Finite Group Databasebeta

Subgroups — #5777

Order 192.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#239	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	#949	C₂	1	1	yes	maximal, characteristic, complemented
5	#942	C₂	1	1	yes	maximal, characteristic, complemented
6	#919	C₂	1	2	yes	maximal, complemented
7	#918	C₂	1	2	yes	maximal, complemented
8	#916	C₂	1	2	yes	maximal, complemented
9	#914	C₂	1	2	yes	maximal, complemented
10	#908	C₂	1	1	yes	maximal, characteristic, complemented
11	D₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#575	—	3	1	no	maximal, Sylow
13	C₃×#41	C₂²	1	1	yes	characteristic
14	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
15	#239	—	2	1	no	—
16	#239	—	2	1	no	—
17	#239	C₂²	1	2	yes	—
18	#239	C₂²	1	2	yes	complemented
19	#239	C₂²	1	1	yes	characteristic, complemented
20	D₃×Q₈	C₂²	1	1	yes	characteristic, complemented
21	#237	—	2	1	no	—
22	#237	C₂²	1	1	yes	characteristic, complemented
23	D₃×D₄	—	2	1	no	—
24	D₃×D₄	C₂²	1	1	yes	characteristic, complemented
25	#235	—	2	1	no	—
26	#235	C₂²	1	1	yes	characteristic
27	C₂×D₁₂	—	2	1	no	—
28	C₂×D₁₂	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₄×D₃	—	2	1	no	—
30	C₂×C₄×D₃	—	2	1	no	—
31	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
32	C₃×Q₁₆	C₂²	1	2	yes	complemented
33	C₃×SD₁₆	C₂²	1	2	yes	complemented
34	C₃×M₁₆	C₂²	1	1	yes	characteristic, complemented
35	#216	C₂²	1	2	yes	complemented
36	#215	C₂²	1	2	yes	complemented
37	#215	C₂²	1	2	yes	complemented
38	#213	C₂²	1	2	yes	complemented
39	#208	C₂²	1	1	yes	characteristic, complemented
40	D₂₄	C₂²	1	2	yes	complemented
41	#204	C₂²	1	2	yes	complemented
42	ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
43	D₃×C₈	C₂²	1	2	yes	complemented
44	C₂×#41	—	3	1	no	—
45	C₂×#41	—	3	1	no	—
46	#137	—	3	1	no	—
47	#137	—	3	2	no	—
48	#137	D₃	1	1	yes	characteristic, complemented
49	AGL(1,ℤ/8)	—	3	1	no	—
50	AGL(1,ℤ/8)	—	3	2	no	—
51	AGL(1,ℤ/8)	—	3	1	no	—
52	#135	—	3	2	no	—
53	#135	—	3	2	no	—
54	C₂×M₁₆	—	3	1	no	—
55	C₂²×D₃	—	2	1	no	—
56	C₂²×D₃	—	2	1	no	—
57	C₂²×D₃	D₄	1	1	yes	characteristic
58	C₃×Q₈	—	2	1	no	—
59	C₃×Q₈	C₂³	1	1	yes	characteristic
60	C₃×Q₈	C₂³	1	2	yes	—
61	C₃×D₄	—	2	1	no	—
62	C₃×D₄	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₄	—	2	1	no	—
64	C₂×C₃×C₄	—	2	1	no	—
65	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
66	#67	—	2	1	no	—
67	#67	—	2	1	no	—
68	#67	—	2	1	no	—
69	C₂×Dic₃	—	2	1	no	—
70	C₂×Dic₃	D₄	1	1	yes	characteristic
71	D₁₂	—	2	2	no	—
72	D₁₂	—	2	2	no	—
73	D₁₂	—	2	1	no	—
74	D₁₂	—	2	1	no	—
75	D₁₂	C₂³	1	2	yes	—
76	D₁₂	C₂³	1	1	yes	characteristic
77	C₄×D₃	—	2	1	no	—
78	C₄×D₃	—	2	1	no	—
79	C₄×D₃	—	2	1	no	—
80	C₄×D₃	—	2	2	no	—
81	C₄×D₃	—	2	2	no	—
82	C₄×D₃	D₄	1	1	yes	characteristic
83	C₄×D₃	D₄	1	1	yes	characteristic
84	C₄×D₃	C₂³	1	1	yes	characteristic
85	C₄×D₃	C₂³	1	1	yes	characteristic
86	Dic₆	—	2	1	no	—
87	Dic₆	C₂³	1	1	yes	characteristic
88	C₃×C₈	C₂³	1	2	yes	—
89	C₃⋊C₈	C₂³	1	2	yes	—
90	#41	—	3	1	no	—
91	#41	—	3	2	no	—
92	#41	—	3	2	no	—
93	#41	—	3	1	no	—
94	#41	—	3	1	no	—
95	#41	—	6	1	no	—
96	#41	—	6	1	no	—
97	#41	—	6	1	no	—
98	#41	—	6	1	no	—
99	#41	C₂×D₃	1	1	yes	characteristic
100	C₂×Q₈	—	3	1	no	—
101	C₂×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
102	C₂×D₄	—	3	1	no	—
103	C₂×D₄	—	3	1	no	—
104	C₂×D₄	—	6	1	no	—
105	C₂×D₄	—	6	1	no	—
106	C₂²×C₄	—	3	1	no	—
107	C₂²×C₄	—	6	1	no	—
108	C₂²×C₄	—	6	1	no	—
109	Q₁₆	—	3	2	no	—
110	Q₁₆	C₂×D₃	1	2	yes	complemented
111	SD₁₆	—	3	2	no	—
112	SD₁₆	—	3	2	no	—
113	SD₁₆	—	3	2	no	—
114	SD₁₆	C₂×D₃	1	2	yes	complemented
115	D₈	—	3	2	no	—
116	D₈	—	3	2	no	—
117	M₁₆	—	3	1	no	—
118	M₁₆	—	3	2	no	—
119	M₁₆	C₂×D₃	1	1	yes	characteristic, complemented
120	C₂×C₈	—	3	2	no	—
121	C₂²×C₃	—	2	1	no	—
122	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
123	C₂×D₃	—	2	2	no	—
124	C₂×D₃	—	2	1	no	—
125	C₂×D₃	—	4	1	no	—
126	C₂×D₃	—	4	2	no	—
127	C₂×D₃	—	4	2	no	—
128	C₂×D₃	C₂×D₄	1	1	yes	characteristic
129	C₂×D₃	C₂×D₄	1	1	yes	characteristic
130	C₃×C₄	—	2	1	no	—
131	C₃×C₄	—	2	2	no	—
132	C₃×C₄	C₂×D₄	1	1	yes	characteristic
133	C₃×C₄	C₂⁴	1	1	yes	characteristic
134	Dic₃	—	2	1	no	—
135	Dic₃	C₂×D₄	1	1	yes	characteristic
136	Dic₃	C₂×D₄	1	1	yes	characteristic
137	C₂³	—	3	1	no	—
138	C₂³	—	6	1	no	—
139	C₂³	—	6	1	no	—
140	Q₈	—	2	1	no	—
141	Q₈	—	3	1	no	—
142	Q₈	—	6	1	no	—
143	Q₈	C₂²×D₃	1	1	yes	characteristic
144	Q₈	C₂²×D₃	1	2	yes	—
145	D₄	—	2	1	no	—
146	D₄	—	3	2	no	—
147	D₄	—	3	1	no	—
148	D₄	—	6	1	no	—
149	D₄	—	6	2	no	—
150	D₄	—	6	2	no	—
151	D₄	—	6	1	no	—
152	D₄	—	6	1	no	—
153	D₄	—	6	1	no	—
154	D₄	—	6	1	no	—
155	D₄	C₂²×D₃	1	1	yes	characteristic
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	3	1	no	—
159	C₂×C₄	—	3	1	no	—
160	C₂×C₄	—	3	1	no	—
161	C₂×C₄	—	3	1	no	—
162	C₂×C₄	—	3	1	no	—
163	C₂×C₄	—	6	1	no	—
164	C₂×C₄	—	6	2	no	—
165	C₂×C₄	—	6	2	no	—
166	C₂×C₄	—	6	1	no	—
167	C₂×C₄	—	6	1	no	—
168	C₂×C₄	—	6	1	no	—
169	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
170	C₈	—	3	2	no	—
171	C₈	C₂²×D₃	1	2	yes	—
172	C₂×C₃	—	2	1	no	—
173	C₂×C₃	—	4	1	no	—
174	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
175	D₃	—	2	1	no	—
176	D₃	—	2	1	no	—
177	D₃	—	4	2	no	—
178	D₃	—	4	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	3	1	no	—
181	C₂²	—	3	1	no	—
182	C₂²	—	6	2	no	—
183	C₂²	—	6	1	no	—
184	C₂²	—	12	1	no	—
185	C₂²	—	12	2	no	—
186	C₂²	—	12	2	no	—
187	C₂²	D₃×D₄	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	2	no	—
190	C₄	—	3	1	no	—
191	C₄	—	3	1	no	—
192	C₄	—	6	1	no	—
193	C₄	D₃×D₄	1	1	yes	characteristic
194	C₄	C₂³×D₃	1	1	yes	characteristic
195	C₃	#575	1	1	yes	characteristic, complemented, Sylow
196	C₂	—	2	1	no	—
197	C₂	—	4	1	no	—
198	C₂	—	6	1	no	—
199	C₂	—	6	1	no	—
200	C₂	—	12	2	no	—
201	C₂	—	12	1	no	—
202	C₂	C₂×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁴
Frattini	C₄	C₂³×D₃
Fitting	C₃×#137	C₂
socle	C₂×C₃	C₂²×D₄
hypercenter	#137	D₃
hypocenter	C₃	#575
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. D₃×#41	C₂	semidirect
2. C₂×#239	C₂	semidirect
3. C₃×#137	C₂	semidirect
4. #949	C₂	semidirect
5. #942	C₂	semidirect
6. #919	C₂	semidirect
7. #918	C₂	semidirect
8. #916	C₂	semidirect
9. #914	C₂	semidirect
10. #908	C₂	semidirect
11. D₃×M₁₆	C₂	semidirect
14. C₂×C₃×Q₈	C₂²	semidirect
18. #239	C₂²	semidirect
19. #239	C₂²	semidirect
20. D₃×Q₈	C₂²	semidirect
22. #237	C₂²	semidirect
24. D₃×D₄	C₂²	semidirect
28. C₂×D₁₂	C₂²	semidirect
32. C₃×Q₁₆	C₂²	semidirect
33. C₃×SD₁₆	C₂²	semidirect
34. C₃×M₁₆	C₂²	semidirect
35. #216	C₂²	semidirect
36. #215	C₂²	semidirect
37. #215	C₂²	semidirect
38. #213	C₂²	semidirect
39. #208	C₂²	semidirect
40. D₂₄	C₂²	semidirect
41. #204	C₂²	semidirect
42. ΓL(1,𝔽₂₅)	C₂²	semidirect
43. D₃×C₈	C₂²	semidirect
48. #137	D₃	semidirect
101. C₂×Q₈	C₂×D₃	semidirect
110. Q₁₆	C₂×D₃	semidirect
114. SD₁₆	C₂×D₃	semidirect
119. M₁₆	C₂×D₃	semidirect
195. C₃	#575	semidirect

Maximal subgroups

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

subgroup	index
D₃×#41	2
C₂×#239	2
C₃×#137	2
#949	2
#942	2
#919	2
#918	2
#916	2
#914	2
#908	2
D₃×M₁₆	2
#575	3