Finite Group Databasebeta

Subgroups — #5774

Order 192.

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

normal subgroups only

This group has 268 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	#1009	C₂	1	1	yes	maximal, characteristic, complemented
2	#1004	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
4	#950	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#213	C₂	1	1	yes	maximal, characteristic, complemented
6	#916	C₂	1	2	yes	maximal, complemented
7	#914	C₂	1	2	yes	maximal, complemented
8	#911	C₂	1	2	yes	maximal, complemented
9	D₃×D₈	C₂	1	2	yes	maximal, complemented
10	#908	C₂	1	1	yes	maximal, characteristic, complemented
11	#907	C₂	1	1	yes	maximal, characteristic, complemented
12	#576	—	3	1	no	maximal, Sylow
13	C₃×#41	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₃×D₄	C₂²	1	1	yes	characteristic, complemented
15	C₂×#67	—	2	2	no	—
16	#239	C₂²	1	2	yes	complemented
17	#237	—	2	1	no	—
18	#237	C₂²	1	2	yes	complemented
19	D₃×D₄	—	2	2	no	—
20	D₃×D₄	—	2	1	no	—
21	D₃×D₄	C₂²	1	2	yes	complemented
22	D₃×D₄	C₂²	1	2	yes	complemented
23	#235	—	2	1	no	—
24	#235	C₂²	1	1	yes	characteristic
25	#235	C₂²	1	1	yes	characteristic, complemented
26	C₂×D₁₂	—	2	1	no	—
27	C₂×D₁₂	C₂²	1	1	yes	characteristic, complemented
28	C₃×SD₁₆	C₂²	1	2	yes	complemented
29	C₃×D₈	C₂²	1	2	yes	complemented
30	C₃×M₁₆	C₂²	1	1	yes	characteristic, complemented
31	#216	C₂²	1	1	yes	characteristic, complemented
32	#215	C₂²	1	1	yes	characteristic, complemented
33	#214	C₂²	1	1	yes	characteristic, complemented
34	#213	C₂²	1	1	yes	characteristic, complemented
35	#213	C₂²	1	2	yes	complemented
36	#213	C₂²	1	2	yes	complemented
37	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
38	D₂₄	C₂²	1	2	yes	complemented
39	#204	C₂²	1	2	yes	complemented
40	ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
41	D₃×C₈	C₂²	1	2	yes	complemented
42	H(5,𝔽₂)	—	3	1	no	—
43	H(5,𝔽₂)	—	3	1	no	—
44	AGL(1,ℤ/8)	—	3	2	no	—
45	AGL(1,ℤ/8)	—	3	2	no	—
46	AGL(1,ℤ/8)	—	3	1	no	—
47	AGL(1,ℤ/8)	D₃	1	1	yes	characteristic, complemented
48	#135	—	3	1	no	—
49	#135	—	3	2	no	—
50	C₂×D₈	—	3	1	no	—
51	C₂×D₈	—	3	2	no	—
52	#131	—	3	1	no	—
53	C₂³×C₃	—	2	1	no	—
54	C₂²×D₃	—	2	1	no	—
55	C₂²×D₃	—	2	2	no	—
56	C₂²×D₃	—	2	2	no	—
57	C₃×Q₈	C₂³	1	1	yes	characteristic
58	C₃×D₄	—	2	1	no	—
59	C₃×D₄	—	2	1	no	—
60	C₃×D₄	C₂³	1	2	yes	—
61	C₃×D₄	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₄	—	2	1	no	—
63	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
64	#67	—	2	1	no	—
65	#67	—	2	2	no	—
66	#67	—	2	2	no	—
67	#67	—	2	2	no	—
68	#67	D₄	1	2	yes	—
69	C₂×Dic₃	—	2	2	no	—
70	D₁₂	—	2	2	no	—
71	D₁₂	—	2	1	no	—
72	D₁₂	D₄	1	1	yes	characteristic
73	D₁₂	C₂³	1	2	yes	—
74	D₁₂	C₂³	1	1	yes	characteristic
75	C₄×D₃	—	2	1	no	—
76	C₄×D₃	—	2	2	no	—
77	C₄×D₃	C₂³	1	2	yes	—
78	Dic₆	D₄	1	1	yes	characteristic
79	Dic₆	C₂³	1	1	yes	characteristic
80	C₃×C₈	C₂³	1	2	yes	—
81	C₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₃⋊C₈	C₂³	1	1	yes	characteristic
83	#41	—	3	1	no	—
84	#41	—	3	2	no	—
85	#41	—	3	2	no	—
86	#41	—	3	1	no	—
87	#41	—	6	1	no	—
88	#41	—	6	1	no	—
89	#41	C₂×D₃	1	1	yes	characteristic, complemented
90	C₂×D₄	—	3	2	no	—
91	C₂×D₄	—	3	2	no	—
92	C₂×D₄	—	3	1	no	—
93	C₂×D₄	—	6	1	no	—
94	C₂×D₄	—	6	2	no	—
95	C₂×D₄	—	6	1	no	—
96	C₂×D₄	—	6	2	no	—
97	C₂×D₄	C₂×D₃	1	1	yes	characteristic, complemented
98	Q₁₆	—	3	1	no	—
99	SD₁₆	—	3	1	no	—
100	SD₁₆	—	3	1	no	—
101	SD₁₆	—	3	2	no	—
102	SD₁₆	C₂×D₃	1	2	yes	complemented
103	D₈	—	3	2	no	—
104	D₈	—	3	1	no	—
105	D₈	—	3	2	no	—
106	D₈	—	3	2	no	—
107	D₈	C₂×D₃	1	2	yes	complemented
108	M₁₆	—	3	2	no	—
109	M₁₆	C₂×D₃	1	1	yes	characteristic, complemented
110	C₂×C₈	—	3	1	no	—
111	C₂×C₈	—	3	2	no	—
112	C₂²×C₃	—	2	2	no	—
113	C₂²×C₃	—	2	1	no	—
114	C₂²×C₃	—	8	1	no	—
115	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
116	C₂×D₃	—	2	2	no	—
117	C₂×D₃	—	2	1	no	—
118	C₂×D₃	—	8	2	no	—
119	C₂×D₃	—	8	2	no	—
120	C₂×D₃	—	8	1	no	—
121	C₂×D₃	C₂×D₄	1	2	yes	—
122	C₃×C₄	—	2	1	no	—
123	C₃×C₄	C₂×D₄	1	1	yes	characteristic
124	C₃×C₄	C₂⁴	1	1	yes	characteristic
125	Dic₃	—	2	1	no	—
126	Dic₃	C₂×D₄	1	2	yes	—
127	C₂³	—	2	1	no	—
128	C₂³	—	6	2	no	—
129	C₂³	—	6	2	no	—
130	C₂³	—	6	1	no	—
131	Q₈	—	3	1	no	—
132	Q₈	—	3	1	no	—
133	Q₈	C₂²×D₃	1	1	yes	characteristic
134	D₄	—	2	1	no	—
135	D₄	—	2	1	no	—
136	D₄	—	3	2	no	—
137	D₄	—	3	1	no	—
138	D₄	—	3	2	no	—
139	D₄	—	3	1	no	—
140	D₄	—	6	2	no	—
141	D₄	—	6	2	no	—
142	D₄	—	6	2	no	—
143	D₄	—	6	2	no	—
144	D₄	—	6	1	no	—
145	D₄	—	6	1	no	—
146	D₄	C₂²×D₃	1	2	yes	—
147	D₄	C₂²×D₃	1	1	yes	characteristic
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	3	2	no	—
150	C₂×C₄	—	6	2	no	—
151	C₂×C₄	—	6	2	no	—
152	C₂×C₄	—	6	1	no	—
153	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
154	C₈	—	3	1	no	—
155	C₈	—	3	1	no	—
156	C₈	C₂²×D₃	1	2	yes	—
157	C₂×C₃	—	2	1	no	—
158	C₂×C₃	—	4	1	no	—
159	C₂×C₃	—	4	2	no	—
160	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
161	D₃	—	2	2	no	—
162	D₃	—	4	2	no	—
163	D₃	—	4	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	3	2	no	—
167	C₂²	—	6	2	no	—
168	C₂²	—	6	1	no	—
169	C₂²	—	8	1	no	—
170	C₂²	—	24	1	no	—
171	C₂²	—	24	2	no	—
172	C₂²	—	24	2	no	—
173	C₂²	D₃×D₄	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	3	2	no	—
176	C₄	—	6	1	no	—
177	C₄	D₃×D₄	1	1	yes	characteristic
178	C₄	C₂³×D₃	1	1	yes	characteristic
179	C₃	#576	1	1	yes	characteristic, complemented, Sylow
180	C₂	—	2	1	no	—
181	C₂	—	4	2	no	—
182	C₂	—	4	1	no	—
183	C₂	—	6	2	no	—
184	C₂	—	12	2	no	—
185	C₂	—	12	1	no	—
186	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	#576
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. #1009	C₂	semidirect
2. #1004	C₂	semidirect
3. C₃×AGL(1,ℤ/8)	C₂	semidirect
4. #950	C₂	semidirect
5. C₂×#213	C₂	semidirect
6. #916	C₂	semidirect
7. #914	C₂	semidirect
8. #911	C₂	semidirect
9. D₃×D₈	C₂	semidirect
10. #908	C₂	semidirect
11. #907	C₂	semidirect
13. C₃×#41	C₂²	semidirect
14. C₂×C₃×D₄	C₂²	semidirect
16. #239	C₂²	semidirect
18. #237	C₂²	semidirect
21. D₃×D₄	C₂²	semidirect
22. D₃×D₄	C₂²	semidirect
25. #235	C₂²	semidirect
27. C₂×D₁₂	C₂²	semidirect
28. C₃×SD₁₆	C₂²	semidirect
29. C₃×D₈	C₂²	semidirect
30. C₃×M₁₆	C₂²	semidirect
31. #216	C₂²	semidirect
32. #215	C₂²	semidirect
33. #214	C₂²	semidirect
34. #213	C₂²	semidirect
35. #213	C₂²	semidirect
36. #213	C₂²	semidirect
37. C₂×C₃⋊C₈	C₂²	semidirect
38. D₂₄	C₂²	semidirect
39. #204	C₂²	semidirect
40. ΓL(1,𝔽₂₅)	C₂²	semidirect
41. D₃×C₈	C₂²	semidirect
47. AGL(1,ℤ/8)	D₃	semidirect
89. #41	C₂×D₃	semidirect
97. C₂×D₄	C₂×D₃	semidirect
102. SD₁₆	C₂×D₃	semidirect
107. D₈	C₂×D₃	semidirect
109. M₁₆	C₂×D₃	semidirect
179. C₃	#576	semidirect

Maximal subgroups

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

subgroup	index
#1009	2
#1004	2
C₃×AGL(1,ℤ/8)	2
#950	2
C₂×#213	2
#916	2
#914	2
#911	2
D₃×D₈	2
#908	2
#907	2
#576	3