Finite Group Databasebeta

← #5547

Subgroups — #5547

Order 192.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#235C211yesmaximal, characteristic, complemented
2C22×C4×D3C211yesmaximal, characteristic, complemented
3C3×C4×D4C211yesmaximal, characteristic, complemented
4#939C211yesmaximal, characteristic, complemented
5#937C211yesmaximal, characteristic, complemented
6C4×#67C211yesmaximal, characteristic, complemented
7C4×#67C211yesmaximal, characteristic, complemented
8#896C211yesmaximal, characteristic, complemented
9#894C211yesmaximal, characteristic, complemented
10#884C211yesmaximal, characteristic, complemented
11#883C211yesmaximal, characteristic, complemented
12#882C211yesmaximal, characteristic, complemented
13#878C211yesmaximal, characteristic, complemented
14C4×D12C211yesmaximal, characteristic, complemented
15#872C211yesmaximal, characteristic, complemented
16#525—31nomaximal, Sylow
17C23×D3C2211yescharacteristic
18C2×C3×D4C2211yescharacteristic
19C22×C3×C4C2211yescharacteristic
20C22×C3×C4C2211yescharacteristic, complemented
21C2×#67C2211yescharacteristic
22C2×#67C2211yescharacteristic
23C2×#67C2211yescharacteristic
24C2×#67C2211yescharacteristic, complemented
25C22×Dic3C2211yescharacteristic
26#235—24no—
27C2×D12C2211yescharacteristic
28C2×D12C2211yescharacteristic
29C2×C4×D3—24no—
30C2×C4×D3C2211yescharacteristic
31C2×C4×D3C2211yescharacteristic
32C2×C4×D3C2211yescharacteristic
33C2×C4×D3C2211yescharacteristic
34C2×C4×D3C2211yescharacteristic, complemented
35C2×Dic6C2211yescharacteristic, complemented
36C3×C4⋊C4C2211yescharacteristic, complemented
37C3×C22⋊C4C2211yescharacteristic
38C3×C22⋊C4C2211yescharacteristic, complemented
39C3×C42C2211yescharacteristic, complemented
40#217C2211yescharacteristic
41#217C2211yescharacteristic, complemented
42(C2×D3)⋊C4C2211yescharacteristic
43(C2×D3)⋊C4C2211yescharacteristic
44(C2×D3)⋊C4C2211yescharacteristic
45(C2×D3)⋊C4C2211yescharacteristic
46(C2×D3)⋊C4C2211yescharacteristic, complemented
47(C2×D3)⋊C4C2211yescharacteristic, complemented
48C4⋊Dic3C2211yescharacteristic, complemented
49Dic3⋊C4C2211yescharacteristic, complemented
50Dic3⋊C4C2211yescharacteristic, complemented
51Dic3⋊C4C2211yescharacteristic, complemented
52Dic3⋊C4C2211yescharacteristic, complemented
53C4×Dic3C2211yescharacteristic, complemented
54C2×#41—31no—
55C23×C4—31no—
56#123—31no—
57#123—31no—
58C22⋊Q8—31no—
59C22⋊Q8—31no—
60#121—31no—
61#121—31no—
62#120—31no—
63#120—31no—
64C4×D4—31no—
65C4×D4—31no—
66C4×D4—31no—
67C4×D4D311yescharacteristic, complemented
68#117—31no—
69C23×C3C2311yescharacteristic
70C23×C3C2311yescharacteristic
71C22×D3—22no—
72C22×D3—22no—
73C22×D3—22no—
74C22×D3C2311yescharacteristic
75C22×D3C2311yescharacteristic
76C22×D3C2311yescharacteristic
77C3×D4—22no—
78C2×C3×C4—22no—
79C2×C3×C4—22no—
80C2×C3×C4C2311yescharacteristic
81C2×C3×C4C2311yescharacteristic
82C2×C3×C4C2311yescharacteristic
83C2×C3×C4C2311yescharacteristic
84C2×C3×C4C2311yescharacteristic
85#67—22no—
86#67—22no—
87#67—22no—
88#67—22no—
89C2×Dic3—22no—
90C2×Dic3C2311yescharacteristic
91C2×Dic3C2311yescharacteristic
92C2×Dic3C2311yescharacteristic
93C2×Dic3C2311yescharacteristic
94C2×Dic3C2311yescharacteristic
95D12—22no—
96D12—22no—
97C4×D3—22no—
98C4×D3—22no—
99C4×D3—22no—
100C4×D3—22no—
101C4×D3D414yescomplemented
102Dic6—22no—
103C24—31no—
104#41—64no—
105C2×Q8—31no—
106C2×D4—31no—
107C2×D4—31no—
108C2×D4—31no—
109C2×D4—31no—
110C2×D4—31no—
111C2×D4—31no—
112C2×D4C2×D311yescharacteristic
113C22×C4—31no—
114C22×C4—31no—
115C22×C4—31no—
116C22×C4—31no—
117C22×C4—31no—
118C22×C4—31no—
119C22×C4—64no—
120C22×C4C2×D311yescharacteristic
121C22×C4C2×D311yescharacteristic, complemented
122C4⋊C4—31no—
123C4⋊C4—31no—
124C4⋊C4—31no—
125C4⋊C4—31no—
126C4⋊C4—31no—
127C4⋊C4C2×D311yescharacteristic, complemented
128C22⋊C4—31no—
129C22⋊C4—31no—
130C22⋊C4—31no—
131C22⋊C4—31no—
132C22⋊C4—31no—
133C22⋊C4—31no—
134C22⋊C4—31no—
135C22⋊C4—31no—
136C22⋊C4C2×D311yescharacteristic
137C22⋊C4C2×D311yescharacteristic, complemented
138C42—31no—
139C42C2×D311yescharacteristic, complemented
140C22×C3—21no—
141C22×C3—21no—
142C22×C3—21no—
143C22×C3—21no—
144C22×C3—21no—
145C22×C3#4112yescomplemented
146C22×C3C2411yescharacteristic
147C2×D3—28no—
148C2×D3—21no—
149C2×D3—21no—
150C2×D3—21no—
151C2×D3—21no—
152C2×D3—21no—
153C2×D3—21no—
154C2×D3—21no—
155C2×D3C2×D412yes—
156C2×D3#4112yescomplemented
157C3×C4—21no—
158C3×C4—21no—
159C3×C4—21no—
160C3×C4—21no—
161C3×C4C2×D412yes—
162Dic3—21no—
163Dic3—21no—
164Dic3—21no—
165Dic3—21no—
166Dic3C2×D412yes—
167C23—31no—
168C23—31no—
169C23—31no—
170C23—62no—
171C23—62no—
172C23—62no—
173C23C22×D311yescharacteristic
174C23C22×D311yescharacteristic
175Q8—62no—
176D4—22no—
177D4—62no—
178D4—62no—
179D4—62no—
180D4—62no—
181D4—62no—
182D4—62no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—31no—
186C2×C4—34no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—31no—
190C2×C4—31no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4C22×D311yescharacteristic
197C2×C4C22×D311yescharacteristic
198C2×C4C22×D311yescharacteristic
199C2×C4C22×D311yescharacteristic
200C2×C4C22×D311yescharacteristic
201C2×C3—22no—
202C2×C3—41no—
203C2×C3C22×D411yescharacteristic
204C2×C3C2×#4111yescharacteristic
205C2×C3C2×#4111yescharacteristic
206D3—22no—
207D3—22no—
208D3—41no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—32no—
215C22—32no—
216C22—68no—
217C22—61no—
218C22—61no—
219C22—61no—
220C22—61no—
221C22—61no—
222C22—61no—
223C22—61no—
224C22#23512yescomplemented
225C22C23×D311yescharacteristic
226C4—21no—
227C4—21no—
228C4—21no—
229C4—21no—
230C4—32no—
231C4—61no—
232C4—61no—
233C4—61no—
234C4—61no—
235C4D3×D412yes—
236C3#52511yescharacteristic, complemented, Sylow
237C2—22no—
238C2—41no—
239C2—62no—
240C2—62no—
241C2—121no—
242C2C2×#23511yescharacteristic
243C2C2×D3×D411yescharacteristic
244C2D3×#4111yescharacteristic
subgroupG/H
centerC2×C4C22×D3
commutator, socle, cosocleC22×C3C24
FrattiniC22C23×D3
FittingC3×C4×D4C2
hypercenterC4×D4D3
hypocenterC3#525
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#235C2semidirect
2. C22×C4×D3C2semidirect
3. C3×C4×D4C2semidirect
4. #939C2semidirect
5. #937C2semidirect
6. C4×#67C2semidirect
7. C4×#67C2semidirect
8. #896C2semidirect
9. #894C2semidirect
10. #884C2semidirect
11. #883C2semidirect
12. #882C2semidirect
13. #878C2semidirect
14. C4×D12C2semidirect
15. #872C2semidirect
20. C22×C3×C4C22semidirect
24. C2×#67C22semidirect
34. C2×C4×D3C22semidirect
35. C2×Dic6C22semidirect
36. C3×C4⋊C4C22semidirect
38. C3×C22⋊C4C22semidirect
39. C3×C42C22semidirect
41. #217C22semidirect
46. (C2×D3)⋊C4C22semidirect
47. (C2×D3)⋊C4C22semidirect
48. C4⋊Dic3C22semidirect
49. Dic3⋊C4C22semidirect
50. Dic3⋊C4C22semidirect
51. Dic3⋊C4C22semidirect
52. Dic3⋊C4C22semidirect
53. C4×Dic3C22semidirect
67. C4×D4D3semidirect
101. C4×D3D4semidirect
121. C22×C4C2×D3semidirect
127. C4⋊C4C2×D3semidirect
137. C22⋊C4C2×D3semidirect
139. C42C2×D3semidirect
145. C22×C3#41semidirect
156. C2×D3#41semidirect
224. C22#235semidirect
236. C3#525semidirect

Maximal subgroups

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

subgroupindex
C2×#2352
C22×C4×D32
C3×C4×D42
#9392
#9372
C4×#672
#8962
#8942
#8842
#8832
#8822
#8782
C4×D122
#8722
#5253