Subgroups — C₃×#33630 #10652237

Order 768.

542 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₄×#135	C₂	1	1	yes	maximal, characteristic
3	C₃×#2229	C₂	1	4	yes	maximal, complemented
4	C₃×#2228	C₂	1	4	yes	maximal, complemented
5	C₃×#2159	C₂	1	2	yes	maximal, complemented
6	C₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#2139	C₂	1	2	yes	maximal
8	#33630	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×#135	C₂²	1	1	yes	characteristic
10	C₂×C₃×#125	—	2	1	no	—
11	C₃×C₄×#41	—	2	1	no	—
12	C₂×C₃×M₃₂	C₂²	1	2	yes	—
13	C₃×#499	—	2	2	no	—
14	C₃×#499	C₂²	1	4	yes	—
15	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
16	C₃×C₄×SD₁₆	—	2	2	no	—
17	C₃×C₄×D₈	C₂²	1	2	yes	—
18	C₃×#427	C₂²	1	1	yes	characteristic
19	C₂×C₃×#107	C₂²	1	2	yes	—
20	C₂×C₃×#106	—	2	1	no	—
21	C₃×#416	—	2	1	no	—
22	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃×#367	C₂²	1	4	yes	—
24	C₃×#366	C₂²	1	4	yes	—
25	C₃×C₄⋊C₁₆	C₂²	1	4	yes	—
26	C₃×#358	C₂²	1	4	yes	—
27	C₃×#357	C₂²	1	4	yes	—
28	C₂×#499	C₂×C₃	1	1	yes	characteristic, complemented
29	C₄×#135	C₂×C₃	1	1	yes	characteristic
30	#2229	C₂×C₃	1	4	yes	complemented
31	#2228	C₂×C₃	1	4	yes	complemented
32	#2159	C₂×C₃	1	2	yes	complemented
33	#2150	C₂×C₃	1	1	yes	characteristic, complemented
34	#2139	C₂×C₃	1	2	yes	—
35	C₂×C₃×#41	—	2	1	no	—
36	C₃×#135	—	2	2	no	—
37	C₃×#135	Q₈	1	4	yes	—
38	C₂×C₃×Q₁₆	C₂³	1	1	yes	characteristic
39	C₂×C₃×SD₁₆	—	2	1	no	—
40	C₂×C₃×D₈	C₂³	1	1	yes	characteristic
41	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
43	C₃×#125	—	2	4	no	—
44	C₃×#125	—	4	1	no	—
45	C₃×#125	—	4	1	no	—
46	C₃×C₄×Q₈	—	2	2	no	—
47	C₃×C₄×D₄	—	2	2	no	—
48	C₃×C₄×D₄	—	4	2	no	—
49	C₃×#117	—	2	1	no	—
50	C₃×#117	—	4	1	no	—
51	C₂×C₃×C₄⋊C₄	—	2	2	no	—
52	C₂×C₃×C₄⋊C₄	—	4	2	no	—
53	C₂×C₃×C₄²	—	4	1	no	—
54	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
55	C₃×M₃₂	—	2	4	no	—
56	C₂×C₃×C₁₆	C₂³	1	4	yes	—
57	C₃×#107	—	2	2	no	—
58	C₃×#107	C₂³	1	2	yes	—
59	C₃×#107	C₂³	1	4	yes	—
60	C₃×#106	—	2	4	no	—
61	C₃×#106	—	2	1	no	—
62	C₃×#103	—	2	2	no	—
63	C₃×#102	—	2	2	no	—
64	C₃×C₄×C₈	D₄	1	2	yes	—
65	C₃×C₄×C₈	C₂³	1	2	yes	—
66	C₂×#135	C₂²×C₃	1	1	yes	characteristic
67	C₂×#125	—	2	1	no	—
68	C₄×#41	—	2	1	no	—
69	C₂×M₃₂	C₂²×C₃	1	2	yes	—
70	#499	—	2	2	no	—
71	#499	C₂²×C₃	1	4	yes	—
72	C₄×Q₁₆	C₂²×C₃	1	2	yes	—
73	C₄×SD₁₆	—	2	2	no	—
74	C₄×D₈	C₂²×C₃	1	2	yes	—
75	#427	C₂²×C₃	1	1	yes	characteristic
76	C₂×#107	C₂²×C₃	1	2	yes	—
77	C₂×#106	—	2	1	no	—
78	#416	—	2	1	no	—
79	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
80	#367	C₂²×C₃	1	4	yes	—
81	#366	C₂²×C₃	1	4	yes	—
82	C₄⋊C₁₆	C₂²×C₃	1	4	yes	—
83	#358	C₂²×C₃	1	4	yes	—
84	#357	C₂²×C₃	1	4	yes	—
85	C₃×#41	—	2	4	no	—
86	C₃×#41	—	4	2	no	—
87	C₂×C₃×Q₈	—	2	1	no	—
88	C₂×C₃×D₄	—	2	1	no	—
89	C₂×C₃×D₄	—	4	1	no	—
90	C₂²×C₃×C₄	—	4	1	no	—
91	C₂²×C₃×C₄	—	4	2	no	—
92	C₂²×C₃×C₄	—	4	2	no	—
93	C₂²×C₃×C₄	D₈	1	2	yes	—
94	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
95	C₃×Q₁₆	—	2	1	no	—
96	C₃×Q₁₆	C₂×Q₈	1	2	yes	—
97	C₃×SD₁₆	—	2	4	no	—
98	C₃×D₈	—	2	1	no	—
99	C₃×D₈	C₂×Q₈	1	2	yes	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
102	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
104	C₂×C₃×C₈	#41	1	2	yes	—
105	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
106	C₃×C₄⋊C₄	—	2	2	no	—
107	C₃×C₄⋊C₄	—	2	4	no	—
108	C₃×C₄⋊C₄	—	4	2	no	—
109	C₃×C₄⋊C₄	—	4	4	no	—
110	C₃×C₄⋊C₄	—	4	4	no	—
111	C₃×C₄⋊C₄	—	4	2	no	—
112	C₃×C₂²⋊C₄	—	4	1	no	—
113	C₃×C₂²⋊C₄	—	4	2	no	—
114	C₃×C₄²	—	4	1	no	—
115	C₃×C₄²	—	4	2	no	—
116	C₃×C₄²	D₈	1	1	yes	characteristic
117	C₃×C₄²	D₈	1	1	yes	characteristic
118	C₃×C₄²	C₂×D₄	1	2	yes	—
119	C₃×C₁₆	—	2	4	no	—
120	C₂×#41	—	2	1	no	—
121	#135	—	2	2	no	—
122	#135	C₃×Q₈	1	4	yes	—
123	C₂×Q₁₆	C₂³×C₃	1	1	yes	characteristic
124	C₂×SD₁₆	—	2	1	no	—
125	C₂×D₈	C₂³×C₃	1	1	yes	characteristic
126	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
127	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
128	#125	—	2	4	no	—
129	#125	—	4	1	no	—
130	#125	—	4	1	no	—
131	C₄×Q₈	—	2	2	no	—
132	C₄×D₄	—	2	2	no	—
133	C₄×D₄	—	4	2	no	—
134	#117	—	2	1	no	—
135	#117	—	4	1	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	4	2	no	—
138	C₂×C₄²	—	4	1	no	—
139	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
140	M₃₂	—	2	4	no	—
141	C₂×C₁₆	C₂³×C₃	1	4	yes	—
142	#107	—	2	2	no	—
143	#107	C₂³×C₃	1	4	yes	—
144	#107	C₂³×C₃	1	2	yes	—
145	#106	—	2	4	no	—
146	#106	—	2	1	no	—
147	#103	—	2	2	no	—
148	#102	—	2	2	no	—
149	C₄×C₈	C₃×D₄	1	2	yes	—
150	C₄×C₈	C₂³×C₃	1	2	yes	—
151	C₂³×C₃	—	4	1	no	—
152	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
153	C₃×Q₈	—	2	2	no	—
154	C₃×Q₈	—	4	1	no	—
155	C₃×D₄	—	2	2	no	—
156	C₃×D₄	—	4	4	no	—
157	C₃×D₄	—	4	1	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	—	4	1	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	—	8	4	no	—
164	C₂×C₃×C₄	—	8	4	no	—
165	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
166	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
167	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
168	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
169	C₂×C₃×C₄	#137	1	2	yes	—
170	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
171	C₃×C₈	—	2	2	no	—
172	C₃×C₈	C₂²⋊Q₈	1	2	yes	—
173	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
174	C₃×C₈	C₂×#41	1	1	yes	characteristic
175	#41	—	2	4	no	—
176	#41	—	4	2	no	—
177	C₂×Q₈	—	2	1	no	—
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	4	1	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₃×D₈	1	2	yes	—
184	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
185	Q₁₆	—	2	1	no	—
186	Q₁₆	C₂×C₃×Q₈	1	2	yes	—
187	SD₁₆	—	2	4	no	—
188	D₈	—	2	1	no	—
189	D₈	C₂×C₃×Q₈	1	2	yes	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
192	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
193	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
194	C₂×C₈	C₃×#41	1	2	yes	—
195	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	4	1	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₄²	—	4	1	no	—
205	C₄²	—	4	2	no	—
206	C₄²	C₃×D₈	1	1	yes	characteristic
207	C₄²	C₃×D₈	1	1	yes	characteristic
208	C₄²	C₂×C₃×D₄	1	2	yes	—
209	C₁₆	—	2	4	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	4	2	no	—
212	C₂²×C₃	—	8	2	no	—
213	C₂²×C₃	#474	1	2	yes	—
214	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	—	4	2	no	—
218	C₃×C₄	—	8	2	no	—
219	C₃×C₄	—	8	4	no	—
220	C₃×C₄	#474	1	2	yes	—
221	C₃×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
222	C₃×C₄	C₂×#137	1	1	yes	characteristic
223	C₂³	—	4	1	no	—
224	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
225	Q₈	—	2	2	no	—
226	Q₈	—	4	1	no	—
227	D₄	—	2	2	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	8	4	no	—
237	C₂×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
238	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
239	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
240	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
241	C₂×C₄	C₃×#137	1	2	yes	—
242	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	C₃×C₂²⋊Q₈	1	2	yes	—
245	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
246	C₈	C₂×C₃×#41	1	1	yes	characteristic
247	C₂×C₃	—	2	2	no	—
248	C₂×C₃	—	8	2	no	—
249	C₂×C₃	C₂×#474	1	1	yes	characteristic
250	C₂×C₃	#3414	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	C₃×#474	1	2	yes	—
255	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	8	2	no	—
260	C₄	—	8	4	no	—
261	C₄	C₃×#474	1	2	yes	—
262	C₄	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
263	C₄	C₂×C₃×#137	1	1	yes	characteristic
264	C₃	#33630	1	1	yes	characteristic, direct product factor, Sylow
265	C₂	—	2	2	no	—
266	C₂	—	8	2	no	—
267	C₂	C₂×C₃×#474	1	1	yes	characteristic
268	C₂	C₃×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂×C₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#499	C₂	semidirect
3. C₃×#2229	C₂	semidirect
4. C₃×#2228	C₂	semidirect
5. C₃×#2159	C₂	semidirect
6. C₃×#2150	C₂	semidirect
8. #33630	C₃	direct (N × H)
28. C₂×#499	C₂×C₃	semidirect
30. #2229	C₂×C₃	semidirect
31. #2228	C₂×C₃	semidirect
32. #2159	C₂×C₃	semidirect
33. #2150	C₂×C₃	semidirect
264. C₃	#33630	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#499	2
C₃×C₄×#135	2
C₃×#2229	2
C₃×#2228	2
C₃×#2159	2
C₃×#2150	2
C₃×#2139	2
#33630	3

Subgroups — C3×#33630 #10652237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33630 #10652237