Finite Group Databasebeta

← #44485490

Subgroups — #44485490

Order 30592.

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#29505325C211yesmaximal, characteristic
2M16×Dic239C211yesmaximal, characteristic
3C4×Dic239⋊C4C211yesmaximal, characteristic
4#29504533C211yesmaximal, characteristic
5#29504532C211yesmaximal, characteristic
6#29504531C211yesmaximal, characteristic
7#339×C239C211yesmaximal, characteristic
8C2×#16838532C2211yescharacteristic
9#21439959C411yescharacteristic, complemented
10#21439959C411yescharacteristic, complemented
11#21439951C411yescharacteristic, complemented
12#21439951C411yescharacteristic, complemented
13#21439937—21no—
14#21439937—21no—
15C4×Dic478—21no—
16C4×Dic478—21no—
17#21439919C2211yescharacteristic
18C2×Dic239⋊C4—21no—
19C2×C4×Dic239C2211yescharacteristic
20C2×C4×Dic239C2211yescharacteristic
21#21439870—21no—
22C8×Dic239—21no—
23C239⋊#95—21no—
24C2×M16×C239C2211yescharacteristic
25#117×C239C2211yescharacteristic
26C2×C42×C239C2211yescharacteristic
27C2×Dic478—21no—
28C2×Dic478—21no—
29C22×Dic239—21no—
30C22×Dic239D411yescharacteristic
31C22×Dic239D411yescharacteristic
32#16838532—21no—
33#16838532Q812yescomplemented
34C2×C239⋊C8—21no—
35C4⋊Dic239—21no—
36C4⋊Dic239—21no—
37C4⋊Dic239C2×C411yescharacteristic
38C4⋊Dic239C2×C411yescharacteristic
39Dic239⋊C4—24no—
40Dic239⋊C4—21no—
41Dic239⋊C4—21no—
42Dic239⋊C4—21no—
43Dic239⋊C4—21no—
44C22⋊Dic239—21no—
45C4×Dic239—21no—
46C4×Dic239—21no—
47C4×Dic239—21no—
48C4×Dic239C2×C411yescharacteristic
49C4×Dic239C2×C411yescharacteristic
50C4×Dic239D411yescharacteristic
51C4×Dic239D411yescharacteristic
52C22×C4×C239—21no—
53C22×C4×C239C2311yescharacteristic
54M16×C239—21no—
55M16×C239Q812yescomplemented
56C2×C8×C239—21no—
57C4⋊C4×C239C2×C411yescharacteristic
58C4⋊C4×C239C2×C411yescharacteristic
59C22⋊C4×C239—21no—
60C42×C239—21no—
61C42×C239—21no—
62C42×C239—21no—
63Dic478—42no—
64Dic478—42no—
65C2×Dic239—22no—
66C2×Dic239—22no—
67C2×Dic239—21no—
68C2×Dic239—21no—
69C2×Dic239—41no—
70C2×Dic239—41no—
71C2×Dic239—41no—
72C2×Dic239C22⋊C411yescharacteristic
73C2×Dic239C22⋊C411yescharacteristic
74C2×Dic239C22⋊C411yescharacteristic
75C2×Dic239C22⋊C411yescharacteristic
76C239⋊C8—22no—
77C23×C239C2×D411yescharacteristic
78C2×C4×C239—22no—
79C2×C4×C239—21no—
80C2×C4×C239—21no—
81C2×C4×C239—41no—
82C2×C4×C239—41no—
83C2×C4×C239C22×C411yescharacteristic
84C2×C4×C239C2×D411yescharacteristic
85C2×C4×C239C2×Q811yescharacteristic
86C2×C4×C239C2×Q811yescharacteristic
87C2×C4×C239#4111yescharacteristic
88C2×C4×C239#4111yescharacteristic
89C8×C239—22no—
90C22×C239—21no—
91C22×C239—21no—
92C22×C239C2×C22⋊C411yescharacteristic
93C22×C239#12411yescharacteristic
94C22×C239C4⋊Q811yescharacteristic
95C4×C239—42no—
96C4×C239—41no—
97C4×C239—41no—
98C4×C239C4×Q811yescharacteristic
99C4×C239C4×Q811yescharacteristic
100C4×C239C22⋊Q811yescharacteristic
101C4×C239C22⋊Q811yescharacteristic
102Dic239—21no—
103Dic239—21no—
104Dic239—21no—
105Dic239—42no—
106Dic239—41no—
107Dic239—41no—
108Dic239#10412yes—
109C2×C239—21no—
110C2×C239—21no—
111C2×C239#39111yescharacteristic
112C2×C239C2×#10411yescharacteristic
113C2×C239#42111yescharacteristic
114C239#198611yescharacteristic, complemented, Sylow
115#1986—2391nomaximal, Sylow
116#533—2391no—
117C4×M16—2391no—
118C4×C4⋊C4—2391no—
119#339—2391no—
120#339—2391no—
121#339—2391no—
122#339D23911yescharacteristic
123C2×M16—2391no—
124C2×M16C2×D23911yescharacteristic
125C4⋊Q8—2391no—
126C4⋊Q8—2391no—
127#125—2391no—
128#125—2391no—
129C22⋊Q8—4781no—
130C22⋊Q8—4781no—
131C4×Q8—4781no—
132C4×Q8—4781no—
133#117—2391no—
134#117C2×D23911yescharacteristic
135C2×C4⋊C4—4781no—
136C2×C42—2391no—
137C2×C42—2391no—
138C2×C42C2×D23911yescharacteristic
139#97—4781no—
140C4×C8—4781no—
141#95—4781no—
142C2×Q8—4781no—
143C2×Q8—4781no—
144C22×C4—21no—
145C22×C4—4781no—
146C22×C4—2391no—
147C22×C4—2391no—
148C22×C4C22×D23911yescharacteristic
149M16—21no—
150M16—4781no—
151M16—2392no—
152M16Dic47812yescomplemented
153C2×C8—21no—
154C2×C8—4781no—
155C4⋊C4—4784no—
156C4⋊C4—4781no—
157C4⋊C4—4781no—
158C4⋊C4—4781no—
159C4⋊C4—4781no—
160C4⋊C4—4781no—
161C4⋊C4—4781no—
162C4⋊C4—2391no—
163C4⋊C4—2391no—
164C4⋊C4C4×D23911yescharacteristic
165C4⋊C4C4×D23911yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—4781no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—4781no—
172C42—4781no—
173C42—4781no—
174C42—2391no—
175C42—2391no—
176C42—2391no—
177C42—2391no—
178C23D4×D23911yescharacteristic
179Q8—9562no—
180Q8—9562no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—9561no—
187C2×C4—9561no—
188C2×C4—9561no—
189C2×C4—4782no—
190C2×C4—4781no—
191C2×C4—4781no—
192C2×C4—4782no—
193C2×C4—2391no—
194C2×C4—2391no—
195C2×C4—2391no—
196C2×C4—2391no—
197C2×C4C2×C4×D23911yescharacteristic
198C2×C4D4×D23911yescharacteristic
199C2×C4Q8×D23911yescharacteristic
200C2×C4C2×Dic47811yescharacteristic
201C2×C4#1683854711yescharacteristic
202C2×C4#1683854811yescharacteristic
203C8—22no—
204C8—4782no—
205C22—21no—
206C22—21no—
207C22C22⋊C4×D23911yescharacteristic
208C22#2143994611yescharacteristic
209C22#2143995911yescharacteristic
210C4—42no—
211C4—41no—
212C4—41no—
213C4—9562no—
214C4—9561no—
215C4—9561no—
216C4—4781no—
217C4—4781no—
218C4—4781no—
219C4—2392no—
220C4#2143992511yescharacteristic
221C4C4×Dic47811yescharacteristic
222C4#2143993711yescharacteristic
223C4#2143993811yescharacteristic
224C2—21no—
225C2—21no—
226C2#2950467611yescharacteristic
227C2#104×D23911yescharacteristic
228C2#2950478711yescharacteristic
subgroupG/H
centerC2×C4D4×D239
commutatorC2×C4×C239C22×C4
FrattiniC22×C4C22×D239
Fitting#339×C239C2
socleC22×C239C2×C22⋊C4
prime radicalC23×C239C2×D4
cosocleC22×C4×C239C23
hypercenter#339D239
hypocenterC239#1986
LCSG ▷ C2×C4×C239 ▷ C2×C239 ▷ C239
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
9. #21439959C4semidirect
10. #21439959C4semidirect
11. #21439951C4semidirect
12. #21439951C4semidirect
33. #16838532Q8semidirect
55. M16×C239Q8semidirect
114. C239#1986semidirect
152. M16Dic478semidirect

Maximal subgroups

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

subgroupindex
#295053252
M16×Dic2392
C4×Dic239⋊C42
#295045332
#295045322
#295045312
#339×C2392
#1986239