Finite Group Databasebeta

← #27180913

Subgroups — #27180913

Order 12928.

388 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 131 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#20265732C211yesmaximal, characteristic, complemented
2C4×#13108073C211yesmaximal, characteristic
3#20265483C212yesmaximal, complemented
4#20265475C212yesmaximal, complemented
5#20265473C211yesmaximal, characteristic, complemented
6#20265467C211yesmaximal, characteristic, complemented
7#20265459C211yesmaximal, characteristic, complemented
8#20265214C212yesmaximal, complemented
9C4⋊C8×D101C211yesmaximal, characteristic, complemented
10#20265170C212yesmaximal
11#474×C101C211yesmaximal, characteristic, complemented
12C2×#13108073C2211yescharacteristic
13C2×C8×D101C2212yes—
14#15692255C2211yescharacteristic
15#15692246C2212yes—
16#15692235—21no—
17#15692231—21no—
18Q8×Dic101—21no—
19C4×Dic202C2211yescharacteristic
20#15692220—21no—
21#15692220C2211yescharacteristic
22D4×Dic101C2211yescharacteristic
23C4×C22⋊D101—21no—
24#15692215—21no—
25#15692213C2212yes—
26#15692213C2211yescharacteristic
27#15692211—21no—
28C2×C4×Dic101—21no—
29C42×D101C2211yescharacteristic
30#15692189C2212yes—
31#15692186C2212yes—
32#15692186C2212yes—
33C4⋊#11717579C2211yescharacteristic
34#15692184C2212yes—
35#15692180C2212yes—
36#15692178C2212yes—
37#15692176C2212yes—
38C4⋊Q8×C101C2211yescharacteristic
39C4×D4×C101C2211yescharacteristic
40#107×C101C2212yes—
41C4⋊C8×C101C2211yescharacteristic
42#102×C101C2212yes—
43#13108073—22no—
44#13108073Q814yescomplemented
45C2×Dic202—21no—
46C2×Dic202C2311yescharacteristic
47C2×C22⋊D101—21no—
48C22×Dic101—21no—
49C22×Dic101—21no—
50C2×C4×D101D411yescharacteristic
51C2×C4×D101D411yescharacteristic
52C2×C4×D101C2311yescharacteristic
53C2×#11717579C2312yes—
54C8×D101—24no—
55C4⋊Dic101—21no—
56C4⋊Dic101—22no—
57C4⋊Dic101C2312yes—
58C4⋊Dic101C2311yescharacteristic
59#13108053—21no—
60#13108053—21no—
61#13108053—22no—
62C22⋊Dic101—21no—
63#13108051—21no—
64#13108051—22no—
65C4×Dic101—21no—
66C4×Dic101—21no—
67C4×Dic101—21no—
68C4×Dic101—22no—
69C4×Dic101D411yescharacteristic
70C4×Dic101D411yescharacteristic
71C4×Dic101C2311yescharacteristic
72C2×Q8×C101—21no—
73C2×D4×C101C2311yescharacteristic
74C22×C4×C101—21no—
75C2×C8×C101C2312yes—
76C4⋊C4×C101—21no—
77C4⋊C4×C101C2312yes—
78C4⋊C4×C101C2311yescharacteristic
79C22⋊C4×C101—21no—
80C42×C101C2311yescharacteristic
81C23×C101—21no—
82C22×D101C2×D411yescharacteristic
83Q8×C101—42no—
84D4×C101—21no—
85D4×C101C2×Q812yes—
86C2×C4×C101—21no—
87C2×C4×C101—22no—
88C2×C4×C101—42no—
89C2×C4×C101C2×D411yescharacteristic
90C2×C4×C101C2×D411yescharacteristic
91C2×C4×C101C2411yescharacteristic
92C22⋊D101—24no—
93C2×Dic101—21no—
94C2×Dic101—24no—
95C2×Dic101—21no—
96C2×Dic101—22no—
97C2×Dic101—42no—
98C2×Dic101C2×D411yescharacteristic
99C2×Dic101C2×D411yescharacteristic
100C2×Dic101C2×D411yescharacteristic
101C4×D101—22no—
102C4×D101—41no—
103C4×D101C2×Q812yes—
104C4×D101#4112yes—
105Dic202—21no—
106Dic202—42no—
107Dic202C2×Q812yes—
108C8×C101—22no—
109#11717579—22no—
110C22×C101—22no—
111C22×C101—41no—
112C22×C101—41no—
113C22×C101C22×D411yescharacteristic
114C2×D101—21no—
115C2×D101—21no—
116C2×D101C22⋊Q812yes—
117C4×C101—21no—
118C4×C101—41no—
119C4×C101—42no—
120C4×C101#13512yes—
121C4×C101C22×Q811yescharacteristic
122C4×C101C2×#4111yescharacteristic
123Dic101—22no—
124Dic101—21no—
125Dic101—21no—
126Dic101—42no—
127Dic101—41no—
128Dic101C22⋊Q812yes—
129C2×C101—42no—
130C2×C101C2×C22⋊Q811yescharacteristic
131C2×C101C2×#13511yescharacteristic
132C2×C101#57511yescharacteristic
133D101—22no—
134#3076—1011nomaximal, Sylow
135C101#307611yescharacteristic, complemented, Sylow
136#533—1011no—
137C4×#41—1011no—
138#479—1012no—
139#478—1012no—
140#477—1011no—
141#476—1011no—
142#475—1011no—
143#474D10111yescharacteristic, complemented
144#427—1012no—
145C2×C4⋊C8—1011no—
146#416—1012no—
147C2×#41—1011no—
148C22×C8—1012no—
149C4⋊Q8—1011no—
150C4⋊Q8C2×D10111yescharacteristic
151#125—1012no—
152C22⋊Q8—2021no—
153C22⋊Q8—2021no—
154C4×Q8—1011no—
155C4×Q8—1011no—
156C4×Q8—2021no—
157C4×Q8—2021no—
158C4×D4—1011no—
159C4×D4—2021no—
160C4×D4—2021no—
161C4×D4C2×D10111yescharacteristic
162#117—1011no—
163#117—1012no—
164#117—2021no—
165C2×C42—1011no—
166C2×C42—2021no—
167#107—1012no—
168#107C2×D10112yes—
169#106—1012no—
170#106—1012no—
171C4⋊C8—1012no—
172C4⋊C8—1011no—
173C4⋊C8C2×D10111yescharacteristic
174#103—1012no—
175#103—1012no—
176#102—1012no—
177#102C2×D10112yes—
178#41—1014no—
179#41—2022no—
180C2×Q8—21no—
181C2×Q8—1011no—
182C2×Q8—2021no—
183C2×D4—2021no—
184C2×D4C22×D10111yescharacteristic
185C22×C4—21no—
186C22×C4—1011no—
187C22×C4—1011no—
188C22×C4—1011no—
189C22×C4—2021no—
190C22×C4—2021no—
191C2×C8—1012no—
192C2×C8—2024no—
193C2×C8C22×D10112yes—
194C4⋊C4—21no—
195C4⋊C4—1012no—
196C4⋊C4—1011no—
197C4⋊C4—2021no—
198C4⋊C4—2022no—
199C4⋊C4—2022no—
200C4⋊C4—2021no—
201C4⋊C4—2021no—
202C4⋊C4C22×D10112yes—
203C4⋊C4C22×D10111yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—2021no—
206C22⋊C4—2022no—
207C22⋊C4—2021no—
208C42—1011no—
209C42—1011no—
210C42—1011no—
211C42—2021no—
212C42—2021no—
213C42—2022no—
214C42—2021no—
215C42C22×D10111yescharacteristic
216C23—21no—
217C23—1011no—
218Q8—42no—
219Q8—1012no—
220Q8—4042no—
221Q8—2021no—
222D4—21no—
223D4—2024no—
224D4Q8×D10112yes—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—1011no—
229C2×C4—1012no—
230C2×C4—1012no—
231C2×C4—1011no—
232C2×C4—1011no—
233C2×C4—4041no—
234C2×C4—4042no—
235C2×C4—2024no—
236C2×C4—2021no—
237C2×C4—2022no—
238C2×C4—2021no—
239C2×C4—2022no—
240C2×C4D4×D10111yescharacteristic
241C2×C4D4×D10111yescharacteristic
242C2×C4C23×D10111yescharacteristic
243C8—22no—
244C8—2022no—
245C22—22no—
246C22—41no—
247C22—41no—
248C22—1012no—
249C22—2021no—
250C22—2021no—
251C22C2×D4×D10111yescharacteristic
252C4—21no—
253C4—41no—
254C4—42no—
255C4—1012no—
256C4—4041no—
257C4—4042no—
258C4—2022no—
259C4—2021no—
260C4—2021no—
261C4#1569227412yes—
262C4C2×Q8×D10111yescharacteristic
263C4#41×D10111yescharacteristic
264C2—42no—
265C2—2022no—
266C2C22⋊Q8×D10111yescharacteristic
267C2C2×#1569227411yescharacteristic
268C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#474×C101C2
socleC22×C101C22×D4
prime radicalC2×#13108073C22
hypercenter#474D101
hypocenterC101#3076
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #20265732C2semidirect
3. #20265483C2semidirect
4. #20265475C2semidirect
5. #20265473C2semidirect
6. #20265467C2semidirect
7. #20265459C2semidirect
8. #20265214C2semidirect
9. C4⋊C8×D101C2semidirect
11. #474×C101C2semidirect
44. #13108073Q8semidirect
135. C101#3076semidirect
143. #474D101semidirect

Maximal subgroups

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

subgroupindex
#202657322
C4×#131080732
#202654832
#202654752
#202654732
#202654672
#202654592
#202652142
C4⋊C8×D1012
#202651702
#474×C1012
#3076101