Subgroups — #82060

Order 384.

568 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
2	#5212	C₂	1	2	yes	maximal
3	#5211	C₂	1	2	yes	maximal
4	C₄×#217	C₂	1	2	yes	maximal
5	#4937	C₂	1	2	yes	maximal, complemented
6	#4936	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊Dic₃	C₂	1	2	yes	maximal, complemented
8	#4932	C₂	1	2	yes	maximal, complemented
9	#2283	—	3	1	no	maximal, Sylow
10	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄²	C₄	1	8	yes	complemented
14	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
18	C₂×#217	C₂²	1	4	yes	—
19	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
20	C₂×C₄×Dic₃	C₂²	1	4	yes	—
21	C₃⋊#95	C₂²	1	4	yes	—
22	C₃⋊#95	C₂²	1	4	yes	—
23	C₃⋊#95	C₂²	1	4	yes	—
24	C₃⋊#95	C₂²	1	4	yes	—
25	C₂²×C₄²	D₃	1	1	yes	characteristic
26	#383	—	3	2	no	—
27	#382	—	3	2	no	—
28	#381	—	3	2	no	—
29	#380	—	3	2	no	—
30	C₄×C₄⋊C₄	—	3	2	no	—
31	C₄×C₂²⋊C₄	—	3	2	no	—
32	#376	—	3	2	no	—
33	C₂⁴×C₃	C₂³	1	1	yes	characteristic
34	C₂²×C₃×C₄	—	2	4	no	—
35	C₂²×C₃×C₄	—	2	4	no	—
36	C₂²×C₃×C₄	—	2	4	no	—
37	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
38	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
39	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
40	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
41	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
43	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
44	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
46	C₂²×Dic₃	C₂³	1	8	yes	—
47	C₃×C₄²	C₂×C₄	1	4	yes	—
48	C₃×C₄²	C₂×C₄	1	4	yes	—
49	C₃×C₄²	C₂×C₄	1	4	yes	—
50	C₃×C₄²	C₂×C₄	1	4	yes	—
51	#217	—	2	8	no	—
52	C₄⋊Dic₃	—	2	8	no	—
53	C₄×Dic₃	—	2	8	no	—
54	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
55	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
56	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
57	C₂×C₄⋊C₄	—	3	4	no	—
58	C₂×C₂²⋊C₄	—	3	4	no	—
59	C₂×C₄²	—	3	4	no	—
60	C₂×C₄²	Dic₃	1	8	yes	complemented
61	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
62	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
63	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
64	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
65	#95	—	3	4	no	—
66	#95	—	3	4	no	—
67	#95	—	3	4	no	—
68	#95	—	3	4	no	—
69	C₂³×C₃	—	2	2	no	—
70	C₂³×C₃	—	2	2	no	—
71	C₂³×C₃	C₂²×C₄	1	2	yes	—
72	C₂³×C₃	#41	1	4	yes	—
73	C₂³×C₃	C₂⁴	1	1	yes	characteristic
74	C₂×C₃×C₄	—	2	8	no	—
75	C₂×C₃×C₄	—	2	2	no	—
76	C₂×C₃×C₄	—	2	2	no	—
77	C₂×C₃×C₄	—	2	2	no	—
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	—	2	8	no	—
80	C₂×C₃×C₄	—	2	8	no	—
81	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
82	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
83	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
84	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
85	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
86	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
87	C₂×C₃×C₄	#41	1	4	yes	—
88	C₂×C₃×C₄	#41	1	4	yes	—
89	C₂×Dic₃	—	2	8	no	—
90	C₂×Dic₃	—	2	8	no	—
91	C₂×Dic₃	—	2	8	no	—
92	C₂⁴	C₂²×D₃	1	1	yes	characteristic
93	C₂²×C₄	—	2	4	no	—
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	—	2	4	no	—
96	C₂²×C₄	—	3	8	no	—
97	C₂²×C₄	C₂×Dic₃	1	4	yes	—
98	C₂²×C₄	C₂×Dic₃	1	4	yes	—
99	C₂²×C₄	C₂×Dic₃	1	4	yes	—
100	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
101	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
102	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₄⋊C₄	—	6	8	no	—
107	C₂²⋊C₄	—	6	8	no	—
108	C₄²	—	6	8	no	—
109	C₄²	C₂×Dic₃	1	4	yes	—
110	C₄²	C₂×Dic₃	1	4	yes	—
111	C₄²	C₂×Dic₃	1	4	yes	—
112	C₄²	C₂×Dic₃	1	4	yes	—
113	C₂²×C₃	—	2	2	no	—
114	C₂²×C₃	—	2	2	no	—
115	C₂²×C₃	—	2	8	no	—
116	C₂²×C₃	#117	1	4	yes	—
117	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
118	C₂²×C₃	C₂×#41	1	2	yes	—
119	C₂²×C₃	C₂×#41	1	2	yes	—
120	C₂²×C₃	C₂×#41	1	2	yes	—
121	C₃×C₄	—	2	2	no	—
122	C₃×C₄	—	2	2	no	—
123	C₃×C₄	—	2	2	no	—
124	C₃×C₄	—	2	2	no	—
125	C₃×C₄	#117	1	4	yes	—
126	C₃×C₄	#117	1	4	yes	—
127	Dic₃	—	4	8	no	—
128	C₂³	—	2	2	no	—
129	C₂³	—	2	2	no	—
130	C₂³	#235	1	4	yes	—
131	C₂³	C₂²×Dic₃	1	2	yes	—
132	C₂³	C₂³×D₃	1	1	yes	characteristic
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	6	8	no	—
141	C₂×C₄	—	6	8	no	—
142	C₂×C₄	—	6	8	no	—
143	C₂×C₄	#235	1	4	yes	—
144	C₂×C₄	#235	1	4	yes	—
145	C₂×C₄	C₂²×Dic₃	1	2	yes	—
146	C₂×C₄	C₂²×Dic₃	1	2	yes	—
147	C₂×C₄	C₂²×Dic₃	1	2	yes	—
148	C₂×C₄	C₂²×Dic₃	1	2	yes	—
149	C₂×C₄	C₂²×Dic₃	1	2	yes	—
150	C₂×C₄	C₂²×Dic₃	1	2	yes	—
151	C₂×C₃	—	2	4	no	—
152	C₂×C₃	C₂×#117	1	1	yes	characteristic
153	C₂×C₃	C₂×#117	1	1	yes	characteristic
154	C₂×C₃	C₂×#117	1	1	yes	characteristic
155	C₂×C₃	#529	1	4	yes	—
156	C₂²	—	2	8	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	#926	1	4	yes	—
160	C₂²	C₂×#235	1	2	yes	—
161	C₂²	C₂×#235	1	2	yes	—
162	C₂²	C₂×#235	1	2	yes	—
163	C₂²	C₂³×Dic₃	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	2	2	no	—
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	12	8	no	—
169	C₄	#926	1	4	yes	—
170	C₄	#926	1	4	yes	—
171	C₃	#2283	1	1	yes	characteristic, complemented, Sylow
172	C₂	—	2	4	no	—
173	C₂	#5479	1	4	yes	—
174	C₂	C₂×#926	1	1	yes	characteristic
175	C₂	C₂×#926	1	1	yes	characteristic
176	C₂	C₂×#926	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂³×C₄
Frattini	C₂³	C₂³×D₃
Fitting	C₂²×C₃×C₄²	C₂
socle, cosocle	C₂³×C₃	C₂⁴
prime radical	C₂⁴×C₃	C₂³
hypercenter	C₂²×C₄²	D₃
hypocenter	C₃	#2283
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
5. #4937	C₂	semidirect
6. #4936	C₂	semidirect
7. C₄×C₄⋊Dic₃	C₂	semidirect
8. #4932	C₂	semidirect
13. C₂×C₃×C₄²	C₄	semidirect
60. C₂×C₄²	Dic₃	semidirect
171. C₃	#2283	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄²	2
#5212	2
#5211	2
C₄×#217	2
#4937	2
#4936	2
C₄×C₄⋊Dic₃	2
#4932	2
#2283	3