Finite Group Databasebeta

← #27181609

Subgroups — #27181609

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×#15692285C211yesmaximal, characteristic, complemented
2C2×#15692280C211yesmaximal, characteristic, complemented
3C22×#13108063C211yesmaximal, characteristic, complemented
4#20265748C211yesmaximal, characteristic, complemented
5#20265741C211yesmaximal, characteristic, complemented
6#20265570C212yesmaximal, complemented
7#20265565C212yesmaximal, complemented
8#20265558C212yesmaximal, complemented
9#20265548C212yesmaximal, complemented
10#20265132C211yesmaximal, characteristic, complemented
11C2×#135×C101C211yesmaximal, characteristic, complemented
12C22×Dic202C2211yescharacteristic
13C22×D404C2211yescharacteristic
14#15692285—24no—
15#15692280—24no—
16C2×#13108064C2212yescomplemented
17C2×#13108063—24no—
18C2×#13108063C2212yes—
19C2×#13108063C2212yes—
20C2×#13108062C2212yescomplemented
21C2×#13108061C2212yes—
22C2×#13108059C2212yes—
23C2×#13108058C2211yescharacteristic
24#15692253C2212yescomplemented
25#15692251C2212yes—
26#15692244C2212yes—
27#15692242C2212yescomplemented
28#15692236—22no—
29#15692234—22no—
30#15692229—22no—
31#15692223—22no—
32#15692214C2211yescharacteristic
33#15692165C2212yescomplemented
34C8×Dic101C2212yescomplemented
35C2×#41×C101C2211yescharacteristic
36C2×#41×C101C2211yescharacteristic
37#135×C101—24no—
38C2×Q16×C101C2211yescharacteristic, complemented
39C2×SD16×C101C2211yescharacteristic, complemented
40C2×SD16×C101C2211yescharacteristic, complemented
41C2×D8×C101C2211yescharacteristic, complemented
42C22×C8×C101C2211yescharacteristic
43C23×D101—21no—
44C2×Dic202—22no—
45C2×Dic202—22no—
46C2×Dic202—21no—
47C2×Dic202C2312yes—
48C2×D404—22no—
49C2×D404—22no—
50C2×D404—21no—
51C2×D404C2312yes—
52C2×C22⋊D101—24no—
53C22×Dic101—21no—
54#13108064—24no—
55#13108063—24no—
56#13108063—24no—
57#13108062—24no—
58#13108061—24no—
59#13108059—24no—
60#13108058D414yescomplemented
61C2×#11717579C2312yes—
62C4⋊Dic101D412yes—
63#13108053—24no—
64C22⋊Dic101—24no—
65C22⋊Dic101D412yes—
66#13108051—24no—
67C4×Dic101C2312yes—
68#41×C101—22no—
69#41×C101—22no—
70#41×C101—22no—
71#41×C101—22no—
72C2×Q8×C101C2311yescharacteristic
73C2×Q8×C101C2311yescharacteristic
74C2×D4×C101—21no—
75C2×D4×C101—21no—
76C2×D4×C101C2311yescharacteristic
77C2×D4×C101C2311yescharacteristic
78C22×C4×C101—21no—
79C22×C4×C101—21no—
80C22×C4×C101C2311yescharacteristic
81Q16×C101—22no—
82SD16×C101—22no—
83SD16×C101—22no—
84D8×C101—22no—
85C2×C8×C101D414yescomplemented
86C2×C8×C101C2311yescharacteristic
87C2×C8×C101C2311yescharacteristic
88C23×C101—21no—
89C23×C101—21no—
90C23×C101C2×D411yescharacteristic
91C22×D101—22no—
92C22×D101—42no—
93C22×D101—82no—
94Q8×C101—21no—
95Q8×C101—21no—
96Q8×C101—21no—
97Q8×C101—21no—
98D4×C101—21no—
99D4×C101—21no—
100D4×C101—21no—
101D4×C101—21no—
102D4×C101—42no—
103D4×C101—42no—
104C2×C4×C101—21no—
105C2×C4×C101—21no—
106C2×C4×C101—42no—
107C2×C4×C101—42no—
108C2×C4×C101C2×D411yescharacteristic
109C2×C4×C101C2×D412yes—
110C2×C4×C101C2×D412yes—
111C2×C4×C101C2411yescharacteristic
112C22⋊D101—84no—
113C2×Dic101—22no—
114C2×Dic101—42no—
115C2×Dic101C2×D414yes—
116D404—22no—
117D404—22no—
118D404—42no—
119Dic202—22no—
120Dic202—22no—
121Dic202—42no—
122C8×C101C2×D412yes—
123C8×C101C2×D412yes—
124#11717579C2×D414yes—
125C22×C101—22no—
126C22×C101—41no—
127C22×C101—41no—
128C22×C101—42no—
129C22×C101—42no—
130C22×C101#12712yes—
131C22×C101C22×D411yescharacteristic
132C2×D101—42no—
133C2×D101—44no—
134C2×D101—84no—
135C4×C101—41no—
136C4×C101—41no—
137C4×C101#12712yes—
138C4×C101C22×D411yescharacteristic
139C4×C101C22×D411yescharacteristic
140Dic101—24no—
141Dic101—42no—
142C2×C101—22no—
143C2×C101—81no—
144C2×C101—81no—
145C2×C101C2×#12711yescharacteristic
146C2×C101#57712yes—
147D101—82no—
148#3154—1011nomaximal, Sylow
149C101#315411yescharacteristic, complemented, Sylow
150C2×#137—1011no—
151C2×AGL(1,ℤ/8)—1011no—
152C2×#135D10111yescharacteristic, complemented
153C22×SD16—1011no—
154#536—1011no—
155#535—1011no—
156#497—1012no—
157#496—1012no—
158#495—1012no—
159#492—1012no—
160#405—1011no—
161C2×#41C2×D10111yescharacteristic
162C2×#41C2×D10111yescharacteristic
163C22×Q8—1011no—
164C22×D4—1011no—
165#137—2024no—
166AGL(1,ℤ/8)—2024no—
167#135—24no—
168C2×Q16—1012no—
169C2×Q16C2×D10111yescharacteristic, complemented
170C2×SD16—1012no—
171C2×SD16—1012no—
172C2×SD16—1012no—
173C2×SD16—1012no—
174C2×SD16—2024no—
175C2×SD16C2×D10111yescharacteristic, complemented
176C2×SD16C2×D10111yescharacteristic, complemented
177C2×D8—1012no—
178C2×D8C2×D10111yescharacteristic, complemented
179C2×M16—1011no—
180C22×C8C2×D10111yescharacteristic
181C4⋊Q8—1012no—
182#127—1012no—
183#124—1012no—
184#124—1012no—
185#123—2022no—
186C22⋊Q8—2022no—
187#121—2022no—
188#120—2022no—
189#117—1011no—
190#97—1012no—
191C4×C8—1012no—
192C24—2021no—
193#41—22no—
194#41—22no—
195#41—22no—
196#41—22no—
197C2×Q8—1012no—
198C2×Q8—2022no—
199C2×Q8—2022no—
200C2×Q8—2021no—
201C2×Q8C22×D10111yescharacteristic
202C2×Q8C22×D10111yescharacteristic
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—1012no—
206C2×D4—2024no—
207C2×D4—2021no—
208C2×D4—2022no—
209C2×D4—2022no—
210C2×D4C22×D10111yescharacteristic
211C2×D4C22×D10111yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—2021no—
215C22×C4C22×D10111yescharacteristic
216Q16—22no—
217Q16—2024no—
218SD16—22no—
219SD16—22no—
220SD16—2024no—
221SD16—2024no—
222SD16—2024no—
223SD16—2024no—
224D8—22no—
225D8—2024no—
226M16—1014no—
227C2×C8—1012no—
228C2×C8C22⋊D10114yescomplemented
229C2×C8C22×D10111yescharacteristic
230C2×C8C22×D10111yescharacteristic
231C4⋊C4—1012no—
232C4⋊C4—2024no—
233C22⋊C4—1012no—
234C22⋊C4—2024no—
235C22⋊C4—2024no—
236C42—1012no—
237C23—21no—
238C23—21no—
239C23—8082no—
240C23—4042no—
241C23—2022no—
242C23D4×D10111yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—21no—
246Q8—21no—
247Q8—4042no—
248Q8—2022no—
249Q8—2022no—
250D4—21no—
251D4—21no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—42no—
256D4—8084no—
257D4—4042no—
258D4—2022no—
259D4—2022no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—1014no—
265C2×C4—4042no—
266C2×C4—2022no—
267C2×C4D4×D10112yes—
268C2×C4D4×D10111yescharacteristic
269C2×C4C2×C22⋊D10112yes—
270C2×C4C23×D10111yescharacteristic
271C8—1014no—
272C8C2×C22⋊D10112yes—
273C8C2×C22⋊D10112yes—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—8084no—
280C22—4044no—
281C22—4042no—
282C22#1569225112yes—
283C22C2×D4×D10111yescharacteristic
284C4—41no—
285C4—41no—
286C4—4042no—
287C4—2024no—
288C4#1569225112yes—
289C4C2×D4×D10111yescharacteristic
290C4C22×C22⋊D10111yescharacteristic
291C2—22no—
292C2—81no—
293C2—81no—
294C2—8082no—
295C2C2×#1569225111yescharacteristic
296C2#2026598412yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#135×C101C2
socleC22×C101C22×D4
hypercenterC2×#135D101
hypocenterC101#3154
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#15692285C2semidirect
2. C2×#15692280C2semidirect
3. C22×#13108063C2semidirect
4. #20265748C2semidirect
5. #20265741C2semidirect
6. #20265570C2semidirect
7. #20265565C2semidirect
8. #20265558C2semidirect
9. #20265548C2semidirect
10. #20265132C2semidirect
11. C2×#135×C101C2semidirect
16. C2×#13108064C22semidirect
20. C2×#13108062C22semidirect
24. #15692253C22semidirect
27. #15692242C22semidirect
33. #15692165C22semidirect
34. C8×Dic101C22semidirect
38. C2×Q16×C101C22semidirect
39. C2×SD16×C101C22semidirect
40. C2×SD16×C101C22semidirect
41. C2×D8×C101C22semidirect
60. #13108058D4semidirect
85. C2×C8×C101D4semidirect
149. C101#3154semidirect
152. C2×#135D101semidirect
169. C2×Q16C2×D101semidirect
175. C2×SD16C2×D101semidirect
176. C2×SD16C2×D101semidirect
178. C2×D8C2×D101semidirect
228. C2×C8C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922852
C2×#156922802
C22×#131080632
#202657482
#202657412
#202655702
#202655652
#202655582
#202655482
#202651322
C2×#135×C1012
#3154101