Finite Group Databasebeta

← #5591

Subgroups — #5591

Order 192.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C22×#67C211yesmaximal, characteristic, complemented
2C2×#237C211yesmaximal, characteristic, complemented
3C3×#120C211yesmaximal, characteristic, complemented
4C2×#217C211yesmaximal, characteristic, complemented
5#939C211yesmaximal, characteristic, complemented
6#939C211yesmaximal, characteristic, complemented
7#938C211yesmaximal, characteristic, complemented
8#937C211yesmaximal, characteristic, complemented
9D4×Dic3C211yesmaximal, characteristic, complemented
10#886C211yesmaximal, characteristic, complemented
11#885C211yesmaximal, characteristic, complemented
12#883C211yesmaximal, characteristic, complemented
13#67⋊C4C211yesmaximal, characteristic, complemented
14D3×C22⋊C4C211yesmaximal, characteristic, complemented
15#878C211yesmaximal, characteristic, complemented
16#546—31nomaximal, Sylow
17C24×C3C2211yescharacteristic
18C23×D3C2211yescharacteristic
19C2×C3×D4C2211yescharacteristic
20C2×C3×D4C2211yescharacteristic, complemented
21C2×C3×D4C2211yescharacteristic, complemented
22C2×#67—24no—
23C2×#67C2211yescharacteristic
24C2×#67C2211yescharacteristic
25C2×#67C2211yescharacteristic
26C2×#67C2211yescharacteristic
27C2×#67C2211yescharacteristic, complemented
28C2×#67C2211yescharacteristic, complemented
29C22×Dic3C2211yescharacteristic
30C22×Dic3C2211yescharacteristic, complemented
31C22×Dic3C2211yescharacteristic, complemented
32C22×Dic3C2211yescharacteristic, complemented
33#237—24no—
34C2×C4×D3C2211yescharacteristic, complemented
35C2×C4×D3C2211yescharacteristic, complemented
36C2×Dic6C2211yescharacteristic, complemented
37C3×C22⋊C4C2211yescharacteristic
38C3×C22⋊C4C2211yescharacteristic, complemented
39C3×C22⋊C4C2211yescharacteristic, complemented
40#217C2211yescharacteristic
41#217C2211yescharacteristic
42#217C2211yescharacteristic, complemented
43#217C2211yescharacteristic, complemented
44#217C2211yescharacteristic, complemented
45(C2×D3)⋊C4C2211yescharacteristic, complemented
46(C2×D3)⋊C4C2211yescharacteristic, complemented
47(C2×D3)⋊C4C2211yescharacteristic, complemented
48(C2×D3)⋊C4C2211yescharacteristic, complemented
49C4⋊Dic3C2211yescharacteristic, complemented
50C4⋊Dic3C2211yescharacteristic, complemented
51Dic3⋊C4C2211yescharacteristic, complemented
52Dic3⋊C4C2211yescharacteristic, complemented
53C4×Dic3C2211yescharacteristic, complemented
54C2×#41—31no—
55C22×D4—31no—
56#124—31no—
57#123—31no—
58#123—31no—
59C22⋊Q8—31no—
60#121—31no—
61#121—31no—
62#121—31no—
63#120—31no—
64#120D311yescharacteristic, complemented
65C4×D4—31no—
66C4×D4—31no—
67C2×C22⋊C4—31no—
68C2×C22⋊C4—31no—
69C23×C3—22no—
70C23×C3—22no—
71C23×C3—41no—
72C23×C3C2311yescharacteristic
73C23×C3C2311yescharacteristic
74C23×C3C2311yescharacteristic
75C23×C3C2311yescharacteristic
76C22×D3—22no—
77C22×D3—22no—
78C22×D3—41no—
79C22×D3C2311yescharacteristic
80C22×D3C2311yescharacteristic
81C3×D4—22no—
82C3×D4—22no—
83C3×D4—41no—
84C2×C3×C4C2311yescharacteristic
85C2×C3×C4C2311yescharacteristic
86C2×C3×C4C2311yescharacteristic
87#67—22no—
88#67—22no—
89#67—22no—
90#67—22no—
91#67—41no—
92#67D414yescomplemented
93C2×Dic3—22no—
94C2×Dic3—22no—
95C2×Dic3—22no—
96C2×Dic3—41no—
97C2×Dic3C2311yescharacteristic
98C2×Dic3C2311yescharacteristic
99C2×Dic3C2311yescharacteristic
100C2×Dic3C2311yescharacteristic
101C2×Dic3C2311yescharacteristic
102C2×Dic3C2311yescharacteristic
103C4×D3—22no—
104C4×D3—41no—
105Dic6—22no—
106C24—31no—
107C24C2×D311yescharacteristic
108#41—64no—
109C2×Q8—31no—
110C2×D4—31no—
111C2×D4—31no—
112C2×D4—31no—
113C2×D4—31no—
114C2×D4—31no—
115C2×D4—31no—
116C2×D4—64no—
117C2×D4C2×D311yescharacteristic
118C2×D4C2×D311yescharacteristic, complemented
119C2×D4C2×D311yescharacteristic, complemented
120C22×C4—31no—
121C22×C4—31no—
122C22×C4—31no—
123C22×C4—31no—
124C22×C4—31no—
125C22×C4—31no—
126C4⋊C4—31no—
127C4⋊C4—31no—
128C4⋊C4—31no—
129C4⋊C4—31no—
130C22⋊C4—31no—
131C22⋊C4—31no—
132C22⋊C4—31no—
133C22⋊C4—31no—
134C22⋊C4—31no—
135C22⋊C4—31no—
136C22⋊C4—31no—
137C22⋊C4—31no—
138C22⋊C4—31no—
139C22⋊C4C2×D311yescharacteristic
140C22⋊C4C2×D311yescharacteristic, complemented
141C22⋊C4C2×D311yescharacteristic, complemented
142C42—31no—
143C22×C3—21no—
144C22×C3—21no—
145C22×C3—21no—
146C22×C3—21no—
147C22×C3—21no—
148C22×C3—21no—
149C22×C3—21no—
150C22×C3—21no—
151C22×C3—21no—
152C22×C3—21no—
153C22×C3—44no—
154C22×C3C2×D412yes—
155C22×C3#4112yescomplemented
156C22×C3C2411yescharacteristic
157C2×D3—21no—
158C2×D3—21no—
159C2×D3—21no—
160C2×D3—21no—
161C2×D3—21no—
162C2×D3—44no—
163C2×D3C2×D412yes—
164C3×C4—21no—
165C3×C4—21no—
166C3×C4—21no—
167Dic3—21no—
168Dic3—21no—
169Dic3—21no—
170Dic3—21no—
171Dic3—21no—
172Dic3C2×D412yes—
173C23—22no—
174C23—22no—
175C23—31no—
176C23—31no—
177C23—41no—
178C23—62no—
179C23—62no—
180C23—121no—
181C23C22×D311yescharacteristic
182C23C22×D311yescharacteristic
183C23C22×D311yescharacteristic
184C23C22×D311yescharacteristic
185Q8—62no—
186D4—22no—
187D4—22no—
188D4—34no—
189D4—41no—
190D4—62no—
191D4—62no—
192D4—62no—
193D4—62no—
194D4—121no—
195C2×C4—31no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—62no—
205C2×C4—121no—
206C2×C4—121no—
207C2×C4C22×D311yescharacteristic
208C2×C4C22×D311yescharacteristic
209C2×C4C22×D311yescharacteristic
210C2×C3—22no—
211C2×C3—22no—
212C2×C3—41no—
213C2×C3—41no—
214C2×C3C22×D411yescharacteristic
215C2×C3C2×#4111yescharacteristic
216C2×C3H(5,𝔽2)11yescharacteristic
217D3—22no—
218D3—41no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—32no—
230C22—44no—
231C22—61no—
232C22—61no—
233C22—61no—
234C22—61no—
235C22—61no—
236C22—124no—
237C22D3×D412yes—
238C22#23712yescomplemented
239C22C23×D311yescharacteristic
240C4—21no—
241C4—21no—
242C4—21no—
243C4—32no—
244C4—61no—
245C4—61no—
246C4—61no—
247C4—61no—
248C4—61no—
249C3#54611yescharacteristic, complemented, Sylow
250C2—22no—
251C2—22no—
252C2—41no—
253C2—41no—
254C2—62no—
255C2—121no—
256C2C2×D3×D411yescharacteristic
257C2C2×#23711yescharacteristic
258C2#100411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3
commutator, socle, cosocleC22×C3C24
FittingC3×#120C2
hypercenter#120D3
hypocenterC3#546
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×#67C2semidirect
2. C2×#237C2semidirect
3. C3×#120C2semidirect
4. C2×#217C2semidirect
5. #939C2semidirect
6. #939C2semidirect
7. #938C2semidirect
8. #937C2semidirect
9. D4×Dic3C2semidirect
10. #886C2semidirect
11. #885C2semidirect
12. #883C2semidirect
13. #67⋊C4C2semidirect
14. D3×C22⋊C4C2semidirect
15. #878C2semidirect
20. C2×C3×D4C22semidirect
21. C2×C3×D4C22semidirect
27. C2×#67C22semidirect
28. C2×#67C22semidirect
30. C22×Dic3C22semidirect
31. C22×Dic3C22semidirect
32. C22×Dic3C22semidirect
34. C2×C4×D3C22semidirect
35. C2×C4×D3C22semidirect
36. C2×Dic6C22semidirect
38. C3×C22⋊C4C22semidirect
39. C3×C22⋊C4C22semidirect
42. #217C22semidirect
43. #217C22semidirect
44. #217C22semidirect
45. (C2×D3)⋊C4C22semidirect
46. (C2×D3)⋊C4C22semidirect
47. (C2×D3)⋊C4C22semidirect
48. (C2×D3)⋊C4C22semidirect
49. C4⋊Dic3C22semidirect
50. C4⋊Dic3C22semidirect
51. Dic3⋊C4C22semidirect
52. Dic3⋊C4C22semidirect
53. C4×Dic3C22semidirect
64. #120D3semidirect
92. #67D4semidirect
118. C2×D4C2×D3semidirect
119. C2×D4C2×D3semidirect
140. C22⋊C4C2×D3semidirect
141. C22⋊C4C2×D3semidirect
155. C22×C3#41semidirect
238. C22#237semidirect
249. C3#546semidirect

Maximal subgroups

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

subgroupindex
C22×#672
C2×#2372
C3×#1202
C2×#2172
#9392
#9382
#9372
D4×Dic32
#8862
#8852
#8832
#67⋊C42
D3×C22⋊C42
#8782
#5463