Subgroups — C₃×#3284 #85148

Order 384.

494 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#135	C₂	1	2	yes	maximal, complemented
2	C₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#465	C₂	1	2	yes	maximal, complemented
6	C₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#452	C₂	1	2	yes	maximal, complemented
9	C₃×#449	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#3284	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	2	yes	—
14	C₂×C₃×#41	C₂²	1	2	yes	—
15	C₃×#135	—	2	8	no	—
16	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
17	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
18	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
19	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₃×D₈	C₂²	1	2	yes	—
21	C₂²×C₃×C₈	C₂²	1	2	yes	—
22	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₃×#127	—	2	1	no	—
24	C₃×#127	C₂²	1	1	yes	characteristic
25	C₃×#123	—	2	2	no	—
26	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₃×#121	—	2	1	no	—
28	C₃×#121	—	2	2	no	—
29	C₃×#121	C₂²	1	2	yes	—
30	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
31	C₃×C₄×D₄	—	2	1	no	—
32	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
33	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₄⋊C₄	—	2	1	no	—
35	C₃×#106	C₂²	1	1	yes	characteristic
36	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₃×#103	C₂²	1	1	yes	characteristic
38	C₃×#103	C₂²	1	2	yes	—
39	C₃×#102	C₂²	1	1	yes	characteristic
40	C₃×#102	C₂²	1	2	yes	—
41	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
42	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
43	C₂×#135	C₂×C₃	1	2	yes	complemented
44	#550	C₂×C₃	1	1	yes	characteristic, complemented
45	#547	C₂×C₃	1	1	yes	characteristic, complemented
46	#492	C₂×C₃	1	1	yes	characteristic, complemented
47	#465	C₂×C₃	1	2	yes	complemented
48	#462	C₂×C₃	1	1	yes	characteristic, complemented
49	#459	C₂×C₃	1	1	yes	characteristic, complemented
50	#452	C₂×C₃	1	2	yes	complemented
51	#449	C₂×C₃	1	2	yes	complemented
52	C₄×SD₁₆	C₂×C₃	1	1	yes	characteristic, complemented
53	C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
54	C₃×#41	—	2	4	no	—
55	C₃×#41	—	2	4	no	—
56	C₃×#41	—	4	2	no	—
57	C₃×#41	—	4	2	no	—
58	C₂×C₃×Q₈	C₂³	1	2	yes	—
59	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
60	C₂×C₃×D₄	—	2	2	no	—
61	C₂×C₃×D₄	—	2	2	no	—
62	C₂×C₃×D₄	—	2	2	no	—
63	C₂×C₃×D₄	—	2	1	no	—
64	C₂×C₃×D₄	—	2	1	no	—
65	C₂×C₃×D₄	D₄	1	1	yes	characteristic
66	C₂×C₃×D₄	C₂³	1	2	yes	—
67	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
68	C₂²×C₃×C₄	—	2	1	no	—
69	C₂²×C₃×C₄	—	2	2	no	—
70	C₂²×C₃×C₄	—	2	2	no	—
71	C₂²×C₃×C₄	C₂³	1	2	yes	—
72	C₃×Q₁₆	—	2	4	no	—
73	C₃×SD₁₆	—	2	4	no	—
74	C₃×SD₁₆	—	4	2	no	—
75	C₃×SD₁₆	D₄	1	4	yes	complemented
76	C₃×D₈	—	2	4	no	—
77	C₂×C₃×C₈	—	2	4	no	—
78	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂³	1	2	yes	—
80	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
81	C₃×C₄⋊C₄	—	2	1	no	—
82	C₃×C₄⋊C₄	—	2	2	no	—
83	C₃×C₄⋊C₄	—	2	1	no	—
84	C₃×C₄⋊C₄	—	2	1	no	—
85	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
86	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
87	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₃×C₂²⋊C₄	—	2	2	no	—
89	C₃×C₂²⋊C₄	—	2	2	no	—
90	C₃×C₂²⋊C₄	—	2	1	no	—
91	C₃×C₂²⋊C₄	D₄	1	2	yes	—
92	C₃×C₄²	—	2	1	no	—
93	C₃×C₄²	C₂³	1	1	yes	characteristic
94	C₂×#41	C₂²×C₃	1	2	yes	—
95	C₂×#41	C₂²×C₃	1	2	yes	—
96	#135	—	2	8	no	—
97	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
98	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
99	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
100	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic
101	C₂×D₈	C₂²×C₃	1	2	yes	—
102	C₂²×C₈	C₂²×C₃	1	2	yes	—
103	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
104	#127	—	2	1	no	—
105	#127	C₂²×C₃	1	1	yes	characteristic
106	#123	—	2	2	no	—
107	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
108	#121	—	2	1	no	—
109	#121	—	2	2	no	—
110	#121	C₂²×C₃	1	2	yes	—
111	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
112	C₄×D₄	—	2	1	no	—
113	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
114	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	1	no	—
116	#106	C₂²×C₃	1	1	yes	characteristic
117	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
118	#103	C₂²×C₃	1	1	yes	characteristic
119	#103	C₂²×C₃	1	2	yes	—
120	#102	C₂²×C₃	1	1	yes	characteristic
121	#102	C₂²×C₃	1	2	yes	—
122	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
123	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
124	C₂³×C₃	—	2	1	no	—
125	C₂³×C₃	—	2	2	no	—
126	C₂³×C₃	C₂×D₄	1	2	yes	—
127	C₃×Q₈	—	2	1	no	—
128	C₃×Q₈	—	2	2	no	—
129	C₃×Q₈	—	2	2	no	—
130	C₃×Q₈	C₂×D₄	1	2	yes	—
131	C₃×D₄	—	2	2	no	—
132	C₃×D₄	—	2	2	no	—
133	C₃×D₄	—	2	1	no	—
134	C₃×D₄	—	2	2	no	—
135	C₃×D₄	—	4	4	no	—
136	C₃×D₄	—	4	4	no	—
137	C₃×D₄	—	4	2	no	—
138	C₃×D₄	—	8	1	no	—
139	C₃×D₄	—	8	2	no	—
140	C₃×D₄	C₂×D₄	1	2	yes	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	4	4	no	—
148	C₂×C₃×C₄	—	4	4	no	—
149	C₂×C₃×C₄	—	4	2	no	—
150	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
152	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
154	C₃×C₈	—	2	2	no	—
155	C₃×C₈	—	2	1	no	—
156	C₃×C₈	C₂×D₄	1	2	yes	—
157	#41	—	2	4	no	—
158	#41	—	2	4	no	—
159	#41	—	4	2	no	—
160	#41	—	4	2	no	—
161	C₂×Q₈	C₂³×C₃	1	2	yes	—
162	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
163	C₂×D₄	—	2	2	no	—
164	C₂×D₄	—	2	2	no	—
165	C₂×D₄	—	2	2	no	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	C₃×D₄	1	1	yes	characteristic
169	C₂×D₄	C₂³×C₃	1	2	yes	—
170	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	C₂³×C₃	1	2	yes	—
175	Q₁₆	—	2	4	no	—
176	SD₁₆	—	2	4	no	—
177	SD₁₆	—	4	2	no	—
178	SD₁₆	C₃×D₄	1	4	yes	complemented
179	D₈	—	2	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
182	C₂×C₈	C₂³×C₃	1	2	yes	—
183	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
189	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
190	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	1	no	—
194	C₂²⋊C₄	C₃×D₄	1	2	yes	—
195	C₄²	—	2	1	no	—
196	C₄²	C₂³×C₃	1	1	yes	characteristic
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	4	1	no	—
202	C₂²×C₃	—	4	1	no	—
203	C₂²×C₃	—	4	2	no	—
204	C₂²×C₃	—	4	2	no	—
205	C₂²×C₃	—	4	2	no	—
206	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	—	2	2	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	4	2	no	—
211	C₃×C₄	—	4	1	no	—
212	C₃×C₄	—	4	1	no	—
213	C₃×C₄	#135	1	2	yes	complemented
214	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
215	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
216	C₂³	—	2	1	no	—
217	C₂³	—	2	2	no	—
218	C₂³	C₂×C₃×D₄	1	2	yes	—
219	Q₈	—	2	1	no	—
220	Q₈	—	2	2	no	—
221	Q₈	—	2	2	no	—
222	Q₈	C₂×C₃×D₄	1	2	yes	—
223	D₄	—	2	2	no	—
224	D₄	—	2	2	no	—
225	D₄	—	2	1	no	—
226	D₄	—	2	2	no	—
227	D₄	—	4	4	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	2	no	—
230	D₄	—	8	1	no	—
231	D₄	—	8	2	no	—
232	D₄	C₂×C₃×D₄	1	2	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
244	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
245	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	2	1	no	—
248	C₈	C₂×C₃×D₄	1	2	yes	—
249	C₂×C₃	—	4	2	no	—
250	C₂×C₃	—	4	2	no	—
251	C₂×C₃	—	8	2	no	—
252	C₂×C₃	D₄²	1	1	yes	characteristic
253	C₂×C₃	C₂×#135	1	1	yes	characteristic
254	C₂×C₃	#577	1	1	yes	characteristic
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	1	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	2	no	—
271	C₄	C₃×#135	1	2	yes	complemented
272	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
273	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
274	C₃	#3284	1	1	yes	characteristic, direct product factor, Sylow
275	C₂	—	4	2	no	—
276	C₂	—	4	2	no	—
277	C₂	—	8	2	no	—
278	C₂	C₃×D₄²	1	1	yes	characteristic
279	C₂	C₂×C₃×#135	1	1	yes	characteristic
280	C₂	C₃×#577	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#135	C₂	semidirect
2. C₃×#550	C₂	semidirect
3. C₃×#547	C₂	semidirect
4. C₃×#492	C₂	semidirect
5. C₃×#465	C₂	semidirect
6. C₃×#462	C₂	semidirect
7. C₃×#459	C₂	semidirect
8. C₃×#452	C₂	semidirect
9. C₃×#449	C₂	semidirect
10. C₃×C₄×SD₁₆	C₂	semidirect
11. C₃×C₈×D₄	C₂	semidirect
12. #3284	C₃	direct (N × H)
43. C₂×#135	C₂×C₃	semidirect
44. #550	C₂×C₃	semidirect
45. #547	C₂×C₃	semidirect
46. #492	C₂×C₃	semidirect
47. #465	C₂×C₃	semidirect
48. #462	C₂×C₃	semidirect
49. #459	C₂×C₃	semidirect
50. #452	C₂×C₃	semidirect
51. #449	C₂×C₃	semidirect
52. C₄×SD₁₆	C₂×C₃	semidirect
53. C₈×D₄	C₂×C₃	semidirect
75. C₃×SD₁₆	D₄	semidirect
178. SD₁₆	C₃×D₄	semidirect
213. C₃×C₄	#135	semidirect
271. C₄	C₃×#135	semidirect
274. C₃	#3284	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×#135	2
C₃×#550	2
C₃×#547	2
C₃×#492	2
C₃×#465	2
C₃×#462	2
C₃×#459	2
C₃×#452	2
C₃×#449	2
C₃×C₄×SD₁₆	2
C₃×C₈×D₄	2
#3284	3

Subgroups — C3×#3284 #85148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3284 #85148