Subgroups — C₃×#29222 #10647829

Order 768.

654 conjugacy classes of subgroups.

The subgroup lattice has 276 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×#2101	C₂	1	4	yes	maximal
4	C₃×#1991	C₂	1	2	yes	maximal
5	C₃×#1988	C₂	1	2	yes	maximal
6	C₃×#1987	C₂	1	2	yes	maximal
7	C₃×#1835	C₂	1	2	yes	maximal
8	C₃×C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	#29222	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×#125	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
15	C₃×#391	—	2	2	no	—
16	C₃×#391	—	2	2	no	—
17	C₃×#389	C₂²	1	2	yes	—
18	C₃×#387	—	2	2	no	—
19	C₃×#382	C₂²	1	2	yes	—
20	C₃×#382	C₂²	1	1	yes	characteristic
21	C₃×#382	C₂²	1	2	yes	—
22	C₃×C₄×C₄⋊C₄	—	2	2	no	—
23	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
24	C₃×C₄³	C₂²	1	1	yes	characteristic
25	C₃×#340	C₂²	1	4	yes	—
26	C₃×#340	C₂²	1	4	yes	—
27	C₃×#340	C₂²	1	4	yes	—
28	C₃×#340	C₂²	1	4	yes	—
29	C₃×#336	C₂²	1	2	yes	—
30	#2862	C₂×C₃	1	1	yes	characteristic
31	C₄×#125	C₂×C₃	1	1	yes	characteristic
32	#2101	C₂×C₃	1	4	yes	—
33	#1991	C₂×C₃	1	2	yes	—
34	#1988	C₂×C₃	1	2	yes	—
35	#1987	C₂×C₃	1	2	yes	—
36	#1835	C₂×C₃	1	2	yes	—
37	C₄×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
38	C₂²×C₃×Q₈	—	2	1	no	—
39	C₂²×C₃×Q₈	—	2	1	no	—
40	C₂²×C₃×C₈	C₂³	1	4	yes	—
41	C₃×C₄⋊Q₈	—	2	4	no	—
42	C₃×C₄⋊Q₈	—	2	4	no	—
43	C₃×#125	—	2	2	no	—
44	C₃×#125	Q₈	1	4	yes	complemented
45	C₂×C₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₃×C₄⋊C₄	—	2	1	no	—
48	C₂×C₃×C₄⋊C₄	—	2	1	no	—
49	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
52	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₃×C₄²	—	2	2	no	—
54	C₂×C₃×C₄²	D₄	1	2	yes	—
55	C₂×C₃×C₄²	D₄	1	2	yes	—
56	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
59	C₃×C₄⋊C₈	—	2	4	no	—
60	C₃×C₄×C₈	—	2	4	no	—
61	C₃×#95	—	2	2	no	—
62	C₃×#95	—	2	2	no	—
63	C₃×#95	—	2	2	no	—
64	C₃×#95	—	2	4	no	—
65	C₃×#95	—	2	4	no	—
66	C₂×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
68	C₂×#125	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
70	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
71	#391	—	2	2	no	—
72	#391	—	2	2	no	—
73	#389	C₂²×C₃	1	2	yes	—
74	#387	—	2	2	no	—
75	#382	C₂²×C₃	1	1	yes	characteristic
76	#382	C₂²×C₃	1	2	yes	—
77	#382	C₂²×C₃	1	2	yes	—
78	C₄×C₄⋊C₄	—	2	2	no	—
79	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₄³	C₂²×C₃	1	1	yes	characteristic
81	#340	C₂²×C₃	1	4	yes	—
82	#340	C₂²×C₃	1	4	yes	—
83	#340	C₂²×C₃	1	4	yes	—
84	#340	C₂²×C₃	1	4	yes	—
85	#336	C₂²×C₃	1	2	yes	—
86	C₂×C₃×Q₈	—	4	4	no	—
87	C₂×C₃×Q₈	—	4	2	no	—
88	C₂×C₃×Q₈	—	4	2	no	—
89	C₂×C₃×Q₈	—	4	4	no	—
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	—	2	2	no	—
92	C₂²×C₃×C₄	—	2	2	no	—
93	C₂²×C₃×C₄	—	2	2	no	—
94	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
95	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
96	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
99	C₂×C₃×C₈	—	2	4	no	—
100	C₂×C₃×C₈	—	2	4	no	—
101	C₂×C₃×C₈	—	2	4	no	—
102	C₃×C₄⋊C₄	—	2	8	no	—
103	C₃×C₄⋊C₄	—	2	2	no	—
104	C₃×C₄⋊C₄	—	2	4	no	—
105	C₃×C₄⋊C₄	—	2	4	no	—
106	C₃×C₄⋊C₄	—	2	4	no	—
107	C₃×C₄⋊C₄	—	4	4	no	—
108	C₃×C₄⋊C₄	—	4	2	no	—
109	C₃×C₄⋊C₄	—	4	2	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	#41	1	2	yes	—
117	C₃×C₄²	#41	1	4	yes	—
118	C₂²×Q₈	—	2	1	no	—
119	C₂²×Q₈	—	2	1	no	—
120	C₂²×C₈	C₂³×C₃	1	4	yes	—
121	C₄⋊Q₈	—	2	4	no	—
122	C₄⋊Q₈	—	2	4	no	—
123	#125	—	2	2	no	—
124	#125	C₃×Q₈	1	4	yes	complemented
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
130	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
131	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
132	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
133	C₂×C₄²	—	2	2	no	—
134	C₂×C₄²	C₃×D₄	1	2	yes	—
135	C₂×C₄²	C₃×D₄	1	2	yes	—
136	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
137	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
138	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
139	C₄⋊C₈	—	2	4	no	—
140	C₄×C₈	—	2	4	no	—
141	#95	—	2	2	no	—
142	#95	—	2	2	no	—
143	#95	—	2	2	no	—
144	#95	—	2	4	no	—
145	#95	—	2	4	no	—
146	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
147	C₃×Q₈	—	8	4	no	—
148	C₃×Q₈	—	8	4	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	4	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	4	2	no	—
157	C₂×C₃×C₄	—	4	2	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	—	4	2	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	—	4	2	no	—
162	C₂×C₃×C₄	—	4	4	no	—
163	C₂×C₃×C₄	—	4	2	no	—
164	C₂×C₃×C₄	—	4	4	no	—
165	C₂×C₃×C₄	C₂²⋊Q₈	1	4	yes	—
166	C₂×C₃×C₄	#124	1	2	yes	—
167	C₂×C₃×C₄	#124	1	2	yes	—
168	C₂×C₃×C₄	#135	1	4	yes	—
169	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×#41	1	2	yes	—
172	C₂×C₃×C₄	2₋¹⁺⁴	1	2	yes	—
173	C₃×C₈	—	4	4	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	4	4	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
183	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
184	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
186	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
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	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	C₂×C₃×Q₈	1	4	yes	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	4	2	no	—
202	C₄²	—	4	4	no	—
203	C₄²	C₂×C₃×Q₈	1	2	yes	—
204	C₄²	C₃×#41	1	2	yes	—
205	C₄²	C₃×#41	1	4	yes	—
206	C₂²×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
207	C₂²×C₃	C₂×#124	1	1	yes	characteristic
208	C₂²×C₃	#536	1	1	yes	characteristic
209	C₂²×C₃	C₂×#135	1	2	yes	—
210	C₂²×C₃	#575	1	2	yes	—
211	C₃×C₄	—	2	4	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	4	4	no	—
214	C₃×C₄	—	4	2	no	—
215	C₃×C₄	—	8	2	no	—
216	C₃×C₄	—	8	2	no	—
217	C₃×C₄	—	8	2	no	—
218	C₃×C₄	#488	1	4	yes	—
219	C₃×C₄	#556	1	2	yes	—
220	C₃×C₄	#557	1	2	yes	—
221	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
222	Q₈	—	8	4	no	—
223	Q₈	—	8	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	C₃×C₂²⋊Q₈	1	4	yes	—
241	C₂×C₄	C₃×#124	1	2	yes	—
242	C₂×C₄	C₃×#124	1	2	yes	—
243	C₂×C₄	C₃×#135	1	4	yes	—
244	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
245	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
246	C₂×C₄	C₂×C₃×#41	1	2	yes	—
247	C₂×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
248	C₈	—	4	4	no	—
249	C₂×C₃	#2568	1	1	yes	characteristic
250	C₂×C₃	#3076	1	2	yes	—
251	C₂×C₃	C₂×#488	1	1	yes	characteristic
252	C₂×C₃	#3140	1	1	yes	characteristic
253	C₂×C₃	#3233	1	2	yes	—
254	C₂²	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
255	C₂²	C₂×C₃×#124	1	1	yes	characteristic
256	C₂²	C₃×#536	1	1	yes	characteristic
257	C₂²	C₂×C₃×#135	1	2	yes	—
258	C₂²	C₃×#575	1	2	yes	—
259	C₄	—	2	2	no	—
260	C₄	—	2	4	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	4	no	—
263	C₄	—	8	2	no	—
264	C₄	—	8	2	no	—
265	C₄	—	8	2	no	—
266	C₄	C₃×#488	1	4	yes	—
267	C₄	C₃×#556	1	2	yes	—
268	C₄	C₃×#557	1	2	yes	—
269	C₃	#29222	1	1	yes	characteristic, direct product factor, Sylow
270	C₂	C₃×#2568	1	1	yes	characteristic
271	C₂	C₃×#3076	1	2	yes	—
272	C₂	C₂×C₃×#488	1	1	yes	characteristic
273	C₂	C₃×#3140	1	1	yes	characteristic
274	C₂	C₃×#3233	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. #29222	C₃	direct (N × H)
44. C₃×#125	Q₈	semidirect
124. #125	C₃×Q₈	semidirect
269. C₃	#29222	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×#2101	2
C₃×#1991	2
C₃×#1988	2
C₃×#1987	2
C₃×#1835	2
C₃×C₄×C₄⋊C₈	2
#29222	3

Subgroups — C3×#29222 #10647829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#29222 #10647829