Subgroups — D₄×Q₈×3₋¹⁺² #13154631

Order 1728.

1120 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	C₂²×Q₈×3₋¹⁺²	C₂	1	2	yes	maximal, complemented
2	3₋¹⁺²×C₄⋊Q₈	C₂	1	3	yes	maximal, complemented
3	3₋¹⁺²×C₂²⋊Q₈	C₂	1	6	yes	maximal, complemented
4	C₄×Q₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₄×3₋¹⁺²	C₂	1	3	yes	maximal
6	C₃²×D₄×Q₈	C₃	1	1	yes	maximal, characteristic
7	D₄×Q₈×C₉	C₃	1	3	yes	maximal, complemented
8	C₂×Q₈×3₋¹⁺²	—	2	8	no	—
9	C₂×Q₈×3₋¹⁺²	C₂²	1	6	yes	—
10	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₂×D₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
12	C₂²×C₄×3₋¹⁺²	C₂²	1	6	yes	—
13	C₄⋊C₄×3₋¹⁺²	C₂²	1	3	yes	—
14	C₄⋊C₄×3₋¹⁺²	C₂²	1	3	yes	—
15	C₄⋊C₄×3₋¹⁺²	C₂²	1	6	yes	—
16	C₂²⋊C₄×3₋¹⁺²	C₂²	1	6	yes	—
17	C₄²×3₋¹⁺²	C₂²	1	3	yes	—
18	C₂²×C₃²×Q₈	C₂×C₃	1	2	yes	—
19	C₃²×C₄⋊Q₈	C₂×C₃	1	3	yes	—
20	C₃²×C₂²⋊Q₈	C₂×C₃	1	6	yes	—
21	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
22	C₃²×C₄×D₄	C₂×C₃	1	3	yes	—
23	C₂²×Q₈×C₉	C₂×C₃	1	6	yes	complemented
24	C₉×C₄⋊Q₈	C₂×C₃	1	9	yes	complemented
25	C₉×C₂²⋊Q₈	C₂×C₃	1	18	yes	complemented
26	C₄×Q₈×C₉	C₂×C₃	1	3	yes	complemented
27	C₄×D₄×C₉	C₂×C₃	1	9	yes	—
28	C₂³×3₋¹⁺²	C₂³	1	2	yes	—
29	Q₈×3₋¹⁺²	—	2	12	no	—
30	Q₈×3₋¹⁺²	D₄	1	4	yes	direct product factor
31	D₄×3₋¹⁺²	Q₈	1	4	yes	direct product factor
32	C₂×C₄×3₋¹⁺²	—	2	12	no	—
33	C₂×C₄×3₋¹⁺²	C₂³	1	6	yes	—
34	C₂×C₄×3₋¹⁺²	C₂³	1	3	yes	—
35	C₂×C₄×3₋¹⁺²	C₂³	1	3	yes	—
36	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
37	C₃×D₄×Q₈	—	3	1	no	—
38	C₃×D₄×Q₈	C₃²	1	1	yes	characteristic
39	C₂×C₃²×Q₈	—	2	8	no	—
40	C₂×C₃²×Q₈	C₂²×C₃	1	6	yes	—
41	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
43	C₂²×C₃²×C₄	C₂²×C₃	1	6	yes	—
44	C₃²×C₄⋊C₄	C₂²×C₃	1	3	yes	—
45	C₃²×C₄⋊C₄	C₂²×C₃	1	3	yes	—
46	C₃²×C₄⋊C₄	C₂²×C₃	1	6	yes	—
47	C₃²×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
48	C₃²×C₄²	C₂²×C₃	1	3	yes	—
49	C₂×Q₈×C₉	—	2	24	no	—
50	C₂×Q₈×C₉	C₂²×C₃	1	18	yes	—
51	C₂×Q₈×C₉	C₂²×C₃	1	3	yes	—
52	C₂×D₄×C₉	C₂²×C₃	1	3	yes	—
53	C₂²×C₄×C₉	C₂²×C₃	1	18	yes	—
54	C₉×C₄⋊C₄	C₂²×C₃	1	9	yes	—
55	C₉×C₄⋊C₄	C₂²×C₃	1	9	yes	—
56	C₉×C₄⋊C₄	C₂²×C₃	1	18	yes	—
57	C₉×C₂²⋊C₄	C₂²×C₃	1	18	yes	—
58	C₄²×C₉	C₂²×C₃	1	9	yes	—
59	C₂²×3₋¹⁺²	—	2	2	no	—
60	C₂²×3₋¹⁺²	—	2	2	no	—
61	C₂²×3₋¹⁺²	C₂×Q₈	1	4	yes	complemented
62	C₂²×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
63	C₄×3₋¹⁺²	—	2	6	no	—
64	C₄×3₋¹⁺²	—	2	3	no	—
65	C₄×3₋¹⁺²	C₂×D₄	1	6	yes	—
66	C₄×3₋¹⁺²	C₂×Q₈	1	2	yes	complemented
67	C₂²×C₃×Q₈	—	3	2	no	—
68	C₂²×C₃×Q₈	C₂×C₃²	1	2	yes	—
69	C₃×C₄⋊Q₈	—	3	3	no	—
70	C₃×C₄⋊Q₈	C₂×C₃²	1	3	yes	—
71	C₃×C₂²⋊Q₈	—	3	6	no	—
72	C₃×C₂²⋊Q₈	C₂×C₃²	1	6	yes	—
73	C₃×C₄×Q₈	—	3	1	no	—
74	C₃×C₄×Q₈	C₂×C₃²	1	1	yes	characteristic
75	C₃×C₄×D₄	—	3	3	no	—
76	C₃×C₄×D₄	C₂×C₃²	1	3	yes	—
77	C₂³×C₃²	C₂³×C₃	1	2	yes	—
78	C₃²×Q₈	—	2	12	no	—
79	C₃²×Q₈	C₃×D₄	1	4	yes	—
80	C₃²×D₄	C₃×Q₈	1	4	yes	—
81	C₂×C₃²×C₄	—	2	12	no	—
82	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
83	C₂×C₃²×C₄	C₂³×C₃	1	6	yes	—
84	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
85	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
86	C₂³×C₉	C₂³×C₃	1	6	yes	—
87	Q₈×C₉	—	2	36	no	—
88	Q₈×C₉	C₃×D₄	1	12	yes	complemented
89	D₄×C₉	C₃×Q₈	1	12	yes	complemented
90	C₂×C₄×C₉	—	2	36	no	—
91	C₂×C₄×C₉	C₂³×C₃	1	9	yes	—
92	C₂×C₄×C₉	C₂³×C₃	1	18	yes	—
93	C₂×C₄×C₉	C₂³×C₃	1	9	yes	—
94	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
95	D₄×Q₈	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×3₋¹⁺²	—	2	4	no	—
97	C₂×3₋¹⁺²	C₂²×D₄	1	1	yes	characteristic
98	C₂×3₋¹⁺²	C₂²×Q₈	1	1	yes	characteristic
99	C₂×3₋¹⁺²	2₋¹⁺⁴	1	1	yes	characteristic
100	C₂×C₃×Q₈	—	2	8	no	—
101	C₂×C₃×Q₈	—	3	1	no	—
102	C₂×C₃×Q₈	—	3	6	no	—
103	C₂×C₃×Q₈	—	6	8	no	—
104	C₂×C₃×Q₈	C₂²×C₃²	1	6	yes	—
105	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
106	C₂×C₃×D₄	—	3	1	no	—
107	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
108	C₂²×C₃×C₄	—	3	6	no	—
109	C₂²×C₃×C₄	C₂²×C₃²	1	6	yes	—
110	C₃×C₄⋊C₄	—	3	3	no	—
111	C₃×C₄⋊C₄	—	3	3	no	—
112	C₃×C₄⋊C₄	—	3	6	no	—
113	C₃×C₄⋊C₄	C₂²×C₃²	1	3	yes	—
114	C₃×C₄⋊C₄	C₂²×C₃²	1	3	yes	—
115	C₃×C₄⋊C₄	C₂²×C₃²	1	6	yes	—
116	C₃×C₂²⋊C₄	—	3	6	no	—
117	C₃×C₂²⋊C₄	C₂²×C₃²	1	6	yes	—
118	C₃×C₄²	—	3	3	no	—
119	C₃×C₄²	C₂²×C₃²	1	3	yes	—
120	C₂²×C₃²	—	2	2	no	—
121	C₂²×C₃²	—	2	2	no	—
122	C₂²×C₃²	C₂×C₃×Q₈	1	4	yes	—
123	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
124	C₃²×C₄	—	2	6	no	—
125	C₃²×C₄	—	2	3	no	—
126	C₃²×C₄	C₂×C₃×D₄	1	6	yes	—
127	C₃²×C₄	C₂×C₃×Q₈	1	2	yes	—
128	C₂²×C₉	—	2	6	no	—
129	C₂²×C₉	—	2	6	no	—
130	C₂²×C₉	C₂×C₃×Q₈	1	12	yes	complemented
131	C₂²×C₉	C₂⁴×C₃	1	3	yes	—
132	C₄×C₉	—	2	18	no	—
133	C₄×C₉	—	2	9	no	—
134	C₄×C₉	C₂×C₃×D₄	1	18	yes	—
135	C₄×C₉	C₂×C₃×Q₈	1	6	yes	complemented
136	C₂²×Q₈	C₂×3₋¹⁺²	1	2	yes	complemented
137	C₄⋊Q₈	C₂×3₋¹⁺²	1	3	yes	complemented
138	C₂²⋊Q₈	C₂×3₋¹⁺²	1	6	yes	complemented
139	C₄×Q₈	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
140	C₄×D₄	C₂×3₋¹⁺²	1	3	yes	—
141	3₋¹⁺²	D₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
142	C₂³×C₃	—	3	2	no	—
143	C₂³×C₃	C₂³×C₃²	1	2	yes	—
144	C₃×Q₈	—	2	12	no	—
145	C₃×Q₈	—	3	4	no	—
146	C₃×Q₈	—	6	12	no	—
147	C₃×Q₈	C₃²×D₄	1	4	yes	—
148	C₃×D₄	—	3	4	no	—
149	C₃×D₄	C₃²×Q₈	1	4	yes	—
150	C₂×C₃×C₄	—	2	12	no	—
151	C₂×C₃×C₄	—	3	1	no	—
152	C₂×C₃×C₄	—	3	3	no	—
153	C₂×C₃×C₄	—	3	3	no	—
154	C₂×C₃×C₄	—	3	6	no	—
155	C₂×C₃×C₄	—	6	12	no	—
156	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
157	C₂×C₃×C₄	C₂³×C₃²	1	6	yes	—
158	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
159	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
160	C₂×C₃²	—	2	4	no	—
161	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
162	C₂×C₃²	C₂²×C₃×Q₈	1	1	yes	characteristic
163	C₂×C₃²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
164	C₂×C₉	—	2	12	no	—
165	C₂×C₉	C₂²×C₃×D₄	1	3	yes	—
166	C₂×C₉	C₂²×C₃×Q₈	1	3	yes	—
167	C₂×C₉	C₃×2₋¹⁺⁴	1	3	yes	—
168	C₂×Q₈	—	2	8	no	—
169	C₂×Q₈	C₂²×3₋¹⁺²	1	6	yes	—
170	C₂×Q₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
171	C₂×D₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
172	C₂²×C₄	C₂²×3₋¹⁺²	1	6	yes	—
173	C₄⋊C₄	C₂²×3₋¹⁺²	1	3	yes	—
174	C₄⋊C₄	C₂²×3₋¹⁺²	1	3	yes	—
175	C₄⋊C₄	C₂²×3₋¹⁺²	1	6	yes	—
176	C₂²⋊C₄	C₂²×3₋¹⁺²	1	6	yes	—
177	C₄²	C₂²×3₋¹⁺²	1	3	yes	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	3	1	no	—
181	C₂²×C₃	—	3	4	no	—
182	C₂²×C₃	—	6	2	no	—
183	C₂²×C₃	—	6	2	no	—
184	C₂²×C₃	C₂×C₃²×Q₈	1	4	yes	—
185	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
186	C₃×C₄	—	2	6	no	—
187	C₃×C₄	—	2	3	no	—
188	C₃×C₄	—	3	6	no	—
189	C₃×C₄	—	3	2	no	—
190	C₃×C₄	—	6	6	no	—
191	C₃×C₄	—	6	3	no	—
192	C₃×C₄	C₂×C₃²×D₄	1	6	yes	—
193	C₃×C₄	C₂×C₃²×Q₈	1	2	yes	—
194	C₃²	C₃×D₄×Q₈	1	1	yes	characteristic
195	C₉	C₃×D₄×Q₈	1	3	yes	complemented
196	C₂³	C₂³×3₋¹⁺²	1	2	yes	—
197	Q₈	—	2	12	no	—
198	Q₈	D₄×3₋¹⁺²	1	4	yes	direct product factor
199	D₄	Q₈×3₋¹⁺²	1	4	yes	direct product factor
200	C₂×C₄	—	2	12	no	—
201	C₂×C₄	C₂³×3₋¹⁺²	1	3	yes	—
202	C₂×C₄	C₂³×3₋¹⁺²	1	6	yes	—
203	C₂×C₄	C₂³×3₋¹⁺²	1	3	yes	—
204	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
205	C₂×C₃	—	2	4	no	—
206	C₂×C₃	—	3	1	no	—
207	C₂×C₃	—	3	1	no	—
208	C₂×C₃	—	3	1	no	—
209	C₂×C₃	—	6	4	no	—
210	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
211	C₂×C₃	C₂²×C₃²×Q₈	1	1	yes	characteristic
212	C₂×C₃	C₃²×2₋¹⁺⁴	1	1	yes	characteristic
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	C₂×Q₈×3₋¹⁺²	1	4	yes	complemented
216	C₂²	C₂⁴×3₋¹⁺²	1	1	yes	characteristic
217	C₄	—	2	6	no	—
218	C₄	—	2	3	no	—
219	C₄	C₂×D₄×3₋¹⁺²	1	6	yes	—
220	C₄	C₂×Q₈×3₋¹⁺²	1	2	yes	complemented
221	C₃	—	3	1	no	—
222	C₃	C₃²×D₄×Q₈	1	1	yes	characteristic
223	C₂	—	2	4	no	—
224	C₂	C₂²×D₄×3₋¹⁺²	1	1	yes	characteristic
225	C₂	C₂²×Q₈×3₋¹⁺²	1	1	yes	characteristic
226	C₂	3₋¹⁺²×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²×C₃	C₂⁴×C₃²
prime radical	C₂×C₃²×D₄	C₂²×C₃
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×3₋¹⁺²	C₂	semidirect
2. 3₋¹⁺²×C₄⋊Q₈	C₂	semidirect
3. 3₋¹⁺²×C₂²⋊Q₈	C₂	semidirect
4. C₄×Q₈×3₋¹⁺²	C₂	semidirect
7. D₄×Q₈×C₉	C₃	semidirect
23. C₂²×Q₈×C₉	C₂×C₃	semidirect
24. C₉×C₄⋊Q₈	C₂×C₃	semidirect
25. C₉×C₂²⋊Q₈	C₂×C₃	semidirect
26. C₄×Q₈×C₉	C₂×C₃	semidirect
30. Q₈×3₋¹⁺²	D₄	direct (N × H)
31. D₄×3₋¹⁺²	Q₈	direct (N × H)
61. C₂²×3₋¹⁺²	C₂×Q₈	semidirect
66. C₄×3₋¹⁺²	C₂×Q₈	semidirect
88. Q₈×C₉	C₃×D₄	semidirect
89. D₄×C₉	C₃×Q₈	semidirect
95. D₄×Q₈	3₋¹⁺²	direct (N × H)
130. C₂²×C₉	C₂×C₃×Q₈	semidirect
135. C₄×C₉	C₂×C₃×Q₈	semidirect
136. C₂²×Q₈	C₂×3₋¹⁺²	semidirect
137. C₄⋊Q₈	C₂×3₋¹⁺²	semidirect
138. C₂²⋊Q₈	C₂×3₋¹⁺²	semidirect
139. C₄×Q₈	C₂×3₋¹⁺²	semidirect
141. 3₋¹⁺²	D₄×Q₈	direct (N × H)
195. C₉	C₃×D₄×Q₈	semidirect
198. Q₈	D₄×3₋¹⁺²	direct (N × H)
199. D₄	Q₈×3₋¹⁺²	direct (N × H)
215. C₂²	C₂×Q₈×3₋¹⁺²	semidirect
220. C₄	C₂×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
C₂²×Q₈×3₋¹⁺²	2
3₋¹⁺²×C₄⋊Q₈	2
3₋¹⁺²×C₂²⋊Q₈	2
C₄×Q₈×3₋¹⁺²	2
C₄×D₄×3₋¹⁺²	2
C₃²×D₄×Q₈	3
D₄×Q₈×C₉	3

Subgroups — D4×Q8×3−1+2 #13154631

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Q₈×3₋¹⁺² #13154631