#13108061—22no—
62#13108061—24no—
63#13108059—24no—
64#13108059—22no—
65#13108058D414yescomplemented
66C2×#11717579D414yescomplemented
67C2×#11717579C2311yescharacteristic
68C2×#11717579C2312yes—
69C2×#11717579C2311yescharacteristic
70#13108051—24no—
71#41×C101—22no—
72#41×C101—22no—
73C2×Q8×C101—21no—
74C2×Q8×C101—22no—
75C2×Q8×C101—22no—
76C2×Q8×C101C2312yes—
77C2×Q8×C101C2311yescharacteristic
78C2×D4×C101—21no—
79C2×D4×C101C2311yescharacteristic
80C22×C4×C101—21no—
81C22×C4×C101—21no—
82C22×C4×C101C2311yescharacteristic
83C4⋊C4×C101—24no—
84C4⋊C4×C101D412yes—
85C22⋊C4×C101—24no—
86C22⋊C4×C101D412yes—
87C42×C101C2312yes—
88C23×C101—21no—
89C23×C101C2×D411yescharacteristic
90C22×D101—22no—
91C22×D101—21no—
92C22×D101—42no—
93C22×D101—82no—
94Q8×C101—22no—
95Q8×C101—22no—
96Q8×C101—21no—
97Q8×C101—21no—
98Q8×C101—42no—
99D4×C101—21no—
100D4×C101—21no—
101D4×C101—42no—
102C2×C4×C101—22no—
103C2×C4×C101—21no—
104C2×C4×C101—42no—
105C2×C4×C101—42no—
106C2×C4×C101C2×D412yes—
107C2×C4×C101C2×D412yes—
108C2×C4×C101C2×D414yes—
109C2×C4×C101C2×D411yescharacteristic
110C2×C4×C101C2411yescharacteristic
111C22⋊D101—42no—
112C2×Dic101—21no—
113D404—22no—
114D404—21no—
115D404—22no—
116D404—21no—
117D404—42no—
118D404—84no—
119C4×D101—42no—
120Dic202—21no—
121Dic202—21no—
122#11717579C2×D412yes—
123#11717579C2×D412yes—
124#11717579C2×D414yes—
125C22×C101—22no—
126C22×C101—41no—
127C22×C101—42no—
128C22×C101#12712yes—
129C22×C101C22×D411yescharacteristic
130C2×D101—41no—
131C2×D101—42no—
132C2×D101—42no—
133C2×D101—44no—
134C2×D101—84no—
135C4×C101—24no—
136C4×C101—41no—
137C4×C101—42no—
138C4×C101#12712yes—
139C4×C101C22×D411yescharacteristic
140C4×C101C22×D411yescharacteristic
141Dic101—41no—
142C2×C101—22no—
143C2×C101—81no—
144C2×C101C2×#12711yescharacteristic
145C2×C101#57712yes—
146D101—81no—
147D101—82no—
148#3154—1011nomaximal, Sylow
149C101#315411yescharacteristic, complemented, Sylow
150C2×#137—1011no—
151C2×AGL(1,ℤ/8)—1011no—
152C2×#135—1011no—
153C22×SD16—1011no—
154#536D10111yescharacteristic, complemented
155#535—1011no—
156#497—1012no—
157#496—1012no—
158#495—1012no—
159#492—1012no—
160#405—1011no—
161C2×#41—1011no—
162C2×#41C2×D10111yescharacteristic
163C22×Q8C2×D10111yescharacteristic
164C22×D4—1011no—
165#137—2024no—
166AGL(1,ℤ/8)—2024no—
167#135—2024no—
168C2×Q16—1011no—
169C2×Q16—1012no—
170C2×SD16—1012no—
171C2×SD16—1012no—
172C2×SD16—1011no—
173C2×SD16—1012no—
174C2×SD16—1012no—
175C2×SD16—1011no—
176C2×SD16—2024no—
177C2×D8—1011no—
178C2×D8—1012no—
179C2×M16—1011no—
180C22×C8—1011no—
181C4⋊Q8C2×D10112yescomplemented
182#127—1012no—
183#124—1012no—
184#124C2×D10112yescomplemented
185#123—22no—
186C22⋊Q8—22no—
187#121—2022no—
188#120—2022no—
189#117C2×D10111yescharacteristic
190#97—1012no—
191C4×C8—1012no—
192C24—2021no—
193#41—22no—
194#41—22no—
195#41—2022no—
196#41—2022no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—1011no—
201C2×Q8C22×D10112yes—
202C2×Q8C22×D10111yescharacteristic
203C2×D4—21no—
204C2×D4—1012no—
205C2×D4—1011no—
206C2×D4—2021no—
207C2×D4—2022no—
208C2×D4—2022no—
209C2×D4—2024no—
210C2×D4—2021no—
211C2×D4C22×D10111yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—2021no—
215C22×C4C22×D10111yescharacteristic
216Q16—2022no—
217Q16—2024no—
218SD16—2022no—
219SD16—2024no—
220SD16—2024no—
221SD16—2022no—
222SD16—2024no—
223SD16—2024no—
224D8—2022no—
225D8—2024no—
226M16—1014no—
227C2×C8—1014no—
228C2×C8—1011no—
229C2×C8—1012no—
230C2×C8—1011no—
231C4⋊C4—24no—
232C4⋊C4C22⋊D10112yes—
233C22⋊C4—24no—
234C22⋊C4—2024no—
235C22⋊C4C22⋊D10112yes—
236C42C22×D10112yes—
237C23—21no—
238C23—8082no—
239C23—4042no—
240C23—2022no—
241C23—2021no—
242C23C2×C22⋊D10111yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—2021no—
249Q8—2021no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—8084no—
254D4—4042no—
255D4—4042no—
256D4—2022no—
257D4—2021no—
258D4—2022no—
259D4—2021no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—4042no—
265C2×C4—2021no—
266C2×C4D4×D10112yes—
267C2×C4D4×D10112yes—
268C2×C4C2×C22⋊D10114yes—
269C2×C4C2×C22⋊D10111yescharacteristic
270C2×C4C23×D10111yescharacteristic
271C8—1012no—
272C8—1012no—
273C8—1014no—
274C22—22no—
275C22—42no—
276C22—41no—
277C22—8084no—
278C22—4042no—
279C22—4044no—
280C22—4041no—
281C22—4042no—
282C22#1569225112yes—
283C22C22×C22⋊D10111yescharacteristic
284C4—24no—
285C4—41no—
286C4—42no—
287C4—4041no—
288C4#1569225112yes—
289C4C2×D4×D10111yescharacteristic
290C4C2×D4×D10111yescharacteristic
291C2—22no—
292C2—81no—
293C2—8082no—
294C2—8081no—
295C2C2×#1569225111yescharacteristic
296C2#2026598512yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#536×C101C2
socleC22×C101C22×D4
hypercenter#536D101
hypocenterC101#3154
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#15692286C2semidirect
2. C2×#15692280C2semidirect
3. C2×#15692275C2semidirect
4. C22×#13108061C2semidirect
5. #20265746C2semidirect
6. #20265569C2semidirect
7. #20265567C2semidirect
8. #20265559C2semidirect
9. #20265551C2semidirect
10. #20265133C2semidirect
11. #536×C101C2semidirect
17. C2×#13108064C22semidirect
18. C2×#13108064C22semidirect
19. C2×#13108062C22semidirect
24. C2×#13108061C22semidirect
25. C2×#13108061C22semidirect
27. C2×#13108059C22semidirect
34. #15692166C22semidirect
35. C4×#11717579C22semidirect
38. C4⋊Q8×C101C22semidirect
39. #124×C101C22semidirect
65. #13108058D4semidirect
66. C2×#11717579D4semidirect
149. C101#3154semidirect
154. #536D101semidirect
181. C4⋊Q8C2×D101semidirect
184. #124C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922862
C2×#156922802
C2×#156922752
C22×#131080612
#202657462
#202655692
#202655672
#202655592
#202655512
#202651332
#536×C1012
#3154101