Finite Group Databasebeta

← #27180675

Subgroups — #27180675

Order 12928.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15692275C211yesmaximal, characteristic, complemented
2#20265744C211yesmaximal, characteristic, complemented
3#20265660C211yesmaximal, characteristic, complemented
4#20265397C212yesmaximal, complemented
5#20265390C212yesmaximal, complemented
6#20265375C212yesmaximal, complemented
7#20265370C212yesmaximal, complemented
8#20265205C211yesmaximal, characteristic, complemented
9#20265179C211yesmaximal, characteristic, complemented
10C2×#15692176C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13#15692275—24no—
14C2×#13108064C2211yescharacteristic, complemented
15C2×#13108062C2211yescharacteristic, complemented
16C2×#13108061C2211yescharacteristic, complemented
17C2×#13108059C2211yescharacteristic, complemented
18C22×#11717579C2211yescharacteristic
19#15692251C2212yes—
20#15692242C2212yes—
21#15692226—21no—
22#15692226—21no—
23#15692224—22no—
24Q8×Dic101C2212yescomplemented
25D4×Dic101—22no—
26D4×Dic101C2212yescomplemented
27#15692214—21no—
28#15692214C2211yescharacteristic
29C2×C4⋊Dic101—21no—
30C2×C4⋊Dic101C2211yescharacteristic
31#15692184C2212yescomplemented
32#15692184C2212yescomplemented
33#15692180C2212yescomplemented
34#15692177C2212yescomplemented
35#15692176C2212yes—
36#15692176C2212yes—
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×Dic202C2311yescharacteristic
46C2×D404C2311yescharacteristic
47C2×C22⋊D101—21no—
48C2×C22⋊D101—22no—
49C2×C22⋊D101—22no—
50C22×Dic101—21no—
51C22×Dic101—22no—
52C2×C4×D101—21no—
53#13108064—22no—
54#13108062—22no—
55#13108061—22no—
56#13108059—22no—
57C2×#11717579—22no—
58C2×#11717579C2311yescharacteristic
59C2×#11717579C2311yescharacteristic
60C4⋊Dic101—21no—
61C4⋊Dic101—22no—
62C4⋊Dic101D411yescharacteristic
63C4⋊Dic101D411yescharacteristic
64C4⋊Dic101C2312yes—
65C22⋊Dic101—22no—
66C22⋊Dic101—22no—
67C22⋊Dic101—22no—
68C22⋊Dic101D412yes—
69C4×Dic101—22no—
70C4×Dic101C2312yes—
71C24×C101—21no—
72#41×C101—22no—
73#41×C101D414yescomplemented
74C2×Q8×C101C2311yescharacteristic
75C2×D4×C101—21no—
76C2×D4×C101—22no—
77C2×D4×C101—21no—
78C2×D4×C101—22no—
79C2×D4×C101C2312yes—
80C2×D4×C101C2311yescharacteristic
81C22×C4×C101—21no—
82C22×C4×C101C2311yescharacteristic
83SD16×C101—24no—
84D8×C101—24no—
85M16×C101—22no—
86C2×C8×C101C2312yes—
87C23×C101—21no—
88C23×C101—22no—
89C23×C101—42no—
90C23×C101—81no—
91C23×C101—81no—
92C23×C101C2×D411yescharacteristic
93C22×D101—21no—
94Q8×C101—21no—
95Q8×C101C2×D412yes—
96D4×C101—24no—
97D4×C101—22no—
98D4×C101—21no—
99D4×C101—22no—
100D4×C101—42no—
101D4×C101C2×D412yes—
102C2×C4×C101—21no—
103C2×C4×C101—24no—
104C2×C4×C101C2×D411yescharacteristic
105C2×C4×C101C2×D412yes—
106C2×C4×C101#4112yes—
107C2×C4×C101C2411yescharacteristic
108C22⋊D101—42no—
109C22⋊D101—82no—
110C22⋊D101—82no—
111C2×Dic101—22no—
112C2×Dic101—21no—
113C2×Dic101—82no—
114C2×Dic101—81no—
115C2×Dic101C2×D412yes—
116C2×Dic101C2×D412yes—
117D404—21no—
118D404—21no—
119C4×D101—42no—
120Dic202—21no—
121Dic202—21no—
122C8×C101—22no—
123#11717579—21no—
124#11717579—21no—
125C22×C101—21no—
126C22×C101—21no—
127C22×C101—22no—
128C22×C101—41no—
129C22×C101—41no—
130C22×C101—42no—
131C22×C101—42no—
132C22×C101—42no—
133C22×C101—84no—
134C22×C101#12112yes—
135C22×C101C22×D411yescharacteristic
136C2×D101—41no—
137C2×D101—41no—
138C2×D101—41no—
139C4×C101—22no—
140C4×C101#12112yes—
141C4×C101C22×D411yescharacteristic
142C4×C101C2×#4111yescharacteristic
143Dic101—22no—
144Dic101—22no—
145Dic101—41no—
146Dic101—42no—
147C2×C101—22no—
148C2×C101—42no—
149C2×C101—82no—
150C2×C101C2×#12111yescharacteristic
151C2×C101#57611yescharacteristic
152C2×C101#57711yescharacteristic
153D101—81no—
154#3045—1011nomaximal, Sylow
155C101#304511yescharacteristic, complemented, Sylow
156C2×AGL(1,ℤ/8)D10111yescharacteristic, complemented
157C2×#135—1011no—
158#535—1011no—
159#520—1011no—
160#464—1012no—
161#463—1012no—
162#460—1012no—
163#459—1012no—
164#424—1011no—
165#417—1011no—
166C2×#102—1011no—
167C2×#41—1011no—
168C2×#41C2×D10111yescharacteristic
169C22×D4C2×D10111yescharacteristic
170AGL(1,ℤ/8)—24no—
171#135—2024no—
172C2×Q16—1011no—
173C2×SD16—1011no—
174C2×SD16—1011no—
175C2×SD16C2×D10112yes—
176C2×D8—1011no—
177C2×D8C2×D10112yes—
178C2×M16C2×D10111yescharacteristic
179C22×C8—1011no—
180#127—1012no—
181#124—1012no—
182#121—2021no—
183#121—2021no—
184#120—2022no—
185C4×Q8—1012no—
186C4×D4—1012no—
187C4×D4—2022no—
188#117—1011no—
189#117—2021no—
190C2×C4⋊C4—1011no—
191C2×C4⋊C4—2021no—
192C4⋊C8—1012no—
193C4⋊C8—1012no—
194#103—1012no—
195#102—1012no—
196#102—1012no—
197#102—1012no—
198C24—21no—
199#41—22no—
200#41—2022no—
201#41—2022no—
202#41C22⋊D10114yescomplemented
203C2×Q8—1011no—
204C2×Q8C22×D10111yescharacteristic
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—1011no—
210C2×D4—2022no—
211C2×D4—2022no—
212C2×D4—2021no—
213C2×D4C22×D10112yes—
214C2×D4C22×D10111yescharacteristic
215C22×C4—21no—
216C22×C4—2022no—
217C22×C4—2021no—
218C22×C4—2021no—
219C22×C4C22×D10111yescharacteristic
220Q16—2022no—
221SD16—24no—
222SD16—2022no—
223SD16—2022no—
224D8—24no—
225D8—2022no—
226M16—22no—
227C2×C8—1011no—
228C2×C8—1011no—
229C2×C8—2022no—
230C2×C8C22×D10112yes—
231C4⋊C4—1011no—
232C4⋊C4—1011no—
233C4⋊C4—1012no—
234C4⋊C4—2022no—
235C4⋊C4—2021no—
236C22⋊C4—1012no—
237C22⋊C4—2022no—
238C22⋊C4—2022no—
239C22⋊C4—2022no—
240C42—1012no—
241C42—2022no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—2021no—
248C23D4×D10111yescharacteristic
249Q8—21no—
250Q8—2021no—
251Q8—2021no—
252Q8C2×C22⋊D10112yes—
253D4—24no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—8082no—
259D4—8082no—
260D4—4042no—
261D4—2021no—
262D4—2021no—
263D4C2×C22⋊D10112yes—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—8081no—
267C2×C4—8082no—
268C2×C4—1012no—
269C2×C4—1012no—
270C2×C4—4042no—
271C2×C4—2022no—
272C2×C4—2021no—
273C2×C4D4×D10111yescharacteristic
274C2×C4C2×C22⋊D10112yes—
275C2×C4#1310807312yes—
276C2×C4C23×D10111yescharacteristic
277C8—22no—
278C8—2021no—
279C8—2021no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—4041no—
290C22—4041no—
291C22—4041no—
292C22#1569222712yes—
293C22C2×D4×D10111yescharacteristic
294C4—22no—
295C4—4041no—
296C4—4042no—
297C4—2022no—
298C4—2022no—
299C4#1569222712yes—
300C4C22×C22⋊D10111yescharacteristic
301C4C2×#1310807311yescharacteristic
302C2—22no—
303C2—42no—
304C2—82no—
305C2—8081no—
306C2C2×#1569222711yescharacteristic
307C2#2026598111yescharacteristic
308C2#2026598711yescharacteristic
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#3045
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×#15692275C2semidirect
2. #20265744C2semidirect
3. #20265660C2semidirect
4. #20265397C2semidirect
5. #20265390C2semidirect
6. #20265375C2semidirect
7. #20265370C2semidirect
8. #20265205C2semidirect
9. #20265179C2semidirect
10. C2×#15692176C2semidirect
11. C2×AGL(1,ℤ/8)×C101C2semidirect
14. C2×#13108064C22semidirect
15. C2×#13108062C22semidirect
16. C2×#13108061C22semidirect
17. C2×#13108059C22semidirect
24. Q8×Dic101C22semidirect
26. D4×Dic101C22semidirect
31. #15692184C22semidirect
32. #15692184C22semidirect
33. #15692180C22semidirect
34. #15692177C22semidirect
73. #41×C101D4semidirect
155. C101#3045semidirect
156. C2×AGL(1,ℤ/8)D101semidirect
202. #41C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922752
#202657442
#202656602
#202653972
#202653902
#202653752
#202653702
#202652052
#202651792
C2×#156921762
C2×AGL(1,ℤ/8)×C1012
#3045101