Finite Group Databasebeta

← #27181086

Subgroups — #27181086

Order 12928.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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#20265744C211yesmaximal, characteristic
2#20265658C211yesmaximal, characteristic, complemented
3#20265500C212yesmaximal, complemented
4#20265499C212yesmaximal, complemented
5#20265494C212yesmaximal, complemented
6#20265489C212yesmaximal, complemented
7#20265217C211yesmaximal, characteristic, complemented
8#20265215C211yesmaximal, characteristic, complemented
9#20265182C211yesmaximal, characteristic, complemented
10#20265143C211yesmaximal, characteristic, complemented
11C2×#102×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C2×#13108058C2211yescharacteristic
14#15692251—21no—
15#15692242—21no—
16#15692226C2212yes—
17#15692226C2212yes—
18#15692224—22no—
19#15692220—21no—
20#15692217—21no—
21C4×C22⋊D101—22no—
22#15692214C2211yescharacteristic
23#15692214C2211yescharacteristic
24#15692213C2212yescomplemented
25C4⋊C4×D101C2212yescomplemented
26#15692189C2211yescharacteristic, complemented
27#15692189C2211yescharacteristic, complemented
28#15692188C2212yescomplemented
29#15692187C2211yescharacteristic, complemented
30#15692187C2211yescharacteristic, complemented
31#15692186C2212yescomplemented
32#15692176C2212yes—
33#15692176C2212yes—
34#15692167C2212yescomplemented
35#15692167C2212yescomplemented
36C22×D4×C101C2211yescharacteristic
37C22×C8×C101C2211yescharacteristic
38C2×C4⋊C4×C101C2211yescharacteristic
39#102×C101C2212yes—
40#102×C101C2212yes—
41#13108074—22no—
42#13108074—22no—
43C2×Dic202D411yescharacteristic
44C2×D404D411yescharacteristic
45C2×C22⋊D101—22no—
46C2×C22⋊D101—22no—
47C2×C22⋊D101D412yes—
48C2×C4×D101—22no—
49C2×C4×D101C2312yes—
50#13108058—22no—
51C2×#11717579C2312yes—
52C4⋊Dic101C2311yescharacteristic
53C4⋊Dic101C2311yescharacteristic
54C4⋊Dic101C2311yescharacteristic
55C4⋊Dic101C2311yescharacteristic
56#13108053—22no—
57C22⋊Dic101—21no—
58C22⋊Dic101—21no—
59C22⋊Dic101—22no—
60C22⋊Dic101—22no—
61#13108051—22no—
62C4×Dic101—21no—
63C4×Dic101—21no—
64C4×Dic101—22no—
65C24×C101—21no—
66C2×D4×C101—21no—
67C2×D4×C101—22no—
68C2×D4×C101—22no—
69C2×D4×C101C2312yes—
70C22×C4×C101—21no—
71C22×C4×C101C2311yescharacteristic
72C2×C8×C101—22no—
73C2×C8×C101C2311yescharacteristic
74C2×C8×C101C2311yescharacteristic
75C4⋊C4×C101—21no—
76C4⋊C4×C101C2312yes—
77C23×C101—22no—
78C23×C101—42no—
79C23×C101—81no—
80C23×C101—81no—
81C23×C101C2×D411yescharacteristic
82C22×D101C2×D412yes—
83D4×C101—22no—
84D4×C101—22no—
85D4×C101—42no—
86C2×C4×C101—22no—
87C2×C4×C101—81no—
88C2×C4×C101C2×D411yescharacteristic
89C2×C4×C101#4112yes—
90C2×C4×C101#4112yes—
91C2×C4×C101C2411yescharacteristic
92C22⋊D101—24no—
93C22⋊D101—82no—
94C22⋊D101—82no—
95C2×Dic101—21no—
96C2×Dic101—21no—
97C2×Dic101—21no—
98C2×Dic101—21no—
99C2×Dic101C2×D412yes—
100D404—22no—
101C4×D101—22no—
102C4×D101—22no—
103C4×D101—82no—
104Dic202—22no—
105C8×C101—21no—
106C8×C101—21no—
107#11717579—22no—
108C22×C101—21no—
109C22×C101—21no—
110C22×C101—42no—
111C22×C101—42no—
112C22×C101—42no—
113C22×C101—84no—
114C22×C101#12312yes—
115C22×C101C22×D411yescharacteristic
116C2×D101—22no—
117C2×D101—22no—
118C2×D101—22no—
119C4×C101—42no—
120C4×C101#12312yes—
121C4×C101C2×#4111yescharacteristic
122C4×C101C2×#4111yescharacteristic
123Dic101—22no—
124Dic101—41no—
125Dic101—41no—
126Dic101—41no—
127Dic101—41no—
128C2×C101—22no—
129C2×C101—82no—
130C2×C101C2×#12311yescharacteristic
131C2×C101#57611yescharacteristic
132C2×C101#57711yescharacteristic
133D101—42no—
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×#102D10111yescharacteristic, complemented
146#408—1011no—
147C2×#41—1011no—
148C22×D4C2×D10111yescharacteristic
149C2×M16—1011no—
150C22×C8C2×D10111yescharacteristic
151#127—2021no—
152#124—2021no—
153#121—1012no—
154#121—1012no—
155#120—2022no—
156C4×Q8—2021no—
157C4×D4—2021no—
158C4×D4—2022no—
159#117—1011no—
160#117—1011no—
161#117—1012no—
162C2×C4⋊C4—1012no—
163C2×C4⋊C4C2×D10111yescharacteristic
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#102C2×D10112yes—
173#102C2×D10112yes—
174C22⋊C8—1012no—
175C22⋊C8—1012no—
176C24—21no—
177#41—2022no—
178#41—2022no—
179C2×Q8—1011no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—1012no—
184C2×D4—1011no—
185C2×D4—2022no—
186C2×D4—2022no—
187C2×D4C22×D10112yes—
188C22×C4—21no—
189C22×C4—1012no—
190C22×C4—2022no—
191C22×C4C22×D10111yescharacteristic
192M16—2022no—
193C2×C8—22no—
194C2×C8—1012no—
195C2×C8C22×D10111yescharacteristic
196C2×C8C22×D10111yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—1011no—
199C4⋊C4—1011no—
200C4⋊C4—1011no—
201C4⋊C4—1011no—
202C4⋊C4—2022no—
203C4⋊C4C22×D10112yes—
204C22⋊C4—2022no—
205C22⋊C4—2022no—
206C22⋊C4—2022no—
207C22⋊C4—2021no—
208C22⋊C4—2021no—
209C42—2022no—
210C42—2021no—
211C42—2021no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23—1012no—
217C23D4×D10111yescharacteristic
218Q8—2022no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—8082no—
223D4—8082no—
224D4—2024no—
225D4—2022no—
226C2×C4—22no—
227C2×C4—81no—
228C2×C4—8082no—
229C2×C4—1012no—
230C2×C4—2021no—
231C2×C4—2021no—
232C2×C4—2021no—
233C2×C4—2021no—
234C2×C4—2022no—
235C2×C4—2022no—
236C2×C4D4×D10111yescharacteristic
237C2×C4#1310807312yes—
238C2×C4#1310807412yes—
239C2×C4C23×D10111yescharacteristic
240C8—21no—
241C8—21no—
242C8—2022no—
243C22—21no—
244C22—21no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—84no—
249C22—2022no—
250C22—2022no—
251C22—2022no—
252C22#1569223712yes—
253C22C2×D4×D10111yescharacteristic
254C4—42no—
255C4—4041no—
256C4—4041no—
257C4—4041no—
258C4—4041no—
259C4—2022no—
260C4#1569223712yes—
261C4C2×#1310807311yescharacteristic
262C4C2×#1310807411yescharacteristic
263C2—22no—
264C2—82no—
265C2—4042no—
266C2C2×#1569223711yescharacteristic
267C2#2026597811yescharacteristic
268C2#2026598311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#102×C101C2
socleC22×C101C22×D4
prime radical#20265744C2
hypercenterC2×#102D101
hypocenterC101#3096
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. #20265658C2semidirect
3. #20265500C2semidirect
4. #20265499C2semidirect
5. #20265494C2semidirect
6. #20265489C2semidirect
7. #20265217C2semidirect
8. #20265215C2semidirect
9. #20265182C2semidirect
10. #20265143C2semidirect
11. C2×#102×C101C2semidirect
24. #15692213C22semidirect
25. C4⋊C4×D101C22semidirect
26. #15692189C22semidirect
27. #15692189C22semidirect
28. #15692188C22semidirect
29. #15692187C22semidirect
30. #15692187C22semidirect
31. #15692186C22semidirect
34. #15692167C22semidirect
35. #15692167C22semidirect
135. C101#3096semidirect
145. C2×#102D101semidirect

Maximal subgroups

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

subgroupindex
#202657442
#202656582
#202655002
#202654992
#202654942
#202654892
#202652172
#202652152
#202651822
#202651432
C2×#102×C1012
#3096101