Finite Group Databasebeta

← #27180754

Subgroups — #27180754

Order 12928.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#15692280C211yesmaximal, characteristic, complemented
2C2×M16×D101C211yesmaximal, characteristic, complemented
3#20265684C211yesmaximal, characteristic, complemented
4C2×#15692230C211yesmaximal, characteristic, complemented
5#20265429C212yesmaximal, complemented
6#20265426C212yesmaximal, complemented
7#20265412C212yesmaximal, complemented
8#20265409C212yesmaximal, complemented
9#20265217C211yesmaximal, characteristic, complemented
10#20265183C211yesmaximal, characteristic, complemented
11#417×C101C211yesmaximal, characteristic, complemented
12C22×D404—21no—
13C22×D404C2211yescharacteristic
14C22×C4×D101C2211yescharacteristic
15#15692280—24no—
16C2×#13108061C2212yescomplemented
17C2×#13108059C2212yescomplemented
18C2×#13108058C2211yescharacteristic
19C2×#13108057C2212yescomplemented
20M16×D101—24no—
21C2×C8×D101C2212yescomplemented
22#15692236—21no—
23#15692231C2211yescharacteristic, complemented
24#15692231C2211yescharacteristic, complemented
25#15692230—21no—
26#15692230—21no—
27#15692230C2212yes—
28#15692230C2212yes—
29#15692226C2211yescharacteristic, complemented
30#15692226C2211yescharacteristic, complemented
31#15692223—21no—
32C4×C22⋊D101—21no—
33C4×C22⋊D101—21no—
34#15692214C2211yescharacteristic
35C2×#13108051—21no—
36#15692188C2212yescomplemented
37#15692186C2212yescomplemented
38#15692177C2212yescomplemented
39#15692177C2212yescomplemented
40C2×#41×C101C2211yescharacteristic
41C2×M16×C101C2211yescharacteristic
42C2×C4⋊C4×C101C2211yescharacteristic
43#103×C101C2212yescomplemented
44#102×C101C2212yescomplemented
45C23×D101—21no—
46C23×D101D411yescharacteristic
47C2×D404—22no—
48C2×D404—22no—
49C2×D404—22no—
50C2×D404—22no—
51C2×D404—21no—
52C2×D404—44no—
53C2×D404C2312yes—
54C2×C22⋊D101—21no—
55C2×C22⋊D101—21no—
56C22×Dic101D411yescharacteristic
57C2×C4×D101—22no—
58C2×C4×D101—22no—
59C2×C4×D101D411yescharacteristic
60C2×C4×D101D411yescharacteristic
61C2×C4×D101C2311yescharacteristic
62C2×C4×D101C2311yescharacteristic
63#13108061—24no—
64#13108059—24no—
65#13108058D414yescomplemented
66#13108057—24no—
67C2×#11717579C2312yes—
68C8×D101—24no—
69C4⋊Dic101C2311yescharacteristic
70C4⋊Dic101C2311yescharacteristic
71#13108053—21no—
72#13108053—21no—
73C22⋊Dic101—21no—
74C22⋊Dic101—21no—
75C22⋊Dic101—21no—
76#13108051—21no—
77#13108051—21no—
78#13108051—22no—
79#13108051—22no—
80C4×Dic101—21no—
81#41×C101—22no—
82#41×C101—22no—
83C2×Q8×C101C2311yescharacteristic
84C2×D4×C101—21no—
85C2×D4×C101C2311yescharacteristic
86C22×C4×C101—21no—
87C22×C4×C101—21no—
88C22×C4×C101C2311yescharacteristic
89M16×C101—22no—
90C2×C8×C101C2312yes—
91C4⋊C4×C101—21no—
92C4⋊C4×C101C2312yes—
93C23×C101—21no—
94C23×C101C2×D411yescharacteristic
95C22×D101—22no—
96C22×D101—22no—
97C22×D101—22no—
98C22×D101—41no—
99C22×D101—42no—
100C22×D101—42no—
101C22×D101—42no—
102C22×D101C2×D411yescharacteristic
103C22×D101C2×D411yescharacteristic
104Q8×C101—21no—
105Q8×C101—21no—
106D4×C101—21no—
107D4×C101—21no—
108D4×C101—42no—
109C2×C4×C101—22no—
110C2×C4×C101—21no—
111C2×C4×C101—42no—
112C2×C4×C101—42no—
113C2×C4×C101C2×D411yescharacteristic
114C2×C4×C101C2×D412yes—
115C2×C4×C101#4112yes—
116C2×C4×C101C2411yescharacteristic
117C22⋊D101—42no—
118C22⋊D101—42no—
119C2×Dic101—22no—
120C2×Dic101—21no—
121C2×Dic101—21no—
122C2×Dic101C2×D411yescharacteristic
123C2×Dic101C2×D411yescharacteristic
124D404—22no—
125D404—22no—
126D404—44no—
127D404—44no—
128D404—42no—
129C4×D101—22no—
130C4×D101—22no—
131C4×D101—21no—
132C4×D101—21no—
133C4×D101—21no—
134C4×D101—21no—
135C8×C101—22no—
136#11717579C2×D414yescomplemented
137C22×C101—21no—
138C22×C101—21no—
139C22×C101—41no—
140C22×C101—41no—
141C22×C101—41no—
142C22×C101#12112yes—
143C22×C101C22×D411yescharacteristic
144C2×D101—21no—
145C2×D101—21no—
146C2×D101—21no—
147C2×D101—21no—
148C2×D101—21no—
149C2×D101—44no—
150C2×D101—42no—
151C2×D101—42no—
152C2×D101—42no—
153C2×D101—84no—
154C2×D101AGL(1,ℤ/8)12yescomplemented
155C4×C101—41no—
156C4×C101—42no—
157C4×C101#12112yes—
158C4×C101C22×D411yescharacteristic
159C4×C101C2×#4111yescharacteristic
160Dic101—21no—
161Dic101—21no—
162Dic101—41no—
163Dic101—41no—
164C2×C101—22no—
165C2×C101—81no—
166C2×C101C2×#12111yescharacteristic
167C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C101#57511yescharacteristic
169D101—22no—
170D101—41no—
171D101—82no—
172#3055—1011nomaximal, Sylow
173C101#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)—1011no—
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#417D10111yescharacteristic, complemented
185C2×#41C2×D10111yescharacteristic
186C22×D4—1011no—
187C22×D4—2021no—
188C23×C4—1011no—
189AGL(1,ℤ/8)—2024no—
190C2×SD16—1012no—
191C2×D8—1012no—
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—1011no—
201#121—1011no—
202#121—1012no—
203#121—1012no—
204#121—2021no—
205#121—2021no—
206#120—2021no—
207C4×D4—2021no—
208C4×D4—2021no—
209#117—1011no—
210C2×C4⋊C4C2×D10111yescharacteristic
211C2×C22⋊C4—2021no—
212#107—1012no—
213#106—1012no—
214#103C2×D10112yescomplemented
215#102—1012no—
216#102—1012no—
217#102C2×D10112yescomplemented
218C24—1011no—
219C24—2021no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D10111yescharacteristic
223C2×D4—21no—
224C2×D4—1012no—
225C2×D4—4044no—
226C2×D4—2022no—
227C2×D4—2022no—
228C2×D4—2021no—
229C2×D4—2021no—
230C2×D4—2021no—
231C2×D4—2022no—
232C2×D4—2022no—
233C2×D4C22×D10111yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—1011no—
237C22×C4—1011no—
238C22×C4—1011no—
239C22×C4—1011no—
240C22×C4—1011no—
241C22×C4—2022no—
242C22×C4—2022no—
243C22×C4C22×D10111yescharacteristic
244SD16—2024no—
245D8—2024no—
246M16—22no—
247M16—1014no—
248M16—2024no—
249C2×C8—1012no—
250C2×C8—2024no—
251C2×C8C22×D10112yes—
252C4⋊C4—21no—
253C4⋊C4—1011no—
254C4⋊C4—1011no—
255C4⋊C4—2021no—
256C4⋊C4—2021no—
257C4⋊C4C22×D10112yes—
258C22⋊C4—2022no—
259C22⋊C4—2022no—
260C22⋊C4—2021no—
261C22⋊C4—2021no—
262C22⋊C4—2021no—
263C22⋊C4—2021no—
264C22⋊C4—2021no—
265C42—2021no—
266C23—21no—
267C23—1011no—
268C23—1011no—
269C23—4042no—
270C23—4042no—
271C23—4041no—
272C23—4042no—
273C23—2022no—
274C23—2022no—
275C23—2022no—
276C23D4×D10111yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—4044no—
283D4—4044no—
284D4—4042no—
285D4—4042no—
286D4—4042no—
287D4—2022no—
288D4—2022no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—1011no—
294C2×C4—1011no—
295C2×C4—2021no—
296C2×C4—2021no—
297C2×C4—2022no—
298C2×C4—2021no—
299C2×C4—2021no—
300C2×C4—2021no—
301C2×C4—2021no—
302C2×C4—2022no—
303C2×C4—2022no—
304C2×C4D4×D10112yes—
305C2×C4D4×D10111yescharacteristic
306C2×C4#1310807412yes—
307C2×C4C23×D10111yescharacteristic
308C8—22no—
309C8—1014no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—8084no—
316C22—1012no—
317C22—4044no—
318C22—4042no—
319C22—4042no—
320C22—4042no—
321C22—2021no—
322C22—2021no—
323C22—2021no—
324C22—2021no—
325C22—2021no—
326C22#1569222812yes—
327C22C2×D4×D10111yescharacteristic
328C4—41no—
329C4—42no—
330C4—4041no—
331C4—4041no—
332C4—2021no—
333C4—2021no—
334C4#1569222812yes—
335C4C2×D4×D10111yescharacteristic
336C4C2×#1310807411yescharacteristic
337C2—22no—
338C2—81no—
339C2—8082no—
340C2—4041no—
341C2—2022no—
342C2C2×#1569222811yescharacteristic
343C2AGL(1,ℤ/8)×D10111yescharacteristic
344C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#417×C101C2
socleC22×C101C22×D4
hypercenter#417D101
hypocenterC101#3055
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#15692280C2semidirect
2. C2×M16×D101C2semidirect
3. #20265684C2semidirect
4. C2×#15692230C2semidirect
5. #20265429C2semidirect
6. #20265426C2semidirect
7. #20265412C2semidirect
8. #20265409C2semidirect
9. #20265217C2semidirect
10. #20265183C2semidirect
11. #417×C101C2semidirect
16. C2×#13108061C22semidirect
17. C2×#13108059C22semidirect
19. C2×#13108057C22semidirect
21. C2×C8×D101C22semidirect
23. #15692231C22semidirect
24. #15692231C22semidirect
29. #15692226C22semidirect
30. #15692226C22semidirect
36. #15692188C22semidirect
37. #15692186C22semidirect
38. #15692177C22semidirect
39. #15692177C22semidirect
43. #103×C101C22semidirect
44. #102×C101C22semidirect
65. #13108058D4semidirect
136. #11717579C2×D4semidirect
154. C2×D101AGL(1,ℤ/8)semidirect
173. C101#3055semidirect
184. #417D101semidirect
214. #103C2×D101semidirect
217. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922802
C2×M16×D1012
#202656842
C2×#156922302
#202654292
#202654262
#202654122
#202654092
#202652172
#202651832
#417×C1012
#3055101