Subgroups — #10948175

Order 768.

706 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₂²×#812	C₂	1	2	yes	maximal
3	#69882	C₂	1	8	yes	maximal, complemented
4	#69743	C₂	1	4	yes	maximal, complemented
5	#33397	—	3	1	no	maximal, Sylow
6	C₂²×C₃×M₁₆	C₄	1	1	yes	characteristic
7	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×#117	C₄	1	1	yes	characteristic
9	C₃×#405	C₄	1	4	yes	—
10	C₃×#405	C₂²	1	4	yes	—
11	C₂×C₃×#97	C₄	1	2	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
13	C₂×#812	—	2	8	no	—
14	C₂×#812	C₂²	1	4	yes	—
15	C₂×#812	C₂²	1	4	yes	—
16	C₂²×C₃⋊C₁₆	C₂²	1	4	yes	—
17	C₄⋊(C₃⋊C₁₆)	C₂²	1	16	yes	complemented
18	C₂²×M₃₂	—	3	2	no	—
19	C₂×#405	D₃	1	1	yes	characteristic
20	#2151	—	3	8	no	—
21	#2150	—	3	4	no	—
22	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×M₁₆	—	2	4	no	—
24	C₂×C₃×M₁₆	C₂×C₄	1	4	yes	—
25	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₈	D₄	1	4	yes	—
29	C₂²×C₃×C₈	Q₈	1	4	yes	—
30	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₃×C₈	C₂³	1	2	yes	—
32	C₃×#117	C₈	1	4	yes	—
33	C₃×#117	C₂×C₄	1	4	yes	—
34	C₂×C₃×C₄⋊C₄	C₈	1	2	yes	—
35	C₂×C₃×C₂²⋊C₄	C₈	1	2	yes	—
36	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
37	C₃×#97	C₂×C₄	1	8	yes	—
38	C₃×C₄×C₈	C₂×C₄	1	4	yes	—
39	C₃×C₄×C₈	C₂³	1	4	yes	—
40	#812	—	2	8	no	—
41	#812	—	2	8	no	—
42	C₂×C₃⋊C₁₆	—	2	8	no	—
43	C₂×C₃⋊C₁₆	C₂³	1	8	yes	—
44	C₂²×M₁₆	Dic₃	1	1	yes	characteristic
45	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
46	C₂×#117	Dic₃	1	1	yes	characteristic
47	C₂×M₃₂	—	3	4	no	—
48	C₂×M₃₂	—	3	4	no	—
49	C₂×M₃₂	—	6	8	no	—
50	C₂²×C₁₆	—	3	4	no	—
51	#405	Dic₃	1	4	yes	—
52	#405	C₂×D₃	1	4	yes	—
53	C₂×#97	Dic₃	1	2	yes	—
54	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
55	C₄⋊C₁₆	—	3	16	no	—
56	C₂⁴×C₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₄⋊C₄	1	4	yes	—
58	C₂²×C₃×C₄	C₂×C₈	1	4	yes	—
59	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
61	C₂²×C₃×C₄	M₁₆	1	4	yes	—
62	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
64	C₃×M₁₆	—	2	8	no	—
65	C₂×C₃×C₈	—	2	4	no	—
66	C₂×C₃×C₈	C₄⋊C₄	1	4	yes	—
67	C₂×C₃×C₈	C₄⋊C₄	1	8	yes	—
68	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
69	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
72	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
73	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
74	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
75	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
76	C₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
77	C₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
78	C₃×C₄²	C₂×C₈	1	4	yes	—
79	C₃×C₄²	C₂²×C₄	1	4	yes	—
80	C₃⋊C₁₆	—	2	8	no	—
81	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
82	C₂×M₁₆	—	2	4	no	—
83	C₂×M₁₆	C₂×Dic₃	1	4	yes	—
84	C₂²×C₈	Dic₆	1	4	yes	—
85	C₂²×C₈	D₁₂	1	4	yes	—
86	C₂²×C₈	C₂×Dic₃	1	2	yes	—
87	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
88	C₂²×C₈	C₂×Dic₃	1	2	yes	—
89	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₃	1	2	yes	—
91	#117	C₃⋊C₈	1	4	yes	—
92	#117	C₂×Dic₃	1	4	yes	—
93	C₂×C₄⋊C₄	C₃⋊C₈	1	2	yes	—
94	C₂×C₂²⋊C₄	C₃⋊C₈	1	2	yes	—
95	C₂×C₄²	C₂×Dic₃	1	2	yes	—
96	M₃₂	—	6	8	no	—
97	M₃₂	—	6	8	no	—
98	C₂×C₁₆	—	3	8	no	—
99	C₂×C₁₆	—	6	8	no	—
100	#97	C₂×Dic₃	1	8	yes	—
101	C₄×C₈	C₂×Dic₃	1	4	yes	—
102	C₄×C₈	C₂²×D₃	1	4	yes	—
103	C₂³×C₃	—	4	2	no	—
104	C₂³×C₃	C₄⋊C₈	1	4	yes	—
105	C₂³×C₃	C₂²×C₈	1	2	yes	—
106	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₃×C₄	—	4	4	no	—
108	C₂×C₃×C₄	C₄⋊C₈	1	4	yes	—
109	C₂×C₃×C₄	C₄⋊C₈	1	8	yes	—
110	C₂×C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₃×C₄	C₂²×C₈	1	8	yes	—
113	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
115	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
116	C₂×C₃×C₄	C₂×M₁₆	1	4	yes	—
117	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
118	C₃×C₈	—	2	4	no	—
119	C₃×C₈	C₂×C₄⋊C₄	1	2	yes	—
120	C₃×C₈	C₂×C₄⋊C₄	1	4	yes	—
121	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
122	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
123	C₂⁴	C₂×C₃⋊C₈	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₃⋊C₈	1	4	yes	—
125	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
126	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
127	C₂²×C₄	#208	1	4	yes	—
128	C₂²×C₄	C₄⋊Dic₃	1	4	yes	—
129	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
130	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
131	M₁₆	—	2	8	no	—
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	C₄⋊Dic₃	1	4	yes	—
134	C₂×C₈	C₄⋊Dic₃	1	8	yes	—
135	C₂×C₈	C₂×Dic₆	1	2	yes	—
136	C₂×C₈	C₂×Dic₆	1	4	yes	—
137	C₂×C₈	C₂×D₁₂	1	4	yes	—
138	C₂×C₈	C₂×D₁₂	1	2	yes	—
139	C₂×C₈	C₂²×Dic₃	1	4	yes	—
140	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
141	C₂×C₈	C₂²×Dic₃	1	2	yes	—
142	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
143	C₄⋊C₄	C₂×C₃⋊C₈	1	8	yes	—
144	C₂²⋊C₄	C₂×C₃⋊C₈	1	8	yes	—
145	C₄²	C₂×C₃⋊C₈	1	4	yes	—
146	C₄²	C₂²×Dic₃	1	4	yes	—
147	C₁₆	—	6	8	no	—
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	—	2	4	no	—
150	C₂²×C₃	—	4	4	no	—
151	C₂²×C₃	C₂×C₄⋊C₈	1	2	yes	—
152	C₂²×C₃	C₂×C₄⋊C₈	1	4	yes	—
153	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
154	C₃×C₄	—	2	8	no	—
155	C₃×C₄	C₂×C₄⋊C₈	1	2	yes	—
156	C₃×C₄	C₂×C₄⋊C₈	1	4	yes	—
157	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
159	C₂³	—	4	2	no	—
160	C₂³	C₄⋊(C₃⋊C₈)	1	4	yes	—
161	C₂³	C₂²×C₃⋊C₈	1	2	yes	—
162	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	C₄⋊(C₃⋊C₈)	1	4	yes	—
165	C₂×C₄	C₄⋊(C₃⋊C₈)	1	8	yes	—
166	C₂×C₄	C₂²×C₃⋊C₈	1	8	yes	—
167	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
168	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
169	C₂×C₄	C₂×#208	1	2	yes	—
170	C₂×C₄	C₂×#208	1	4	yes	—
171	C₂×C₄	C₂×C₄⋊Dic₃	1	4	yes	—
172	C₂×C₄	C₂×C₄⋊Dic₃	1	2	yes	—
173	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
174	C₈	—	2	4	no	—
175	C₈	C₂×C₄⋊Dic₃	1	4	yes	—
176	C₈	C₂×C₄⋊Dic₃	1	2	yes	—
177	C₈	C₂²×Dic₆	1	1	yes	characteristic
178	C₈	C₂²×D₁₂	1	1	yes	characteristic
179	C₂×C₃	—	2	2	no	—
180	C₂×C₃	—	2	4	no	—
181	C₂×C₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
182	C₂×C₃	#3407	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	4	no	—
185	C₂²	—	4	4	no	—
186	C₂²	C₂×C₄⋊(C₃⋊C₈)	1	2	yes	—
187	C₂²	C₂×C₄⋊(C₃⋊C₈)	1	4	yes	—
188	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
189	C₄	—	2	8	no	—
190	C₄	C₂×C₄⋊(C₃⋊C₈)	1	2	yes	—
191	C₄	C₂×C₄⋊(C₃⋊C₈)	1	4	yes	—
192	C₄	C₂²×#208	1	1	yes	characteristic
193	C₄	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
194	C₃	#33397	1	1	yes	characteristic, complemented, Sylow
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₂²×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
198	C₂	#83780	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₂×C₃×#405	C₂
prime radical	C₂⁴×C₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×#405	D₃
hypocenter	C₃	#33397
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #69882	C₂	semidirect
4. #69743	C₂	semidirect
17. C₄⋊(C₃⋊C₁₆)	C₂²	semidirect
194. C₃	#33397	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#405	2
C₂²×#812	2
#69882	2
#69743	2
#33397	3