Finite Group Databasebeta

← #27181451

Subgroups — #27181451

Order 12928.

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

normal subgroups only

This group has 400 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#20265732C211yesmaximal, characteristic, complemented
2#20265725C211yesmaximal, characteristic
3#20265530C214yesmaximal, complemented
4#20265529C211yesmaximal, characteristic, complemented
5#20265528C211yesmaximal, characteristic, complemented
6#20265523C211yesmaximal, characteristic, complemented
7#20265170C214yesmaximal, complemented
8C4×C8×D101C211yesmaximal, characteristic, complemented
9#486×C101C211yesmaximal, characteristic, complemented
10C2×#13108073C2212yes—
11C2×C8×D101C2212yes—
12#15692255C2211yescharacteristic
13#15692255C2211yescharacteristic
14#15692246C2212yes—
15#15692242C2212yes—
16#15692235—21no—
17#15692231—21no—
18#15692226—21no—
19#15692226—21no—
20Q8×Dic101—21no—
21#15692220—21no—
22D4×Dic101—21no—
23D4×Dic101—21no—
24#15692213C2212yes—
25C42×D101C2211yescharacteristic
26#15692180C2214yes—
27#15692178C2214yes—
28#15692176C2214yes—
29C4×#11717579C2211yescharacteristic
30C8×Dic101C2212yes—
31C4⋊Q8×C101C2211yescharacteristic
32#127×C101C2211yescharacteristic
33#102×C101C2214yes—
34C4×C8×C101C2211yescharacteristic
35#13108073—28no—
36C2×Dic202—21no—
37C2×Dic202C2312yes—
38C2×C22⋊D101—22no—
39C22×Dic101—22no—
40C2×C4×D101D411yescharacteristic
41C2×C4×D101D411yescharacteristic
42C2×C4×D101C2311yescharacteristic
43C2×#11717579C2312yes—
44C8×D101—24no—
45C4⋊Dic101—21no—
46C4⋊Dic101—21no—
47C4⋊Dic101—21no—
48C4⋊Dic101—22no—
49C4⋊Dic101C2312yes—
50#13108053—22no—
51C22⋊Dic101—22no—
52C22⋊Dic101—22no—
53#13108051—22no—
54C4×Dic101—22no—
55C4×Dic101D411yescharacteristic
56C4×Dic101D411yescharacteristic
57C4×Dic101C2311yescharacteristic
58C2×Q8×C101—21no—
59C2×D4×C101—21no—
60C2×D4×C101—21no—
61C2×D4×C101C2312yes—
62C2×C8×C101C2312yes—
63C4⋊C4×C101—21no—
64C4⋊C4×C101C2312yes—
65C42×C101C2311yescharacteristic
66C23×C101—22no—
67C22×D101C2×D411yescharacteristic
68Q8×C101—42no—
69D4×C101—24no—
70D4×C101—42no—
71D4×C101—42no—
72C2×C4×C101—22no—
73C2×C4×C101C2×D411yescharacteristic
74C2×C4×C101C2×D411yescharacteristic
75C2×C4×C101C2411yescharacteristic
76C22⋊D101—44no—
77C2×Dic101—22no—
78C2×Dic101—22no—
79C2×Dic101—44no—
80C2×Dic101C2×D411yescharacteristic
81C2×Dic101C2×D411yescharacteristic
82C2×Dic101C2×D411yescharacteristic
83C4×D101—22no—
84C4×D101—22no—
85C4×D101#4114yes—
86Dic202—24no—
87Dic202—42no—
88C8×C101—22no—
89#11717579—22no—
90C22×C101—42no—
91C22×C101—42no—
92C22×C101—42no—
93C22×C101C22×D411yescharacteristic
94C2×D101—21no—
95C2×D101—21no—
96C2×D101#12412yes—
97C4×C101—42no—
98C4×C101#13512yes—
99C4×C101#13512yes—
100C4×C101C2×#4112yes—
101Dic101—21no—
102Dic101—21no—
103Dic101—42no—
104Dic101—42no—
105Dic101#12412yes—
106C2×C101—82no—
107C2×C101C2×#12411yescharacteristic
108C2×C101C2×#13511yescharacteristic
109C2×C101C2×#13511yescharacteristic
110D101—22no—
111#3131—1011nomaximal, Sylow
112C101#313111yescharacteristic, complemented, Sylow
113#533—1011no—
114#532—1011no—
115#488—1014no—
116#487—1011no—
117#487—1011no—
118#486—1011no—
119#486D10111yescharacteristic, complemented
120#416—1014no—
121C2×C4×C8—1011no—
122C2×#41—1012no—
123C22×C8—1012no—
124C4⋊Q8—1011no—
125C4⋊Q8—1011no—
126C4⋊Q8C2×D10111yescharacteristic
127#127C2×D10111yescharacteristic
128#125—1012no—
129#124—1012no—
130C22⋊Q8—2021no—
131C22⋊Q8—2021no—
132#121—2021no—
133#121—2021no—
134C4×Q8—2021no—
135C4×Q8—2021no—
136C4×D4—2021no—
137C4×D4—2021no—
138#117—1012no—
139C2×C42—1011no—
140#103—1014no—
141#103—1014no—
142#102—1014no—
143#102C2×D10114yes—
144C4×C8—1012no—
145C4×C8—1011no—
146C4×C8C2×D10111yescharacteristic
147#41—2028no—
148C2×Q8—21no—
149C2×Q8—1012no—
150C2×Q8—2021no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—2022no—
154C2×D4C22×D10112yes—
155C22×C4—1011no—
156C22×C4—1011no—
157C22×C4—1011no—
158C22×C4—2022no—
159C2×C8—1012no—
160C2×C8—2024no—
161C2×C8C22×D10112yes—
162C4⋊C4—21no—
163C4⋊C4—1012no—
164C4⋊C4—2022no—
165C4⋊C4—2022no—
166C4⋊C4—2021no—
167C4⋊C4—2021no—
168C4⋊C4—2021no—
169C4⋊C4C22×D10112yes—
170C22⋊C4—2022no—
171C22⋊C4—2022no—
172C22⋊C4—2022no—
173C42—1011no—
174C42—1011no—
175C42—1011no—
176C42—2022no—
177C42C22×D10111yescharacteristic
178C23—22no—
179C23—1011no—
180Q8—42no—
181Q8—4042no—
182Q8—2024no—
183D4—24no—
184D4—42no—
185D4—42no—
186D4—4044no—
187C2×C4—22no—
188C2×C4—1011no—
189C2×C4—1014no—
190C2×C4—1011no—
191C2×C4—1011no—
192C2×C4—4044no—
193C2×C4—2022no—
194C2×C4—2022no—
195C2×C4—2022no—
196C2×C4—2022no—
197C2×C4D4×D10111yescharacteristic
198C2×C4D4×D10111yescharacteristic
199C2×C4C23×D10111yescharacteristic
200C8—22no—
201C8—2022no—
202C22—42no—
203C22—42no—
204C22—42no—
205C22—1012no—
206C22—2021no—
207C22—2021no—
208C22C2×D4×D10111yescharacteristic
209C4—42no—
210C4—1012no—
211C4—4042no—
212C4—4042no—
213C4—2021no—
214C4—2021no—
215C4#1569227412yes—
216C4#1569227612yes—
217C4#41×D10112yes—
218C2—82no—
219C2—2022no—
220C2#124×D10111yescharacteristic
221C2C2×#1569227411yescharacteristic
222C2C2×#1569227611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#486×C101C2
socleC22×C101C22×D4
prime radical#20265725C2
hypercenter#486D101
hypocenterC101#3131
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #20265732C2semidirect
3. #20265530C2semidirect
4. #20265529C2semidirect
5. #20265528C2semidirect
6. #20265523C2semidirect
7. #20265170C2semidirect
8. C4×C8×D101C2semidirect
9. #486×C101C2semidirect
112. C101#3131semidirect
119. #486D101semidirect

Maximal subgroups

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

subgroupindex
#202657322
#202657252
#202655302
#202655292
#202655282
#202655232
#202651702
C4×C8×D1012
#486×C1012
#3131101