Subgroups — #86270

Order 384.

890 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#533	C₂	1	1	yes	maximal, characteristic
2	#5721	C₂	1	4	yes	maximal, complemented
3	C₃⋊#564	C₂	1	4	yes	maximal, complemented
4	#5625	C₂	1	8	yes	maximal
5	#5570	C₂	1	4	yes	maximal, complemented
6	#5567	C₂	1	4	yes	maximal, complemented
7	#5518	C₂	1	4	yes	maximal
8	#5470	C₂	1	2	yes	maximal
9	#3556	—	3	1	no	maximal, Sylow
10	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₃×#125	C₂²	1	2	yes	—
12	C₃×C₂²⋊Q₈	C₂²	1	4	yes	—
13	C₃×C₄×Q₈	C₂²	1	4	yes	—
14	C₃×#117	C₂²	1	2	yes	—
15	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
16	Q₈×Dic₃	C₂²	1	8	yes	—
17	#944	C₂²	1	8	yes	—
18	#926	C₂²	1	4	yes	—
19	#924	C₂²	1	4	yes	—
20	#924	C₂²	1	4	yes	—
21	#890	C₂²	1	16	yes	—
22	#890	C₂²	1	8	yes	—
23	#889	C₂²	1	16	yes	—
24	#889	C₂²	1	16	yes	—
25	#887	C₂²	1	8	yes	—
26	#879	C₂²	1	16	yes	—
27	#870	C₂²	1	4	yes	—
28	#870	C₂²	1	4	yes	—
29	#869	C₂²	1	2	yes	—
30	#869	C₂²	1	2	yes	—
31	#869	C₂²	1	4	yes	—
32	C₄×Dic₆	C₂²	1	4	yes	—
33	C₄×Dic₆	C₂²	1	4	yes	—
34	C₄×Dic₆	C₂²	1	8	yes	—
35	#564	—	3	4	no	—
36	#557	—	3	4	no	—
37	#557	—	3	4	no	—
38	#557	—	3	4	no	—
39	#541	—	3	8	no	—
40	#541	—	3	4	no	—
41	#533	—	3	2	no	—
42	#533	D₃	1	1	yes	characteristic
43	C₂×C₃×Q₈	C₂³	1	4	yes	—
44	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₂²×C₃×C₄	C₂³	1	2	yes	—
46	C₂×Dic₆	C₂³	1	4	yes	—
47	C₂×Dic₆	C₂³	1	4	yes	—
48	C₃×C₄⋊C₄	C₂³	1	4	yes	—
49	C₃×C₄⋊C₄	C₂³	1	4	yes	—
50	C₃×C₄⋊C₄	C₂³	1	8	yes	—
51	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
52	C₃×C₄²	C₂³	1	1	yes	characteristic
53	C₃×C₄²	C₂³	1	2	yes	—
54	C₃×C₄²	C₂³	1	4	yes	—
55	C₃×C₄²	C₂³	1	1	yes	characteristic
56	#217	C₂³	1	8	yes	—
57	C₄⋊Dic₃	C₂³	1	4	yes	—
58	C₄⋊Dic₃	C₂³	1	4	yes	—
59	C₄⋊Dic₃	C₂³	1	4	yes	—
60	C₄⋊Dic₃	C₂³	1	4	yes	—
61	C₄⋊Dic₃	C₂³	1	16	yes	—
62	Dic₃⋊C₄	C₂³	1	8	yes	—
63	Dic₃⋊C₄	C₂³	1	8	yes	—
64	Dic₃⋊C₄	C₂³	1	16	yes	—
65	Dic₃⋊C₄	C₂³	1	16	yes	—
66	C₄×Dic₃	C₂³	1	8	yes	—
67	C₄×Dic₃	C₂³	1	16	yes	—
68	C₄⋊Q₈	—	3	8	no	—
69	C₄⋊Q₈	—	3	2	no	—
70	C₄⋊Q₈	—	3	2	no	—
71	C₄⋊Q₈	—	3	4	no	—
72	C₄⋊Q₈	C₂×D₃	1	2	yes	—
73	#126	—	3	16	no	—
74	#125	—	3	8	no	—
75	#125	—	3	16	no	—
76	#125	—	3	16	no	—
77	#125	—	3	16	no	—
78	#125	—	3	4	no	—
79	#125	—	3	4	no	—
80	#125	C₂×D₃	1	2	yes	—
81	C₂²⋊Q₈	—	3	4	no	—
82	C₂²⋊Q₈	—	3	4	no	—
83	C₂²⋊Q₈	C₂×D₃	1	4	yes	—
84	C₄×Q₈	—	3	8	no	—
85	C₄×Q₈	—	3	8	no	—
86	C₄×Q₈	—	3	4	no	—
87	C₄×Q₈	—	3	4	no	—
88	C₄×Q₈	—	3	8	no	—
89	C₄×Q₈	C₂×D₃	1	4	yes	—
90	#117	—	3	4	no	—
91	#117	C₂×D₃	1	2	yes	—
92	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
93	C₂³×C₃	C₂⁴	1	1	yes	characteristic
94	C₃×Q₈	—	2	8	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	—	2	4	no	—
97	C₂×C₃×C₄	C₂⁴	1	8	yes	—
98	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
99	C₂×C₃×C₄	C₂⁴	1	2	yes	—
100	C₂×C₃×C₄	C₂⁴	1	2	yes	—
101	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
102	C₂×Dic₃	C₂⁴	1	8	yes	—
103	C₂×Dic₃	C₂⁴	1	8	yes	—
104	Dic₆	—	2	8	no	—
105	Dic₆	—	2	8	no	—
106	C₂×Q₈	—	3	4	no	—
107	C₂×Q₈	—	3	4	no	—
108	C₂×Q₈	C₂²×D₃	1	4	yes	—
109	C₂²×C₄	C₂²×D₃	1	2	yes	—
110	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
111	C₄⋊C₄	—	3	4	no	—
112	C₄⋊C₄	—	3	4	no	—
113	C₄⋊C₄	—	3	4	no	—
114	C₄⋊C₄	—	3	4	no	—
115	C₄⋊C₄	—	3	8	no	—
116	C₄⋊C₄	—	3	8	no	—
117	C₄⋊C₄	—	3	16	no	—
118	C₄⋊C₄	—	3	16	no	—
119	C₄⋊C₄	—	3	16	no	—
120	C₄⋊C₄	C₂²×D₃	1	4	yes	—
121	C₄⋊C₄	C₂²×D₃	1	4	yes	—
122	C₄⋊C₄	C₂²×D₃	1	8	yes	—
123	C₂²⋊C₄	—	3	8	no	—
124	C₂²⋊C₄	C₂²×D₃	1	4	yes	—
125	C₄²	—	3	8	no	—
126	C₄²	—	3	16	no	—
127	C₄²	C₂²×D₃	1	1	yes	characteristic
128	C₄²	C₂²×D₃	1	2	yes	—
129	C₄²	C₂²×D₃	1	1	yes	characteristic
130	C₄²	C₂²×D₃	1	4	yes	—
131	C₂²×C₃	—	2	1	no	—
132	C₂²×C₃	—	2	2	no	—
133	C₂²×C₃	C₂⁵	1	1	yes	characteristic
134	C₃×C₄	—	2	1	no	—
135	C₃×C₄	—	2	2	no	—
136	C₃×C₄	—	2	8	no	—
137	C₃×C₄	2₋¹⁺⁴	1	2	yes	—
138	C₃×C₄	2₋¹⁺⁴	1	4	yes	—
139	Dic₃	—	2	8	no	—
140	Dic₃	—	2	8	no	—
141	C₂³	C₂³×D₃	1	1	yes	characteristic
142	Q₈	—	2	8	no	—
143	Q₈	—	6	8	no	—
144	Q₈	—	6	8	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	3	8	no	—
148	C₂×C₄	—	3	8	no	—
149	C₂×C₄	C₂³×D₃	1	8	yes	—
150	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
151	C₂×C₄	C₂³×D₃	1	2	yes	—
152	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
153	C₂×C₄	C₂³×D₃	1	2	yes	—
154	C₂×C₃	—	4	1	no	—
155	C₂×C₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
156	C₂×C₃	C₂×2₋¹⁺⁴	1	2	yes	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	2	no	—
159	C₂²	C₂⁴×D₃	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	2	no	—
162	C₄	—	2	8	no	—
163	C₄	—	6	8	no	—
164	C₄	—	6	8	no	—
165	C₄	#1007	1	2	yes	—
166	C₄	#1010	1	4	yes	—
167	C₃	#3556	1	1	yes	characteristic, complemented, Sylow
168	C₂	—	4	1	no	—
169	C₂	C₂×#1007	1	1	yes	characteristic
170	C₂	C₂×#1010	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁵
Fitting	C₃×#533	C₂
prime radical	C₂³×C₃	C₂⁴
hypercenter	#533	D₃
hypocenter	C₃	#3556
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
2. #5721	C₂	semidirect
3. C₃⋊#564	C₂	semidirect
5. #5570	C₂	semidirect
6. #5567	C₂	semidirect
167. C₃	#3556	semidirect

Maximal subgroups

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

subgroup	index
C₃×#533	2
#5721	2
C₃⋊#564	2
#5625	2
#5570	2
#5567	2
#5518	2
#5470	2
#3556	3