Subgroups — #82233

Order 384.

600 conjugacy classes of subgroups.

The subgroup lattice has 184 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	C₂²×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	#4984	C₂	1	4	yes	maximal, complemented
5	#4950	C₂	1	4	yes	maximal
6	#4936	C₂	1	2	yes	maximal, complemented
7	#4935	C₂	1	2	yes	maximal, complemented
8	#2317	—	3	1	no	maximal, Sylow
9	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂³×Dic₃	C₂²	1	2	yes	—
11	C₃×#117	C₄	1	8	yes	complemented
12	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄²	C₂²	1	2	yes	—
15	C₂×#217	C₂²	1	2	yes	—
16	#926	—	2	4	no	—
17	C₂×C₄⋊Dic₃	—	2	4	no	—
18	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₃	C₂²	1	8	yes	—
23	C₂×C₄×Dic₃	C₂²	1	2	yes	—
24	C₃⋊#95	C₂²	1	8	yes	—
25	C₂×#117	—	3	1	no	—
26	C₂×#117	D₃	1	1	yes	characteristic
27	C₂²×C₄⋊C₄	—	3	1	no	—
28	#389	—	3	4	no	—
29	#385	—	3	4	no	—
30	#384	—	3	2	no	—
31	#382	—	3	2	no	—
32	C₂⁴×C₃	C₂³	1	1	yes	characteristic
33	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
34	C₂²×C₃×C₄	D₄	1	4	yes	—
35	C₂²×C₃×C₄	Q₈	1	4	yes	—
36	C₂²×C₃×C₄	C₂³	1	4	yes	—
37	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂²×Dic₃	—	2	4	no	—
40	C₂²×Dic₃	—	4	4	no	—
41	C₂²×Dic₃	C₂³	1	2	yes	—
42	C₂²×Dic₃	C₂³	1	4	yes	—
43	C₂²×Dic₃	C₂³	1	2	yes	—
44	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	complemented
45	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
46	C₃×C₄²	C₂×C₄	1	8	yes	complemented
47	#217	—	2	4	no	—
48	C₄⋊Dic₃	—	2	4	no	—
49	C₄⋊Dic₃	—	2	4	no	—
50	C₄⋊Dic₃	—	2	2	no	—
51	C₄⋊Dic₃	—	2	2	no	—
52	C₄⋊Dic₃	—	4	8	no	—
53	C₄×Dic₃	—	2	4	no	—
54	C₂³×C₄	—	3	2	no	—
55	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
56	#117	—	6	4	no	—
57	#117	Dic₃	1	8	yes	complemented
58	C₂×C₄⋊C₄	—	3	2	no	—
59	C₂×C₄⋊C₄	—	3	2	no	—
60	C₂×C₄⋊C₄	—	3	1	no	—
61	C₂×C₄⋊C₄	—	3	1	no	—
62	C₂×C₄⋊C₄	—	3	8	no	—
63	C₂×C₄⋊C₄	—	6	4	no	—
64	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
65	C₂×C₂²⋊C₄	—	3	2	no	—
66	C₂×C₂²⋊C₄	C₂×D₃	1	2	yes	—
67	C₂×C₄²	—	3	2	no	—
68	C₂×C₄²	C₂×D₃	1	2	yes	—
69	#95	—	3	8	no	—
70	C₂³×C₃	—	2	2	no	—
71	C₂³×C₃	—	2	2	no	—
72	C₂³×C₃	C₂²×C₄	1	2	yes	—
73	C₂³×C₃	C₂×D₄	1	2	yes	—
74	C₂³×C₃	C₂×Q₈	1	2	yes	—
75	C₂³×C₃	C₂⁴	1	1	yes	characteristic
76	C₂×C₃×C₄	—	2	4	no	—
77	C₂×C₃×C₄	—	2	4	no	—
78	C₂×C₃×C₄	C₄⋊C₄	1	16	yes	complemented
79	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
80	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
81	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
82	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
85	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
86	C₂×Dic₃	—	2	2	no	—
87	C₂×Dic₃	—	2	2	no	—
88	C₂×Dic₃	—	2	4	no	—
89	C₂×Dic₃	—	2	4	no	—
90	C₂×Dic₃	—	2	8	no	—
91	C₂×Dic₃	—	2	4	no	—
92	C₂×Dic₃	—	4	8	no	—
93	C₂⁴	C₂²×D₃	1	1	yes	characteristic
94	C₂²×C₄	—	3	2	no	—
95	C₂²×C₄	—	3	4	no	—
96	C₂²×C₄	—	3	2	no	—
97	C₂²×C₄	—	6	4	no	—
98	C₂²×C₄	—	12	4	no	—
99	C₂²×C₄	Dic₆	1	4	yes	—
100	C₂²×C₄	D₁₂	1	4	yes	—
101	C₂²×C₄	C₂×Dic₃	1	4	yes	—
102	C₂²×C₄	C₂²×D₃	1	4	yes	—
103	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₄⋊C₄	—	6	4	no	—
106	C₄⋊C₄	—	6	4	no	—
107	C₄⋊C₄	—	6	2	no	—
108	C₄⋊C₄	—	6	2	no	—
109	C₄⋊C₄	—	12	8	no	—
110	C₄⋊C₄	C₂×Dic₃	1	8	yes	complemented
111	C₂²⋊C₄	—	6	4	no	—
112	C₂²⋊C₄	C₂×Dic₃	1	8	yes	—
113	C₄²	—	6	4	no	—
114	C₄²	C₂×Dic₃	1	8	yes	complemented
115	C₂²×C₃	—	2	4	no	—
116	C₂²×C₃	—	2	2	no	—
117	C₂²×C₃	—	2	2	no	—
118	C₂²×C₃	—	2	4	no	—
119	C₂²×C₃	C₂×C₄⋊C₄	1	4	yes	—
120	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
121	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
124	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
125	C₃×C₄	—	2	8	no	—
126	C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
127	C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
128	Dic₃	—	4	2	no	—
129	Dic₃	—	4	4	no	—
130	Dic₃	—	4	2	no	—
131	C₂³	—	2	2	no	—
132	C₂³	—	2	2	no	—
133	C₂³	C₂×Dic₆	1	2	yes	—
134	C₂³	C₂×D₁₂	1	2	yes	—
135	C₂³	C₂²×Dic₃	1	2	yes	—
136	C₂³	C₂³×D₃	1	1	yes	characteristic
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	6	2	no	—
140	C₂×C₄	—	6	2	no	—
141	C₂×C₄	—	6	4	no	—
142	C₂×C₄	—	6	4	no	—
143	C₂×C₄	—	6	8	no	—
144	C₂×C₄	—	6	4	no	—
145	C₂×C₄	—	12	8	no	—
146	C₂×C₄	C₄⋊Dic₃	1	16	yes	complemented
147	C₂×C₄	C₂×Dic₆	1	2	yes	—
148	C₂×C₄	C₂×Dic₆	1	2	yes	—
149	C₂×C₄	C₂×D₁₂	1	2	yes	—
150	C₂×C₄	C₂×D₁₂	1	2	yes	—
151	C₂×C₄	C₂²×Dic₃	1	8	yes	—
152	C₂×C₄	C₂²×Dic₃	1	2	yes	—
153	C₂×C₄	C₂²×Dic₃	1	2	yes	—
154	C₂×C₃	—	2	4	no	—
155	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₃	#520	1	2	yes	—
157	C₂×C₃	#535	1	1	yes	characteristic
158	C₂×C₃	#536	1	1	yes	characteristic
159	C₂×C₃	#544	1	2	yes	—
160	C₂²	—	2	4	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	4	no	—
164	C₂²	C₂×C₄⋊Dic₃	1	4	yes	—
165	C₂²	C₂²×Dic₆	1	1	yes	characteristic
166	C₂²	C₂²×D₁₂	1	1	yes	characteristic
167	C₂²	#1009	1	2	yes	—
168	C₂²	#1010	1	2	yes	—
169	C₂²	C₂³×Dic₃	1	1	yes	characteristic
170	C₄	—	2	8	no	—
171	C₄	—	12	4	no	—
172	C₄	—	12	2	no	—
173	C₄	—	12	2	no	—
174	C₄	C₂×C₄⋊Dic₃	1	4	yes	—
175	C₄	C₂×C₄⋊Dic₃	1	4	yes	—
176	C₃	#2317	1	1	yes	characteristic, complemented, Sylow
177	C₂	—	2	4	no	—
178	C₂	#5519	1	2	yes	—
179	C₂	#5525	1	1	yes	characteristic
180	C₂	#5526	1	1	yes	characteristic
181	C₂	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
182	C₂	#5827	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×#117	C₂
socle, cosocle	C₂³×C₃	C₂⁴
prime radical	C₂⁴×C₃	C₂³
hypercenter	C₂×#117	D₃
hypocenter	C₃	#2317
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
4. #4984	C₂	semidirect
6. #4936	C₂	semidirect
7. #4935	C₂	semidirect
11. C₃×#117	C₄	semidirect
44. C₃×C₄⋊C₄	C₂×C₄	semidirect
46. C₃×C₄²	C₂×C₄	semidirect
57. #117	Dic₃	semidirect
78. C₂×C₃×C₄	C₄⋊C₄	semidirect
110. C₄⋊C₄	C₂×Dic₃	semidirect
114. C₄²	C₂×Dic₃	semidirect
146. C₂×C₄	C₄⋊Dic₃	semidirect
176. C₃	#2317	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
C₂²×C₄⋊Dic₃	2
#4984	2
#4950	2
#4936	2
#4935	2
#2317	3