Subgroups — C₃×Q₈×#634 #13179317

Order 1728.

1552 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×Q₈×C₃⋊Dic₃	C₂	1	2	yes	maximal
3	C₃×#64097	C₂	1	6	yes	maximal
4	C₃×#64078	C₂	1	3	yes	maximal
5	C₃×C₄×#634	C₂	1	3	yes	maximal
6	Q₈×#634	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×Q₈×Dic₆	—	3	4	no	maximal
8	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×#634	C₂²	1	6	yes	—
10	C₂×C₃×#634	C₂²	1	1	yes	characteristic
11	C₃³×C₄⋊C₄	C₂²	1	3	yes	—
12	C₃³×C₄²	C₂²	1	3	yes	—
13	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	3	yes	—
14	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	6	yes	—
15	C₃×#3750	C₂²	1	6	yes	—
16	C₃×C₄×C₃⋊Dic₃	C₂²	1	6	yes	—
17	C₃²×C₄×Q₈	—	2	4	no	—
18	C₃²×C₄×Q₈	D₃	1	4	yes	—
19	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
20	Q₈×C₃⋊Dic₃	C₂×C₃	1	2	yes	—
21	#64097	C₂×C₃	1	6	yes	—
22	#64078	C₂×C₃	1	3	yes	—
23	C₄×#634	C₂×C₃	1	3	yes	—
24	C₃×Q₈×Dic₃	—	3	8	no	—
25	C₃×#888	—	3	24	no	—
26	C₃×#869	—	3	12	no	—
27	C₃×C₄×Dic₆	—	3	12	no	—
28	C₃³×Q₈	Q₈	1	4	yes	complemented
29	C₂×C₃³×C₄	C₂³	1	3	yes	—
30	C₂×C₃³×C₄	C₂³	1	3	yes	—
31	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₃⋊Dic₃	C₂³	1	6	yes	—
33	C₂×C₃×C₃⋊Dic₃	C₂³	1	2	yes	—
34	C₃×#634	—	4	6	no	—
35	C₃×#634	Q₈	1	4	yes	direct product factor
36	C₃×Q₈²	—	9	1	no	—
37	Q₈×Dic₆	—	3	4	no	—
38	C₂×C₃²×Q₈	—	2	4	no	—
39	C₂×C₃²×Q₈	C₂×D₃	1	4	yes	—
40	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
41	C₂×#634	C₂²×C₃	1	6	yes	—
42	C₂×#634	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₃×Dic₆	—	3	4	no	—
44	C₂×C₃×Dic₆	—	3	24	no	—
45	C₃²×C₄⋊C₄	—	2	12	no	—
46	C₃²×C₄⋊C₄	C₂×D₃	1	12	yes	—
47	C₃²×C₄⋊C₄	C₂²×C₃	1	3	yes	—
48	C₃²×C₄²	—	2	12	no	—
49	C₃²×C₄²	C₂×D₃	1	12	yes	—
50	C₃²×C₄²	C₂²×C₃	1	3	yes	—
51	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	3	yes	—
52	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	6	yes	—
53	#3750	C₂²×C₃	1	6	yes	—
54	C₄×C₃⋊Dic₃	C₂²×C₃	1	6	yes	—
55	C₃×C₄⋊Dic₃	—	3	24	no	—
56	C₃×C₄⋊Dic₃	—	3	12	no	—
57	C₃×Dic₃⋊C₄	—	3	24	no	—
58	C₃×C₄×Dic₃	—	3	24	no	—
59	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
60	C₃³×C₄	—	2	3	no	—
61	C₃³×C₄	C₂×Q₈	1	6	yes	—
62	C₃³×C₄	C₂×Q₈	1	2	yes	—
63	C₃×C₃⋊Dic₃	—	2	6	no	—
64	C₃×C₃⋊Dic₃	C₂×Q₈	1	4	yes	—
65	C₃×C₄⋊Q₈	—	9	6	no	—
66	C₃×C₄⋊Q₈	—	9	3	no	—
67	C₃×C₄×Q₈	—	2	4	no	—
68	C₃×C₄×Q₈	—	9	2	no	—
69	C₃×C₄×Q₈	—	9	3	no	—
70	C₃×C₄×Q₈	C₃×D₃	1	4	yes	—
71	C₃×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
72	Q₈×Dic₃	—	3	8	no	—
73	#888	—	3	24	no	—
74	#869	—	3	12	no	—
75	C₄×Dic₆	—	3	12	no	—
76	C₃²×Q₈	—	2	16	no	—
77	C₃²×Q₈	Dic₆	1	16	yes	complemented
78	C₃²×Q₈	C₃×Q₈	1	4	yes	complemented
79	C₂×C₃²×C₄	—	2	4	no	—
80	C₂×C₃²×C₄	—	2	12	no	—
81	C₂×C₃²×C₄	—	2	12	no	—
82	C₂×C₃²×C₄	C₂²×D₃	1	12	yes	—
83	C₂×C₃²×C₄	C₂²×D₃	1	12	yes	—
84	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
85	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
86	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
87	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
88	C₂×C₃⋊Dic₃	C₂³×C₃	1	6	yes	—
89	C₂×C₃⋊Dic₃	C₂³×C₃	1	2	yes	—
90	#634	—	4	6	no	—
91	#634	C₃×Q₈	1	4	yes	direct product factor
92	C₂×C₃×Dic₃	—	3	8	no	—
93	C₂×C₃×Dic₃	—	3	24	no	—
94	C₃×Dic₆	—	3	16	no	—
95	C₃×Dic₆	—	12	24	no	—
96	Q₈²	—	9	1	no	Sylow
97	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂×C₃×Q₈	—	2	4	no	—
101	C₂×C₃×Q₈	—	9	1	no	—
102	C₂×C₃×Q₈	—	9	6	no	—
103	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
104	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
105	C₂×Dic₆	—	3	4	no	—
106	C₂×Dic₆	—	3	24	no	—
107	C₃×C₄⋊C₄	—	2	12	no	—
108	C₃×C₄⋊C₄	—	9	6	no	—
109	C₃×C₄⋊C₄	—	9	6	no	—
110	C₃×C₄⋊C₄	—	9	3	no	—
111	C₃×C₄⋊C₄	C₂×C₃×D₃	1	12	yes	—
112	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	3	yes	—
113	C₃×C₄²	—	2	12	no	—
114	C₃×C₄²	—	9	6	no	—
115	C₃×C₄²	C₂×C₃×D₃	1	12	yes	—
116	C₃×C₄²	C₂×C₃⋊D₃	1	3	yes	—
117	C₄⋊Dic₃	—	3	12	no	—
118	C₄⋊Dic₃	—	3	24	no	—
119	Dic₃⋊C₄	—	3	24	no	—
120	C₄×Dic₃	—	3	24	no	—
121	C₂²×C₃²	—	2	4	no	—
122	C₂²×C₃²	C₂³×D₃	1	4	yes	—
123	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
124	C₃²×C₄	—	2	3	no	—
125	C₃²×C₄	—	2	24	no	—
126	C₃²×C₄	—	2	8	no	—
127	C₃²×C₄	—	2	12	no	—
128	C₃²×C₄	—	4	12	no	—
129	C₃²×C₄	C₂×Dic₆	1	24	yes	—
130	C₃²×C₄	D₃×Q₈	1	8	yes	—
131	C₃²×C₄	C₂×C₃×Q₈	1	6	yes	—
132	C₃²×C₄	C₂×C₃×Q₈	1	2	yes	—
133	C₃⋊Dic₃	—	2	6	no	—
134	C₃⋊Dic₃	C₂×C₃×Q₈	1	4	yes	—
135	C₃×Dic₃	—	3	16	no	—
136	C₃×Dic₃	—	6	24	no	—
137	C₄⋊Q₈	—	9	6	no	—
138	C₄⋊Q₈	—	9	3	no	—
139	C₄×Q₈	—	9	2	no	—
140	C₄×Q₈	—	9	3	no	—
141	C₄×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic
142	C₃³	Q₈²	1	1	yes	characteristic, complemented, Sylow
143	C₃×Q₈	—	2	16	no	—
144	C₃×Q₈	—	9	4	no	—
145	C₃×Q₈	—	36	6	no	—
146	C₃×Q₈	C₃×Dic₆	1	16	yes	complemented
147	C₃×Q₈	#634	1	4	yes	direct product factor
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	2	12	no	—
150	C₂×C₃×C₄	—	2	12	no	—
151	C₂×C₃×C₄	—	9	2	no	—
152	C₂×C₃×C₄	—	9	6	no	—
153	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
154	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
155	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
156	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	3	yes	—
157	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	3	yes	—
158	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
159	C₂×Dic₃	—	3	8	no	—
160	C₂×Dic₃	—	3	24	no	—
161	Dic₆	—	3	16	no	—
162	Dic₆	—	12	24	no	—
163	C₂×C₃²	—	2	4	no	—
164	C₂×C₃²	—	2	4	no	—
165	C₂×C₃²	—	2	4	no	—
166	C₂×C₃²	C₂²×Dic₆	1	4	yes	—
167	C₂×C₃²	C₂×D₃×Q₈	1	4	yes	—
168	C₂×C₃²	#1009	1	4	yes	—
169	C₂×C₃²	C₂²×C₃×Q₈	1	1	yes	characteristic
170	C₂×C₃²	C₂²×C₃×Q₈	1	1	yes	characteristic
171	C₂×C₃²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
172	C₂×Q₈	—	9	1	no	—
173	C₂×Q₈	—	9	6	no	—
174	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
175	C₄⋊C₄	—	9	6	no	—
176	C₄⋊C₄	—	9	6	no	—
177	C₄⋊C₄	—	9	3	no	—
178	C₄⋊C₄	C₂×C₃×C₃⋊D₃	1	3	yes	—
179	C₄²	—	9	6	no	—
180	C₄²	C₂×C₃×C₃⋊D₃	1	3	yes	—
181	C₂²×C₃	—	2	4	no	—
182	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
183	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
184	C₃×C₄	—	2	12	no	—
185	C₃×C₄	—	2	3	no	—
186	C₃×C₄	—	2	24	no	—
187	C₃×C₄	—	2	8	no	—
188	C₃×C₄	—	4	12	no	—
189	C₃×C₄	—	9	4	no	—
190	C₃×C₄	—	18	6	no	—
191	C₃×C₄	C₂×C₃×Dic₆	1	24	yes	—
192	C₃×C₄	C₃×D₃×Q₈	1	8	yes	—
193	C₃×C₄	C₂×#634	1	6	yes	—
194	C₃×C₄	Q₈×C₃⋊D₃	1	2	yes	—
195	Dic₃	—	3	16	no	—
196	Dic₃	—	6	24	no	—
197	C₃²	—	2	4	no	—
198	C₃²	Q₈×Dic₆	1	4	yes	complemented
199	C₃²	C₃×Q₈²	1	1	yes	characteristic, complemented
200	Q₈	—	9	4	no	—
201	Q₈	—	36	6	no	—
202	Q₈	C₃×#634	1	4	yes	direct product factor
203	C₂×C₄	—	9	2	no	—
204	C₂×C₄	—	9	6	no	—
205	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	3	yes	—
206	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	3	yes	—
207	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
208	C₂×C₃	—	2	4	no	—
209	C₂×C₃	—	2	4	no	—
210	C₂×C₃	—	2	4	no	—
211	C₂×C₃	C₂²×C₃×Dic₆	1	4	yes	—
212	C₂×C₃	C₂×C₃×D₃×Q₈	1	4	yes	—
213	C₂×C₃	C₃×#1009	1	4	yes	—
214	C₂×C₃	C₂²×#634	1	1	yes	characteristic
215	C₂×C₃	C₂×Q₈×C₃⋊D₃	1	1	yes	characteristic
216	C₂×C₃	#64366	1	1	yes	characteristic
217	C₂²	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
218	C₄	—	2	3	no	—
219	C₄	—	9	4	no	—
220	C₄	—	18	6	no	—
221	C₄	C₂×C₃×#634	1	6	yes	—
222	C₄	C₃×Q₈×C₃⋊D₃	1	2	yes	—
223	C₃	—	2	4	no	—
224	C₃	C₃×Q₈×Dic₆	1	4	yes	complemented
225	C₃	Q₈×#634	1	1	yes	characteristic, direct product factor
226	C₂	C₂²×C₃×#634	1	1	yes	characteristic
227	C₂	C₂×C₃×Q₈×C₃⋊D₃	1	1	yes	characteristic
228	C₂	C₃×#64366	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×C₃⋊D₃
Fitting	C₃³×C₄×Q₈	C₂
socle, prime radical	C₂²×C₃³	C₂⁴
hypercenter	C₃×C₄×Q₈	C₃⋊D₃
hypocenter	C₃²	C₃×Q₈²
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
6. Q₈×#634	C₃	direct (N × H)
28. C₃³×Q₈	Q₈	semidirect
35. C₃×#634	Q₈	direct (N × H)
77. C₃²×Q₈	Dic₆	semidirect
78. C₃²×Q₈	C₃×Q₈	semidirect
91. #634	C₃×Q₈	direct (N × H)
142. C₃³	Q₈²	semidirect
146. C₃×Q₈	C₃×Dic₆	semidirect
147. C₃×Q₈	#634	direct (N × H)
198. C₃²	Q₈×Dic₆	semidirect
199. C₃²	C₃×Q₈²	semidirect
202. Q₈	C₃×#634	direct (N × H)
224. C₃	C₃×Q₈×Dic₆	semidirect
225. C₃	Q₈×#634	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₄×Q₈	2
C₃×Q₈×C₃⋊Dic₃	2
C₃×#64097	2
C₃×#64078	2
C₃×C₄×#634	2
Q₈×#634	3
C₃×Q₈×Dic₆	3

Subgroups — C3×Q8×#634 #13179317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Q₈×#634 #13179317