Subgroups — #82482

Order 384.

484 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
3	#5229	C₂	1	2	yes	maximal
4	#4982	C₂	1	4	yes	maximal
5	#4972	C₂	1	2	yes	maximal
6	#4935	C₂	1	1	yes	maximal, characteristic
7	#4929	C₂	1	2	yes	maximal
8	#4928	C₂	1	2	yes	maximal
9	#2582	—	3	1	no	maximal, Sylow
10	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
11	C₂²×Dic₆	C₂²	1	1	yes	characteristic
12	C₃×C₄×Q₈	—	2	4	no	—
13	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄²	C₂²	1	2	yes	—
16	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
21	C₂×Dic₃⋊C₄	C₂²	1	2	yes	—
22	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
23	C₂×C₄×Dic₃	C₂²	1	2	yes	—
24	C₄×Dic₆	—	2	4	no	—
25	C₃⋊#95	C₂²	1	4	yes	—
26	C₃⋊#95	C₂²	1	4	yes	—
27	C₃⋊#95	C₂²	1	4	yes	—
28	C₂×C₄×Q₈	—	3	1	no	—
29	C₂×C₄×Q₈	D₃	1	1	yes	characteristic
30	#398	—	3	4	no	—
31	#391	—	3	2	no	—
32	#391	—	3	2	no	—
33	#389	—	3	2	no	—
34	#389	—	3	2	no	—
35	#384	—	3	1	no	—
36	C₂×C₃×Q₈	—	2	2	no	—
37	C₂×C₃×Q₈	—	2	2	no	—
38	C₂×C₃×Q₈	Q₈	1	4	yes	complemented
39	C₂²×C₃×C₄	C₂³	1	2	yes	—
40	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
41	C₂²×C₃×C₄	C₂³	1	2	yes	—
42	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
43	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
44	C₂²×Dic₃	C₂³	1	4	yes	—
45	C₂²×Dic₃	C₂³	1	2	yes	—
46	C₂²×Dic₃	C₂³	1	2	yes	—
47	C₂×Dic₆	—	2	2	no	—
48	C₂×Dic₆	—	2	2	no	—
49	C₂×Dic₆	Q₈	1	4	yes	complemented
50	C₃×C₄⋊C₄	—	2	4	no	—
51	C₃×C₄⋊C₄	—	2	2	no	—
52	C₃×C₄²	—	2	4	no	—
53	C₃×C₄²	D₄	1	4	yes	—
54	C₄⋊Dic₃	—	2	2	no	—
55	C₄⋊Dic₃	—	2	4	no	—
56	C₄⋊Dic₃	—	2	4	no	—
57	C₄⋊Dic₃	—	4	2	no	—
58	Dic₃⋊C₄	—	2	4	no	—
59	Dic₃⋊C₄	—	4	4	no	—
60	C₄×Dic₃	—	2	4	no	—
61	C₂²×Q₈	—	3	1	no	—
62	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic
63	C₄×Q₈	—	2	4	no	—
64	C₄×Q₈	—	6	4	no	—
65	C₂×C₄⋊C₄	—	3	1	no	—
66	C₂×C₄⋊C₄	—	3	2	no	—
67	C₂×C₄⋊C₄	—	3	2	no	—
68	C₂×C₄⋊C₄	—	3	2	no	—
69	C₂×C₄⋊C₄	—	3	2	no	—
70	C₂×C₄⋊C₄	—	3	4	no	—
71	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
72	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
73	C₂×C₄²	—	3	2	no	—
74	C₂×C₄²	C₂×D₃	1	2	yes	—
75	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
76	#95	—	3	4	no	—
77	#95	—	3	4	no	—
78	#95	—	3	4	no	—
79	C₂³×C₃	C₂⁴	1	1	yes	characteristic
80	C₃×Q₈	—	2	8	no	—
81	C₂×C₃×C₄	—	2	2	no	—
82	C₂×C₃×C₄	—	2	2	no	—
83	C₂×C₃×C₄	—	2	1	no	—
84	C₂×C₃×C₄	—	2	1	no	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	—	2	2	no	—
88	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
89	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
90	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
91	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
92	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
93	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
94	C₂×C₃×C₄	#41	1	2	yes	—
95	C₂×C₃×C₄	#41	1	2	yes	—
96	C₂×Dic₃	—	2	2	no	—
97	C₂×Dic₃	—	2	2	no	—
98	C₂×Dic₃	—	2	2	no	—
99	C₂×Dic₃	—	2	4	no	—
100	C₂×Dic₃	—	2	4	no	—
101	C₂×Dic₃	—	2	2	no	—
102	C₂×Dic₃	—	2	4	no	—
103	C₂×Dic₃	—	2	2	no	—
104	C₂×Dic₃	C₂×Q₈	1	4	yes	—
105	Dic₆	—	2	8	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	—	2	2	no	—
108	C₂×Q₈	—	3	4	no	—
109	C₂×Q₈	—	6	2	no	—
110	C₂×Q₈	—	6	2	no	—
111	C₂×Q₈	Dic₆	1	4	yes	complemented
112	C₂²×C₄	—	3	2	no	—
113	C₂²×C₄	—	3	4	no	—
114	C₂²×C₄	—	3	2	no	—
115	C₂²×C₄	C₂²×D₃	1	2	yes	—
116	C₂²×C₄	C₂²×D₃	1	2	yes	—
117	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
119	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	6	4	no	—
123	C₄⋊C₄	—	6	4	no	—
124	C₄⋊C₄	—	6	4	no	—
125	C₄⋊C₄	—	6	2	no	—
126	C₄⋊C₄	—	12	4	no	—
127	C₄⋊C₄	—	12	2	no	—
128	C₄²	—	2	4	no	—
129	C₄²	—	6	4	no	—
130	C₄²	#67	1	4	yes	—
131	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
132	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
134	C₂²×C₃	C₂×#41	1	1	yes	characteristic
135	C₂²×C₃	C₂×#41	1	1	yes	characteristic
136	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
138	C₃×C₄	—	2	4	no	—
139	C₃×C₄	—	2	2	no	—
140	C₃×C₄	—	4	2	no	—
141	C₃×C₄	C₂²⋊Q₈	1	4	yes	—
142	C₃×C₄	C₂²⋊Q₈	1	4	yes	—
143	Dic₃	—	2	4	no	—
144	Dic₃	—	4	2	no	—
145	Dic₃	—	4	4	no	—
146	C₂³	C₂³×D₃	1	1	yes	characteristic
147	Q₈	—	2	8	no	—
148	Q₈	—	6	8	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	3	4	no	—
157	C₂×C₄	—	6	2	no	—
158	C₂×C₄	—	6	2	no	—
159	C₂×C₄	—	6	4	no	—
160	C₂×C₄	—	6	4	no	—
161	C₂×C₄	—	6	2	no	—
162	C₂×C₄	—	6	4	no	—
163	C₂×C₄	—	6	2	no	—
164	C₂×C₄	—	6	2	no	—
165	C₂×C₄	C₂×Dic₆	1	4	yes	—
166	C₂×C₄	C₂×Dic₆	1	2	yes	—
167	C₂×C₄	#235	1	2	yes	—
168	C₂×C₄	D₃×Q₈	1	2	yes	—
169	C₂×C₄	#239	1	2	yes	—
170	C₂×C₄	C₂×#67	1	2	yes	—
171	C₂×C₄	C₂×#67	1	2	yes	—
172	C₂×C₄	C₂×#67	1	2	yes	—
173	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
174	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
175	C₂×C₃	#537	1	1	yes	characteristic
176	C₂×C₃	#557	1	1	yes	characteristic
177	C₂×C₃	#557	1	1	yes	characteristic
178	C₂×C₃	Q₈²	1	1	yes	characteristic
179	C₂×C₃	#559	1	1	yes	characteristic
180	C₂²	C₂²×Dic₆	1	1	yes	characteristic
181	C₂²	C₂×#235	1	1	yes	characteristic
182	C₂²	C₂×D₃×Q₈	1	1	yes	characteristic
183	C₂²	C₂×#239	1	1	yes	characteristic
184	C₂²	#1009	1	1	yes	characteristic
185	C₂²	#1010	1	1	yes	characteristic
186	C₂²	C₂²×#67	1	1	yes	characteristic
187	C₄	—	2	4	no	—
188	C₄	—	2	2	no	—
189	C₄	—	4	2	no	—
190	C₄	—	6	4	no	—
191	C₄	—	12	2	no	—
192	C₄	—	12	4	no	—
193	C₄	#924	1	4	yes	—
194	C₄	#946	1	4	yes	—
195	C₃	#2582	1	1	yes	characteristic, complemented, Sylow
196	C₂	Q₈×Dic₆	1	1	yes	characteristic
197	C₂	#5567	1	1	yes	characteristic
198	C₂	#5570	1	1	yes	characteristic
199	C₂	#5585	1	1	yes	characteristic
200	C₂	C₂×#924	1	1	yes	characteristic
201	C₂	C₂×#946	1	1	yes	characteristic
202	C₂	#5833	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₂×C₃×C₄×Q₈	C₂
hypercenter	C₂×C₄×Q₈	D₃
hypocenter	C₃	#2582
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
38. C₂×C₃×Q₈	Q₈	semidirect
49. C₂×Dic₆	Q₈	semidirect
111. C₂×Q₈	Dic₆	semidirect
195. C₃	#2582	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×Q₈	2
C₂×C₄×Dic₆	2
#5229	2
#4982	2
#4972	2
#4935	2
#4929	2
#4928	2
#2582	3