Subgroups — C₂³×C₃×#108 #10678717

Order 768.

2846 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×M₁₆	C₂	1	2	yes	maximal
2	C₂⁴×C₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×#108	C₂	1	28	yes	maximal, direct product factor
4	C₂³×#108	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
6	C₂²×C₃×M₁₆	—	2	14	no	—
7	C₂²×C₃×M₁₆	C₂²	1	28	yes	—
8	C₂³×C₃×C₈	—	2	2	no	—
9	C₂³×C₃×C₈	C₄	1	2	yes	—
10	C₂³×C₃×C₈	C₄	1	14	yes	—
11	C₂³×C₃×C₈	C₂²	1	14	yes	—
12	C₂×C₃×#108	C₂²	1	112	yes	direct product factor
13	C₂³×M₁₆	C₂×C₃	1	2	yes	—
14	C₂⁴×C₈	C₂×C₃	1	1	yes	characteristic
15	C₂²×#108	C₂×C₃	1	28	yes	direct product factor
16	C₂⁵×C₃	Q₈	1	1	yes	characteristic
17	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₃×C₄	C₂×C₄	1	7	yes	—
19	C₂³×C₃×C₄	D₄	1	7	yes	—
20	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
21	C₂³×C₃×C₄	Q₈	1	7	yes	—
22	C₂³×C₃×C₄	C₂³	1	7	yes	—
23	C₂×C₃×M₁₆	—	2	84	no	—
24	C₂×C₃×M₁₆	C₂³	1	56	yes	—
25	C₂²×C₃×C₈	—	2	28	no	—
26	C₂²×C₃×C₈	C₂×C₄	1	84	yes	—
27	C₂²×C₃×C₈	C₂×C₄	1	28	yes	—
28	C₂²×C₃×C₈	C₂³	1	28	yes	—
29	C₃×#108	C₂³	1	64	yes	direct product factor
30	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
31	C₂²×M₁₆	—	2	14	no	—
32	C₂²×M₁₆	C₂²×C₃	1	28	yes	—
33	C₂³×C₈	—	2	2	no	—
34	C₂³×C₈	C₃×C₄	1	14	yes	—
35	C₂³×C₈	C₃×C₄	1	2	yes	—
36	C₂³×C₈	C₂²×C₃	1	14	yes	—
37	C₂×#108	C₂²×C₃	1	112	yes	direct product factor
38	C₂⁴×C₃	—	2	8	no	—
39	C₂⁴×C₃	C₄⋊C₄	1	7	yes	—
40	C₂⁴×C₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂⁴×C₃	C₂×Q₈	1	7	yes	—
42	C₂²×C₃×C₄	C₄⋊C₄	1	42	yes	—
43	C₂²×C₃×C₄	C₄⋊C₄	1	7	yes	—
44	C₂²×C₃×C₄	C₄⋊C₄	1	7	yes	—
45	C₂²×C₃×C₄	C₂²×C₄	1	21	yes	—
46	C₂²×C₃×C₄	C₂²×C₄	1	7	yes	—
47	C₂²×C₃×C₄	C₂×D₄	1	7	yes	—
48	C₂²×C₃×C₄	C₂×D₄	1	21	yes	—
49	C₂²×C₃×C₄	C₂×Q₈	1	21	yes	—
50	C₂²×C₃×C₄	C₂⁴	1	7	yes	—
51	C₃×M₁₆	—	2	56	no	—
52	C₃×M₁₆	C₂⁴	1	16	yes	—
53	C₂×C₃×C₈	—	2	56	no	—
54	C₂×C₃×C₈	C₂²×C₄	1	56	yes	—
55	C₂×C₃×C₈	C₂²×C₄	1	56	yes	—
56	C₂×C₃×C₈	C₂⁴	1	8	yes	—
57	C₂⁵	C₃×Q₈	1	1	yes	characteristic
58	C₂³×C₄	C₂×C₃×C₄	1	7	yes	—
59	C₂³×C₄	C₂×C₃×C₄	1	1	yes	characteristic
60	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
61	C₂³×C₄	C₃×D₄	1	7	yes	—
62	C₂³×C₄	C₃×Q₈	1	7	yes	—
63	C₂³×C₄	C₂³×C₃	1	7	yes	—
64	C₂×M₁₆	—	2	84	no	—
65	C₂×M₁₆	C₂³×C₃	1	56	yes	—
66	C₂²×C₈	—	2	28	no	—
67	C₂²×C₈	C₂×C₃×C₄	1	84	yes	—
68	C₂²×C₈	C₂×C₃×C₄	1	28	yes	—
69	C₂²×C₈	C₂³×C₃	1	28	yes	—
70	#108	C₂³×C₃	1	64	yes	direct product factor
71	C₂³×C₃	—	2	28	no	—
72	C₂³×C₃	—	2	28	no	—
73	C₂³×C₃	#108	1	8	yes	direct product factor
74	C₂³×C₃	C₂×C₄⋊C₄	1	21	yes	—
75	C₂³×C₃	C₂×C₄⋊C₄	1	7	yes	—
76	C₂³×C₃	C₂²×Q₈	1	7	yes	—
77	C₂×C₃×C₄	C₂×C₄⋊C₄	1	42	yes	—
78	C₂×C₃×C₄	C₂×C₄⋊C₄	1	21	yes	—
79	C₂×C₃×C₄	C₂×C₄⋊C₄	1	21	yes	—
80	C₂×C₃×C₄	C₂³×C₄	1	7	yes	—
81	C₂×C₃×C₄	C₂³×C₄	1	7	yes	—
82	C₂×C₃×C₄	C₂²×D₄	1	7	yes	—
83	C₂×C₃×C₄	C₂²×D₄	1	7	yes	—
84	C₂×C₃×C₄	C₂²×Q₈	1	7	yes	—
85	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
86	C₃×C₈	—	2	16	no	—
87	C₃×C₈	C₂³×C₄	1	16	yes	—
88	C₂⁴	—	2	8	no	—
89	C₂⁴	C₃×C₄⋊C₄	1	7	yes	—
90	C₂⁴	C₃×C₄⋊C₄	1	1	yes	characteristic
91	C₂⁴	C₂×C₃×Q₈	1	7	yes	—
92	C₂²×C₄	C₃×C₄⋊C₄	1	42	yes	—
93	C₂²×C₄	C₃×C₄⋊C₄	1	7	yes	—
94	C₂²×C₄	C₃×C₄⋊C₄	1	7	yes	—
95	C₂²×C₄	C₂²×C₃×C₄	1	21	yes	—
96	C₂²×C₄	C₂²×C₃×C₄	1	7	yes	—
97	C₂²×C₄	C₂×C₃×D₄	1	7	yes	—
98	C₂²×C₄	C₂×C₃×D₄	1	21	yes	—
99	C₂²×C₄	C₂×C₃×Q₈	1	21	yes	—
100	C₂²×C₄	C₂⁴×C₃	1	7	yes	—
101	M₁₆	—	2	56	no	—
102	M₁₆	C₂⁴×C₃	1	16	yes	—
103	C₂×C₈	—	2	56	no	—
104	C₂×C₈	C₂²×C₃×C₄	1	56	yes	—
105	C₂×C₈	C₂²×C₃×C₄	1	56	yes	—
106	C₂×C₈	C₂⁴×C₃	1	8	yes	—
107	C₂²×C₃	—	2	42	no	—
108	C₂²×C₃	—	2	14	no	—
109	C₂²×C₃	C₂×#108	1	28	yes	direct product factor
110	C₂²×C₃	C₂²×C₄⋊C₄	1	7	yes	—
111	C₂²×C₃	C₂²×C₄⋊C₄	1	7	yes	—
112	C₂²×C₃	C₂³×Q₈	1	1	yes	characteristic
113	C₃×C₄	C₂²×C₄⋊C₄	1	7	yes	—
114	C₃×C₄	C₂²×C₄⋊C₄	1	7	yes	—
115	C₃×C₄	C₂⁴×C₄	1	1	yes	characteristic
116	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
117	C₂³	—	2	28	no	—
118	C₂³	—	2	28	no	—
119	C₂³	C₃×#108	1	8	yes	direct product factor
120	C₂³	C₂×C₃×C₄⋊C₄	1	21	yes	—
121	C₂³	C₂×C₃×C₄⋊C₄	1	7	yes	—
122	C₂³	C₂²×C₃×Q₈	1	7	yes	—
123	C₂×C₄	C₂×C₃×C₄⋊C₄	1	42	yes	—
124	C₂×C₄	C₂×C₃×C₄⋊C₄	1	21	yes	—
125	C₂×C₄	C₂×C₃×C₄⋊C₄	1	21	yes	—
126	C₂×C₄	C₂³×C₃×C₄	1	7	yes	—
127	C₂×C₄	C₂³×C₃×C₄	1	7	yes	—
128	C₂×C₄	C₂²×C₃×D₄	1	7	yes	—
129	C₂×C₄	C₂²×C₃×D₄	1	7	yes	—
130	C₂×C₄	C₂²×C₃×Q₈	1	7	yes	—
131	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
132	C₈	—	2	16	no	—
133	C₈	C₂³×C₃×C₄	1	16	yes	—
134	C₂×C₃	—	2	7	no	—
135	C₂×C₃	—	2	1	no	—
136	C₂×C₃	C₂²×#108	1	14	yes	direct product factor
137	C₂×C₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
138	C₂²	—	2	42	no	—
139	C₂²	—	2	14	no	—
140	C₂²	C₂×C₃×#108	1	28	yes	direct product factor
141	C₂²	C₂²×C₃×C₄⋊C₄	1	7	yes	—
142	C₂²	C₂²×C₃×C₄⋊C₄	1	7	yes	—
143	C₂²	C₂³×C₃×Q₈	1	1	yes	characteristic
144	C₄	C₂²×C₃×C₄⋊C₄	1	7	yes	—
145	C₄	C₂²×C₃×C₄⋊C₄	1	7	yes	—
146	C₄	C₂⁴×C₃×C₄	1	1	yes	characteristic
147	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
148	C₃	C₂³×#108	1	1	yes	characteristic, direct product factor, Sylow
149	C₂	—	2	1	no	—
150	C₂	—	2	7	no	—
151	C₂	C₂²×C₃×#108	1	14	yes	direct product factor
152	C₂	C₂³×C₃×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃×C₄	D₄
commutator	C₄	C₂⁴×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁵×C₃
socle	C₂⁴×C₃	C₄⋊C₄
prime radical	C₂⁵×C₃	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₄ ◁ C₂⁴×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₃×#108	C₂	direct (N × H)
4. C₂³×#108	C₃	direct (N × H)
12. C₂×C₃×#108	C₂²	direct (N × H)
15. C₂²×#108	C₂×C₃	direct (N × H)
29. C₃×#108	C₂³	direct (N × H)
37. C₂×#108	C₂²×C₃	direct (N × H)
70. #108	C₂³×C₃	direct (N × H)
73. C₂³×C₃	#108	direct (N × H)
109. C₂²×C₃	C₂×#108	direct (N × H)
119. C₂³	C₃×#108	direct (N × H)
136. C₂×C₃	C₂²×#108	direct (N × H)
140. C₂²	C₂×C₃×#108	direct (N × H)
148. C₃	C₂³×#108	direct (N × H)
151. C₂	C₂²×C₃×#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
C₂³×C₃×M₁₆	2
C₂⁴×C₃×C₈	2
C₂²×C₃×#108	2
C₂³×#108	3

Subgroups — C23×C3×#108 #10678717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×#108 #10678717