Finite Group Databasebeta

← #27180595

Subgroups — #27180595

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×#15692283C211yesmaximal, characteristic, complemented
2C2×#15692280C211yesmaximal, characteristic, complemented
3#20265723C211yesmaximal, characteristic, complemented
4C2×#15692217C211yesmaximal, characteristic, complemented
5#20265395C212yesmaximal, complemented
6#20265391C212yesmaximal, complemented
7#20265381C212yesmaximal, complemented
8#20265371C212yesmaximal, complemented
9#20265200C211yesmaximal, characteristic, complemented
10#20265176C211yesmaximal, characteristic, complemented
11#417×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C22×D404C2211yescharacteristic
14C22×C4×D101—21no—
15#15692283—24no—
16#15692280—24no—
17C2×#13108063C2212yes—
18C2×#13108061C2212yes—
19C2×D808C2212yes—
20C2×#13108059C2212yes—
21C2×#13108058C2211yescharacteristic
22#15692253C2211yescharacteristic, complemented
23#15692251C2211yescharacteristic, complemented
24#15692244C2211yescharacteristic, complemented
25#15692242C2211yescharacteristic, complemented
26#15692227—21no—
27#15692227—21no—
28#15692217—21no—
29#15692217—21no—
30#15692217C2212yes—
31#15692217C2212yes—
32C4×C22⋊D101—21no—
33C4×C22⋊D101—21no—
34C2×#13108051—21no—
35C2×C4×Dic101C2211yescharacteristic
36#15692184C2212yescomplemented
37#15692184C2212yescomplemented
38#15692178C2212yescomplemented
39#15692175C2212yescomplemented
40C2×#41×C101C2211yescharacteristic
41C2×M16×C101C2211yescharacteristic
42C2×C4⋊C4×C101C2211yescharacteristic
43#103×C101C2212yescomplemented
44#102×C101C2212yescomplemented
45C23×D101—21no—
46#13108074—22no—
47#13108074—22no—
48C2×Dic202C2311yescharacteristic
49C2×D404—21no—
50C2×D404—22no—
51C2×D404D414yescomplemented
52C2×D404C2311yescharacteristic
53C2×D404C2312yes—
54C2×C22⋊D101—21no—
55C2×C22⋊D101—21no—
56C2×C22⋊D101—21no—
57C22×Dic101D411yescharacteristic
58C22×Dic101D411yescharacteristic
59C2×C4×D101—21no—
60C2×C4×D101—22no—
61C2×C4×D101—22no—
62C2×C4×D101—44no—
63#13108063—24no—
64#13108061—24no—
65D808—24no—
66#13108059—24no—
67#13108058—22no—
68C2×#11717579C2312yes—
69#13108053—21no—
70#13108053—21no—
71C22⋊Dic101—21no—
72C22⋊Dic101—21no—
73#13108051—22no—
74#13108051—22no—
75#13108051—21no—
76#13108051—21no—
77C4×Dic101D411yescharacteristic
78C4×Dic101D411yescharacteristic
79C4×Dic101C2311yescharacteristic
80C4×Dic101C2311yescharacteristic
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—24no—
96C22×D101—22no—
97C22×D101—21no—
98C22×D101—42no—
99C22×D101—42no—
100Q8×C101—21no—
101Q8×C101—21no—
102D4×C101—21no—
103D4×C101—21no—
104D4×C101—42no—
105C2×C4×C101—22no—
106C2×C4×C101—21no—
107C2×C4×C101—42no—
108C2×C4×C101—42no—
109C2×C4×C101C2×D411yescharacteristic
110C2×C4×C101C2×D412yes—
111C2×C4×C101#4112yes—
112C2×C4×C101C2411yescharacteristic
113C22⋊D101—42no—
114C22⋊D101—42no—
115C22⋊D101—42no—
116C2×Dic101—22no—
117C2×Dic101—21no—
118C2×Dic101—41no—
119C2×Dic101C2×D411yescharacteristic
120C2×Dic101C2×D411yescharacteristic
121C2×Dic101C2×D411yescharacteristic
122C2×Dic101C2×D411yescharacteristic
123D404—24no—
124D404—21no—
125D404—22no—
126D404—21no—
127D404C2×D414yes—
128C4×D101—44no—
129C4×D101—42no—
130C4×D101—44no—
131Dic202—21no—
132Dic202—21no—
133C8×C101—22no—
134#11717579—22no—
135C22×C101—21no—
136C22×C101—21no—
137C22×C101—41no—
138C22×C101—41no—
139C22×C101—41no—
140C22×C101#12112yes—
141C22×C101C22×D411yescharacteristic
142C2×D101—24no—
143C2×D101—48no—
144C2×D101—42no—
145C2×D101—42no—
146C2×D101—41no—
147C2×D101—41no—
148C2×D101—41no—
149C4×C101—41no—
150C4×C101—42no—
151C4×C101#12112yes—
152C4×C101C22×D411yescharacteristic
153C4×C101C2×#4111yescharacteristic
154Dic101—21no—
155Dic101—21no—
156Dic101—21no—
157Dic101—41no—
158Dic101AGL(1,ℤ/8)12yescomplemented
159C2×C101—22no—
160C2×C101—81no—
161C2×C101C2×#12111yescharacteristic
162C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
163C2×C101#57511yescharacteristic
164D101—44no—
165D101—81no—
166#3036—1011nomaximal, Sylow
167C101#303611yescharacteristic, complemented, Sylow
168C2×AGL(1,ℤ/8)—1011no—
169C2×AGL(1,ℤ/8)—1011no—
170#532—1011no—
171C2×C4×D4—1011no—
172#463—1012no—
173#463—1012no—
174#461—1012no—
175#459—1012no—
176#423—1011no—
177#417—1011no—
178#417D10111yescharacteristic, complemented
179C2×#41—1011no—
180C2×#41C2×D10111yescharacteristic
181C22×D4—1011no—
182C23×C4—2021no—
183AGL(1,ℤ/8)—2024no—
184AGL(1,ℤ/8)—2024no—
185C2×SD16—1012no—
186C2×SD16—1012no—
187C2×D8—1012no—
188C2×D8—1012no—
189C2×M16—1011no—
190C2×M16C2×D10111yescharacteristic
191C4⋊Q8—1011no—
192#127—1011no—
193#124—1011no—
194#124—1011no—
195#121—2021no—
196#121—2021no—
197C4×D4—1012no—
198C4×D4—1012no—
199C4×D4—2021no—
200C4×D4—2021no—
201C4×D4—2021no—
202C4×D4—2021no—
203C2×C4⋊C4C2×D10111yescharacteristic
204C2×C22⋊C4—2021no—
205C2×C42—1011no—
206C4⋊C8—1012no—
207C4⋊C8—1012no—
208#103—1012no—
209#103C2×D10112yescomplemented
210#102—1012no—
211#102C2×D10112yescomplemented
212C24—2021no—
213#41—22no—
214#41—22no—
215#41—2022no—
216#41—2022no—
217C2×Q8—1011no—
218C2×Q8C22×D10111yescharacteristic
219C2×D4—21no—
220C2×D4—1014no—
221C2×D4—1012no—
222C2×D4—1011no—
223C2×D4—2021no—
224C2×D4—2021no—
225C2×D4—2021no—
226C2×D4—2021no—
227C2×D4—2022no—
228C2×D4C22×D10111yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—1011no—
232C22×C4—1011no—
233C22×C4—4044no—
234C22×C4—2022no—
235C22×C4—2022no—
236C22×C4—2021no—
237C22×C4C22×D10111yescharacteristic
238SD16—2024no—
239SD16—2024no—
240D8—2024no—
241D8—2024no—
242M16—22no—
243M16—2022no—
244C2×C8—1012no—
245C2×C8C22×D10112yes—
246C4⋊C4—21no—
247C4⋊C4—2021no—
248C4⋊C4—2021no—
249C4⋊C4C22×D10112yes—
250C22⋊C4—2022no—
251C22⋊C4—2022no—
252C22⋊C4—2021no—
253C22⋊C4—2021no—
254C22⋊C4—2021no—
255C22⋊C4—2021no—
256C42—1011no—
257C42—1011no—
258C42—1011no—
259C42—1011no—
260C23—21no—
261C23—4042no—
262C23—4042no—
263C23—2021no—
264C23—2022no—
265C23—2024no—
266C23D4×D10111yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—2021no—
270Q8—2021no—
271D4—21no—
272D4—21no—
273D4—42no—
274D4—1014no—
275D4—4042no—
276D4—4042no—
277D4—4042no—
278D4—2021no—
279D4—2021no—
280D4—2022no—
281D4—2024no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—1011no—
287C2×C4—1011no—
288C2×C4—1011no—
289C2×C4—1011no—
290C2×C4—4044no—
291C2×C4—4044no—
292C2×C4—4042no—
293C2×C4—4041no—
294C2×C4—2022no—
295C2×C4—2021no—
296C2×C4D4×D10112yes—
297C2×C4D4×D10111yescharacteristic
298C2×C4#1310807412yes—
299C2×C4C23×D10111yescharacteristic
300C8—22no—
301C8—2022no—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—4048no—
308C22—4041no—
309C22—4041no—
310C22—4041no—
311C22—4042no—
312C22—4042no—
313C22—2024no—
314C22#1569222812yes—
315C22C2×D4×D10111yescharacteristic
316C4—41no—
317C4—42no—
318C4—1012no—
319C4—4041no—
320C4—2021no—
321C4—2021no—
322C4—2021no—
323C4#1569222812yes—
324C4C2×D4×D10111yescharacteristic
325C4C2×#1310807411yescharacteristic
326C2—22no—
327C2—81no—
328C2—8081no—
329C2—4044no—
330C2C2×#1569222811yescharacteristic
331C2AGL(1,ℤ/8)×D10111yescharacteristic
332C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#417×C101C2
socleC22×C101C22×D4
hypercenter#417D101
hypocenterC101#3036
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×#15692283C2semidirect
2. C2×#15692280C2semidirect
3. #20265723C2semidirect
4. C2×#15692217C2semidirect
5. #20265395C2semidirect
6. #20265391C2semidirect
7. #20265381C2semidirect
8. #20265371C2semidirect
9. #20265200C2semidirect
10. #20265176C2semidirect
11. #417×C101C2semidirect
22. #15692253C22semidirect
23. #15692251C22semidirect
24. #15692244C22semidirect
25. #15692242C22semidirect
36. #15692184C22semidirect
37. #15692184C22semidirect
38. #15692178C22semidirect
39. #15692175C22semidirect
43. #103×C101C22semidirect
44. #102×C101C22semidirect
51. C2×D404D4semidirect
158. Dic101AGL(1,ℤ/8)semidirect
167. C101#3036semidirect
178. #417D101semidirect
209. #103C2×D101semidirect
211. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922832
C2×#156922802
#202657232
C2×#156922172
#202653952
#202653912
#202653812
#202653712
#202652002
#202651762
#417×C1012
#3036101