Subgroups — C₂³×#822 #11658875

Order 768.

2846 conjugacy classes of subgroups.

The subgroup lattice has 176 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	1	yes	maximal, characteristic
2	C₂³×#208	C₂	1	1	yes	maximal, characteristic
3	C₂⁴×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×#822	C₂	1	28	yes	maximal, direct product factor
5	C₂³×#108	—	3	1	no	maximal, Sylow
6	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃×M₁₆	—	2	7	no	—
8	C₂²×C₃×M₁₆	C₂²	1	14	yes	—
9	C₂³×C₃×C₈	—	2	1	no	—
10	C₂²×#208	—	2	7	no	—
11	C₂²×#208	C₂²	1	14	yes	—
12	C₂³×C₃⋊C₈	—	2	1	no	—
13	C₂³×C₃⋊C₈	C₄	1	14	yes	—
14	C₂³×C₃⋊C₈	C₄	1	2	yes	—
15	C₂³×C₃⋊C₈	C₂²	1	14	yes	—
16	C₂×#822	C₂²	1	112	yes	direct product factor
17	C₂³×M₁₆	—	3	1	no	—
18	C₂³×M₁₆	D₃	1	1	yes	characteristic
19	C₂⁴×C₈	—	3	1	no	—
20	C₂²×#108	—	3	28	no	—
21	C₂⁵×C₃	Q₈	1	1	yes	characteristic
22	C₂³×C₃×C₄	C₂×C₄	1	7	yes	—
23	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₃×C₄	D₄	1	7	yes	—
25	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
26	C₂³×C₃×C₄	Q₈	1	7	yes	—
27	C₂³×C₃×C₄	C₂³	1	7	yes	—
28	C₂×C₃×M₁₆	—	2	42	no	—
29	C₂×C₃×M₁₆	C₂³	1	28	yes	—
30	C₂²×C₃×C₈	—	2	14	no	—
31	C₂×#208	—	2	42	no	—
32	C₂×#208	C₂³	1	28	yes	—
33	C₂²×C₃⋊C₈	—	2	14	no	—
34	C₂²×C₃⋊C₈	C₂×C₄	1	84	yes	—
35	C₂²×C₃⋊C₈	C₂×C₄	1	28	yes	—
36	C₂²×C₃⋊C₈	C₂³	1	28	yes	—
37	#822	C₂³	1	64	yes	direct product factor
38	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
39	C₂²×M₁₆	—	2	7	no	—
40	C₂²×M₁₆	—	3	14	no	—
41	C₂²×M₁₆	—	6	7	no	—
42	C₂²×M₁₆	C₂×D₃	1	14	yes	—
43	C₂³×C₈	—	2	1	no	—
44	C₂³×C₈	—	3	2	no	—
45	C₂³×C₈	—	3	14	no	—
46	C₂³×C₈	—	3	14	no	—
47	C₂³×C₈	—	6	1	no	—
48	C₂×#108	—	3	112	no	—
49	C₂⁴×C₃	—	2	8	no	—
50	C₂⁴×C₃	C₄⋊C₄	1	7	yes	—
51	C₂⁴×C₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂⁴×C₃	C₂×Q₈	1	7	yes	—
53	C₂²×C₃×C₄	C₄⋊C₄	1	42	yes	—
54	C₂²×C₃×C₄	C₄⋊C₄	1	7	yes	—
55	C₂²×C₃×C₄	C₄⋊C₄	1	7	yes	—
56	C₂²×C₃×C₄	C₂²×C₄	1	21	yes	—
57	C₂²×C₃×C₄	C₂²×C₄	1	7	yes	—
58	C₂²×C₃×C₄	C₂×D₄	1	7	yes	—
59	C₂²×C₃×C₄	C₂×D₄	1	21	yes	—
60	C₂²×C₃×C₄	C₂×Q₈	1	21	yes	—
61	C₂²×C₃×C₄	C₂⁴	1	7	yes	—
62	C₃×M₁₆	—	2	28	no	—
63	C₃×M₁₆	C₂⁴	1	8	yes	—
64	C₂×C₃×C₈	—	2	28	no	—
65	#208	—	2	28	no	—
66	#208	C₂⁴	1	8	yes	—
67	C₂×C₃⋊C₈	—	2	28	no	—
68	C₂×C₃⋊C₈	C₂²×C₄	1	56	yes	—
69	C₂×C₃⋊C₈	C₂²×C₄	1	56	yes	—
70	C₂×C₃⋊C₈	C₂⁴	1	8	yes	—
71	C₂⁵	Dic₆	1	1	yes	characteristic
72	C₂³×C₄	Dic₆	1	7	yes	—
73	C₂³×C₄	C₄×D₃	1	1	yes	characteristic
74	C₂³×C₄	C₄×D₃	1	7	yes	—
75	C₂³×C₄	#67	1	7	yes	—
76	C₂³×C₄	#67	1	1	yes	characteristic
77	C₂³×C₄	C₂²×D₃	1	7	yes	—
78	C₂×M₁₆	—	2	42	no	—
79	C₂×M₁₆	—	3	28	no	—
80	C₂×M₁₆	—	6	42	no	—
81	C₂×M₁₆	C₂²×D₃	1	28	yes	—
82	C₂²×C₈	—	2	14	no	—
83	C₂²×C₈	—	3	84	no	—
84	C₂²×C₈	—	3	28	no	—
85	C₂²×C₈	—	3	28	no	—
86	C₂²×C₈	—	6	14	no	—
87	#108	—	3	64	no	—
88	C₂³×C₃	—	2	28	no	—
89	C₂³×C₃	—	2	28	no	—
90	C₂³×C₃	#108	1	8	yes	complemented
91	C₂³×C₃	C₂×C₄⋊C₄	1	21	yes	—
92	C₂³×C₃	C₂×C₄⋊C₄	1	7	yes	—
93	C₂³×C₃	C₂²×Q₈	1	7	yes	—
94	C₂×C₃×C₄	C₂×C₄⋊C₄	1	42	yes	—
95	C₂×C₃×C₄	C₂×C₄⋊C₄	1	21	yes	—
96	C₂×C₃×C₄	C₂×C₄⋊C₄	1	21	yes	—
97	C₂×C₃×C₄	C₂³×C₄	1	7	yes	—
98	C₂×C₃×C₄	C₂³×C₄	1	7	yes	—
99	C₂×C₃×C₄	C₂²×D₄	1	7	yes	—
100	C₂×C₃×C₄	C₂²×D₄	1	7	yes	—
101	C₂×C₃×C₄	C₂²×Q₈	1	7	yes	—
102	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
103	C₃×C₈	—	2	8	no	—
104	C₃⋊C₈	—	2	8	no	—
105	C₃⋊C₈	C₂³×C₄	1	16	yes	—
106	C₂⁴	—	2	8	no	—
107	C₂⁴	Dic₃⋊C₄	1	7	yes	—
108	C₂⁴	Dic₃⋊C₄	1	1	yes	characteristic
109	C₂⁴	C₂×Dic₆	1	7	yes	—
110	C₂²×C₄	Dic₃⋊C₄	1	42	yes	—
111	C₂²×C₄	Dic₃⋊C₄	1	7	yes	—
112	C₂²×C₄	Dic₃⋊C₄	1	7	yes	—
113	C₂²×C₄	C₂×Dic₆	1	21	yes	—
114	C₂²×C₄	C₂×C₄×D₃	1	21	yes	—
115	C₂²×C₄	C₂×C₄×D₃	1	7	yes	—
116	C₂²×C₄	C₂×#67	1	7	yes	—
117	C₂²×C₄	C₂×#67	1	21	yes	—
118	C₂²×C₄	C₂³×D₃	1	7	yes	—
119	M₁₆	—	2	28	no	—
120	M₁₆	—	3	8	no	—
121	M₁₆	—	6	28	no	—
122	M₁₆	C₂³×D₃	1	8	yes	—
123	C₂×C₈	—	2	28	no	—
124	C₂×C₈	—	3	56	no	—
125	C₂×C₈	—	3	56	no	—
126	C₂×C₈	—	3	8	no	—
127	C₂×C₈	—	6	28	no	—
128	C₂²×C₃	—	2	42	no	—
129	C₂²×C₃	—	2	14	no	—
130	C₂²×C₃	C₂×#108	1	28	yes	complemented
131	C₂²×C₃	C₂²×C₄⋊C₄	1	7	yes	—
132	C₂²×C₃	C₂²×C₄⋊C₄	1	7	yes	—
133	C₂²×C₃	C₂³×Q₈	1	1	yes	characteristic
134	C₃×C₄	C₂²×C₄⋊C₄	1	7	yes	—
135	C₃×C₄	C₂²×C₄⋊C₄	1	7	yes	—
136	C₃×C₄	C₂⁴×C₄	1	1	yes	characteristic
137	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
138	C₂³	—	2	28	no	—
139	C₂³	—	2	28	no	—
140	C₂³	#822	1	8	yes	direct product factor
141	C₂³	C₂×Dic₃⋊C₄	1	21	yes	—
142	C₂³	C₂×Dic₃⋊C₄	1	7	yes	—
143	C₂³	C₂²×Dic₆	1	7	yes	—
144	C₂×C₄	C₂×Dic₃⋊C₄	1	42	yes	—
145	C₂×C₄	C₂×Dic₃⋊C₄	1	21	yes	—
146	C₂×C₄	C₂×Dic₃⋊C₄	1	21	yes	—
147	C₂×C₄	C₂²×Dic₆	1	7	yes	—
148	C₂×C₄	C₂²×C₄×D₃	1	7	yes	—
149	C₂×C₄	C₂²×C₄×D₃	1	7	yes	—
150	C₂×C₄	C₂²×#67	1	7	yes	—
151	C₂×C₄	C₂²×#67	1	7	yes	—
152	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
153	C₈	—	2	8	no	—
154	C₈	—	3	16	no	—
155	C₈	—	6	8	no	—
156	C₂×C₃	—	2	7	no	—
157	C₂×C₃	—	2	1	no	—
158	C₂×C₃	C₂²×#108	1	14	yes	complemented
159	C₂×C₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
160	C₂²	—	2	42	no	—
161	C₂²	—	2	14	no	—
162	C₂²	C₂×#822	1	28	yes	direct product factor
163	C₂²	C₂²×Dic₃⋊C₄	1	7	yes	—
164	C₂²	C₂²×Dic₃⋊C₄	1	7	yes	—
165	C₂²	C₂³×Dic₆	1	1	yes	characteristic
166	C₄	C₂²×Dic₃⋊C₄	1	7	yes	—
167	C₄	C₂²×Dic₃⋊C₄	1	7	yes	—
168	C₄	C₂³×C₄×D₃	1	1	yes	characteristic
169	C₄	C₂³×#67	1	1	yes	characteristic
170	C₃	C₂³×#108	1	1	yes	characteristic, complemented, Sylow
171	C₂	—	2	1	no	—
172	C₂	—	2	7	no	—
173	C₂	C₂²×#822	1	14	yes	direct product factor
174	C₂	C₂³×Dic₃⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	#67
commutator	C₃×C₄	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₃
Fitting	C₂³×C₃×M₁₆	C₂
socle	C₂⁴×C₃	C₄⋊C₄
prime radical	C₂⁵×C₃	Q₈
cosocle	C₂×C₃×C₄	C₂⁵
hypercenter	C₂³×M₁₆	D₃
hypocenter	C₃	C₂³×#108
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₂²×#822	C₂	direct (N × H)
16. C₂×#822	C₂²	direct (N × H)
37. #822	C₂³	direct (N × H)
90. C₂³×C₃	#108	semidirect
130. C₂²×C₃	C₂×#108	semidirect
140. C₂³	#822	direct (N × H)
158. C₂×C₃	C₂²×#108	semidirect
162. C₂²	C₂×#822	direct (N × H)
170. C₃	C₂³×#108	semidirect
173. C₂	C₂²×#822	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
C₂³×#208	2
C₂⁴×C₃⋊C₈	2
C₂²×#822	2
C₂³×#108	3

Subgroups — C23×#822 #11658875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#822 #11658875