Finite Group Databasebeta

← #27181724

Subgroups — #27181724

Order 12928.

408 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 119 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
1#20265725C211yesmaximal, characteristic, complemented
2#20265706C211yesmaximal, characteristic, complemented
3#20265586C211yesmaximal, characteristic, complemented
4#20265541C211yesmaximal, characteristic, complemented
5#20265536C211yesmaximal, characteristic, complemented
6#20265427C212yesmaximal, complemented
7#20265420C212yesmaximal, complemented
8SD16×Dic101C212yesmaximal, complemented
9D8×Dic101C212yesmaximal, complemented
10#20265129C211yesmaximal, characteristic, complemented
11#496×C101C211yesmaximal, characteristic, complemented
12C2×#13108073—21no—
13C2×#13108057C2212yes—
14#15692255C2211yescharacteristic
15#15692248—21no—
16#15692247C2211yescharacteristic
17#15692242—21no—
18#15692237—21no—
19#15692231C2211yescharacteristic
20#15692226—21no—
21#15692226C2212yes—
22#15692226C2211yescharacteristic
23Q8×Dic101C2211yescharacteristic
24D4×Dic101—21no—
25D4×Dic101C2211yescharacteristic
26D4×Dic101C2212yes—
27#15692215—21no—
28#15692211—21no—
29C42×D101C2211yescharacteristic
30#15692189C2212yes—
31#15692187C2212yes—
32#15692179C2212yes—
33#15692176C2212yes—
34#15692176C2212yes—
35#15692176C2212yes—
36#15692166C2211yescharacteristic
37C8×Dic101C2212yes—
38C2×SD16×C101C2212yes—
39C2×D8×C101C2212yes—
40#127×C101C2211yescharacteristic
41#124×C101C2211yescharacteristic
42#97×C101C2211yescharacteristic
43#13108073—44no—
44C2×Dic202—21no—
45C2×C22⋊D101—22no—
46C2×C22⋊D101—21no—
47C22×Dic101—22no—
48C22×Dic101—21no—
49C2×C4×D101D411yescharacteristic
50C2×C4×D101D411yescharacteristic
51C2×C4×D101C2311yescharacteristic
52#13108057—24no—
53C2×#11717579C2312yes—
54C4⋊Dic101—21no—
55C4⋊Dic101—21no—
56C4⋊Dic101C2311yescharacteristic
57C4⋊Dic101C2311yescharacteristic
58C4⋊Dic101C2312yes—
59#13108053—21no—
60#13108053—21no—
61#13108053—21no—
62C22⋊Dic101—21no—
63C22⋊Dic101—22no—
64C22⋊Dic101—22no—
65C22⋊Dic101—21no—
66#13108051—21no—
67C4×Dic101—21no—
68C4×Dic101D411yescharacteristic
69C4×Dic101D411yescharacteristic
70C4×Dic101C2311yescharacteristic
71C2×Q8×C101C2311yescharacteristic
72C2×D4×C101—21no—
73C2×D4×C101—21no—
74C2×D4×C101C2311yescharacteristic
75C2×D4×C101C2312yes—
76SD16×C101—24no—
77D8×C101—24no—
78C2×C8×C101C2312yes—
79C22⋊C4×C101—21no—
80C22⋊C4×C101—21no—
81C42×C101C2311yescharacteristic
82C23×C101—21no—
83C23×C101—22no—
84C22×D101C2×D411yescharacteristic
85Q8×C101—22no—
86D4×C101—22no—
87D4×C101—22no—
88D4×C101—22no—
89D4×C101—42no—
90D4×C101—42no—
91C2×C4×C101—21no—
92C2×C4×C101C2×D411yescharacteristic
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2411yescharacteristic
95C22⋊D101—42no—
96C22⋊D101—44no—
97C2×Dic101—21no—
98C2×Dic101—21no—
99C2×Dic101—22no—
100C2×Dic101—42no—
101C2×Dic101—44no—
102C2×Dic101C2×D411yescharacteristic
103C2×Dic101C2×D411yescharacteristic
104C2×Dic101C2×D411yescharacteristic
105C4×D101—22no—
106C4×D101—22no—
107C4×D101—41no—
108Dic202—42no—
109C8×C101#4114yescomplemented
110#11717579—22no—
111C22×C101—41no—
112C22×C101—42no—
113C22×C101—42no—
114C22×C101—41no—
115C22×C101—41no—
116C22×C101—42no—
117C22×C101C22×D411yescharacteristic
118C2×D101—21no—
119C2×D101—21no—
120C2×D101—21no—
121C4×C101—21no—
122C4×C101—21no—
123C4×C101—41no—
124C4×C101C2×#4112yes—
125Dic101—21no—
126Dic101—21no—
127Dic101—42no—
128Dic101—41no—
129Dic101—41no—
130Dic101AGL(1,ℤ/8)12yes—
131C2×C101—82no—
132C2×C101—81no—
133C2×C101#53211yescharacteristic
134C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
135C2×C101#57511yescharacteristic
136D101—41no—
137#3171—1011nomaximal, Sylow
138C101#317111yescharacteristic, complemented, Sylow
139#532—1011no—
140#529—1011no—
141#501—1011no—
142#496D10111yescharacteristic, complemented
143#490—1011no—
144#489—1011no—
145#469—1012no—
146#468—1012no—
147C4×SD16—1012no—
148C4×D8—1012no—
149C4×M16—1011no—
150C2×#41—2021no—
151C2×SD16C2×D10112yes—
152C2×D8C2×D10112yes—
153C2×M16—1012no—
154C4⋊Q8—1011no—
155#127C2×D10111yescharacteristic
156#126—2021no—
157#125—1011no—
158#124—2021no—
159#124C2×D10111yescharacteristic
160#123—2021no—
161C22⋊Q8—1011no—
162#121—1011no—
163#121—1012no—
164#121—2021no—
165C4×Q8—1011no—
166C4×D4—1011no—
167C4×D4—1012no—
168C4×D4—2021no—
169C4×D4—2021no—
170#117—2021no—
171C2×C42—1011no—
172#107—1012no—
173#106—1012no—
174#103—1012no—
175#102—1012no—
176#102—1012no—
177#102—1012no—
178#97—1011no—
179#97C2×D10111yescharacteristic
180C4×C8—1012no—
181#41—4044no—
182C2×Q8—2021no—
183C2×Q8C22×D10111yescharacteristic
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—2021no—
187C2×D4—2022no—
188C2×D4C22×D10111yescharacteristic
189C2×D4C22×D10112yes—
190C22×C4—1011no—
191C22×C4—1011no—
192C22×C4—1011no—
193C22×C4—2021no—
194C22×C4—2022no—
195SD16—24no—
196D8—24no—
197M16—2024no—
198C2×C8—1012no—
199C2×C8C22×D10112yes—
200C4⋊C4—1011no—
201C4⋊C4—1011no—
202C4⋊C4—1012no—
203C4⋊C4—2021no—
204C4⋊C4—2021no—
205C4⋊C4—2021no—
206C4⋊C4—2021no—
207C4⋊C4—2021no—
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—2021no—
211C22⋊C4—2021no—
212C22⋊C4—2022no—
213C22⋊C4—2021no—
214C22⋊C4—2022no—
215C42—1011no—
216C42—1011no—
217C42—1011no—
218C42—2021no—
219C42C22×D10111yescharacteristic
220C23—21no—
221C23—22no—
222C23—1011no—
223Q8—22no—
224Q8—4042no—
225D4—22no—
226D4—22no—
227D4—22no—
228D4—42no—
229D4—42no—
230D4—4044no—
231D4—4042no—
232C2×C4—21no—
233C2×C4—1011no—
234C2×C4—1011no—
235C2×C4—1011no—
236C2×C4—4041no—
237C2×C4—4044no—
238C2×C4—4042no—
239C2×C4—2022no—
240C2×C4—2021no—
241C2×C4—2021no—
242C2×C4—2022no—
243C2×C4—2022no—
244C2×C4D4×D10111yescharacteristic
245C2×C4D4×D10111yescharacteristic
246C2×C4C23×D10111yescharacteristic
247C8—2022no—
248C8#1310807314yescomplemented
249C22—42no—
250C22—41no—
251C22—42no—
252C22—41no—
253C22—41no—
254C22—42no—
255C22—2021no—
256C22—2021no—
257C22—2021no—
258C22C2×D4×D10111yescharacteristic
259C4—21no—
260C4—21no—
261C4—41no—
262C4—1012no—
263C4—4042no—
264C4—4041no—
265C4—4041no—
266C4—2021no—
267C4—2021no—
268C4C2×#1310807312yes—
269C2—82no—
270C2—81no—
271C2—4041no—
272C2#2026572511yescharacteristic
273C2AGL(1,ℤ/8)×D10111yescharacteristic
274C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#496×C101C2
socleC22×C101C22×D4
hypercenter#496D101
hypocenterC101#3171
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #20265725C2semidirect
2. #20265706C2semidirect
3. #20265586C2semidirect
4. #20265541C2semidirect
5. #20265536C2semidirect
6. #20265427C2semidirect
7. #20265420C2semidirect
8. SD16×Dic101C2semidirect
9. D8×Dic101C2semidirect
10. #20265129C2semidirect
11. #496×C101C2semidirect
109. C8×C101#41semidirect
138. C101#3171semidirect
142. #496D101semidirect
248. C8#13108073semidirect

Maximal subgroups

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

subgroupindex
#202657252
#202657062
#202655862
#202655412
#202655362
#202654272
#202654202
SD16×Dic1012
D8×Dic1012
#202651292
#496×C1012
#3171101