Finite Group Databasebeta

← #27181594

Subgroups — #27181594

Order 12928.

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

normal subgroups only

This group has 572 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×#15692288C211yesmaximal, characteristic, complemented
2C2×#15692285C211yesmaximal, characteristic, complemented
3C2×#15692281C211yesmaximal, characteristic, complemented
4#20265723C211yesmaximal, characteristic, complemented
5C2×#15692242C211yesmaximal, characteristic, complemented
6#20265568C212yesmaximal, complemented
7#20265565C212yesmaximal, complemented
8#20265560C212yesmaximal, complemented
9#20265558C212yesmaximal, complemented
10M16×Dic101C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C22×Dic202C2211yescharacteristic
14#15692288—24no—
15#15692285—24no—
16#15692281—24no—
17C2×Dic404C2212yescomplemented
18C2×#13108064C2212yescomplemented
19C2×#13108063C2212yescomplemented
20C2×#13108062C2212yes—
21C2×#13108062C2212yescomplemented
22C2×#13108059C2212yes—
23C2×#13108058C2211yescharacteristic
24#15692253C2211yescharacteristic, complemented
25#15692251C2211yescharacteristic, complemented
26#15692244C2211yescharacteristic, complemented
27#15692242—22no—
28#15692242C2212yes—
29#15692242C2212yes—
30#15692242C2211yescharacteristic, complemented
31#15692227—22no—
32C4×C22⋊D101—22no—
33C2×C22⋊Dic101—22no—
34C2×C4×Dic101C2211yescharacteristic
35#15692165C2212yescomplemented
36C8×Dic101C2212yescomplemented
37C2×#41×C101C2211yescharacteristic
38C22×D4×C101C2211yescharacteristic
39AGL(1,ℤ/8)×C101—24no—
40C2×SD16×C101C2212yes—
41C2×D8×C101C2212yes—
42C2×M16×C101C2211yescharacteristic
43#13108074—22no—
44#13108074—22no—
45C2×Dic202—22no—
46C2×Dic202—22no—
47C2×Dic202—21no—
48C2×Dic202C2312yes—
49C2×Dic202C2311yescharacteristic
50C2×D404C2311yescharacteristic
51C2×C22⋊D101—21no—
52C2×C22⋊D101—22no—
53C22×Dic101—21no—
54C22×Dic101D412yes—
55C2×C4×D101—21no—
56Dic404—24no—
57#13108064—24no—
58#13108063—24no—
59#13108062—24no—
60#13108062—24no—
61#13108059—24no—
62#13108058D414yescomplemented
63C2×#11717579C2312yes—
64#13108053—22no—
65C22⋊Dic101—24no—
66C22⋊Dic101—24no—
67C22⋊Dic101—22no—
68#13108051—22no—
69C4×Dic101D412yes—
70C4×Dic101C2311yescharacteristic
71C4×Dic101C2311yescharacteristic
72C24×C101—21no—
73#41×C101—22no—
74#41×C101—22no—
75C2×Q8×C101C2311yescharacteristic
76C2×D4×C101—21no—
77C2×D4×C101—22no—
78C2×D4×C101—22no—
79C2×D4×C101—21no—
80C2×D4×C101C2311yescharacteristic
81C2×D4×C101C2312yes—
82C22×C4×C101—21no—
83C22×C4×C101C2311yescharacteristic
84SD16×C101—24no—
85D8×C101—24no—
86M16×C101D414yescomplemented
87C2×C8×C101C2312yes—
88C23×C101—21no—
89C23×C101—22no—
90C23×C101—44no—
91C23×C101—42no—
92C23×C101C2×D411yescharacteristic
93C22×D101—21no—
94Q8×C101—21no—
95Q8×C101—21no—
96D4×C101—21no—
97D4×C101—22no—
98D4×C101—22no—
99D4×C101—21no—
100D4×C101—42no—
101D4×C101—42no—
102C2×C4×C101—21no—
103C2×C4×C101—42no—
104C2×C4×C101C2×D411yescharacteristic
105C2×C4×C101C2×D412yes—
106C2×C4×C101C2×D412yes—
107C2×C4×C101C2411yescharacteristic
108C22⋊D101—44no—
109C22⋊D101—42no—
110C2×Dic101—22no—
111C2×Dic101—21no—
112C2×Dic101—42no—
113C2×Dic101—42no—
114C2×Dic101C2×D412yes—
115C2×Dic101C2×D412yes—
116D404—21no—
117D404—21no—
118C4×D101—42no—
119Dic202—21no—
120Dic202—22no—
121Dic202—22no—
122Dic202—21no—
123Dic202—42no—
124C8×C101C2×D414yes—
125#11717579C2×D414yes—
126C22×C101—22no—
127C22×C101—41no—
128C22×C101—42no—
129C22×C101—42no—
130C22×C101—44no—
131C22×C101—84no—
132C22×C101#12712yes—
133C22×C101C22×D411yescharacteristic
134C2×D101—41no—
135C2×D101—42no—
136C4×C101—41no—
137C4×C101#12712yes—
138C4×C101C22×D411yescharacteristic
139C4×C101C22×D411yescharacteristic
140Dic101—22no—
141Dic101—22no—
142Dic101—42no—
143Dic101—41no—
144C2×C101—22no—
145C2×C101—81no—
146C2×C101—82no—
147C2×C101C2×#12711yescharacteristic
148C2×C101#57512yes—
149D101—81no—
150#3153—1011nomaximal, Sylow
151C101#315311yescharacteristic, complemented, Sylow
152C2×#137—1011no—
153C2×#137—1011no—
154C2×AGL(1,ℤ/8)—1011no—
155C2×AGL(1,ℤ/8)D10111yescharacteristic, complemented
156#532—1011no—
157C2×#124—1011no—
158#497—1012no—
159#496—1012no—
160#495—1012no—
161#495—1012no—
162C4×M16—1011no—
163C2×#41—1011no—
164C2×#41C2×D10111yescharacteristic
165C22×Q8—1011no—
166C22×D4C2×D10111yescharacteristic
167#137—2024no—
168#137—2024no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—2024no—
171C2×Q16—1012no—
172C2×Q16—1012no—
173C2×SD16—1012no—
174C2×SD16—1012no—
175C2×SD16—1012no—
176C2×SD16C2×D10112yes—
177C2×D8—1012no—
178C2×D8C2×D10112yes—
179C2×M16—1011no—
180C2×M16C2×D10111yescharacteristic
181C4⋊Q8—1011no—
182#127—1011no—
183#124—1011no—
184#124—1012no—
185#124—1012no—
186#124—1011no—
187#124—2022no—
188#121—2022no—
189C4×D4—2022no—
190C2×C22⋊C4—2022no—
191C2×C42—1011no—
192#97—1012no—
193C4×C8—1012no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—2022no—
198#41—2022no—
199C2×Q8—1012no—
200C2×Q8—1011no—
201C2×Q8—2022no—
202C2×Q8—2022no—
203C2×Q8—2021no—
204C2×Q8C22×D10111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—1011no—
210C2×D4—2022no—
211C2×D4—2021no—
212C2×D4C22×D10112yes—
213C2×D4C22×D10111yescharacteristic
214C22×C4—21no—
215C22×C4—1012no—
216C22×C4—2021no—
217C22×C4—2021no—
218C22×C4C22×D10111yescharacteristic
219Q16—2024no—
220Q16—2024no—
221SD16—24no—
222SD16—2024no—
223SD16—2024no—
224SD16—2024no—
225D8—24no—
226D8—2024no—
227M16—1014no—
228M16C22⋊D10114yescomplemented
229C2×C8—1012no—
230C2×C8C22×D10112yes—
231C4⋊C4—2022no—
232C22⋊C4—2022no—
233C22⋊C4—2024no—
234C22⋊C4—2024no—
235C22⋊C4—2022no—
236C42—1012no—
237C42—1011no—
238C42—1011no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—2021no—
244C23D4×D10111yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—4042no—
248Q8—2021no—
249Q8—2022no—
250Q8—2022no—
251Q8—2021no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—4042no—
259D4—4044no—
260D4—2021no—
261D4—2021no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—1012no—
265C2×C4—1012no—
266C2×C4—4042no—
267C2×C4—4042no—
268C2×C4—4042no—
269C2×C4—2022no—
270C2×C4—2021no—
271C2×C4D4×D10112yes—
272C2×C4D4×D10111yescharacteristic
273C2×C4C2×C22⋊D10112yes—
274C2×C4C23×D10111yescharacteristic
275C8—1014no—
276C8C2×C22⋊D10114yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—4041no—
284C22—4042no—
285C22#1569225112yes—
286C22C2×D4×D10111yescharacteristic
287C4—41no—
288C4—4041no—
289C4—4042no—
290C4—2022no—
291C4—2022no—
292C4#1569225112yes—
293C4C2×D4×D10111yescharacteristic
294C4C22×C22⋊D10111yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—8081no—
299C2C2×#1569225111yescharacteristic
300C2#2026597312yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×AGL(1,ℤ/8)×C101C2
socleC22×C101C22×D4
hypercenterC2×AGL(1,ℤ/8)D101
hypocenterC101#3153
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#15692288C2semidirect
2. C2×#15692285C2semidirect
3. C2×#15692281C2semidirect
4. #20265723C2semidirect
5. C2×#15692242C2semidirect
6. #20265568C2semidirect
7. #20265565C2semidirect
8. #20265560C2semidirect
9. #20265558C2semidirect
10. M16×Dic101C2semidirect
11. C2×AGL(1,ℤ/8)×C101C2semidirect
17. C2×Dic404C22semidirect
18. C2×#13108064C22semidirect
19. C2×#13108063C22semidirect
21. C2×#13108062C22semidirect
24. #15692253C22semidirect
25. #15692251C22semidirect
26. #15692244C22semidirect
30. #15692242C22semidirect
35. #15692165C22semidirect
36. C8×Dic101C22semidirect
62. #13108058D4semidirect
86. M16×C101D4semidirect
151. C101#3153semidirect
155. C2×AGL(1,ℤ/8)D101semidirect
228. M16C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922882
C2×#156922852
C2×#156922812
#202657232
C2×#156922422
#202655682
#202655652
#202655602
#202655582
M16×Dic1012
C2×AGL(1,ℤ/8)×C1012
#3153101