Finite Group Databasebeta

← #27181255

Subgroups — #27181255

Order 12928.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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#20265727C211yesmaximal, characteristic
2C2×C4×Dic202C211yesmaximal, characteristic, complemented
3#20265491C212yesmaximal, complemented
4#20265473C212yesmaximal, complemented
5#20265378C212yesmaximal, complemented
6#20265318C212yesmaximal, complemented
7#20265279C214yesmaximal, complemented
8#432×C101C211yesmaximal, characteristic, complemented
9C2×#13108074—21no—
10C22×Dic202C2211yescharacteristic
11C2×#13108063C2214yes—
12#15692255C2211yescharacteristic
13#15692252C2211yescharacteristic
14#15692243—21no—
15#15692229—21no—
16#15692229C2212yes—
17C4×Dic202—21no—
18C4×Dic202—21no—
19C4×Dic202C2212yes—
20C4×Dic202C2212yes—
21C4×D404—21no—
22C4×D404C2212yes—
23C2×C4⋊Dic101C2211yescharacteristic
24C2×#13108053—21no—
25C2×C4×Dic101—21no—
26#15692189C2214yes—
27#15692180C2214yes—
28#15692177C2214yes—
29C2×C42×C101C2211yescharacteristic
30C4⋊C8×C101C2212yes—
31C22⋊C8×C101C2212yes—
32#97×C101C2212yes—
33#13108074—44no—
34C2×Dic202—21no—
35C2×Dic202—24no—
36C2×Dic202—21no—
37C2×Dic202C2312yes—
38C2×D404—21no—
39C2×D404—21no—
40C2×D404C2312yes—
41C2×C22⋊D101—22no—
42C22×Dic101—21no—
43C22×Dic101—21no—
44C2×C4×D101—22no—
45#13108063—28no—
46C4⋊Dic101—21no—
47C4⋊Dic101—21no—
48C4⋊Dic101C2312yes—
49C4⋊Dic101C2312yes—
50#13108053—22no—
51#13108053—22no—
52#13108051—22no—
53#13108051—22no—
54C4×Dic101—22no—
55C4×Dic101—22no—
56C22×C4×C101D411yescharacteristic
57C22×C4×C101D411yescharacteristic
58C22×C4×C101C2311yescharacteristic
59C2×C8×C101C2314yes—
60C42×C101D411yescharacteristic
61C42×C101D411yescharacteristic
62C42×C101C2311yescharacteristic
63C42×C101C2311yescharacteristic
64C23×C101C2×D411yescharacteristic
65C22×D101—22no—
66C2×C4×C101—22no—
67C2×C4×C101—22no—
68C2×C4×C101—22no—
69C2×C4×C101C2×D411yescharacteristic
70C2×C4×C101C2×D411yescharacteristic
71C2×C4×C101C2×D411yescharacteristic
72C2×C4×C101C2×D411yescharacteristic
73C2×C4×C101C2×D411yescharacteristic
74C2×C4×C101C2411yescharacteristic
75C22⋊D101—44no—
76C2×Dic101—24no—
77C2×Dic101—22no—
78C2×Dic101—22no—
79C2×Dic101—22no—
80C2×Dic101—42no—
81D404—24no—
82D404—42no—
83D404—42no—
84C4×D101—44no—
85Dic202—24no—
86Dic202—22no—
87Dic202—42no—
88Dic202#4114yescomplemented
89C8×C101—24no—
90C22×C101—21no—
91C22×C101—21no—
92C22×C101AGL(1,ℤ/8)12yes—
93C22×C101C22×D411yescharacteristic
94C2×D101—42no—
95C2×D101—42no—
96C2×D101—42no—
97C4×C101—21no—
98C4×C101—21no—
99C4×C101—21no—
100C4×C101—21no—
101C4×C101#13712yes—
102C4×C101C2×#4112yes—
103Dic101—24no—
104Dic101—42no—
105Dic101—42no—
106C2×C101—22no—
107C2×C101#52511yescharacteristic
108C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
109C2×C101C2×#13711yescharacteristic
110D101—82no—
111#3112—1011nomaximal, Sylow
112C101#311211yescharacteristic, complemented, Sylow
113#532—1011no—
114C2×C4×Q8—1011no—
115#480—1012no—
116#477—1012no—
117#460—1012no—
118#448—1012no—
119#440—1014no—
120#432D10111yescharacteristic, complemented
121C2×#41—2021no—
122C22×Q8—1011no—
123C2×SD16—1014no—
124C4⋊Q8—1011no—
125#127—1011no—
126#124—2021no—
127#121—1012no—
128#121—2021no—
129C4×Q8—1012no—
130C4×Q8—1012no—
131C4×Q8—2021no—
132C4×Q8—2021no—
133C4×D4—1012no—
134C4×D4—2021no—
135C2×C4⋊C4—1011no—
136C2×C4⋊C4—2021no—
137C2×C42—2021no—
138C2×C42C2×D10111yescharacteristic
139#107—1014no—
140C4⋊C8C2×D10112yes—
141#103—1014no—
142#102—1014no—
143C22⋊C8C2×D10112yes—
144#97C2×D10112yes—
145#41—4044no—
146C2×Q8—1012no—
147C2×Q8—2024no—
148C2×Q8—2021no—
149C2×Q8—2021no—
150C2×D4—1012no—
151C2×D4—2022no—
152C2×D4—2021no—
153C2×D4—2021no—
154C22×C4—2021no—
155C22×C4—2022no—
156C22×C4—2021no—
157C22×C4D40411yescharacteristic
158C22×C4D40411yescharacteristic
159C22×C4C22×D10111yescharacteristic
160SD16—2028no—
161C2×C8C22×D10114yes—
162C4⋊C4—1012no—
163C4⋊C4—1012no—
164C4⋊C4—2022no—
165C4⋊C4—2022no—
166C4⋊C4—2021no—
167C4⋊C4—2021no—
168C22⋊C4—2022no—
169C22⋊C4—2022no—
170C42—2022no—
171C42—2022no—
172C42D40411yescharacteristic
173C42D40411yescharacteristic
174C42C22×D10111yescharacteristic
175C42C22×D10111yescharacteristic
176C23—2022no—
177C23C2×D40411yescharacteristic
178Q8—1014no—
179Q8—4042no—
180Q8—2024no—
181Q8—2022no—
182D4—4042no—
183D4—4044no—
184D4—4042no—
185D4—2024no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—4044no—
190C2×C4—4042no—
191C2×C4—2022no—
192C2×C4—2022no—
193C2×C4—2024no—
194C2×C4—2022no—
195C2×C4C2×D40411yescharacteristic
196C2×C4C2×D40411yescharacteristic
197C2×C4C2×D40411yescharacteristic
198C2×C4C2×D40411yescharacteristic
199C2×C4C2×D40411yescharacteristic
200C2×C4C23×D10111yescharacteristic
201C8—24no—
202C22—21no—
203C22—21no—
204C22—4042no—
205C22—4042no—
206C22—4042no—
207C22#1569228312yes—
208C22C22×D40411yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—4042no—
214C4—4042no—
215C4—2024no—
216C4#1569228812yes—
217C4#41×D10112yes—
218C2—22no—
219C2—8082no—
220C2#2026568911yescharacteristic
221C2C2×#1569228311yescharacteristic
222C2C2×#1569228811yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#432×C101C2
socleC22×C101C22×D4
prime radical#20265727C2
hypercenter#432D101
hypocenterC101#3112
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic202C2semidirect
3. #20265491C2semidirect
4. #20265473C2semidirect
5. #20265378C2semidirect
6. #20265318C2semidirect
7. #20265279C2semidirect
8. #432×C101C2semidirect
88. Dic202#41semidirect
112. C101#3112semidirect
120. #432D101semidirect

Maximal subgroups

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

subgroupindex
#202657272
C2×C4×Dic2022
#202654912
#202654732
#202653782
#202653182
#202652792
#432×C1012
#3112101