Subgroups — C₄×C₄⋊C₄×3₋¹⁺² #13144894

Order 1728.

776 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
6. C₄×C₉×C₄⋊C₄	C₃	semidirect
11. C₄⋊C₄×3₋¹⁺²	C₄	direct (N × H)
13. C₄²×3₋¹⁺²	C₄	semidirect
46. C₉×C₄⋊C₄	C₃×C₄	semidirect
48. C₄²×C₉	C₃×C₄	semidirect
56. C₄×3₋¹⁺²	C₄²	semidirect
57. C₄×3₋¹⁺²	C₄⋊C₄	direct (N × H)
88. C₄×C₄⋊C₄	3₋¹⁺²	direct (N × H)
124. C₄×C₉	C₃×C₄²	semidirect
125. C₄×C₉	C₃×C₄⋊C₄	semidirect
130. 3₋¹⁺²	C₄×C₄⋊C₄	direct (N × H)
167. C₄⋊C₄	C₄×3₋¹⁺²	direct (N × H)
169. C₄²	C₄×3₋¹⁺²	semidirect
189. C₉	C₃×C₄×C₄⋊C₄	semidirect
218. C₄	C₄²×3₋¹⁺²	semidirect
219. C₄	C₄⋊C₄×3₋¹⁺²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×3₋¹⁺²	2
C₂×C₄²×3₋¹⁺²	2
3₋¹⁺²×#95	2
C₃²×C₄×C₄⋊C₄	3
C₄×C₉×C₄⋊C₄	3

Subgroups — C4×C4⋊C4×3−1+2 #13144894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄⋊C₄×3₋¹⁺² #13144894