Finite Group Databasebeta

← #5620

Subgroups — #5620

Order 192.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C3×#121C211yesmaximal, characteristic, complemented
2#940C211yesmaximal, characteristic, complemented
3#939C211yesmaximal, characteristic, complemented
4#939C211yesmaximal, characteristic, complemented
5#938C211yesmaximal, characteristic, complemented
6#937C211yesmaximal, characteristic, complemented
7#937C211yesmaximal, characteristic, complemented
8#930C211yesmaximal, characteristic, complemented
9#929C211yesmaximal, characteristic, complemented
10#898C211yesmaximal, characteristic, complemented
11#894C211yesmaximal, characteristic, complemented
12#886C211yesmaximal, characteristic, complemented
13#884C211yesmaximal, characteristic, complemented
14#882C211yesmaximal, characteristic, complemented
15#879C211yesmaximal, characteristic, complemented
16#560—31nomaximal, Sylow
17C23×D3C2211yescharacteristic
18C2×C3×D4C2211yescharacteristic
19C2×C3×D4C2211yescharacteristic
20C2×C3×D4C2211yescharacteristic, complemented
21C22×C3×C4C2211yescharacteristic, complemented
22C2×#67C2211yescharacteristic
23C2×#67C2211yescharacteristic
24C2×#67C2211yescharacteristic
25C2×#67C2211yescharacteristic
26C2×#67C2211yescharacteristic
27C2×#67C2211yescharacteristic, complemented
28C2×#67C2211yescharacteristic, complemented
29C22×Dic3C2211yescharacteristic, complemented
30C2×D12C2211yescharacteristic
31C2×D12C2211yescharacteristic
32C2×C4×D3C2211yescharacteristic, complemented
33C3×C4⋊C4C2211yescharacteristic, complemented
34C3×C22⋊C4C2211yescharacteristic
35C3×C22⋊C4C2211yescharacteristic, complemented
36#217C2211yescharacteristic
37#217C2211yescharacteristic
38#217C2211yescharacteristic, complemented
39(C2×D3)⋊C4C2211yescharacteristic
40(C2×D3)⋊C4C2211yescharacteristic
41(C2×D3)⋊C4C2211yescharacteristic
42(C2×D3)⋊C4C2211yescharacteristic
43(C2×D3)⋊C4C2211yescharacteristic
44(C2×D3)⋊C4C2211yescharacteristic
45(C2×D3)⋊C4C2211yescharacteristic, complemented
46C4⋊Dic3C2211yescharacteristic, complemented
47C4⋊Dic3C2211yescharacteristic, complemented
48Dic3⋊C4C2211yescharacteristic, complemented
49Dic3⋊C4C2211yescharacteristic, complemented
50Dic3⋊C4C2211yescharacteristic, complemented
51C4×Dic3C2211yescharacteristic, complemented
52#127—31no—
53#126—31no—
54#126—31no—
55#123—31no—
56#123—31no—
57#123—31no—
58#121—31no—
59#121—31no—
60#121—31no—
61#121—31no—
62#121—31no—
63#121D311yescharacteristic, complemented
64#120—31no—
65#120—31no—
66#120—31no—
67C23×C3C2311yescharacteristic
68C23×C3C2311yescharacteristic
69C23×C3C2311yescharacteristic
70C22×D3—41no—
71C22×D3—41no—
72C22×D3—41no—
73C22×D3C2311yescharacteristic
74C22×D3C2311yescharacteristic
75C22×D3C2311yescharacteristic
76C3×D4—41no—
77C3×D4—41no—
78C3×D4—41no—
79C2×C3×C4—41no—
80C2×C3×C4C2311yescharacteristic
81C2×C3×C4C2311yescharacteristic
82C2×C3×C4C2311yescharacteristic
83C2×C3×C4C2311yescharacteristic
84#67—41no—
85#67—41no—
86#67—41no—
87#67—41no—
88#67—41no—
89#67—41no—
90#67—41no—
91C2×Dic3—41no—
92C2×Dic3C2311yescharacteristic
93C2×Dic3C2311yescharacteristic
94C2×Dic3C2311yescharacteristic
95C2×Dic3C2311yescharacteristic
96C2×Dic3C2311yescharacteristic
97D12—41no—
98D12—41no—
99C4×D3—41no—
100C24—31no—
101C2×D4—31no—
102C2×D4—31no—
103C2×D4—31no—
104C2×D4—31no—
105C2×D4—31no—
106C2×D4—31no—
107C2×D4—31no—
108C2×D4—31no—
109C2×D4—31no—
110C2×D4C2×D311yescharacteristic
111C2×D4C2×D311yescharacteristic
112C2×D4C2×D311yescharacteristic, complemented
113C22×C4—31no—
114C22×C4—31no—
115C22×C4C2×D311yescharacteristic, complemented
116C4⋊C4—31no—
117C4⋊C4—31no—
118C4⋊C4—31no—
119C4⋊C4—31no—
120C4⋊C4—31no—
121C4⋊C4C2×D311yescharacteristic, complemented
122C22⋊C4—31no—
123C22⋊C4—31no—
124C22⋊C4—31no—
125C22⋊C4—31no—
126C22⋊C4—31no—
127C22⋊C4—31no—
128C22⋊C4—31no—
129C22⋊C4—31no—
130C22⋊C4—31no—
131C22⋊C4—31no—
132C22⋊C4C2×D311yescharacteristic
133C22⋊C4C2×D311yescharacteristic, complemented
134C42—31no—
135C22×C3—21no—
136C22×C3—21no—
137C22×C3—21no—
138C22×C3—21no—
139C22×C3—21no—
140C22×C3—21no—
141C22×C3—21no—
142C22×C3—21no—
143C22×C3—21no—
144C22×C3C2411yescharacteristic
145C2×D3—21no—
146C2×D3—21no—
147C2×D3—21no—
148C2×D3—21no—
149C2×D3—21no—
150C2×D3—21no—
151C2×D3—21no—
152C2×D3—21no—
153C2×D3—21no—
154C2×D3—44no—
155C3×C4—21no—
156C3×C4—21no—
157C3×C4—21no—
158C3×C4—21no—
159Dic3—21no—
160Dic3—21no—
161Dic3—21no—
162Dic3—21no—
163Dic3—21no—
164C23—31no—
165C23—31no—
166C23—31no—
167C23—121no—
168C23—121no—
169C23—121no—
170C23C22×D311yescharacteristic
171C23C22×D311yescharacteristic
172C23C22×D311yescharacteristic
173D4—41no—
174D4—41no—
175D4—41no—
176D4—121no—
177D4—121no—
178D4—121no—
179D4—121no—
180D4—121no—
181D4—121no—
182D4—121no—
183D4—121no—
184D4—121no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—31no—
190C2×C4—41no—
191C2×C4—121no—
192C2×C4—121no—
193C2×C4C22×D311yescharacteristic
194C2×C4C22×D311yescharacteristic
195C2×C4C22×D311yescharacteristic
196C2×C4C22×D311yescharacteristic
197C2×C3—41no—
198C2×C3—41no—
199C2×C3—41no—
200C2×C3H(5,𝔽2)11yescharacteristic
201C2×C3H(5,𝔽2)11yescharacteristic
202C2×C3H(5,𝔽2)11yescharacteristic
203D3—41no—
204D3—41no—
205D3—41no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—61no—
216C22—61no—
217C22—61no—
218C22—61no—
219C22—61no—
220C22—61no—
221C22—61no—
222C22—61no—
223C22—61no—
224C22—124no—
225C22C23×D311yescharacteristic
226C4—21no—
227C4—21no—
228C4—21no—
229C4—21no—
230C4—61no—
231C4—61no—
232C4—61no—
233C4—61no—
234C4—61no—
235C3#56011yescharacteristic, complemented, Sylow
236C2—41no—
237C2—41no—
238C2—41no—
239C2—121no—
240C2—121no—
241C2—121no—
242C2#100411yescharacteristic
243C2#100411yescharacteristic
244C2#100911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3
commutator, socle, cosocleC22×C3C24
FittingC3×#121C2
hypercenter#121D3
hypocenterC3#560
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C3×#121C2semidirect
2. #940C2semidirect
3. #939C2semidirect
4. #939C2semidirect
5. #938C2semidirect
6. #937C2semidirect
7. #937C2semidirect
8. #930C2semidirect
9. #929C2semidirect
10. #898C2semidirect
11. #894C2semidirect
12. #886C2semidirect
13. #884C2semidirect
14. #882C2semidirect
15. #879C2semidirect
20. C2×C3×D4C22semidirect
21. C22×C3×C4C22semidirect
27. C2×#67C22semidirect
28. C2×#67C22semidirect
29. C22×Dic3C22semidirect
32. C2×C4×D3C22semidirect
33. C3×C4⋊C4C22semidirect
35. C3×C22⋊C4C22semidirect
38. #217C22semidirect
45. (C2×D3)⋊C4C22semidirect
46. C4⋊Dic3C22semidirect
47. C4⋊Dic3C22semidirect
48. Dic3⋊C4C22semidirect
49. Dic3⋊C4C22semidirect
50. Dic3⋊C4C22semidirect
51. C4×Dic3C22semidirect
63. #121D3semidirect
112. C2×D4C2×D3semidirect
115. C22×C4C2×D3semidirect
121. C4⋊C4C2×D3semidirect
133. C22⋊C4C2×D3semidirect
235. C3#560semidirect

Maximal subgroups

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

subgroupindex
C3×#1212
#9402
#9392
#9382
#9372
#9302
#9292
#8982
#8942
#8862
#8842
#8822
#8792
#5603