Finite Group Databasebeta

← #27180641

Subgroups — #27180641

Order 12928.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#15692281C211yesmaximal, characteristic
2#20265751C211yesmaximal, characteristic, complemented
3#20265652C211yesmaximal, characteristic, complemented
4#20265394C214yesmaximal, complemented
5#20265380C214yesmaximal, complemented
6#20265203C211yesmaximal, characteristic, complemented
7#20265186C211yesmaximal, characteristic, complemented
8#20265174C211yesmaximal, characteristic, complemented
9C22×SD16×C101C211yesmaximal, characteristic, complemented
10C2×#13108074C2211yescharacteristic
11#15692281—24no—
12C2×#13108062C2212yes—
13C2×#13108059C2212yes—
14C2×#13108058C2211yescharacteristic
15#15692253C2212yescomplemented
16#15692244C2212yescomplemented
17#15692238—22no—
18#15692231—22no—
19D4×Dic101—22no—
20D4×Dic101C2214yes—
21#15692214C2211yescharacteristic
22#15692214C2211yescharacteristic
23C2×C22⋊Dic101—22no—
24#15692184C2214yescomplemented
25#15692180C2212yescomplemented
26#15692179C2214yescomplemented
27#15692177C2212yescomplemented
28C22×Q8×C101C2211yescharacteristic
29C22×D4×C101C2211yescharacteristic
30C2×SD16×C101—24no—
31C2×SD16×C101C2214yes—
32C22×C8×C101C2211yescharacteristic
33#13108074—22no—
34#13108074—22no—
35C2×Dic202C2311yescharacteristic
36C2×D404C2311yescharacteristic
37C2×C22⋊D101—21no—
38C22×Dic101—24no—
39C2×C4×D101—21no—
40#13108062—24no—
41#13108059—24no—
42#13108058—22no—
43C2×#11717579C2312yes—
44C4⋊Dic101D412yes—
45C4⋊Dic101C2312yes—
46#13108053—24no—
47C22⋊Dic101—21no—
48C22⋊Dic101—24no—
49C22⋊Dic101D412yes—
50#13108051—24no—
51C4×Dic101—21no—
52C4×Dic101C2312yes—
53C24×C101—21no—
54C2×Q8×C101—21no—
55C2×Q8×C101—22no—
56C2×Q8×C101—22no—
57C2×Q8×C101C2312yes—
58C2×D4×C101—21no—
59C2×D4×C101—22no—
60C2×D4×C101D414yes—
61C2×D4×C101C2312yes—
62C22×C4×C101—21no—
63C22×C4×C101C2311yescharacteristic
64SD16×C101—28no—
65C2×C8×C101—22no—
66C2×C8×C101C2312yes—
67C23×C101—22no—
68C23×C101—24no—
69C23×C101—82no—
70C23×C101C2×D411yescharacteristic
71C22×D101—21no—
72Q8×C101—22no—
73Q8×C101—22no—
74Q8×C101—42no—
75D4×C101—24no—
76D4×C101—22no—
77D4×C101C2×D414yes—
78C2×C4×C101—22no—
79C2×C4×C101—42no—
80C2×C4×C101C2×D411yescharacteristic
81C2×C4×C101C2×D412yes—
82C2×C4×C101#4112yes—
83C2×C4×C101C2411yescharacteristic
84C22⋊D101—42no—
85C2×Dic101—22no—
86C2×Dic101—21no—
87C2×Dic101—84no—
88C2×Dic101C2×D414yes—
89D404—21no—
90D404—21no—
91C4×D101—42no—
92Dic202—21no—
93Dic202—21no—
94C8×C101—22no—
95#11717579—22no—
96C22×C101—22no—
97C22×C101—24no—
98C22×C101—44no—
99C22×C101—84no—
100C22×C101#12112yes—
101C22×C101C22×D411yescharacteristic
102C2×D101—41no—
103C2×D101—42no—
104C4×C101—42no—
105C4×C101#12112yes—
106C4×C101C22×D411yescharacteristic
107C4×C101C2×#4111yescharacteristic
108Dic101—24no—
109Dic101—41no—
110Dic101—42no—
111C2×C101—22no—
112C2×C101—44no—
113C2×C101C2×#12111yescharacteristic
114C2×C101#57712yes—
115D101—81no—
116#3042—1011nomaximal, Sylow
117C101#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—1011no—
119C22×SD16D10111yescharacteristic, complemented
120#536—1011no—
121#518—1011no—
122#463—1014no—
123#461—1014no—
124#424—1011no—
125#419—1011no—
126#416—1011no—
127C2×#41—1011no—
128C22×Q8C2×D10111yescharacteristic
129C22×D4C2×D10111yescharacteristic
130AGL(1,ℤ/8)—2024no—
131C2×SD16—24no—
132C2×SD16—1012no—
133C2×SD16C2×D10114yes—
134C2×D8—1012no—
135C2×M16—1011no—
136C22×C8C2×D10111yescharacteristic
137C4⋊Q8—1012no—
138#124—1012no—
139#123—2022no—
140C22⋊Q8—2022no—
141C4×D4—1014no—
142C4×D4—2022no—
143#117—1011no—
144#117—1011no—
145C2×C22⋊C4—2022no—
146C4⋊C8—1014no—
147#103—1014no—
148#103—1012no—
149#102—1012no—
150C24—21no—
151#41—2022no—
152#41—2022no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1011no—
157C2×Q8C22×D10112yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—1011no—
161C2×D4—2021no—
162C2×D4C22⋊D10114yes—
163C2×D4C22×D10112yes—
164C22×C4—21no—
165C22×C4—2024no—
166C22×C4—2021no—
167C22×C4C22×D10111yescharacteristic
168SD16—28no—
169SD16—2024no—
170D8—2024no—
171M16—2022no—
172C2×C8—22no—
173C2×C8—1012no—
174C2×C8C22×D10112yes—
175C4⋊C4—1012no—
176C4⋊C4—1012no—
177C4⋊C4—2024no—
178C22⋊C4—1012no—
179C22⋊C4—2024no—
180C22⋊C4—2024no—
181C22⋊C4—2021no—
182C42—1012no—
183C42—2021no—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—2021no—
188C23D4×D10111yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—2021no—
193Q8—2021no—
194D4—22no—
195D4—24no—
196D4—4042no—
197D4—2021no—
198D4—2021no—
199D4C2×C22⋊D10114yes—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—8084no—
203C2×C4—1014no—
204C2×C4—4042no—
205C2×C4—2022no—
206C2×C4—2021no—
207C2×C4D4×D10111yescharacteristic
208C2×C4C2×C22⋊D10112yes—
209C2×C4#1310807312yes—
210C2×C4C23×D10111yescharacteristic
211C8—22no—
212C8—2022no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—4041no—
218C22—4042no—
219C22#1569222712yes—
220C22C2×D4×D10111yescharacteristic
221C4—42no—
222C4—4041no—
223C4—4042no—
224C4—2024no—
225C4#1569222712yes—
226C4C22×C22⋊D10111yescharacteristic
227C4C2×#1310807311yescharacteristic
228C2—22no—
229C2—44no—
230C2—8081no—
231C2C2×#1569222711yescharacteristic
232C2#2026598312yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC22×SD16×C101C2
socleC22×C101C22×D4
prime radicalC2×#15692281C2
hypercenterC22×SD16D101
hypocenterC101#3042
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. #20265751C2semidirect
3. #20265652C2semidirect
4. #20265394C2semidirect
5. #20265380C2semidirect
6. #20265203C2semidirect
7. #20265186C2semidirect
8. #20265174C2semidirect
9. C22×SD16×C101C2semidirect
15. #15692253C22semidirect
16. #15692244C22semidirect
24. #15692184C22semidirect
25. #15692180C22semidirect
26. #15692179C22semidirect
27. #15692177C22semidirect
117. C101#3042semidirect
119. C22×SD16D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922812
#202657512
#202656522
#202653942
#202653802
#202652032
#202651862
#202651742
C22×SD16×C1012
#3042101