Finite Group Databasebeta

← #27181091

Subgroups — #27181091

Order 12928.

418 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 133 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#20265741C211yesmaximal, characteristic
2#20265660C211yesmaximal, characteristic, complemented
3#20265502C212yesmaximal, complemented
4#20265502C212yesmaximal, complemented
5#20265496C212yesmaximal, complemented
6#20265491C212yesmaximal, complemented
7#20265219C211yesmaximal, characteristic, complemented
8#20265215C211yesmaximal, characteristic, complemented
9#20265183C211yesmaximal, characteristic, complemented
10C2×#15692177C211yesmaximal, characteristic, complemented
11#408×C101C211yesmaximal, characteristic, complemented
12C22×D404C2211yescharacteristic
13#15692251—21no—
14#15692244—21no—
15#15692229C2212yes—
16#15692229C2212yes—
17#15692223—22no—
18Q8×Dic101—21no—
19D4×Dic101—22no—
20D4×Dic101—21no—
21#15692214C2211yescharacteristic
22#15692214C2211yescharacteristic
23#15692214C2212yescomplemented
24C2×C4⋊Dic101C2211yescharacteristic
25C2×C4⋊Dic101C2212yescomplemented
26#15692189C2212yescomplemented
27#15692189C2211yescharacteristic, complemented
28#15692189C2211yescharacteristic, complemented
29#15692187C2212yescomplemented
30#15692187C2211yescharacteristic, complemented
31#15692187C2211yescharacteristic, complemented
32#15692177C2212yes—
33#15692177C2212yes—
34#15692177C2212yes—
35#15692177C2212yes—
36C2×#41×C101C2211yescharacteristic
37C2×M16×C101C2211yescharacteristic
38C22×C8×C101C2211yescharacteristic
39C22⋊C8×C101C2212yescomplemented
40C22⋊C8×C101C2212yescomplemented
41C23×D101—21no—
42C2×D404—22no—
43C2×D404—22no—
44C2×D404—21no—
45C2×D404C2312yes—
46C2×C22⋊D101—22no—
47C2×C22⋊D101—22no—
48C22×Dic101—21no—
49C22×Dic101—22no—
50C4⋊Dic101—21no—
51C4⋊Dic101—22no—
52C4⋊Dic101C2312yes—
53C4⋊Dic101C2311yescharacteristic
54C4⋊Dic101C2311yescharacteristic
55C4⋊Dic101C2311yescharacteristic
56C4⋊Dic101C2311yescharacteristic
57C22⋊Dic101—21no—
58C22⋊Dic101—21no—
59C22⋊Dic101—22no—
60#13108051—22no—
61#13108051—22no—
62C4×Dic101—21no—
63C4×Dic101—21no—
64C4×Dic101—22no—
65#41×C101—22no—
66#41×C101—22no—
67C2×Q8×C101D411yescharacteristic
68C2×D4×C101D412yes—
69C2×D4×C101D411yescharacteristic
70C22×C4×C101C2312yes—
71C22×C4×C101C2311yescharacteristic
72M16×C101—22no—
73C2×C8×C101—22no—
74C2×C8×C101C2311yescharacteristic
75C2×C8×C101C2311yescharacteristic
76C2×C8×C101C2312yes—
77C23×C101C2×D412yes—
78C23×C101C2×D411yescharacteristic
79C22×D101—22no—
80C22×D101—42no—
81C22×D101—81no—
82C22×D101—81no—
83Q8×C101—22no—
84D4×C101—24no—
85D4×C101—22no—
86C2×C4×C101—22no—
87C2×C4×C101—22no—
88C2×C4×C101C2×D412yes—
89C2×C4×C101C2×D411yescharacteristic
90C2×C4×C101#4112yes—
91C2×C4×C101#4112yes—
92C2×C4×C101C2411yescharacteristic
93C22⋊D101—82no—
94C22⋊D101—82no—
95C2×Dic101—21no—
96C2×Dic101—21no—
97C2×Dic101—21no—
98C2×Dic101—21no—
99C2×Dic101—22no—
100C2×Dic101—81no—
101C2×Dic101—82no—
102D404—22no—
103D404—22no—
104D404—42no—
105C8×C101—21no—
106C8×C101—21no—
107C8×C101—22no—
108C22×C101—21no—
109C22×C101—21no—
110C22×C101—22no—
111C22×C101—22no—
112C22×C101—22no—
113C22×C101#12312yes—
114C22×C101C22×D411yescharacteristic
115C2×D101—42no—
116C2×D101—42no—
117C2×D101—42no—
118C2×D101—84no—
119C4×C101—22no—
120C4×C101#12312yes—
121C4×C101C2×#4111yescharacteristic
122C4×C101C2×#4111yescharacteristic
123Dic101—41no—
124Dic101—42no—
125Dic101—41no—
126Dic101—41no—
127Dic101—41no—
128C2×C101—22no—
129C2×C101—42no—
130C2×C101C2×#12311yescharacteristic
131C2×C101#57611yescharacteristic
132C2×C101#57711yescharacteristic
133D101—82no—
134#3096—1011nomaximal, Sylow
135C101#309611yescharacteristic, complemented, Sylow
136#535—1011no—
137#520—1011no—
138#482—1012no—
139#482—1012no—
140#481—1012no—
141#480—1012no—
142#428—1011no—
143#427—1011no—
144#418—1011no—
145C2×#102—1011no—
146#408D10111yescharacteristic, complemented
147C2×#41C2×D10111yescharacteristic
148C22×D4—1011no—
149C2×M16C2×D10111yescharacteristic
150C22×C8C2×D10111yescharacteristic
151#127—2021no—
152#124—2021no—
153#121—1012no—
154#121—1012no—
155#120—2022no—
156C4×Q8—2021no—
157C4×D4—2022no—
158C4×D4—2021no—
159#117—1011no—
160#117—1012no—
161#117—1011no—
162C2×C4⋊C4—1011no—
163C2×C4⋊C4—1012no—
164#107—1011no—
165#107—1011no—
166#107—1012no—
167#106—1011no—
168#106—1011no—
169#106—1012no—
170#102—1012no—
171#102—1012no—
172#102—1012no—
173#102—1012no—
174C22⋊C8C2×D10112yescomplemented
175C22⋊C8C2×D10112yescomplemented
176C24—2021no—
177#41—22no—
178#41—22no—
179C2×Q8D40411yescharacteristic
180C2×D4—1012no—
181C2×D4—2022no—
182C2×D4—2022no—
183C2×D4—2021no—
184C2×D4—2022no—
185C2×D4—2022no—
186C2×D4D40412yes—
187C2×D4D40411yescharacteristic
188C22×C4—2022no—
189C22×C4—2021no—
190C22×C4C22×D10112yes—
191C22×C4C22×D10111yescharacteristic
192M16—22no—
193C2×C8—22no—
194C2×C8C22×D10111yescharacteristic
195C2×C8C22×D10112yes—
196C2×C8C22×D10111yescharacteristic
197C4⋊C4—1012no—
198C4⋊C4—1011no—
199C4⋊C4—1011no—
200C4⋊C4—1011no—
201C4⋊C4—1011no—
202C4⋊C4—2022no—
203C4⋊C4—2021no—
204C22⋊C4—2022no—
205C22⋊C4—2021no—
206C22⋊C4—2021no—
207C22⋊C4—2022no—
208C22⋊C4—2022no—
209C42—2022no—
210C42—2021no—
211C42—2021no—
212C23—8081no—
213C23—8081no—
214C23—4042no—
215C23—2022no—
216C23C2×D40412yes—
217C23C2×D40411yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—8082no—
222D4—8082no—
223D4—4042no—
224D4—2022no—
225D4—2022no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—8081no—
229C2×C4—8082no—
230C2×C4—2021no—
231C2×C4—2021no—
232C2×C4—2021no—
233C2×C4—2022no—
234C2×C4—2021no—
235C2×C4C2×D40412yes—
236C2×C4C2×D40411yescharacteristic
237C2×C4#1310807312yes—
238C2×C4#1310807312yes—
239C2×C4C23×D10111yescharacteristic
240C8—21no—
241C8—21no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—8084no—
249C22—4042no—
250C22—4042no—
251C22—4042no—
252C22#1569223912yes—
253C22C22×D40411yescharacteristic
254C4—22no—
255C4—4041no—
256C4—4041no—
257C4—4042no—
258C4—4041no—
259C4—4041no—
260C4#1569223912yes—
261C4C2×#1310807311yescharacteristic
262C4C2×#1310807311yescharacteristic
263C2—22no—
264C2—42no—
265C2—8082no—
266C2C2×#1569223911yescharacteristic
267C2#2026598211yescharacteristic
268C2#2026598911yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#408×C101C2
socleC22×C101C22×D4
prime radical#20265741C2
hypercenter#408D101
hypocenterC101#3096
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #20265660C2semidirect
3. #20265502C2semidirect
4. #20265502C2semidirect
5. #20265496C2semidirect
6. #20265491C2semidirect
7. #20265219C2semidirect
8. #20265215C2semidirect
9. #20265183C2semidirect
10. C2×#15692177C2semidirect
11. #408×C101C2semidirect
23. #15692214C22semidirect
25. C2×C4⋊Dic101C22semidirect
26. #15692189C22semidirect
27. #15692189C22semidirect
28. #15692189C22semidirect
29. #15692187C22semidirect
30. #15692187C22semidirect
31. #15692187C22semidirect
39. C22⋊C8×C101C22semidirect
40. C22⋊C8×C101C22semidirect
135. C101#3096semidirect
146. #408D101semidirect
174. C22⋊C8C2×D101semidirect
175. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657412
#202656602
#202655022
#202654962
#202654912
#202652192
#202652152
#202651832
C2×#156921772
#408×C1012
#3096101