Finite Group Databasebeta

Subgroups — #5773

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×#237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
4	#951	C₂	1	1	yes	maximal, characteristic, complemented
5	#932	C₂	1	1	yes	maximal, characteristic, complemented
6	#916	C₂	1	2	yes	maximal, complemented
7	#915	C₂	1	2	yes	maximal, complemented
8	#912	C₂	1	2	yes	maximal, complemented
9	#911	C₂	1	2	yes	maximal, complemented
10	#909	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×D₄	C₂²	1	1	yes	characteristic, complemented
15	C₂×#67	—	2	1	no	—
16	C₂²×Dic₃	—	2	1	no	—
17	#239	C₂²	1	1	yes	characteristic, complemented
18	D₃×Q₈	C₂²	1	1	yes	characteristic, complemented
19	#237	—	2	1	no	—
20	#237	—	2	1	no	—
21	#237	—	2	1	no	—
22	#237	C₂²	1	2	yes	—
23	#237	C₂²	1	1	yes	characteristic, complemented
24	#237	C₂²	1	2	yes	complemented
25	D₃×D₄	—	2	1	no	—
26	D₃×D₄	C₂²	1	1	yes	characteristic, complemented
27	#235	—	2	1	no	—
28	#235	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₃	—	2	1	no	—
30	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
31	C₂×Dic₆	C₂²	1	1	yes	characteristic, complemented
32	C₃×SD₁₆	C₂²	1	2	yes	complemented
33	C₃×D₈	C₂²	1	2	yes	complemented
34	C₃×M₁₆	C₂²	1	1	yes	characteristic, complemented
35	#216	C₂²	1	2	yes	complemented
36	#214	C₂²	1	2	yes	complemented
37	#214	C₂²	1	2	yes	complemented
38	#213	C₂²	1	2	yes	complemented
39	#208	C₂²	1	1	yes	characteristic, complemented
40	Dic₁₂	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	—	3	1	no	—
49	AGL(1,ℤ/8)	—	3	1	no	—
50	AGL(1,ℤ/8)	—	3	2	no	—
51	AGL(1,ℤ/8)	D₃	1	1	yes	characteristic, complemented
52	#135	—	3	2	no	—
53	#135	—	3	2	no	—
54	C₂×M₁₆	—	3	1	no	—
55	C₂³×C₃	—	2	1	no	—
56	C₂²×D₃	—	2	1	no	—
57	C₂²×D₃	D₄	1	1	yes	characteristic
58	C₃×Q₈	C₂³	1	1	yes	characteristic
59	C₃×D₄	—	2	1	no	—
60	C₃×D₄	—	2	1	no	—
61	C₃×D₄	C₂³	1	2	yes	—
62	C₃×D₄	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₄	—	2	1	no	—
64	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
65	#67	—	2	1	no	—
66	#67	—	2	1	no	—
67	#67	—	2	2	no	—
68	#67	—	2	2	no	—
69	#67	—	2	1	no	—
70	C₂×Dic₃	—	2	1	no	—
71	C₂×Dic₃	—	2	1	no	—
72	C₂×Dic₃	—	2	2	no	—
73	C₂×Dic₃	—	2	2	no	—
74	C₂×Dic₃	D₄	1	1	yes	characteristic
75	D₁₂	—	2	1	no	—
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	D₄	1	1	yes	characteristic
81	C₄×D₃	D₄	1	1	yes	characteristic
82	C₄×D₃	C₂³	1	1	yes	characteristic
83	C₄×D₃	C₂³	1	1	yes	characteristic
84	Dic₆	—	2	1	no	—
85	Dic₆	—	2	1	no	—
86	Dic₆	C₂³	1	2	yes	—
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	1	no	—
93	#41	—	3	2	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	—	3	1	no	—
102	C₂×D₄	—	3	1	no	—
103	C₂×D₄	—	6	1	no	—
104	C₂×D₄	—	6	1	no	—
105	C₂×D₄	C₂×D₃	1	1	yes	characteristic, complemented
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	—	3	2	no	—
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	C₂×D₃	1	2	yes	complemented
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	2	no	—
122	C₂²×C₃	—	2	1	no	—
123	C₂²×C₃	—	4	2	no	—
124	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
125	C₂×D₃	—	2	1	no	—
126	C₂×D₃	—	4	1	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	C₂×D₄	1	1	yes	characteristic
132	C₃×C₄	C₂⁴	1	1	yes	characteristic
133	Dic₃	—	2	1	no	—
134	Dic₃	—	2	2	no	—
135	Dic₃	C₂×D₄	1	1	yes	characteristic
136	Dic₃	C₂×D₄	1	1	yes	characteristic
137	C₂³	—	2	1	no	—
138	C₂³	—	3	1	no	—
139	C₂³	—	6	1	no	—
140	Q₈	—	3	1	no	—
141	Q₈	—	3	2	no	—
142	Q₈	—	6	1	no	—
143	Q₈	—	6	1	no	—
144	Q₈	C₂²×D₃	1	1	yes	characteristic
145	D₄	—	2	1	no	—
146	D₄	—	2	1	no	—
147	D₄	—	3	1	no	—
148	D₄	—	6	1	no	—
149	D₄	—	6	1	no	—
150	D₄	—	6	1	no	—
151	D₄	—	6	2	no	—
152	D₄	—	6	2	no	—
153	D₄	—	6	1	no	—
154	D₄	C₂²×D₃	1	2	yes	—
155	D₄	C₂²×D₃	1	1	yes	characteristic
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	3	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	—	6	1	no	—
163	C₂×C₄	—	6	1	no	—
164	C₂×C₄	—	6	1	no	—
165	C₂×C₄	—	6	2	no	—
166	C₂×C₄	—	6	2	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	—	4	2	no	—
175	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
176	D₃	—	2	1	no	—
177	D₃	—	2	1	no	—
178	D₃	—	4	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	3	1	no	—
182	C₂²	—	3	1	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	6	1	no	—
185	C₂²	—	12	1	no	—
186	C₂²	—	12	2	no	—
187	C₂²	D₃×D₄	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	3	1	no	—
190	C₄	—	3	1	no	—
191	C₄	—	6	1	no	—
192	C₄	—	6	2	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	2	no	—
198	C₂	—	4	1	no	—
199	C₂	—	6	1	no	—
200	C₂	—	6	1	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×AGL(1,ℤ/8)	C₂
socle	C₂×C₃	C₂²×D₄
hypercenter	AGL(1,ℤ/8)	D₃
hypocenter	C₃	#575
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. D₃×#41	C₂	semidirect
2. C₂×#237	C₂	semidirect
3. C₃×AGL(1,ℤ/8)	C₂	semidirect
4. #951	C₂	semidirect
5. #932	C₂	semidirect
6. #916	C₂	semidirect
7. #915	C₂	semidirect
8. #912	C₂	semidirect
9. #911	C₂	semidirect
10. #909	C₂	semidirect
11. D₃×M₁₆	C₂	semidirect
14. C₂×C₃×D₄	C₂²	semidirect
17. #239	C₂²	semidirect
18. D₃×Q₈	C₂²	semidirect
23. #237	C₂²	semidirect
24. #237	C₂²	semidirect
26. D₃×D₄	C₂²	semidirect
31. C₂×Dic₆	C₂²	semidirect
32. C₃×SD₁₆	C₂²	semidirect
33. C₃×D₈	C₂²	semidirect
34. C₃×M₁₆	C₂²	semidirect
35. #216	C₂²	semidirect
36. #214	C₂²	semidirect
37. #214	C₂²	semidirect
38. #213	C₂²	semidirect
39. #208	C₂²	semidirect
40. Dic₁₂	C₂²	semidirect
41. #204	C₂²	semidirect
42. ΓL(1,𝔽₂₅)	C₂²	semidirect
43. D₃×C₈	C₂²	semidirect
51. AGL(1,ℤ/8)	D₃	semidirect
105. C₂×D₄	C₂×D₃	semidirect
114. SD₁₆	C₂×D₃	semidirect
116. D₈	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×#237	2
C₃×AGL(1,ℤ/8)	2
#951	2
#932	2
#916	2
#915	2
#912	2
#911	2
#909	2
D₃×M₁₆	2
#575	3