Finite Group Databasebeta

Subgroups — #5779

Order 192.

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

normal subgroups only

This group has 248 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	#1010	C₂	1	1	yes	maximal, characteristic, complemented
2	#1007	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	#950	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#216	C₂	1	1	yes	maximal, characteristic, complemented
6	#918	C₂	1	2	yes	maximal, complemented
7	D₃×Q₁₆	C₂	1	2	yes	maximal, complemented
8	#916	C₂	1	2	yes	maximal, complemented
9	#915	C₂	1	2	yes	maximal, complemented
10	#909	C₂	1	1	yes	maximal, characteristic, complemented
11	#907	C₂	1	1	yes	maximal, characteristic, complemented
12	#578	—	3	1	no	maximal, Sylow
13	C₃×#41	C₂²	1	1	yes	characteristic
14	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
15	#239	—	2	1	no	—
16	#239	C₂²	1	2	yes	—
17	D₃×Q₈	—	2	1	no	—
18	D₃×Q₈	C₂²	1	2	yes	—
19	D₃×Q₈	C₂²	1	2	yes	—
20	#237	—	2	2	no	—
21	#237	C₂²	1	2	yes	—
22	#235	—	2	2	no	—
23	#235	—	2	1	no	—
24	#235	C₂²	1	1	yes	characteristic
25	#235	C₂²	1	1	yes	characteristic
26	C₂×Dic₆	—	2	1	no	—
27	C₂×Dic₆	C₂²	1	1	yes	characteristic
28	C₃×Q₁₆	C₂²	1	2	yes	—
29	C₃×SD₁₆	C₂²	1	2	yes	—
30	C₃×M₁₆	C₂²	1	1	yes	characteristic
31	#216	C₂²	1	2	yes	—
32	#216	C₂²	1	2	yes	—
33	#216	C₂²	1	1	yes	characteristic
34	#215	C₂²	1	1	yes	characteristic
35	#214	C₂²	1	1	yes	characteristic
36	#213	C₂²	1	1	yes	characteristic
37	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
38	Dic₁₂	C₂²	1	2	yes	—
39	#204	C₂²	1	2	yes	—
40	ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
41	D₃×C₈	C₂²	1	2	yes	—
42	2₋¹⁺⁴	—	3	1	no	—
43	2₋¹⁺⁴	—	3	1	no	—
44	#137	—	3	2	no	—
45	#137	—	3	2	no	—
46	#137	—	3	1	no	—
47	#137	D₃	1	1	yes	characteristic, complemented
48	#135	—	3	1	no	—
49	#135	—	3	2	no	—
50	C₂×Q₁₆	—	3	1	no	—
51	C₂×Q₁₆	—	3	2	no	—
52	#131	—	3	1	no	—
53	C₃×Q₈	—	2	1	no	—
54	C₃×Q₈	C₂³	1	1	yes	characteristic
55	C₃×Q₈	C₂³	1	2	yes	—
56	C₃×D₄	—	2	1	no	—
57	C₃×D₄	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₄	—	2	1	no	—
59	C₂×C₃×C₄	—	2	1	no	—
60	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
61	#67	—	2	1	no	—
62	#67	—	2	2	no	—
63	#67	D₄	1	2	yes	—
64	C₂×Dic₃	—	2	1	no	—
65	C₂×Dic₃	—	2	2	no	—
66	D₁₂	—	2	2	no	—
67	D₁₂	D₄	1	1	yes	characteristic
68	D₁₂	C₂³	1	1	yes	characteristic
69	C₄×D₃	—	2	2	no	—
70	C₄×D₃	—	2	2	no	—
71	C₄×D₃	—	2	1	no	—
72	C₄×D₃	—	2	2	no	—
73	C₄×D₃	C₂³	1	2	yes	—
74	Dic₆	—	2	1	no	—
75	Dic₆	—	2	2	no	—
76	Dic₆	—	2	2	no	—
77	Dic₆	D₄	1	1	yes	characteristic
78	Dic₆	C₂³	1	2	yes	—
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	2	no	—
88	#41	—	6	1	no	—
89	#41	—	6	2	no	—
90	#41	—	6	1	no	—
91	#41	C₂×D₃	1	1	yes	characteristic
92	C₂×Q₈	—	3	2	no	—
93	C₂×Q₈	—	3	2	no	—
94	C₂×Q₈	—	3	1	no	—
95	C₂×Q₈	—	6	1	no	—
96	C₂×Q₈	—	6	1	no	—
97	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
98	Q₁₆	—	3	1	no	—
99	Q₁₆	—	3	2	no	—
100	Q₁₆	—	3	2	no	—
101	Q₁₆	—	3	2	no	—
102	Q₁₆	C₂×D₃	1	2	yes	—
103	SD₁₆	—	3	1	no	—
104	SD₁₆	—	3	1	no	—
105	SD₁₆	—	3	2	no	—
106	SD₁₆	C₂×D₃	1	2	yes	—
107	D₈	—	3	1	no	—
108	M₁₆	—	3	2	no	—
109	M₁₆	C₂×D₃	1	1	yes	characteristic
110	C₂×C₈	—	3	1	no	—
111	C₂×C₈	—	3	2	no	—
112	C₂²×C₃	—	2	1	no	—
113	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
114	C₂×D₃	—	2	1	no	—
115	C₂×D₃	C₂×D₄	1	2	yes	—
116	C₃×C₄	—	2	1	no	—
117	C₃×C₄	—	2	2	no	—
118	C₃×C₄	C₂×D₄	1	1	yes	characteristic
119	C₃×C₄	C₂⁴	1	1	yes	characteristic
120	Dic₃	—	2	1	no	—
121	Dic₃	—	2	2	no	—
122	Dic₃	C₂×D₄	1	2	yes	—
123	Q₈	—	2	1	no	—
124	Q₈	—	3	1	no	—
125	Q₈	—	3	1	no	—
126	Q₈	—	3	2	no	—
127	Q₈	—	6	2	no	—
128	Q₈	—	6	2	no	—
129	Q₈	—	6	1	no	—
130	Q₈	C₂²×D₃	1	1	yes	characteristic
131	Q₈	C₂²×D₃	1	2	yes	—
132	D₄	—	2	1	no	—
133	D₄	—	3	2	no	—
134	D₄	—	3	1	no	—
135	D₄	—	3	1	no	—
136	D₄	—	6	2	no	—
137	D₄	—	6	2	no	—
138	D₄	—	6	1	no	—
139	D₄	C₂²×D₃	1	1	yes	characteristic
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	3	2	no	—
143	C₂×C₄	—	6	2	no	—
144	C₂×C₄	—	6	2	no	—
145	C₂×C₄	—	6	2	no	—
146	C₂×C₄	—	6	2	no	—
147	C₂×C₄	—	6	1	no	—
148	C₂×C₄	—	6	1	no	—
149	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
150	C₈	—	3	1	no	—
151	C₈	—	3	1	no	—
152	C₈	C₂²×D₃	1	2	yes	—
153	C₂×C₃	—	2	1	no	—
154	C₂×C₃	—	4	1	no	—
155	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
156	D₃	—	2	2	no	—
157	D₃	—	4	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	3	2	no	—
160	C₂²	—	6	1	no	—
161	C₂²	D₃×D₄	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	2	no	—
164	C₄	—	3	2	no	—
165	C₄	—	6	1	no	—
166	C₄	—	6	2	no	—
167	C₄	D₃×D₄	1	1	yes	characteristic
168	C₄	C₂³×D₃	1	1	yes	characteristic
169	C₃	#578	1	1	yes	characteristic, complemented, Sylow
170	C₂	—	2	1	no	—
171	C₂	—	4	1	no	—
172	C₂	—	6	2	no	—
173	C₂	—	12	1	no	—
174	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	#578
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. #1010	C₂	semidirect
2. #1007	C₂	semidirect
3. C₃×#137	C₂	semidirect
4. #950	C₂	semidirect
5. C₂×#216	C₂	semidirect
6. #918	C₂	semidirect
7. D₃×Q₁₆	C₂	semidirect
8. #916	C₂	semidirect
9. #915	C₂	semidirect
10. #909	C₂	semidirect
11. #907	C₂	semidirect
47. #137	D₃	semidirect
169. C₃	#578	semidirect

Maximal subgroups

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

subgroup	index
#1010	2
#1007	2
C₃×#137	2
#950	2
C₂×#216	2
#918	2
D₃×Q₁₆	2
#916	2
#915	2
#909	2
#907	2
#578	3