Finite Group Databasebeta

← C₃×#2194 #72909

Subgroups — C₃×#2194 #72909

Order 384.

216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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	C₃×#576	C₂	1	1	yes	maximal, characteristic
2	C₃×#510	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#473	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#359	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#350	C₂	1	1	yes	maximal, characteristic, complemented
8	#2194	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₃×H(5,𝔽₂)	—	2	1	no	—
10	C₃×#137	—	2	1	no	—
11	C₃×AGL(1,ℤ/8)	—	2	2	no	—
12	C₃×AGL(1,ℤ/8)	—	2	1	no	—
13	C₃×#135	—	2	1	no	—
14	C₃×#135	C₂²	1	1	yes	characteristic
15	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₃×D₈	—	2	1	no	—
17	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
18	C₃×#131	C₂²	1	1	yes	characteristic
19	C₃×Q₃₂	—	2	1	no	—
20	C₃×SD₃₂	—	2	2	no	—
21	C₃×SD₃₂	—	2	1	no	—
22	C₃×M₃₂	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic, complemented
24	C₃×#108	C₂²	1	1	yes	characteristic, complemented
25	C₃×#101	—	2	1	no	—
26	#576	C₂×C₃	1	1	yes	characteristic
27	#510	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×SD₃₂	C₂×C₃	1	1	yes	characteristic, complemented
29	#473	C₂×C₃	1	1	yes	characteristic, complemented
30	#361	C₂×C₃	1	1	yes	characteristic, complemented
31	#359	C₂×C₃	1	1	yes	characteristic, complemented
32	#350	C₂×C₃	1	1	yes	characteristic, complemented
33	C₃×#41	—	2	2	no	—
34	C₃×#41	—	2	1	no	—
35	C₃×#41	—	4	1	no	—
36	C₃×#41	D₄	1	1	yes	characteristic
37	C₂×C₃×Q₈	—	2	1	no	—
38	C₂×C₃×D₄	—	2	2	no	—
39	C₂×C₃×D₄	—	2	1	no	—
40	C₂×C₃×D₄	—	4	2	no	—
41	C₂×C₃×D₄	—	4	1	no	—
42	C₃×Q₁₆	—	2	1	no	—
43	C₃×Q₁₆	—	2	1	no	—
44	C₃×Q₁₆	—	4	1	no	—
45	C₃×Q₁₆	D₄	1	1	yes	characteristic
46	C₃×SD₁₆	—	2	2	no	—
47	C₃×SD₁₆	—	2	1	no	—
48	C₃×SD₁₆	—	4	1	no	—
49	C₃×D₈	—	2	2	no	—
50	C₃×D₈	—	2	1	no	—
51	C₃×D₈	D₄	1	1	yes	characteristic
52	C₃×D₈	D₄	1	2	yes	—
53	C₃×M₁₆	—	2	1	no	—
54	C₃×M₁₆	—	2	1	no	—
55	C₃×M₁₆	D₄	1	1	yes	characteristic
56	C₂×C₃×C₈	—	2	1	no	—
57	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
58	C₃×C₁₆	—	2	1	no	—
59	C₃×C₁₆	—	2	1	no	—
60	H(5,𝔽₂)	—	2	1	no	—
61	#137	—	2	1	no	—
62	AGL(1,ℤ/8)	—	2	2	no	—
63	AGL(1,ℤ/8)	—	2	1	no	—
64	#135	—	2	1	no	—
65	#135	C₂²×C₃	1	1	yes	characteristic
66	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
67	C₂×D₈	—	2	1	no	—
68	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
69	#131	C₂²×C₃	1	1	yes	characteristic
70	Q₃₂	—	2	1	no	—
71	SD₃₂	—	2	2	no	—
72	SD₃₂	—	2	1	no	—
73	M₃₂	C₂²×C₃	1	1	yes	characteristic, complemented
74	C₂×C₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
75	#108	C₂²×C₃	1	1	yes	characteristic, complemented
76	#101	—	2	1	no	—
77	C₂³×C₃	—	4	2	no	—
78	C₂³×C₃	—	4	1	no	—
79	C₃×Q₈	—	2	1	no	—
80	C₃×Q₈	—	4	1	no	—
81	C₃×Q₈	—	4	1	no	—
82	C₃×Q₈	D₈	1	1	yes	characteristic
83	C₃×D₄	—	2	1	no	—
84	C₃×D₄	—	2	1	no	—
85	C₃×D₄	—	2	2	no	—
86	C₃×D₄	—	4	2	no	—
87	C₃×D₄	—	4	2	no	—
88	C₃×D₄	—	4	1	no	—
89	C₃×D₄	—	4	1	no	—
90	C₃×D₄	D₈	1	1	yes	characteristic
91	C₂×C₃×C₄	—	2	1	no	—
92	C₂×C₃×C₄	—	4	2	no	—
93	C₂×C₃×C₄	—	4	1	no	—
94	C₂×C₃×C₄	—	4	1	no	—
95	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
96	C₃×C₈	—	2	1	no	—
97	C₃×C₈	—	4	1	no	—
98	C₃×C₈	C₂×D₄	1	1	yes	characteristic
99	C₃×C₈	C₂×D₄	1	1	yes	characteristic
100	#41	—	2	2	no	—
101	#41	—	2	1	no	—
102	#41	—	4	1	no	—
103	#41	C₃×D₄	1	1	yes	characteristic
104	C₂×Q₈	—	2	1	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	4	2	no	—
108	C₂×D₄	—	4	1	no	—
109	Q₁₆	—	2	1	no	—
110	Q₁₆	—	2	1	no	—
111	Q₁₆	—	4	1	no	—
112	Q₁₆	C₃×D₄	1	1	yes	characteristic
113	SD₁₆	—	2	2	no	—
114	SD₁₆	—	2	1	no	—
115	SD₁₆	—	4	1	no	—
116	D₈	—	2	2	no	—
117	D₈	—	2	1	no	—
118	D₈	C₃×D₄	1	1	yes	characteristic
119	D₈	C₃×D₄	1	2	yes	—
120	M₁₆	—	2	1	no	—
121	M₁₆	—	2	1	no	—
122	M₁₆	C₃×D₄	1	1	yes	characteristic
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	2	1	no	—
127	C₂²×C₃	—	2	1	no	—
128	C₂²×C₃	—	4	2	no	—
129	C₂²×C₃	—	4	1	no	—
130	C₂²×C₃	—	16	1	no	—
131	C₂²×C₃	—	16	2	no	—
132	C₂²×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₃×C₄	—	2	1	no	—
134	C₃×C₄	—	4	1	no	—
135	C₃×C₄	—	8	1	no	—
136	C₃×C₄	#120	1	1	yes	characteristic
137	C₃×C₄	C₂×D₈	1	1	yes	characteristic
138	C₂³	—	4	2	no	—
139	C₂³	—	4	1	no	—
140	Q₈	—	2	1	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	4	1	no	—
143	Q₈	C₃×D₈	1	1	yes	characteristic
144	D₄	—	2	1	no	—
145	D₄	—	2	1	no	—
146	D₄	—	2	2	no	—
147	D₄	—	4	2	no	—
148	D₄	—	4	2	no	—
149	D₄	—	4	1	no	—
150	D₄	—	4	1	no	—
151	D₄	C₃×D₈	1	1	yes	characteristic
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	4	1	no	—
159	C₈	C₂×C₃×D₄	1	1	yes	characteristic
160	C₈	C₂×C₃×D₄	1	1	yes	characteristic
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	4	1	no	—
163	C₂×C₃	—	8	2	no	—
164	C₂×C₃	—	8	1	no	—
165	C₂×C₃	#447	1	1	yes	characteristic
166	C₂²	—	2	1	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	16	1	no	—
170	C₂²	—	16	2	no	—
171	C₂²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	4	1	no	—
174	C₄	—	8	1	no	—
175	C₄	C₃×#120	1	1	yes	characteristic
176	C₄	C₂×C₃×D₈	1	1	yes	characteristic
177	C₃	#2194	1	1	yes	characteristic, direct product factor, Sylow
178	C₂	—	2	1	no	—
179	C₂	—	4	1	no	—
180	C₂	—	8	2	no	—
181	C₂	—	8	1	no	—
182	C₂	C₃×#447	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	#447
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃
prime radical	C₃×#576	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#510	C₂	semidirect
3. C₂×C₃×SD₃₂	C₂	semidirect
4. C₃×#473	C₂	semidirect
5. C₃×#361	C₂	semidirect
6. C₃×#359	C₂	semidirect
7. C₃×#350	C₂	semidirect
8. #2194	C₃	direct (N × H)
15. C₂×C₃×Q₁₆	C₂²	semidirect
22. C₃×M₃₂	C₂²	semidirect
23. C₂×C₃×C₁₆	C₂²	semidirect
24. C₃×#108	C₂²	semidirect
27. #510	C₂×C₃	semidirect
28. C₂×SD₃₂	C₂×C₃	semidirect
29. #473	C₂×C₃	semidirect
30. #361	C₂×C₃	semidirect
31. #359	C₂×C₃	semidirect
32. #350	C₂×C₃	semidirect
66. C₂×Q₁₆	C₂²×C₃	semidirect
73. M₃₂	C₂²×C₃	semidirect
74. C₂×C₁₆	C₂²×C₃	semidirect
75. #108	C₂²×C₃	semidirect
177. C₃	#2194	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#576	2
C₃×#510	2
C₂×C₃×SD₃₂	2
C₃×#473	2
C₃×#361	2
C₃×#359	2
C₃×#350	2
#2194	3