Subgroups — 3₋¹⁺²×#477 #13145191

Order 1728.

384 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²×#107	C₂	1	2	yes	maximal
4	3₋¹⁺²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	3₋¹⁺²×#103	C₂	1	2	yes	maximal
6	C₃²×#477	C₃	1	1	yes	maximal, characteristic
7	C₉×#477	C₃	1	3	yes	maximal, complemented
8	C₂×Q₈×3₋¹⁺²	—	2	1	no	—
9	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
11	C₄⋊C₄×3₋¹⁺²	—	2	1	no	—
12	C₄⋊C₄×3₋¹⁺²	—	2	1	no	—
13	C₄⋊C₄×3₋¹⁺²	C₂²	1	2	yes	—
14	C₄⋊C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
15	C₄²×3₋¹⁺²	—	2	1	no	—
16	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
17	C₃²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
18	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
19	C₃²×#107	C₂×C₃	1	2	yes	—
20	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
21	C₃²×#103	C₂×C₃	1	2	yes	—
22	C₉×C₄⋊Q₈	C₂×C₃	1	3	yes	—
23	C₄×Q₈×C₉	C₂×C₃	1	3	yes	—
24	C₉×#107	C₂×C₃	1	6	yes	—
25	C₉×C₄⋊C₈	C₂×C₃	1	3	yes	—
26	C₉×#103	C₂×C₃	1	6	yes	—
27	Q₈×3₋¹⁺²	—	2	1	no	—
28	Q₈×3₋¹⁺²	—	4	2	no	—
29	Q₈×3₋¹⁺²	Q₈	1	2	yes	complemented
30	C₂×C₄×3₋¹⁺²	—	2	1	no	—
31	C₂×C₄×3₋¹⁺²	—	2	2	no	—
32	C₂×C₄×3₋¹⁺²	—	2	1	no	—
33	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
34	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
35	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
36	C₈×3₋¹⁺²	—	2	2	no	—
37	C₃×#477	—	3	1	no	—
38	C₃×#477	C₃²	1	1	yes	characteristic
39	C₂×C₃²×Q₈	—	2	1	no	—
40	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
42	C₃²×C₄⋊C₄	—	2	1	no	—
43	C₃²×C₄⋊C₄	—	2	1	no	—
44	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
45	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
46	C₃²×C₄²	—	2	1	no	—
47	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
48	C₂×Q₈×C₉	—	2	3	no	—
49	C₂×Q₈×C₉	C₂²×C₃	1	3	yes	—
50	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
51	C₉×C₄⋊C₄	—	2	3	no	—
52	C₉×C₄⋊C₄	—	2	3	no	—
53	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
54	C₉×C₄⋊C₄	C₂²×C₃	1	6	yes	—
55	C₄²×C₉	—	2	3	no	—
56	C₄²×C₉	C₂²×C₃	1	3	yes	—
57	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
58	C₄×3₋¹⁺²	—	2	2	no	—
59	C₄×3₋¹⁺²	—	2	1	no	—
60	C₄×3₋¹⁺²	—	4	1	no	—
61	C₄×3₋¹⁺²	—	4	2	no	—
62	C₄×3₋¹⁺²	Q₁₆	1	2	yes	—
63	C₄×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
64	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
65	C₃×C₄⋊Q₈	—	3	1	no	—
66	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
67	C₃×C₄×Q₈	—	3	1	no	—
68	C₃×C₄×Q₈	C₂×C₃²	1	1	yes	characteristic
69	C₃×#107	—	3	2	no	—
70	C₃×#107	C₂×C₃²	1	2	yes	—
71	C₃×C₄⋊C₈	—	3	1	no	—
72	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
73	C₃×#103	—	3	2	no	—
74	C₃×#103	C₂×C₃²	1	2	yes	—
75	C₃²×Q₈	—	2	1	no	—
76	C₃²×Q₈	—	4	2	no	—
77	C₃²×Q₈	C₃×Q₈	1	2	yes	—
78	C₂×C₃²×C₄	—	2	1	no	—
79	C₂×C₃²×C₄	—	2	1	no	—
80	C₂×C₃²×C₄	—	2	2	no	—
81	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
84	C₃²×C₈	—	2	2	no	—
85	Q₈×C₉	—	2	3	no	—
86	Q₈×C₉	—	4	6	no	—
87	Q₈×C₉	C₃×Q₈	1	6	yes	complemented
88	C₂×C₄×C₉	—	2	3	no	—
89	C₂×C₄×C₉	—	2	3	no	—
90	C₂×C₄×C₉	—	2	6	no	—
91	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
92	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
93	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
94	C₈×C₉	—	2	6	no	—
95	#477	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×3₋¹⁺²	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×3₋¹⁺²	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×3₋¹⁺²	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₃×Q₈	—	2	1	no	—
100	C₂×C₃×Q₈	—	3	1	no	—
101	C₂×C₃×Q₈	—	6	1	no	—
102	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
103	C₂×C₃×C₈	—	3	2	no	—
104	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
105	C₃×C₄⋊C₄	—	2	1	no	—
106	C₃×C₄⋊C₄	—	2	1	no	—
107	C₃×C₄⋊C₄	—	3	2	no	—
108	C₃×C₄⋊C₄	—	3	1	no	—
109	C₃×C₄⋊C₄	—	6	1	no	—
110	C₃×C₄⋊C₄	—	6	1	no	—
111	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
112	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
113	C₃×C₄²	—	2	1	no	—
114	C₃×C₄²	—	3	1	no	—
115	C₃×C₄²	—	6	1	no	—
116	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
117	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
118	C₃²×C₄	—	2	2	no	—
119	C₃²×C₄	—	2	1	no	—
120	C₃²×C₄	—	4	1	no	—
121	C₃²×C₄	—	4	2	no	—
122	C₃²×C₄	C₃×Q₁₆	1	2	yes	—
123	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
124	C₃²×C₄	C₃×#41	1	1	yes	characteristic
125	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
126	C₄×C₉	—	2	6	no	—
127	C₄×C₉	—	2	3	no	—
128	C₄×C₉	—	4	3	no	—
129	C₄×C₉	—	4	6	no	—
130	C₄×C₉	C₃×Q₁₆	1	6	yes	—
131	C₄×C₉	C₂×C₃×Q₈	1	3	yes	—
132	C₄×C₉	C₃×#41	1	3	yes	—
133	C₄⋊Q₈	C₂×3₋¹⁺²	1	1	yes	characteristic
134	C₄×Q₈	C₂×3₋¹⁺²	1	1	yes	characteristic
135	#107	C₂×3₋¹⁺²	1	2	yes	—
136	C₄⋊C₈	C₂×3₋¹⁺²	1	1	yes	characteristic
137	#103	C₂×3₋¹⁺²	1	2	yes	—
138	3₋¹⁺²	#477	1	1	yes	characteristic, direct product factor, Sylow
139	C₃×Q₈	—	2	1	no	—
140	C₃×Q₈	—	3	2	no	—
141	C₃×Q₈	—	4	2	no	—
142	C₃×Q₈	—	6	1	no	—
143	C₃×Q₈	—	12	2	no	—
144	C₃×Q₈	C₃²×Q₈	1	2	yes	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	3	1	no	—
149	C₂×C₃×C₄	—	3	1	no	—
150	C₂×C₃×C₄	—	3	1	no	—
151	C₂×C₃×C₄	—	6	1	no	—
152	C₂×C₃×C₄	—	6	1	no	—
153	C₂×C₃×C₄	—	6	2	no	—
154	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
157	C₃×C₈	—	2	2	no	—
158	C₃×C₈	—	6	2	no	—
159	C₂×C₃²	C₃×C₂²⋊Q₈	1	1	yes	characteristic
160	C₂×C₃²	C₂×C₃×Q₁₆	1	1	yes	characteristic
161	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×C₉	C₃×C₂²⋊Q₈	1	3	yes	—
163	C₂×C₉	C₂×C₃×Q₁₆	1	3	yes	—
164	C₂×C₉	C₃×AGL(1,ℤ/8)	1	3	yes	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
167	C₂×C₈	C₂²×3₋¹⁺²	1	2	yes	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	C₂²×3₋¹⁺²	1	2	yes	—
171	C₄⋊C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
172	C₄²	—	2	1	no	—
173	C₄²	C₂²×3₋¹⁺²	1	1	yes	characteristic
174	C₂²×C₃	—	3	1	no	—
175	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
176	C₃×C₄	—	2	2	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	3	2	no	—
179	C₃×C₄	—	3	1	no	—
180	C₃×C₄	—	3	1	no	—
181	C₃×C₄	—	4	2	no	—
182	C₃×C₄	—	4	1	no	—
183	C₃×C₄	—	6	1	no	—
184	C₃×C₄	—	6	2	no	—
185	C₃×C₄	—	12	2	no	—
186	C₃×C₄	—	12	1	no	—
187	C₃×C₄	C₃²×Q₁₆	1	2	yes	—
188	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
189	C₃×C₄	C₃²×#41	1	1	yes	characteristic
190	C₃²	C₃×#477	1	1	yes	characteristic
191	C₉	C₃×#477	1	3	yes	complemented
192	Q₈	—	2	1	no	—
193	Q₈	—	4	2	no	—
194	Q₈	Q₈×3₋¹⁺²	1	2	yes	complemented
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
199	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
200	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₂×C₃	—	3	1	no	—
203	C₂×C₃	—	3	1	no	—
204	C₂×C₃	—	3	1	no	—
205	C₂×C₃	C₃²×C₂²⋊Q₈	1	1	yes	characteristic
206	C₂×C₃	C₂×C₃²×Q₁₆	1	1	yes	characteristic
207	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic
208	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	2	1	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	1	no	—
213	C₄	Q₁₆×3₋¹⁺²	1	2	yes	—
214	C₄	C₂×Q₈×3₋¹⁺²	1	1	yes	characteristic
215	C₄	#41×3₋¹⁺²	1	1	yes	characteristic
216	C₃	—	3	1	no	—
217	C₃	C₃²×#477	1	1	yes	characteristic
218	C₂	3₋¹⁺²×C₂²⋊Q₈	1	1	yes	characteristic
219	C₂	C₂×Q₁₆×3₋¹⁺²	1	1	yes	characteristic
220	C₂	3₋¹⁺²×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	C₂²×C₃²	C₂×C₃×D₄
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₄²×3₋¹⁺² ◁ G

Complements

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

N	H ≅ G/N	type
7. C₉×#477	C₃	semidirect
29. Q₈×3₋¹⁺²	Q₈	semidirect
87. Q₈×C₉	C₃×Q₈	semidirect
95. #477	3₋¹⁺²	direct (N × H)
138. 3₋¹⁺²	#477	direct (N × H)
191. C₉	C₃×#477	semidirect
194. Q₈	Q₈×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
3₋¹⁺²×C₄⋊Q₈	2
C₄×Q₈×3₋¹⁺²	2
3₋¹⁺²×#107	2
3₋¹⁺²×C₄⋊C₈	2
3₋¹⁺²×#103	2
C₃²×#477	3
C₉×#477	3

Subgroups — 3−1+2×#477 #13145191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#477 #13145191