Finite Group Databasebeta

← #27180745

Subgroups — #27180745

Order 12928.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×M16×D101C211yesmaximal, characteristic, complemented
2#20265684C211yesmaximal, characteristic, complemented
3C2×#15692226C211yesmaximal, characteristic, complemented
4#20265427C212yesmaximal, complemented
5#20265420C212yesmaximal, complemented
6#20265410C212yesmaximal, complemented
7#20265403C212yesmaximal, complemented
8#20265219C211yesmaximal, characteristic, complemented
9#20265182C211yesmaximal, characteristic, complemented
10#20265178C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C101C211yesmaximal, characteristic, complemented
12C22×C22⋊D101—21no—
13C22×C4×D101C2211yescharacteristic
14C2×#13108058C2211yescharacteristic
15C2×#13108057C2212yescomplemented
16M16×D101—24no—
17C2×C8×D101C2212yescomplemented
18#15692236—21no—
19#15692231C2211yescharacteristic, complemented
20#15692231C2211yescharacteristic, complemented
21#15692226—21no—
22#15692226—21no—
23#15692226C2212yes—
24#15692226C2212yes—
25#15692226C2211yescharacteristic, complemented
26#15692226C2211yescharacteristic, complemented
27#15692223—21no—
28C4×C22⋊D101—21no—
29C4×C22⋊D101—21no—
30#15692214C2211yescharacteristic
31C2×C4⋊Dic101C2211yescharacteristic
32C2×C22⋊Dic101—21no—
33#15692189C2212yescomplemented
34#15692187C2212yescomplemented
35#15692179C2212yescomplemented
36#15692176C2212yescomplemented
37#15692176C2212yescomplemented
38#15692176C2212yescomplemented
39C2×#41×C101C2211yescharacteristic
40C22×D4×C101C2211yescharacteristic
41AGL(1,ℤ/8)×C101—24no—
42C2×SD16×C101C2212yescomplemented
43C2×D8×C101C2212yescomplemented
44C2×M16×C101C2211yescharacteristic
45C23×D101D411yescharacteristic
46C2×C22⋊D101—22no—
47C2×C22⋊D101—22no—
48C2×C22⋊D101—21no—
49C2×C22⋊D101—21no—
50C2×C22⋊D101—44no—
51C22×Dic101—21no—
52C22×Dic101D411yescharacteristic
53C2×C4×D101—22no—
54C2×C4×D101—22no—
55C2×C4×D101D411yescharacteristic
56C2×C4×D101D411yescharacteristic
57C2×C4×D101C2311yescharacteristic
58C2×C4×D101C2311yescharacteristic
59#13108058—22no—
60#13108057—24no—
61C2×#11717579C2312yes—
62C8×D101—24no—
63C4⋊Dic101—21no—
64C4⋊Dic101C2311yescharacteristic
65C4⋊Dic101C2311yescharacteristic
66C4⋊Dic101C2312yes—
67#13108053—21no—
68#13108053—21no—
69C22⋊Dic101—21no—
70C22⋊Dic101—22no—
71C22⋊Dic101—22no—
72C22⋊Dic101—21no—
73C22⋊Dic101—21no—
74#13108051—21no—
75#13108051—21no—
76C4×Dic101—21no—
77C24×C101—21no—
78#41×C101—22no—
79#41×C101—22no—
80C2×Q8×C101C2311yescharacteristic
81C2×D4×C101—21no—
82C2×D4×C101—22no—
83C2×D4×C101—21no—
84C2×D4×C101—22no—
85C2×D4×C101C2311yescharacteristic
86C2×D4×C101C2312yes—
87C22×C4×C101—21no—
88C22×C4×C101C2311yescharacteristic
89SD16×C101—24no—
90D8×C101—24no—
91M16×C101D414yescomplemented
92C2×C8×C101C2312yes—
93C23×C101—21no—
94C23×C101—22no—
95C23×C101—42no—
96C23×C101—42no—
97C23×C101—42no—
98C23×C101C2×D411yescharacteristic
99C22×D101—22no—
100C22×D101—22no—
101C22×D101—41no—
102C22×D101C2×D411yescharacteristic
103C22×D101C2×D411yescharacteristic
104Q8×C101—21no—
105Q8×C101—21no—
106D4×C101—21no—
107D4×C101—22no—
108D4×C101—22no—
109D4×C101—21no—
110D4×C101—42no—
111D4×C101—42no—
112C2×C4×C101—21no—
113C2×C4×C101—42no—
114C2×C4×C101C2×D411yescharacteristic
115C2×C4×C101C2×D412yes—
116C2×C4×C101#4112yes—
117C2×C4×C101C2411yescharacteristic
118C22⋊D101—42no—
119C22⋊D101—44no—
120C22⋊D101—42no—
121C22⋊D101—44no—
122C2×Dic101—22no—
123C2×Dic101—21no—
124C2×Dic101—21no—
125C2×Dic101—22no—
126C2×Dic101—42no—
127C2×Dic101C2×D411yescharacteristic
128C2×Dic101C2×D411yescharacteristic
129C4×D101—22no—
130C4×D101—22no—
131C4×D101—21no—
132C4×D101—21no—
133C4×D101—21no—
134C4×D101—21no—
135C8×C101C2×D414yescomplemented
136#11717579—22no—
137C22×C101—21no—
138C22×C101—21no—
139C22×C101—41no—
140C22×C101—42no—
141C22×C101—42no—
142C22×C101—41no—
143C22×C101—41no—
144C22×C101—42no—
145C22×C101—84no—
146C22×C101#12112yes—
147C22×C101C22×D411yescharacteristic
148C2×D101—21no—
149C2×D101—21no—
150C2×D101—21no—
151C2×D101—21no—
152C2×D101—21no—
153C2×D101—44no—
154C2×D101AGL(1,ℤ/8)12yescomplemented
155C4×C101—41no—
156C4×C101#12112yes—
157C4×C101C22×D411yescharacteristic
158C4×C101C2×#4111yescharacteristic
159Dic101—21no—
160Dic101—21no—
161Dic101—42no—
162Dic101—41no—
163Dic101—41no—
164C2×C101—22no—
165C2×C101—81no—
166C2×C101—82no—
167C2×C101C2×#12111yescharacteristic
168C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
169C2×C101#57511yescharacteristic
170D101—22no—
171D101—41no—
172#3055—1011nomaximal, Sylow
173C101#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)D10111yescharacteristic, complemented
175C22×M16—1011no—
176#525—1011no—
177C2×#121—1011no—
178#469—1012no—
179#468—1012no—
180#466—1012no—
181#465—1012no—
182#428—1011no—
183#418—1011no—
184#417—1011no—
185C2×#41C2×D10111yescharacteristic
186C22×D4—2021no—
187C22×D4C2×D10111yescharacteristic
188C23×C4—1011no—
189AGL(1,ℤ/8)—24no—
190C2×SD16C2×D10112yescomplemented
191C2×D8C2×D10112yescomplemented
192C2×M16—1011no—
193C2×M16—1012no—
194C2×M16—2024no—
195C2×M16C2×D10111yescharacteristic
196C22×C8—1012no—
197#123—2021no—
198C22⋊Q8—1011no—
199C22⋊Q8—1011no—
200#121—1012no—
201#121—1012no—
202#121—1011no—
203#121—1011no—
204#121—2021no—
205#121—2021no—
206#120—2021no—
207C4×D4—2021no—
208C4×D4—2021no—
209#117—1011no—
210C2×C4⋊C4—1011no—
211C2×C22⋊C4—2021no—
212#107—1012no—
213#106—1012no—
214#103—1012no—
215#102—1012no—
216#102—1012no—
217#102—1012no—
218C24—21no—
219C24—1011no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D10111yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—4044no—
228C2×D4—2021no—
229C2×D4—2021no—
230C2×D4—2022no—
231C2×D4—2022no—
232C2×D4C22×D10111yescharacteristic
233C2×D4C22×D10112yes—
234C22×C4—21no—
235C22×C4—1011no—
236C22×C4—1011no—
237C22×C4—1011no—
238C22×C4—1011no—
239C22×C4—1011no—
240C22×C4—2021no—
241C22×C4—2022no—
242C22×C4—2022no—
243C22×C4C22×D10111yescharacteristic
244SD16—24no—
245D8—24no—
246M16—2022no—
247M16—2024no—
248M16C22⋊D10114yescomplemented
249C2×C8—1012no—
250C2×C8—2024no—
251C2×C8C22×D10112yes—
252C4⋊C4—1011no—
253C4⋊C4—1011no—
254C4⋊C4—1012no—
255C4⋊C4—2021no—
256C4⋊C4—2021no—
257C4⋊C4—2021no—
258C22⋊C4—2021no—
259C22⋊C4—2021no—
260C22⋊C4—2022no—
261C22⋊C4—2022no—
262C22⋊C4—2021no—
263C22⋊C4—2021no—
264C22⋊C4—2021no—
265C42—2021no—
266C23—21no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—1011no—
272C23—1011no—
273C23—4041no—
274C23—2022no—
275C23—2022no—
276C23D4×D10111yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—22no—
281D4—22no—
282D4—21no—
283D4—42no—
284D4—42no—
285D4—4044no—
286D4—4042no—
287D4—4044no—
288D4—4042no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—1011no—
292C2×C4—1011no—
293C2×C4—4042no—
294C2×C4—2021no—
295C2×C4—2021no—
296C2×C4—2022no—
297C2×C4—2022no—
298C2×C4—2021no—
299C2×C4—2021no—
300C2×C4—2021no—
301C2×C4—2021no—
302C2×C4—2022no—
303C2×C4—2022no—
304C2×C4D4×D10111yescharacteristic
305C2×C4C2×C22⋊D10112yes—
306C2×C4#1310807312yes—
307C2×C4C23×D10111yescharacteristic
308C8—2022no—
309C8C2×C22⋊D10114yescomplemented
310C22—21no—
311C22—21no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—1012no—
320C22—4044no—
321C22—2021no—
322C22—2021no—
323C22—2021no—
324C22—2021no—
325C22—2021no—
326C22#1569222612yes—
327C22C2×D4×D10111yescharacteristic
328C4—41no—
329C4—4042no—
330C4—4041no—
331C4—4041no—
332C4—2021no—
333C4—2021no—
334C4#1569222612yes—
335C4C22×C22⋊D10111yescharacteristic
336C4C2×#1310807311yescharacteristic
337C2—22no—
338C2—82no—
339C2—81no—
340C2—4041no—
341C2—2022no—
342C2C2×#1569222611yescharacteristic
343C2AGL(1,ℤ/8)×D10111yescharacteristic
344C2#2026597311yescharacteristic
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#3055
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×M16×D101C2semidirect
2. #20265684C2semidirect
3. C2×#15692226C2semidirect
4. #20265427C2semidirect
5. #20265420C2semidirect
6. #20265410C2semidirect
7. #20265403C2semidirect
8. #20265219C2semidirect
9. #20265182C2semidirect
10. #20265178C2semidirect
11. C2×AGL(1,ℤ/8)×C101C2semidirect
15. C2×#13108057C22semidirect
17. C2×C8×D101C22semidirect
19. #15692231C22semidirect
20. #15692231C22semidirect
25. #15692226C22semidirect
26. #15692226C22semidirect
33. #15692189C22semidirect
34. #15692187C22semidirect
35. #15692179C22semidirect
36. #15692176C22semidirect
37. #15692176C22semidirect
38. #15692176C22semidirect
42. C2×SD16×C101C22semidirect
43. C2×D8×C101C22semidirect
91. M16×C101D4semidirect
135. C8×C101C2×D4semidirect
154. C2×D101AGL(1,ℤ/8)semidirect
173. C101#3055semidirect
174. C2×AGL(1,ℤ/8)D101semidirect
190. C2×SD16C2×D101semidirect
191. C2×D8C2×D101semidirect
248. M16C22⋊D101semidirect
309. C8C2×C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1012
#202656842
C2×#156922262
#202654272
#202654202
#202654102
#202654032
#202652192
#202651822
#202651782
C2×AGL(1,ℤ/8)×C1012
#3055101