Subgroups — C₃×#29209 #10647816

Order 768.

686 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2862	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#125	C₂	1	1	yes	maximal, characteristic
3	C₃×#2058	C₂	1	2	yes	maximal
4	C₃×#2057	C₂	1	2	yes	maximal
5	C₃×#1991	C₂	1	2	yes	maximal
6	C₃×#1949	C₂	1	4	yes	maximal
7	C₃×#1846	C₂	1	2	yes	maximal
8	C₃×C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	#29209	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₃×#125	C₂²	1	1	yes	characteristic
12	C₂×C₃×#107	C₂²	1	4	yes	—
13	C₂×C₃×#106	C₂²	1	4	yes	—
14	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
16	C₃×#391	—	2	2	no	—
17	C₃×#391	—	2	2	no	—
18	C₃×#389	C₂²	1	2	yes	—
19	C₃×#387	—	2	2	no	—
20	C₃×#382	C₂²	1	2	yes	—
21	C₃×#382	C₂²	1	1	yes	characteristic
22	C₃×#382	C₂²	1	2	yes	—
23	C₃×C₄×C₄⋊C₄	—	2	2	no	—
24	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
25	C₃×C₄³	C₂²	1	1	yes	characteristic
26	C₃×#340	C₂²	1	4	yes	—
27	C₃×#340	C₂²	1	4	yes	—
28	C₃×#336	C₂²	1	1	yes	characteristic
29	C₃×#336	C₂²	1	1	yes	characteristic
30	#2862	C₂×C₃	1	1	yes	characteristic
31	C₄×#125	C₂×C₃	1	1	yes	characteristic
32	#2058	C₂×C₃	1	2	yes	—
33	#2057	C₂×C₃	1	2	yes	—
34	#1991	C₂×C₃	1	2	yes	—
35	#1949	C₂×C₃	1	4	yes	—
36	#1846	C₂×C₃	1	2	yes	—
37	C₄×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
38	C₂²×C₃×Q₈	—	2	2	no	—
39	C₂²×C₃×C₈	C₂³	1	2	yes	—
40	C₂²×C₃×C₈	C₂³	1	2	yes	—
41	C₃×C₄⋊Q₈	—	4	4	no	—
42	C₃×#125	—	2	2	no	—
43	C₃×#125	Q₈	1	4	yes	complemented
44	C₂×C₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
49	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₃×C₄²	—	2	2	no	—
51	C₂×C₃×C₄²	D₄	1	2	yes	—
52	C₂×C₃×C₄²	D₄	1	2	yes	—
53	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
55	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
56	C₃×#107	—	4	4	no	—
57	C₃×#106	—	4	4	no	—
58	C₃×C₄⋊C₈	Q₈	1	8	yes	complemented
59	C₃×C₄×C₈	—	2	4	no	—
60	C₃×#95	—	2	2	no	—
61	C₃×#95	—	2	2	no	—
62	C₃×#95	—	2	2	no	—
63	C₃×#95	—	2	4	no	—
64	C₃×#95	—	2	2	no	—
65	C₃×#95	—	2	2	no	—
66	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
67	C₂×#125	C₂²×C₃	1	1	yes	characteristic
68	C₂×#107	C₂²×C₃	1	4	yes	—
69	C₂×#106	C₂²×C₃	1	4	yes	—
70	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
71	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
72	#391	—	2	2	no	—
73	#391	—	2	2	no	—
74	#389	C₂²×C₃	1	2	yes	—
75	#387	—	2	2	no	—
76	#382	C₂²×C₃	1	1	yes	characteristic
77	#382	C₂²×C₃	1	2	yes	—
78	#382	C₂²×C₃	1	2	yes	—
79	C₄×C₄⋊C₄	—	2	2	no	—
80	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
81	C₄³	C₂²×C₃	1	1	yes	characteristic
82	#340	C₂²×C₃	1	4	yes	—
83	#340	C₂²×C₃	1	4	yes	—
84	#336	C₂²×C₃	1	1	yes	characteristic
85	#336	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₃×Q₈	—	4	4	no	—
87	C₂×C₃×Q₈	—	4	4	no	—
88	C₂×C₃×Q₈	—	4	4	no	—
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	—	2	4	no	—
92	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
93	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
96	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
97	C₂×C₃×C₈	—	2	2	no	—
98	C₂×C₃×C₈	—	2	2	no	—
99	C₂×C₃×C₈	—	2	2	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
102	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
103	C₃×C₄⋊C₄	—	2	8	no	—
104	C₃×C₄⋊C₄	—	2	2	no	—
105	C₃×C₄⋊C₄	—	2	4	no	—
106	C₃×C₄⋊C₄	—	2	4	no	—
107	C₃×C₄⋊C₄	—	2	4	no	—
108	C₃×C₄⋊C₄	—	4	4	no	—
109	C₃×C₄⋊C₄	—	4	4	no	—
110	C₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
111	C₃×C₄²	—	2	4	no	—
112	C₃×C₄²	—	2	2	no	—
113	C₃×C₄²	—	4	2	no	—
114	C₃×C₄²	—	4	4	no	—
115	C₃×C₄²	C₂×Q₈	1	2	yes	—
116	C₃×C₄²	C₂×Q₈	1	4	yes	—
117	C₃×C₄²	#41	1	2	yes	—
118	C₂²×Q₈	—	2	2	no	—
119	C₂²×C₈	C₂³×C₃	1	2	yes	—
120	C₂²×C₈	C₂³×C₃	1	2	yes	—
121	C₄⋊Q₈	—	4	4	no	—
122	#125	—	2	2	no	—
123	#125	C₃×Q₈	1	4	yes	complemented
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
128	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
129	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
130	C₂×C₄²	—	2	2	no	—
131	C₂×C₄²	C₃×D₄	1	2	yes	—
132	C₂×C₄²	C₃×D₄	1	2	yes	—
133	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
134	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
135	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
136	#107	—	4	4	no	—
137	#106	—	4	4	no	—
138	C₄⋊C₈	C₃×Q₈	1	8	yes	complemented
139	C₄×C₈	—	2	4	no	—
140	#95	—	2	2	no	—
141	#95	—	2	2	no	—
142	#95	—	2	2	no	—
143	#95	—	2	4	no	—
144	#95	—	2	2	no	—
145	#95	—	2	2	no	—
146	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
147	C₃×Q₈	—	8	8	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	—	4	2	no	—
156	C₂×C₃×C₄	—	4	2	no	—
157	C₂×C₃×C₄	—	4	4	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	—	4	2	no	—
160	C₂×C₃×C₄	—	4	4	no	—
161	C₂×C₃×C₄	—	4	2	no	—
162	C₂×C₃×C₄	—	4	4	no	—
163	C₂×C₃×C₄	C₂²⋊Q₈	1	4	yes	—
164	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
165	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
166	C₂×C₃×C₄	#135	1	4	yes	—
167	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
168	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
170	C₂×C₃×C₄	H(5,𝔽₂)	1	2	yes	—
171	C₃×C₈	—	2	4	no	—
172	C₃×C₈	—	2	4	no	—
173	C₂×Q₈	—	4	4	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
180	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
183	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
189	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
190	C₄⋊C₄	—	2	8	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	C₂×C₃×Q₈	1	4	yes	—
198	C₄²	—	2	4	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	4	2	no	—
201	C₄²	—	4	4	no	—
202	C₄²	C₂×C₃×Q₈	1	2	yes	—
203	C₄²	C₂×C₃×Q₈	1	4	yes	—
204	C₄²	C₃×#41	1	2	yes	—
205	C₂²×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
206	C₂²×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
207	C₂²×C₃	#535	1	1	yes	characteristic
208	C₂²×C₃	C₂×#135	1	2	yes	—
209	C₂²×C₃	#575	1	2	yes	—
210	C₃×C₄	—	2	4	no	—
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	—	4	4	no	—
215	C₃×C₄	—	8	2	no	—
216	C₃×C₄	—	8	4	no	—
217	C₃×C₄	#554	1	2	yes	—
218	C₃×C₄	Q₈²	1	2	yes	—
219	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
220	Q₈	—	8	8	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₃×C₂²⋊Q₈	1	4	yes	—
237	C₂×C₄	C₃×C₄⋊Q₈	1	2	yes	—
238	C₂×C₄	C₃×C₄⋊Q₈	1	2	yes	—
239	C₂×C₄	C₃×#135	1	4	yes	—
240	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₃×Q₈	1	2	yes	—
242	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
243	C₂×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
244	C₈	—	2	4	no	—
245	C₈	—	2	4	no	—
246	C₂×C₃	#2567	1	1	yes	characteristic
247	C₂×C₃	#3076	1	2	yes	—
248	C₂×C₃	#3160	1	1	yes	characteristic
249	C₂×C₃	#3166	1	1	yes	characteristic
250	C₂×C₃	#3211	1	2	yes	—
251	C₂²	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
252	C₂²	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
253	C₂²	C₃×#535	1	1	yes	characteristic
254	C₂²	C₂×C₃×#135	1	2	yes	—
255	C₂²	C₃×#575	1	2	yes	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	4	no	—
260	C₄	—	4	4	no	—
261	C₄	—	8	4	no	—
262	C₄	—	8	2	no	—
263	C₄	C₃×#554	1	2	yes	—
264	C₄	C₃×Q₈²	1	2	yes	—
265	C₃	#29209	1	1	yes	characteristic, direct product factor, Sylow
266	C₂	C₃×#2567	1	1	yes	characteristic
267	C₂	C₃×#3076	1	2	yes	—
268	C₂	C₃×#3160	1	1	yes	characteristic
269	C₂	C₃×#3166	1	1	yes	characteristic
270	C₂	C₃×#3211	1	2	yes	—

	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
9. #29209	C₃	direct (N × H)
43. C₃×#125	Q₈	semidirect
58. C₃×C₄⋊C₈	Q₈	semidirect
123. #125	C₃×Q₈	semidirect
138. C₄⋊C₈	C₃×Q₈	semidirect
265. C₃	#29209	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×#2862	2
C₃×C₄×#125	2
C₃×#2058	2
C₃×#2057	2
C₃×#1991	2
C₃×#1949	2
C₃×#1846	2
C₃×C₄×C₄⋊C₈	2
#29209	3

Subgroups — C3×#29209 #10647816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#29209 #10647816