Finite Group Databasebeta

← #27181726

Subgroups — #27181726

Order 12928.

408 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 115 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#20265712C211yesmaximal, characteristic, complemented
3#20265587C211yesmaximal, characteristic, complemented
4#20265564C211yesmaximal, characteristic, complemented
5#20265537C211yesmaximal, characteristic, complemented
6#20265429C212yesmaximal, complemented
7#20265422C212yesmaximal, complemented
8#20265268C212yesmaximal, complemented
9#20265257C212yesmaximal, complemented
10#20265129C211yesmaximal, characteristic, complemented
11#490×C101C211yesmaximal, characteristic, complemented
12C2×#13108073—21no—
13C2×#13108062C2212yes—
14C2×#13108059C2212yes—
15C2×#13108057C2212yes—
16#15692255C2211yescharacteristic
17#15692250—21no—
18#15692243C2211yescharacteristic
19#15692242—21no—
20#15692240—21no—
21#15692235C2211yescharacteristic
22#15692230C2211yescharacteristic
23#15692226—21no—
24#15692226C2212yes—
25#15692220C2211yescharacteristic
26D4×Dic101—21no—
27D4×Dic101C2212yes—
28#15692217—21no—
29#15692217C2211yescharacteristic
30#15692213—21no—
31C42×D101C2211yescharacteristic
32#15692188C2212yes—
33#15692186C2212yes—
34#15692180C2212yes—
35#15692177C2212yes—
36#15692166C2211yescharacteristic
37C8×Dic101C2212yes—
38#127×C101C2211yescharacteristic
39#125×C101C2211yescharacteristic
40#102×C101C2212yes—
41#102×C101C2212yes—
42#97×C101C2211yescharacteristic
43#13108073—44no—
44C2×Dic202—21no—
45C2×Dic202C2311yescharacteristic
46C2×D404—21no—
47C2×D404C2311yescharacteristic
48C2×C22⋊D101—22no—
49C22×Dic101—22no—
50C2×C4×D101—21no—
51C2×C4×D101D411yescharacteristic
52C2×C4×D101D411yescharacteristic
53C2×C4×D101C2311yescharacteristic
54#13108062—24no—
55#13108059—24no—
56#13108057—24no—
57C2×#11717579C2312yes—
58C4⋊Dic101—21no—
59C4⋊Dic101—21no—
60C4⋊Dic101C2312yes—
61#13108053—21no—
62C22⋊Dic101—22no—
63C22⋊Dic101—22no—
64#13108051—21no—
65#13108051—21no—
66#13108051—21no—
67#13108051—21no—
68#13108051—21no—
69C4×Dic101—21no—
70C4×Dic101D411yescharacteristic
71C4×Dic101D411yescharacteristic
72C4×Dic101C2311yescharacteristic
73C2×D4×C101—21no—
74C2×D4×C101—21no—
75C2×D4×C101C2312yes—
76C2×C8×C101C2312yes—
77C4⋊C4×C101—21no—
78C4⋊C4×C101—21no—
79C4⋊C4×C101C2311yescharacteristic
80C4⋊C4×C101C2311yescharacteristic
81C42×C101C2311yescharacteristic
82C23×C101—22no—
83C22×D101—21no—
84C22×D101C2×D411yescharacteristic
85D4×C101—22no—
86D4×C101—22no—
87D4×C101—42no—
88D4×C101—42no—
89C2×C4×C101—21no—
90C2×C4×C101—21no—
91C2×C4×C101C2×D411yescharacteristic
92C2×C4×C101C2×D411yescharacteristic
93C2×C4×C101C2411yescharacteristic
94C22⋊D101—44no—
95C2×Dic101—21no—
96C2×Dic101—22no—
97C2×Dic101—44no—
98C2×Dic101C2×D411yescharacteristic
99C2×Dic101C2×D411yescharacteristic
100C2×Dic101C2×D411yescharacteristic
101D404—22no—
102D404—42no—
103C4×D101—22no—
104C4×D101—22no—
105C4×D101—41no—
106C4×D101—42no—
107Dic202—22no—
108Dic202—42no—
109C8×C101—22no—
110#11717579#4114yescomplemented
111C22×C101—42no—
112C22×C101—42no—
113C22×C101—42no—
114C22×C101C22×D411yescharacteristic
115C2×D101—21no—
116C2×D101—21no—
117C2×D101—21no—
118C2×D101—41no—
119C2×D101—41no—
120C2×D101—41no—
121C4×C101—21no—
122C4×C101—21no—
123C4×C101—41no—
124C4×C101—41no—
125C4×C101C2×#4112yes—
126Dic101—21no—
127Dic101—21no—
128Dic101—41no—
129Dic101—42no—
130Dic101AGL(1,ℤ/8)12yes—
131C2×C101—82no—
132C2×C101#53211yescharacteristic
133C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
134C2×C101#57511yescharacteristic
135D101—41no—
136D101—81no—
137#3171—1011nomaximal, Sylow
138C101#317111yescharacteristic, complemented, Sylow
139#532—1011no—
140#529—1011no—
141#501—1011no—
142#496—1011no—
143#490D10111yescharacteristic, complemented
144#489—1011no—
145#469—1012no—
146#468—1012no—
147C4×SD16—1012no—
148C4×D8—1012no—
149C4×M16—1011no—
150C2×#41—2021no—
151C2×SD16—1012no—
152C2×D8—1012no—
153C2×M16—1012no—
154C4⋊Q8—1011no—
155#127C2×D10111yescharacteristic
156#126—2021no—
157#125C2×D10111yescharacteristic
158#124—1011no—
159#124—2021no—
160#123—2021no—
161C22⋊Q8—1011no—
162#121—1012no—
163#121—1011no—
164#121—2021no—
165C4×Q8—1011no—
166C4×D4—1012no—
167C4×D4—1011no—
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#102C2×D10112yes—
177#102C2×D10112yes—
178#97—1011no—
179#97C2×D10111yescharacteristic
180C4×C8—1012no—
181#41—4044no—
182C2×Q8—1011no—
183C2×Q8—2021no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—1011no—
187C2×D4—2021no—
188C2×D4—2022no—
189C2×D4C22×D10112yes—
190C22×C4—1011no—
191C22×C4—1011no—
192C22×C4—1011no—
193C22×C4—2021no—
194C22×C4—2022no—
195SD16—2024no—
196D8—2024no—
197M16—2024no—
198C2×C8—1012no—
199C2×C8C22×D10112yes—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—1012no—
203C4⋊C4—2021no—
204C4⋊C4—2021no—
205C4⋊C4—2021no—
206C4⋊C4C22×D10111yescharacteristic
207C4⋊C4C22×D10111yescharacteristic
208C22⋊C4—2021no—
209C22⋊C4—2021no—
210C22⋊C4—2021no—
211C22⋊C4—2022no—
212C22⋊C4—2022no—
213C22⋊C4—2021no—
214C22⋊C4—2021no—
215C42—1011no—
216C42—1011no—
217C42—1011no—
218C42—2021no—
219C42C22×D10111yescharacteristic
220C23—22no—
221C23—1011no—
222C23—2021no—
223Q8—4042no—
224Q8—2022no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—42no—
229D4—4042no—
230D4—4044no—
231D4—2022no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—1011no—
235C2×C4—1011no—
236C2×C4—1011no—
237C2×C4—4041no—
238C2×C4—4042no—
239C2×C4—4044no—
240C2×C4—2022no—
241C2×C4—2021no—
242C2×C4—2022no—
243C2×C4—2022no—
244C2×C4D4×D10111yescharacteristic
245C2×C4D4×D10111yescharacteristic
246C2×C4C23×D10111yescharacteristic
247C8—22no—
248C8—1014no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—4041no—
253C22—4041no—
254C22—4041no—
255C22—2021no—
256C22—2021no—
257C22—2021no—
258C22C2×D4×D10111yescharacteristic
259C4—21no—
260C4—21no—
261C4—41no—
262C4—41no—
263C4—1012no—
264C4—4042no—
265C4—4041no—
266C4—2021no—
267C4—2021no—
268C4#41×D10112yes—
269C2—82no—
270C2—8081no—
271C2—4041no—
272C2#2026572611yescharacteristic
273C2AGL(1,ℤ/8)×D10111yescharacteristic
274C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#490×C101C2
socleC22×C101C22×D4
hypercenter#490D101
hypocenterC101#3171
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #20265725C2semidirect
2. #20265712C2semidirect
3. #20265587C2semidirect
4. #20265564C2semidirect
5. #20265537C2semidirect
6. #20265429C2semidirect
7. #20265422C2semidirect
8. #20265268C2semidirect
9. #20265257C2semidirect
10. #20265129C2semidirect
11. #490×C101C2semidirect
110. #11717579#41semidirect
138. C101#3171semidirect
143. #490D101semidirect

Maximal subgroups

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

subgroupindex
#202657252
#202657122
#202655872
#202655642
#202655372
#202654292
#202654222
#202652682
#202652572
#202651292
#490×C1012
#3171101