Subgroups — C₂×D₃×#108 #79250

Order 384.

480 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×M₁₆	C₂	1	2	yes	maximal
2	C₂²×D₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#822	C₂	1	2	yes	maximal, complemented
5	C₂×#819	C₂	1	1	yes	maximal, characteristic, complemented
6	D₃×#108	C₂	1	8	yes	maximal, direct product factor
7	C₂²×#108	—	3	1	no	maximal, Sylow
8	C₂²×C₄×D₃	C₂²	1	1	yes	characteristic
9	C₂×C₃×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×#208	C₂²	1	2	yes	—
12	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	D₃×M₁₆	—	2	4	no	—
14	D₃×M₁₆	C₂²	1	8	yes	—
15	C₂×ΓL(1,𝔽₂₅)	—	2	2	no	—
16	C₂×D₃×C₈	—	2	2	no	—
17	C₂×D₃×C₈	C₄	1	2	yes	—
18	C₂×D₃×C₈	C₄	1	4	yes	—
19	C₂×D₃×C₈	C₄	1	2	yes	—
20	C₂×D₃×C₈	C₂²	1	4	yes	—
21	C₃×#108	C₂²	1	4	yes	complemented
22	#822	C₂²	1	8	yes	complemented
23	#819	C₂²	1	4	yes	complemented
24	C₂²×M₁₆	—	3	2	no	—
25	C₂³×C₈	—	3	1	no	—
26	C₂×#108	—	3	2	no	—
27	C₂×#108	—	3	1	no	—
28	C₂×#108	—	3	8	no	—
29	C₂×#108	D₃	1	1	yes	characteristic, direct product factor
30	C₂³×D₃	Q₈	1	1	yes	characteristic
31	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂²×Dic₃	Q₈	1	1	yes	characteristic
33	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₃	D₄	1	1	yes	characteristic
37	C₂×C₄×D₃	D₄	1	1	yes	characteristic
38	C₂×C₄×D₃	D₄	1	2	yes	—
39	C₂×C₄×D₃	Q₈	1	2	yes	—
40	C₂×C₄×D₃	C₂³	1	2	yes	—
41	C₃×M₁₆	—	2	2	no	—
42	C₃×M₁₆	C₂³	1	4	yes	—
43	C₂×C₃×C₈	—	2	2	no	—
44	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
45	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
46	C₂×C₃×C₈	C₂³	1	2	yes	—
47	#208	—	2	2	no	—
48	#208	C₂³	1	4	yes	—
49	C₂×C₃⋊C₈	—	2	2	no	—
50	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
51	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
52	C₂×C₃⋊C₈	C₂³	1	2	yes	—
53	ΓL(1,𝔽₂₅)	—	2	8	no	—
54	D₃×C₈	—	2	8	no	—
55	D₃×C₈	C₂×C₄	1	8	yes	—
56	D₃×C₈	C₂×C₄	1	8	yes	—
57	C₂³×C₄	—	3	1	no	—
58	C₂×M₁₆	—	3	2	no	—
59	C₂×M₁₆	—	3	8	no	—
60	C₂×M₁₆	—	6	2	no	—
61	C₂×M₁₆	—	6	4	no	—
62	C₂×M₁₆	C₂×D₃	1	2	yes	—
63	C₂²×C₈	—	3	1	no	—
64	C₂²×C₈	—	3	2	no	—
65	C₂²×C₈	—	3	2	no	—
66	C₂²×C₈	—	3	4	no	—
67	C₂²×C₈	—	3	4	no	—
68	C₂²×C₈	—	6	2	no	—
69	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
70	#108	—	3	8	no	—
71	#108	—	3	4	no	—
72	#108	C₂×D₃	1	4	yes	direct product factor
73	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
74	C₂²×D₃	—	2	4	no	—
75	C₂²×D₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×D₃	C₄⋊C₄	1	2	yes	—
77	C₂²×D₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×D₃	C₂×Q₈	1	2	yes	—
79	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
85	C₂×Dic₃	C₄⋊C₄	1	2	yes	—
86	C₂×Dic₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂×Dic₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂×Dic₃	C₂×Q₈	1	2	yes	—
89	C₄×D₃	C₄⋊C₄	1	2	yes	—
90	C₄×D₃	C₄⋊C₄	1	2	yes	—
91	C₄×D₃	C₄⋊C₄	1	2	yes	—
92	C₄×D₃	C₄⋊C₄	1	2	yes	—
93	C₄×D₃	C₂²×C₄	1	2	yes	—
94	C₄×D₃	C₂²×C₄	1	2	yes	—
95	C₄×D₃	C₂×D₄	1	2	yes	—
96	C₄×D₃	C₂×D₄	1	2	yes	—
97	C₃×C₈	—	2	4	no	—
98	C₃×C₈	C₂²×C₄	1	4	yes	—
99	C₃⋊C₈	—	2	4	no	—
100	C₃⋊C₈	C₂²×C₄	1	4	yes	—
101	C₂⁴	—	3	1	no	—
102	C₂²×C₄	—	3	1	no	—
103	C₂²×C₄	—	3	1	no	—
104	C₂²×C₄	—	3	2	no	—
105	C₂²×C₄	—	3	1	no	—
106	C₂²×C₄	—	3	2	no	—
107	C₂²×C₄	—	3	1	no	—
108	C₂²×C₄	—	3	1	no	—
109	C₂²×C₄	—	3	2	no	—
110	C₂²×C₄	—	3	2	no	—
111	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	—	3	4	no	—
114	M₁₆	—	6	8	no	—
115	M₁₆	—	6	2	no	—
116	M₁₆	C₂²×D₃	1	4	yes	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	3	2	no	—
119	C₂×C₈	—	3	2	no	—
120	C₂×C₈	—	3	2	no	—
121	C₂×C₈	—	3	8	no	—
122	C₂×C₈	—	3	8	no	—
123	C₂×C₈	—	6	8	no	—
124	C₂×C₈	—	6	2	no	—
125	C₂×C₈	C₄×D₃	1	2	yes	—
126	C₂×C₈	C₄×D₃	1	2	yes	—
127	C₂×C₈	C₂²×D₃	1	2	yes	—
128	C₂²×C₃	—	2	2	no	—
129	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
132	C₂×D₃	—	2	2	no	—
133	C₂×D₃	—	2	4	no	—
134	C₂×D₃	—	2	4	no	—
135	C₂×D₃	#108	1	4	yes	direct product factor
136	C₂×D₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×D₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
141	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
142	Dic₃	C₂×C₄⋊C₄	1	2	yes	—
143	Dic₃	C₂×C₄⋊C₄	1	2	yes	—
144	C₂³	—	3	1	no	—
145	C₂³	—	3	2	no	—
146	C₂³	—	3	1	no	—
147	C₂³	—	3	2	no	—
148	C₂³	—	6	4	no	—
149	C₂³	D₃×Q₈	1	1	yes	characteristic
150	C₂×C₄	—	3	2	no	—
151	C₂×C₄	—	3	1	no	—
152	C₂×C₄	—	3	1	no	—
153	C₂×C₄	—	3	2	no	—
154	C₂×C₄	—	3	2	no	—
155	C₂×C₄	—	3	2	no	—
156	C₂×C₄	—	3	2	no	—
157	C₂×C₄	—	3	2	no	—
158	C₂×C₄	—	3	2	no	—
159	C₂×C₄	—	3	2	no	—
160	C₂×C₄	—	3	2	no	—
161	C₂×C₄	—	3	2	no	—
162	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
164	C₂×C₄	D₃×D₄	1	1	yes	characteristic
165	C₂×C₄	D₃×D₄	1	1	yes	characteristic
166	C₂×C₄	D₃×Q₈	1	1	yes	characteristic
167	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
168	C₈	—	2	4	no	—
169	C₈	—	3	4	no	—
170	C₈	—	6	4	no	—
171	C₈	C₂×C₄×D₃	1	4	yes	—
172	C₂×C₃	—	2	1	no	—
173	C₂×C₃	—	2	1	no	—
174	C₂×C₃	C₂×#108	1	2	yes	complemented
175	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
176	D₃	—	2	2	no	—
177	D₃	C₂×#108	1	4	yes	direct product factor
178	C₂²	—	2	2	no	—
179	C₂²	—	3	4	no	—
180	C₂²	—	3	2	no	—
181	C₂²	—	3	2	no	—
182	C₂²	—	6	4	no	—
183	C₂²	—	6	4	no	—
184	C₂²	—	6	2	no	—
185	C₂²	D₃×C₄⋊C₄	1	1	yes	characteristic
186	C₂²	D₃×C₄⋊C₄	1	1	yes	characteristic
187	C₂²	C₂×D₃×Q₈	1	1	yes	characteristic
188	C₄	—	3	2	no	—
189	C₄	—	3	2	no	—
190	C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
191	C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
192	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
193	C₄	C₂×D₃×D₄	1	1	yes	characteristic
194	C₃	C₂²×#108	1	1	yes	characteristic, complemented, Sylow
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	—	3	4	no	—
198	C₂	—	6	2	no	—
199	C₂	D₃×#108	1	2	yes	direct product factor
200	C₂	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×#108	C₂
socle	C₂²×C₃	C₂×C₄⋊C₄
prime radical	C₂³×D₃	Q₈
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×#108	D₃
hypocenter	C₃	C₂²×#108
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#108	C₂	semidirect
4. C₂×#822	C₂	semidirect
5. C₂×#819	C₂	semidirect
6. D₃×#108	C₂	direct (N × H)
21. C₃×#108	C₂²	semidirect
22. #822	C₂²	semidirect
23. #819	C₂²	semidirect
29. C₂×#108	D₃	direct (N × H)
72. #108	C₂×D₃	direct (N × H)
135. C₂×D₃	#108	direct (N × H)
174. C₂×C₃	C₂×#108	semidirect
177. D₃	C₂×#108	direct (N × H)
194. C₃	C₂²×#108	semidirect
199. C₂	D₃×#108	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₃×M₁₆	2
C₂²×D₃×C₈	2
C₂×C₃×#108	2
C₂×#822	2
C₂×#819	2
D₃×#108	2
C₂²×#108	3

Subgroups — C2×D3×#108 #79250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×#108 #79250