Finite Group Databasebeta

← #27172074

Subgroups — #27172074

Order 12928.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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#20265730C211yesmaximal, characteristic
2M16×Dic101C211yesmaximal, characteristic
3C4×#13108053C211yesmaximal, characteristic
4#20264938C211yesmaximal, characteristic
5#20264937C211yesmaximal, characteristic
6#20264936C211yesmaximal, characteristic
7#339×C101C211yesmaximal, characteristic
8C2×#13108058C2211yescharacteristic
9#15692254C411yescharacteristic, complemented
10#15692254C411yescharacteristic, complemented
11#15692246C411yescharacteristic, complemented
12#15692246C411yescharacteristic, complemented
13#15692232—21no—
14#15692232—21no—
15C4×Dic202—21no—
16C4×Dic202—21no—
17#15692214C2211yescharacteristic
18C2×#13108053—21no—
19C2×C4×Dic101C2211yescharacteristic
20C2×C4×Dic101C2211yescharacteristic
21#15692165—21no—
22C8×Dic101—21no—
23#15692162—21no—
24C2×M16×C101C2211yescharacteristic
25#117×C101C2211yescharacteristic
26C2×C42×C101C2211yescharacteristic
27C2×Dic202—21no—
28C2×Dic202—21no—
29C22×Dic101—21no—
30C22×Dic101D411yescharacteristic
31C22×Dic101D411yescharacteristic
32#13108058—21no—
33#13108058Q812yescomplemented
34C2×#11717579—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101—21no—
37C4⋊Dic101C2×C411yescharacteristic
38C4⋊Dic101C2×C411yescharacteristic
39#13108053—24no—
40#13108053—21no—
41#13108053—21no—
42#13108053—21no—
43#13108053—21no—
44C22⋊Dic101—21no—
45C4×Dic101—21no—
46C4×Dic101—21no—
47C4×Dic101—21no—
48C4×Dic101C2×C411yescharacteristic
49C4×Dic101C2×C411yescharacteristic
50C4×Dic101D411yescharacteristic
51C4×Dic101D411yescharacteristic
52C22×C4×C101—21no—
53C22×C4×C101C2311yescharacteristic
54M16×C101—21no—
55M16×C101Q812yescomplemented
56C2×C8×C101—21no—
57C4⋊C4×C101C2×C411yescharacteristic
58C4⋊C4×C101C2×C411yescharacteristic
59C22⋊C4×C101—21no—
60C42×C101—21no—
61C42×C101—21no—
62C42×C101—21no—
63C23×C101C2×D411yescharacteristic
64C2×C4×C101—22no—
65C2×C4×C101—21no—
66C2×C4×C101—21no—
67C2×C4×C101—41no—
68C2×C4×C101—41no—
69C2×C4×C101C22×C411yescharacteristic
70C2×C4×C101C2×D411yescharacteristic
71C2×C4×C101C2×Q811yescharacteristic
72C2×C4×C101C2×Q811yescharacteristic
73C2×C4×C101#4111yescharacteristic
74C2×C4×C101#4111yescharacteristic
75C2×Dic101—22no—
76C2×Dic101—22no—
77C2×Dic101—21no—
78C2×Dic101—21no—
79C2×Dic101—41no—
80C2×Dic101—41no—
81C2×Dic101—41no—
82C2×Dic101C22⋊C411yescharacteristic
83C2×Dic101C22⋊C411yescharacteristic
84C2×Dic101C22⋊C411yescharacteristic
85C2×Dic101C22⋊C411yescharacteristic
86Dic202—42no—
87Dic202—42no—
88C8×C101—22no—
89#11717579—22no—
90C22×C101—21no—
91C22×C101—21no—
92C22×C101C2×C22⋊C411yescharacteristic
93C22×C101#12411yescharacteristic
94C22×C101C4⋊Q811yescharacteristic
95C4×C101—42no—
96C4×C101—41no—
97C4×C101—41no—
98C4×C101C4×Q811yescharacteristic
99C4×C101C4×Q811yescharacteristic
100C4×C101C22⋊Q811yescharacteristic
101C4×C101C22⋊Q811yescharacteristic
102Dic101—21no—
103Dic101—21no—
104Dic101—21no—
105Dic101—42no—
106Dic101—41no—
107Dic101—41no—
108Dic101#10412yes—
109C2×C101—21no—
110C2×C101—21no—
111C2×C101#39111yescharacteristic
112C2×C101C2×#10411yescharacteristic
113C2×C101#42111yescharacteristic
114#1986—1011nomaximal, Sylow
115C101#198611yescharacteristic, complemented, Sylow
116#533—1011no—
117C4×M16—1011no—
118C4×C4⋊C4—1011no—
119#339—1011no—
120#339—1011no—
121#339—1011no—
122#339D10111yescharacteristic
123C2×M16—1011no—
124C2×M16C2×D10111yescharacteristic
125C4⋊Q8—1011no—
126C4⋊Q8—1011no—
127#125—1011no—
128#125—1011no—
129C22⋊Q8—2021no—
130C22⋊Q8—2021no—
131C4×Q8—2021no—
132C4×Q8—2021no—
133#117—1011no—
134#117C2×D10111yescharacteristic
135C2×C4⋊C4—2021no—
136C2×C42—1011no—
137C2×C42—1011no—
138C2×C42C2×D10111yescharacteristic
139#97—2021no—
140C4×C8—2021no—
141#95—2021no—
142C2×Q8—2021no—
143C2×Q8—2021no—
144C22×C4—21no—
145C22×C4—1011no—
146C22×C4—1011no—
147C22×C4—2021no—
148C22×C4C22×D10111yescharacteristic
149M16—21no—
150M16—1012no—
151M16—2021no—
152M16Dic20212yescomplemented
153C2×C8—21no—
154C2×C8—2021no—
155C4⋊C4—1011no—
156C4⋊C4—1011no—
157C4⋊C4—2021no—
158C4⋊C4—2021no—
159C4⋊C4—2021no—
160C4⋊C4—2021no—
161C4⋊C4—2021no—
162C4⋊C4—2021no—
163C4⋊C4—2024no—
164C4⋊C4C4×D10111yescharacteristic
165C4⋊C4C4×D10111yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—2021no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—1011no—
172C42—1011no—
173C42—1011no—
174C42—1011no—
175C42—2021no—
176C42—2021no—
177C42—2021no—
178C23D4×D10111yescharacteristic
179Q8—4042no—
180Q8—4042no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—1011no—
187C2×C4—1011no—
188C2×C4—1011no—
189C2×C4—1011no—
190C2×C4—4041no—
191C2×C4—4041no—
192C2×C4—4041no—
193C2×C4—2022no—
194C2×C4—2022no—
195C2×C4—2021no—
196C2×C4—2021no—
197C2×C4C2×C4×D10111yescharacteristic
198C2×C4D4×D10111yescharacteristic
199C2×C4Q8×D10111yescharacteristic
200C2×C4C2×Dic20211yescharacteristic
201C2×C4#1310807311yescharacteristic
202C2×C4#1310807411yescharacteristic
203C8—22no—
204C8—2022no—
205C22—21no—
206C22—21no—
207C22C22⋊C4×D10111yescharacteristic
208C22#1569224111yescharacteristic
209C22#1569225411yescharacteristic
210C4—42no—
211C4—41no—
212C4—41no—
213C4—1012no—
214C4—4042no—
215C4—4041no—
216C4—4041no—
217C4—2021no—
218C4—2021no—
219C4—2021no—
220C4#1569222011yescharacteristic
221C4C4×Dic20211yescharacteristic
222C4#1569223211yescharacteristic
223C4#1569223311yescharacteristic
224C2—21no—
225C2—21no—
226C2#2026508111yescharacteristic
227C2#104×D10111yescharacteristic
228C2#2026519211yescharacteristic
subgroupG/H
centerC2×C4D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#339×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC23×C101C2×D4
cosocleC22×C4×C101C23
hypercenter#339D101
hypocenterC101#1986
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
9. #15692254C4semidirect
10. #15692254C4semidirect
11. #15692246C4semidirect
12. #15692246C4semidirect
33. #13108058Q8semidirect
55. M16×C101Q8semidirect
115. C101#1986semidirect
152. M16Dic202semidirect

Maximal subgroups

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

subgroupindex
#202657302
M16×Dic1012
C4×#131080532
#202649382
#202649372
#202649362
#339×C1012
#1986101