Subgroups — C₂³×#832 #11658695

Order 768.

7646 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#832	C₂	1	28	yes	maximal, direct product factor
5	C₂³×#102	—	3	1	no	maximal, Sylow
6	C₂⁶×C₃	—	2	1	no	—
7	C₂³×C₃×D₄	—	2	8	no	—
8	C₂³×C₃×D₄	—	2	7	no	—
9	C₂³×C₃×D₄	C₄	1	16	yes	complemented
10	C₂³×C₃×D₄	C₂²	1	14	yes	—
11	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂⁴×Dic₃	—	2	1	no	—
13	C₂²×C₄⋊Dic₃	—	2	7	no	—
14	C₂²×C₄⋊Dic₃	C₂²	1	14	yes	complemented
15	C₂³×C₃⋊C₈	—	2	8	no	—
16	C₂³×C₃⋊C₈	C₂²	1	14	yes	complemented
17	C₂×#832	C₂²	1	112	yes	direct product factor
18	C₂⁴×D₄	D₃	1	1	yes	characteristic
19	C₂⁴×C₈	—	3	1	no	—
20	C₂³×C₄⋊C₄	—	3	1	no	—
21	C₂²×#102	—	3	28	no	—
22	C₂⁵×C₃	—	2	16	no	—
23	C₂⁵×C₃	—	2	14	no	—
24	C₂⁵×C₃	—	4	8	no	—
25	C₂⁵×C₃	—	4	8	no	—
26	C₂⁵×C₃	D₄	1	1	yes	characteristic
27	C₂²×C₃×D₄	—	2	112	no	—
28	C₂²×C₃×D₄	—	2	42	no	—
29	C₂²×C₃×D₄	—	2	56	no	—
30	C₂²×C₃×D₄	C₂×C₄	1	112	yes	complemented
31	C₂²×C₃×D₄	C₂³	1	28	yes	—
32	C₂³×C₃×C₄	C₂×C₄	1	8	yes	—
33	C₂³×C₃×C₄	D₄	1	7	yes	—
34	C₂³×C₃×C₄	D₄	1	8	yes	—
35	C₂³×C₃×C₄	C₂³	1	7	yes	—
36	C₂³×Dic₃	—	2	14	no	—
37	C₂³×Dic₃	—	4	8	no	—
38	C₂×C₄⋊Dic₃	—	2	42	no	—
39	C₂×C₄⋊Dic₃	C₂³	1	28	yes	complemented
40	C₂²×C₃⋊C₈	—	2	56	no	—
41	C₂²×C₃⋊C₈	C₂³	1	28	yes	complemented
42	#832	C₂³	1	64	yes	direct product factor
43	C₂⁶	—	2	1	no	—
44	C₂³×D₄	—	2	7	no	—
45	C₂³×D₄	—	2	8	no	—
46	C₂³×D₄	Dic₃	1	16	yes	complemented
47	C₂³×D₄	C₂×D₃	1	14	yes	—
48	C₂⁴×C₄	—	6	1	no	—
49	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
50	C₂³×C₈	—	3	14	no	—
51	C₂³×C₈	—	6	8	no	—
52	C₂²×C₄⋊C₄	—	3	14	no	—
53	C₂²×C₄⋊C₄	—	6	7	no	—
54	C₂×#102	—	3	112	no	—
55	C₂⁴×C₃	—	2	112	no	—
56	C₂⁴×C₃	—	2	28	no	—
57	C₂⁴×C₃	—	4	128	no	—
58	C₂⁴×C₃	—	4	56	no	—
59	C₂⁴×C₃	—	4	56	no	—
60	C₂⁴×C₃	C₂²⋊C₄	1	8	yes	—
61	C₂⁴×C₃	D₈	1	8	yes	—
62	C₂⁴×C₃	SD₁₆	1	8	yes	—
63	C₂⁴×C₃	C₂×D₄	1	7	yes	—
64	C₂×C₃×D₄	—	2	336	no	—
65	C₂×C₃×D₄	—	2	28	no	—
66	C₂×C₃×D₄	—	2	56	no	—
67	C₂×C₃×D₄	C₂²×C₄	1	112	yes	complemented
68	C₂×C₃×D₄	C₂⁴	1	8	yes	—
69	C₂²×C₃×C₄	C₂²⋊C₄	1	56	yes	—
70	C₂²×C₃×C₄	C₂²×C₄	1	28	yes	—
71	C₂²×C₃×C₄	C₂×D₄	1	21	yes	—
72	C₂²×C₃×C₄	C₂×D₄	1	28	yes	—
73	C₂²×C₃×C₄	C₂⁴	1	7	yes	—
74	C₂²×Dic₃	—	2	28	no	—
75	C₂²×Dic₃	—	4	56	no	—
76	C₄⋊Dic₃	—	2	28	no	—
77	C₄⋊Dic₃	C₂⁴	1	8	yes	complemented
78	C₂×C₃⋊C₈	—	2	56	no	—
79	C₂×C₃⋊C₈	C₂⁴	1	8	yes	complemented
80	C₂⁵	—	2	16	no	—
81	C₂⁵	—	2	14	no	—
82	C₂⁵	—	4	8	no	—
83	C₂⁵	—	4	8	no	—
84	C₂⁵	#67	1	1	yes	characteristic
85	C₂²×D₄	—	2	112	no	—
86	C₂²×D₄	—	2	42	no	—
87	C₂²×D₄	—	2	56	no	—
88	C₂²×D₄	C₂×Dic₃	1	112	yes	complemented
89	C₂²×D₄	C₂²×D₃	1	28	yes	—
90	C₂³×C₄	—	6	14	no	—
91	C₂³×C₄	—	12	8	no	—
92	C₂³×C₄	C₂×Dic₃	1	8	yes	—
93	C₂³×C₄	#67	1	7	yes	—
94	C₂³×C₄	#67	1	8	yes	—
95	C₂³×C₄	C₂²×D₃	1	7	yes	—
96	C₂²×C₈	—	3	28	no	—
97	C₂²×C₈	—	6	56	no	—
98	C₂×C₄⋊C₄	—	3	28	no	—
99	C₂×C₄⋊C₄	—	6	42	no	—
100	#102	—	3	64	no	—
101	C₂³×C₃	—	2	112	no	—
102	C₂³×C₃	—	2	8	no	—
103	C₂³×C₃	—	4	56	no	—
104	C₂³×C₃	—	4	56	no	—
105	C₂³×C₃	—	4	448	no	—
106	C₂³×C₃	#102	1	64	yes	complemented
107	C₂³×C₃	C₂×C₂²⋊C₄	1	28	yes	—
108	C₂³×C₃	C₂×D₈	1	28	yes	—
109	C₂³×C₃	C₂×SD₁₆	1	28	yes	—
110	C₂³×C₃	C₂²×D₄	1	7	yes	—
111	C₃×D₄	—	2	112	no	—
112	C₃×D₄	—	2	8	no	—
113	C₃×D₄	C₂³×C₄	1	16	yes	complemented
114	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	84	yes	—
115	C₂×C₃×C₄	C₂³×C₄	1	14	yes	—
116	C₂×C₃×C₄	C₂²×D₄	1	7	yes	—
117	C₂×C₃×C₄	C₂²×D₄	1	14	yes	—
118	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
119	C₂×Dic₃	—	2	8	no	—
120	C₂×Dic₃	—	4	56	no	—
121	C₃⋊C₈	—	2	8	no	—
122	C₂⁴	—	2	112	no	—
123	C₂⁴	—	2	28	no	—
124	C₂⁴	—	4	128	no	—
125	C₂⁴	—	4	56	no	—
126	C₂⁴	—	4	56	no	—
127	C₂⁴	#213	1	8	yes	—
128	C₂⁴	#214	1	8	yes	—
129	C₂⁴	#217	1	8	yes	—
130	C₂⁴	C₂×#67	1	7	yes	—
131	C₂×D₄	—	2	336	no	—
132	C₂×D₄	—	2	56	no	—
133	C₂×D₄	—	2	28	no	—
134	C₂×D₄	C₂²×Dic₃	1	112	yes	complemented
135	C₂×D₄	C₂³×D₃	1	8	yes	—
136	C₂²×C₄	—	6	28	no	—
137	C₂²×C₄	—	12	56	no	—
138	C₂²×C₄	#217	1	56	yes	—
139	C₂²×C₄	C₂²×Dic₃	1	28	yes	—
140	C₂²×C₄	C₂×#67	1	21	yes	—
141	C₂²×C₄	C₂×#67	1	28	yes	—
142	C₂²×C₄	C₂³×D₃	1	7	yes	—
143	C₂×C₈	—	3	8	no	—
144	C₂×C₈	—	6	56	no	—
145	C₄⋊C₄	—	3	8	no	—
146	C₄⋊C₄	—	6	28	no	—
147	C₂²×C₃	—	2	16	no	—
148	C₂²×C₃	—	4	224	no	—
149	C₂²×C₃	—	4	8	no	—
150	C₂²×C₃	—	4	8	no	—
151	C₂²×C₃	C₂×#102	1	112	yes	complemented
152	C₂²×C₃	C₂²×C₂²⋊C₄	1	14	yes	—
153	C₂²×C₃	C₂²×D₈	1	14	yes	—
154	C₂²×C₃	C₂²×SD₁₆	1	14	yes	—
155	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
156	C₃×C₄	C₂²×C₂²⋊C₄	1	14	yes	—
157	C₃×C₄	C₂⁴×C₄	1	1	yes	characteristic
158	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
159	Dic₃	—	4	8	no	—
160	C₂³	—	2	8	no	—
161	C₂³	—	2	112	no	—
162	C₂³	—	4	448	no	—
163	C₂³	—	4	56	no	—
164	C₂³	—	4	56	no	—
165	C₂³	#832	1	64	yes	direct product factor
166	C₂³	C₂×#213	1	28	yes	—
167	C₂³	C₂×#214	1	28	yes	—
168	C₂³	C₂×#217	1	28	yes	—
169	C₂³	C₂²×#67	1	7	yes	—
170	D₄	—	2	8	no	—
171	D₄	—	2	112	no	—
172	D₄	C₂³×Dic₃	1	16	yes	complemented
173	C₂×C₄	—	6	8	no	—
174	C₂×C₄	—	12	56	no	—
175	C₂×C₄	C₂×#217	1	84	yes	—
176	C₂×C₄	C₂³×Dic₃	1	14	yes	—
177	C₂×C₄	C₂²×#67	1	14	yes	—
178	C₂×C₄	C₂²×#67	1	7	yes	—
179	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
180	C₈	—	6	8	no	—
181	C₂×C₃	—	4	16	no	—
182	C₂×C₃	C₂²×#102	1	28	yes	complemented
183	C₂×C₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
184	C₂×C₃	C₂³×D₈	1	1	yes	characteristic
185	C₂×C₃	C₂³×SD₁₆	1	1	yes	characteristic
186	C₂²	—	2	16	no	—
187	C₂²	—	4	224	no	—
188	C₂²	—	4	8	no	—
189	C₂²	—	4	8	no	—
190	C₂²	C₂×#832	1	112	yes	direct product factor
191	C₂²	C₂²×#213	1	14	yes	—
192	C₂²	C₂²×#214	1	14	yes	—
193	C₂²	C₂²×#217	1	14	yes	—
194	C₂²	C₂³×#67	1	1	yes	characteristic
195	C₄	—	12	8	no	—
196	C₄	C₂²×#217	1	14	yes	—
197	C₄	C₂⁴×Dic₃	1	1	yes	characteristic
198	C₄	C₂³×#67	1	1	yes	characteristic
199	C₃	C₂³×#102	1	1	yes	characteristic, complemented, Sylow
200	C₂	—	4	16	no	—
201	C₂	C₂²×#832	1	28	yes	direct product factor
202	C₂	C₂³×#213	1	1	yes	characteristic
203	C₂	C₂³×#214	1	1	yes	characteristic
204	C₂	C₂³×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	#67
commutator	C₃×C₄	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₃
Fitting, prime radical	C₂⁴×C₃×D₄	C₂
socle	C₂⁵×C₃	D₄
cosocle	C₂×C₃×C₄	C₂⁵
hypercenter	C₂⁴×D₄	D₃
hypocenter	C₃	C₂³×#102
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄⋊Dic₃	C₂	semidirect
3. C₂⁴×C₃⋊C₈	C₂	semidirect
4. C₂²×#832	C₂	direct (N × H)
9. C₂³×C₃×D₄	C₄	semidirect
14. C₂²×C₄⋊Dic₃	C₂²	semidirect
16. C₂³×C₃⋊C₈	C₂²	semidirect
17. C₂×#832	C₂²	direct (N × H)
30. C₂²×C₃×D₄	C₂×C₄	semidirect
39. C₂×C₄⋊Dic₃	C₂³	semidirect
41. C₂²×C₃⋊C₈	C₂³	semidirect
42. #832	C₂³	direct (N × H)
46. C₂³×D₄	Dic₃	semidirect
67. C₂×C₃×D₄	C₂²×C₄	semidirect
77. C₄⋊Dic₃	C₂⁴	semidirect
79. C₂×C₃⋊C₈	C₂⁴	semidirect
88. C₂²×D₄	C₂×Dic₃	semidirect
106. C₂³×C₃	#102	semidirect
113. C₃×D₄	C₂³×C₄	semidirect
134. C₂×D₄	C₂²×Dic₃	semidirect
151. C₂²×C₃	C₂×#102	semidirect
165. C₂³	#832	direct (N × H)
172. D₄	C₂³×Dic₃	semidirect
182. C₂×C₃	C₂²×#102	semidirect
190. C₂²	C₂×#832	direct (N × H)
199. C₃	C₂³×#102	semidirect
201. C₂	C₂²×#832	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×D₄	2
C₂³×C₄⋊Dic₃	2
C₂⁴×C₃⋊C₈	2
C₂²×#832	2
C₂³×#102	3

Subgroups — C23×#832 #11658695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#832 #11658695