Subgroups — #71580

Order 384.

424 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#926	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#4529	C₂	1	4	yes	maximal, complemented
5	#1735	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₃×#117	—	2	2	no	—
8	C₃×#117	C₄	1	4	yes	complemented
9	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
11	C₂×C₃×C₄²	—	2	1	no	—
12	C₂×#217	—	2	1	no	—
13	#926	—	2	2	no	—
14	#926	C₄	1	4	yes	complemented
15	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₃	—	2	1	no	—
17	C₂²×C₃⋊C₈	—	2	2	no	—
18	C₂²×C₃⋊C₈	—	2	2	no	—
19	C₂²×C₃⋊C₈	C₄	1	4	yes	complemented
20	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
21	C₂³×C₈	—	3	1	no	—
22	C₂×#117	—	3	1	no	—
23	C₂×#117	D₃	1	1	yes	characteristic
24	#340	—	3	4	no	—
25	C₂⁴×C₃	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄	—	2	2	no	—
27	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄	D₄	1	2	yes	—
31	C₂²×C₃×C₄	D₄	1	2	yes	—
32	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
33	C₂²×C₃×C₄	Q₈	1	2	yes	—
34	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂²×Dic₃	—	2	2	no	—
36	C₃×C₄⋊C₄	—	2	2	no	—
37	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
38	C₃×C₂²⋊C₄	—	2	4	no	—
39	C₃×C₄²	—	2	4	no	—
40	#217	—	2	4	no	—
41	C₄⋊Dic₃	—	2	2	no	—
42	C₄⋊Dic₃	C₂×C₄	1	4	yes	—
43	C₄×Dic₃	—	2	4	no	—
44	C₂×C₃⋊C₈	—	2	8	no	—
45	C₂×C₃⋊C₈	—	2	2	no	—
46	C₂×C₃⋊C₈	—	2	2	no	—
47	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
48	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₈	—	3	4	no	—
50	C₂²×C₈	—	3	2	no	—
51	C₂²×C₈	—	6	2	no	—
52	C₂²×C₈	—	6	2	no	—
53	#117	—	2	2	no	—
54	#117	—	3	4	no	—
55	#117	—	6	2	no	—
56	#117	Dic₃	1	4	yes	complemented
57	C₂×C₄⋊C₄	—	3	2	no	—
58	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
59	C₂×C₂²⋊C₄	—	2	1	no	—
60	C₂×C₂²⋊C₄	—	6	1	no	—
61	C₂×C₄²	—	2	1	no	—
62	C₂×C₄²	—	6	1	no	—
63	C₂³×C₃	—	2	4	no	—
64	C₂³×C₃	C₂²⋊C₄	1	2	yes	—
65	C₂³×C₃	C₂²⋊C₄	1	2	yes	—
66	C₂³×C₃	C₄⋊C₄	1	2	yes	—
67	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄	—	2	4	no	—
69	C₂×C₃×C₄	—	4	4	no	—
70	C₂×C₃×C₄	C₄²	1	4	yes	—
71	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
72	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
73	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
74	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
75	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
76	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
77	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
83	C₂×Dic₃	—	2	4	no	—
84	C₂×Dic₃	—	4	4	no	—
85	C₃⋊C₈	—	2	4	no	—
86	C₂⁴	#67	1	1	yes	characteristic
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	6	2	no	—
89	C₂²×C₄	Dic₆	1	2	yes	—
90	C₂²×C₄	C₄×D₃	1	2	yes	—
91	C₂²×C₄	C₄×D₃	1	2	yes	—
92	C₂²×C₄	D₁₂	1	2	yes	—
93	C₂²×C₄	C₂×Dic₃	1	2	yes	—
94	C₂²×C₄	#67	1	2	yes	—
95	C₂²×C₄	#67	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂×C₈	—	3	4	no	—
98	C₂×C₈	—	6	8	no	—
99	C₂×C₈	—	6	2	no	—
100	C₂×C₈	—	6	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	3	4	no	—
103	C₄⋊C₄	—	6	2	no	—
104	C₄⋊C₄	C₂×Dic₃	1	4	yes	—
105	C₂²⋊C₄	—	2	4	no	—
106	C₂²⋊C₄	—	6	4	no	—
107	C₄²	—	2	4	no	—
108	C₄²	—	6	4	no	—
109	C₂²×C₃	—	2	4	no	—
110	C₂²×C₃	—	2	4	no	—
111	C₂²×C₃	—	2	4	no	—
112	C₂²×C₃	#95	1	4	yes	—
113	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₃	#135	1	4	yes	—
117	C₃×C₄	—	4	4	no	—
118	C₃×C₄	#95	1	4	yes	—
119	C₃×C₄	C₂×C₄²	1	1	yes	characteristic
120	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
123	Dic₃	—	4	4	no	—
124	C₂³	—	2	4	no	—
125	C₂³	Dic₃⋊C₄	1	2	yes	—
126	C₂³	(C₂×D₃)⋊C₄	1	2	yes	—
127	C₂³	#217	1	2	yes	—
128	C₂³	C₂×#67	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	—	6	4	no	—
132	C₂×C₄	—	12	4	no	—
133	C₂×C₄	C₄×Dic₃	1	4	yes	—
134	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
135	C₂×C₄	Dic₃⋊C₄	1	4	yes	—
136	C₂×C₄	C₄⋊Dic₃	1	4	yes	—
137	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
138	C₂×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
139	C₂×C₄	#217	1	2	yes	—
140	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
143	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
144	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
145	C₂×C₄	C₂×#67	1	1	yes	characteristic
146	C₈	—	6	4	no	—
147	C₂×C₃	—	2	4	no	—
148	C₂×C₃	C₂×#95	1	1	yes	characteristic
149	C₂×C₃	#416	1	2	yes	—
150	C₂×C₃	#416	1	2	yes	—
151	C₂×C₃	#427	1	2	yes	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	C₃⋊#95	1	4	yes	—
156	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
157	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
158	C₂²	#950	1	4	yes	—
159	C₂²	C₂×#217	1	1	yes	characteristic
160	C₄	—	4	4	no	—
161	C₄	—	12	4	no	—
162	C₄	C₃⋊#95	1	4	yes	—
163	C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
164	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
165	C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
166	C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
167	C₃	#1735	1	1	yes	characteristic, complemented, Sylow
168	C₂	—	2	4	no	—
169	C₂	#4998	1	2	yes	—
170	C₂	#5002	1	2	yes	—
171	C₂	C₂×C₃⋊#95	1	1	yes	characteristic
172	C₂	#5233	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#67
commutator	C₃×C₄	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×#117	C₂
socle	C₂³×C₃	C₂×D₄
prime radical	C₂⁴×C₃	D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#117	D₃
hypocenter	C₃	#1735
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
4. #4529	C₂	semidirect
8. C₃×#117	C₄	semidirect
14. #926	C₄	semidirect
19. C₂²×C₃⋊C₈	C₄	semidirect
56. #117	Dic₃	semidirect
167. C₃	#1735	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#117	2
C₂×#926	2
C₂³×C₃⋊C₈	2
#4529	2
#1735	3