Subgroups — C₃×#29286 #10647893

Order 768.

756 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	C₃×C₄×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₃×#2086	C₂	1	4	yes	maximal
3	C₃×#1988	C₂	1	4	yes	maximal
4	C₃×#1947	C₂	1	4	yes	maximal
5	C₃×#1842	C₂	1	1	yes	maximal, characteristic
6	#29286	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄×Q₈	—	2	2	no	—
9	C₂×C₃×#107	C₂²	1	4	yes	—
10	C₂×C₃×C₄⋊C₈	C₂²	1	4	yes	—
11	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₃×#391	—	2	2	no	—
13	C₃×#389	—	2	2	no	—
14	C₃×#389	C₂²	1	4	yes	—
15	C₃×#384	C₂²	1	2	yes	—
16	C₃×C₄×C₄⋊C₄	—	2	2	no	—
17	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	—
18	C₃×C₄³	C₂²	1	1	yes	characteristic
19	C₃×#340	C₂²	1	8	yes	—
20	C₃×#340	C₂²	1	4	yes	—
21	C₄×C₄⋊Q₈	C₂×C₃	1	2	yes	—
22	#2086	C₂×C₃	1	4	yes	—
23	#1988	C₂×C₃	1	4	yes	—
24	#1947	C₂×C₃	1	4	yes	—
25	#1842	C₂×C₃	1	1	yes	characteristic
26	C₂²×C₃×Q₈	—	2	2	no	—
27	C₂²×C₃×C₈	C₂³	1	4	yes	—
28	C₃×C₄⋊Q₈	—	2	4	no	—
29	C₃×C₄⋊Q₈	Q₈	1	8	yes	complemented
30	C₃×C₄×Q₈	—	4	8	no	—
31	C₂×C₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₃×C₄⋊C₄	—	2	4	no	—
35	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
36	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
37	C₂×C₃×C₄²	—	2	4	no	—
38	C₂×C₃×C₄²	D₄	1	2	yes	—
39	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
41	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂³	1	2	yes	—
43	C₃×#107	—	4	4	no	—
44	C₃×C₄⋊C₈	—	2	8	no	—
45	C₃×C₄⋊C₈	—	4	2	no	—
46	C₃×#95	—	2	4	no	—
47	C₃×#95	—	2	4	no	—
48	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
49	C₂×C₄×Q₈	—	2	2	no	—
50	C₂×#107	C₂²×C₃	1	4	yes	—
51	C₂×C₄⋊C₈	C₂²×C₃	1	4	yes	—
52	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
53	#391	—	2	2	no	—
54	#389	—	2	2	no	—
55	#389	C₂²×C₃	1	4	yes	—
56	#384	C₂²×C₃	1	2	yes	—
57	C₄×C₄⋊C₄	—	2	2	no	—
58	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	—
59	C₄³	C₂²×C₃	1	1	yes	characteristic
60	#340	C₂²×C₃	1	8	yes	—
61	#340	C₂²×C₃	1	4	yes	—
62	C₂×C₃×Q₈	—	2	8	no	—
63	C₂×C₃×Q₈	—	4	4	no	—
64	C₂×C₃×Q₈	—	4	4	no	—
65	C₂²×C₃×C₄	—	2	4	no	—
66	C₂²×C₃×C₄	—	2	4	no	—
67	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
69	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
71	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
72	C₂×C₃×C₈	—	2	8	no	—
73	C₂×C₃×C₈	—	2	4	no	—
74	C₃×C₄⋊C₄	—	2	8	no	—
75	C₃×C₄⋊C₄	—	2	4	no	—
76	C₃×C₄⋊C₄	—	2	4	no	—
77	C₃×C₄⋊C₄	—	2	4	no	—
78	C₃×C₄⋊C₄	—	4	4	no	—
79	C₃×C₄⋊C₄	—	4	8	no	—
80	C₃×C₄⋊C₄	—	4	4	no	—
81	C₃×C₄⋊C₄	C₂×Q₈	1	8	yes	—
82	C₃×C₄²	—	2	4	no	—
83	C₃×C₄²	—	2	1	no	—
84	C₃×C₄²	—	2	1	no	—
85	C₃×C₄²	—	4	1	no	—
86	C₃×C₄²	—	4	8	no	—
87	C₃×C₄²	D₈	1	4	yes	—
88	C₃×C₄²	C₂×Q₈	1	4	yes	—
89	C₃×C₄²	#41	1	4	yes	—
90	C₂²×Q₈	—	2	2	no	—
91	C₂²×C₈	C₂³×C₃	1	4	yes	—
92	C₄⋊Q₈	—	2	4	no	—
93	C₄⋊Q₈	C₃×Q₈	1	8	yes	complemented
94	C₄×Q₈	—	4	8	no	—
95	C₂×C₄⋊C₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	2	4	no	—
98	C₂×C₄⋊C₄	—	2	2	no	—
99	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
100	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
101	C₂×C₄²	—	2	4	no	—
102	C₂×C₄²	C₃×D₄	1	2	yes	—
103	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
104	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
105	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
106	C₂×C₄²	C₂³×C₃	1	2	yes	—
107	#107	—	4	4	no	—
108	C₄⋊C₈	—	2	8	no	—
109	C₄⋊C₈	—	4	2	no	—
110	#95	—	2	4	no	—
111	#95	—	2	4	no	—
112	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
113	C₃×Q₈	—	4	16	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	1	no	—
118	C₂×C₃×C₄	—	2	8	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	4	4	no	—
122	C₂×C₃×C₄	—	4	4	no	—
123	C₂×C₃×C₄	—	4	4	no	—
124	C₂×C₃×C₄	—	4	4	no	—
125	C₂×C₃×C₄	—	4	4	no	—
126	C₂×C₃×C₄	C₂²⋊Q₈	1	4	yes	—
127	C₂×C₃×C₄	C₂²⋊Q₈	1	4	yes	—
128	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
129	C₂×C₃×C₄	C₂×D₈	1	4	yes	—
130	C₂×C₃×C₄	#137	1	4	yes	—
131	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
132	C₂×C₃×C₄	C₂×#41	1	2	yes	—
133	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂×C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
135	C₃×C₈	—	4	4	no	—
136	C₂×Q₈	—	2	8	no	—
137	C₂×Q₈	—	4	4	no	—
138	C₂×Q₈	—	4	4	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
143	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
145	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
146	C₂×C₈	—	2	8	no	—
147	C₂×C₈	—	2	4	no	—
148	C₄⋊C₄	—	2	8	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	4	4	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	C₂×C₃×Q₈	1	8	yes	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	4	1	no	—
160	C₄²	—	4	8	no	—
161	C₄²	C₃×D₈	1	4	yes	—
162	C₄²	C₂×C₃×Q₈	1	4	yes	—
163	C₄²	C₃×#41	1	4	yes	—
164	C₂²×C₃	C₂×C₂²⋊Q₈	1	2	yes	—
165	C₂²×C₃	#537	1	1	yes	characteristic
166	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
167	C₂²×C₃	C₂×#137	1	2	yes	—
168	C₂²×C₃	#575	1	1	yes	characteristic
169	C₃×C₄	—	2	4	no	—
170	C₃×C₄	—	2	2	no	—
171	C₃×C₄	—	4	1	no	—
172	C₃×C₄	—	4	8	no	—
173	C₃×C₄	—	8	4	no	—
174	C₃×C₄	#474	1	8	yes	—
175	C₃×C₄	#557	1	2	yes	—
176	C₃×C₄	Q₈²	1	1	yes	characteristic
177	C₃×C₄	#559	1	1	yes	characteristic
178	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
179	Q₈	—	4	16	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	C₃×C₂²⋊Q₈	1	4	yes	—
193	C₂×C₄	C₃×C₂²⋊Q₈	1	4	yes	—
194	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
195	C₂×C₄	C₂×C₃×D₈	1	4	yes	—
196	C₂×C₄	C₃×#137	1	4	yes	—
197	C₂×C₄	C₂²×C₃×Q₈	1	2	yes	—
198	C₂×C₄	C₂×C₃×#41	1	2	yes	—
199	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
200	C₂×C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
201	C₈	—	4	4	no	—
202	C₂×C₃	#2582	1	1	yes	characteristic
203	C₂×C₃	C₂×#474	1	2	yes	—
204	C₂×C₃	#3078	1	2	yes	—
205	C₂×C₃	#3245	1	1	yes	characteristic
206	C₂×C₃	#3255	1	1	yes	characteristic
207	C₂²	C₂×C₃×C₂²⋊Q₈	1	2	yes	—
208	C₂²	C₃×#537	1	1	yes	characteristic
209	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
210	C₂²	C₂×C₃×#137	1	2	yes	—
211	C₂²	C₃×#575	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	4	no	—
214	C₄	—	4	1	no	—
215	C₄	—	4	8	no	—
216	C₄	—	8	4	no	—
217	C₄	C₃×#474	1	8	yes	—
218	C₄	C₃×#557	1	2	yes	—
219	C₄	C₃×Q₈²	1	1	yes	characteristic
220	C₄	C₃×#559	1	1	yes	characteristic
221	C₃	#29286	1	1	yes	characteristic, direct product factor, Sylow
222	C₂	C₃×#2582	1	1	yes	characteristic
223	C₂	C₂×C₃×#474	1	2	yes	—
224	C₂	C₃×#3078	1	2	yes	—
225	C₂	C₃×#3245	1	1	yes	characteristic
226	C₂	C₃×#3255	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×C₄³ ◁ G

Complements

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

N	H ≅ G/N	type
6. #29286	C₃	direct (N × H)
29. C₃×C₄⋊Q₈	Q₈	semidirect
93. C₄⋊Q₈	C₃×Q₈	semidirect
221. C₃	#29286	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×C₄⋊Q₈	2
C₃×#2086	2
C₃×#1988	2
C₃×#1947	2
C₃×#1842	2
#29286	3

Subgroups — C3×#29286 #10647893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#29286 #10647893