Subgroups — C₃×#3324 #85188

Order 384.

448 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×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₄²	C₂	1	1	yes	maximal, characteristic
3	C₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#482	C₂	1	2	yes	maximal, complemented
5	C₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#468	C₂	1	2	yes	maximal, complemented
7	C₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#450	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#418	C₂	1	2	yes	maximal, complemented
12	#3324	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×D₄	—	2	1	no	—
14	C₂²×C₃×D₄	C₂²	1	2	yes	—
15	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
17	C₂×C₃×M₁₆	C₂²	1	2	yes	complemented
18	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₃×#127	C₂²	1	1	yes	characteristic
20	C₃×#126	—	2	2	no	—
21	C₃×#124	—	2	1	no	—
22	C₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
23	C₃×#121	—	2	1	no	—
24	C₃×#121	C₂²	1	2	yes	—
25	C₃×#120	—	2	2	no	—
26	C₃×C₄×Q₈	—	2	1	no	—
27	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
28	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
29	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
30	C₃×#117	C₂²	1	2	yes	complemented
31	C₃×#107	C₂²	1	2	yes	complemented
32	C₃×#106	C₂²	1	1	yes	characteristic, complemented
33	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
34	C₃×#103	C₂²	1	1	yes	characteristic, complemented
35	C₃×#102	C₂²	1	1	yes	characteristic
36	C₃×#102	C₂²	1	2	yes	—
37	C₃×#102	C₂²	1	2	yes	complemented
38	C₃×#102	C₂²	1	2	yes	complemented
39	C₃×#102	C₂²	1	2	yes	complemented
40	C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
42	#556	C₂×C₃	1	1	yes	characteristic, complemented
43	D₄²	C₂×C₃	1	1	yes	characteristic
44	#486	C₂×C₃	1	1	yes	characteristic, complemented
45	#482	C₂×C₃	1	2	yes	complemented
46	#474	C₂×C₃	1	1	yes	characteristic, complemented
47	#468	C₂×C₃	1	2	yes	complemented
48	#460	C₂×C₃	1	1	yes	characteristic, complemented
49	#450	C₂×C₃	1	2	yes	complemented
50	C₄×SD₁₆	C₂×C₃	1	1	yes	characteristic, complemented
51	#436	C₂×C₃	1	1	yes	characteristic, complemented
52	#418	C₂×C₃	1	2	yes	complemented
53	C₂⁴×C₃	—	2	2	no	—
54	C₂×C₃×Q₈	—	2	1	no	—
55	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
56	C₂×C₃×D₄	—	2	4	no	—
57	C₂×C₃×D₄	—	2	2	no	—
58	C₂×C₃×D₄	—	2	2	no	—
59	C₂×C₃×D₄	—	2	1	no	—
60	C₂×C₃×D₄	—	2	1	no	—
61	C₂×C₃×D₄	—	4	2	no	—
62	C₂×C₃×D₄	—	4	4	no	—
63	C₂×C₃×D₄	D₄	1	1	yes	characteristic
64	C₂×C₃×D₄	C₂³	1	2	yes	—
65	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
66	C₂²×C₃×C₄	—	2	1	no	—
67	C₂²×C₃×C₄	C₂³	1	2	yes	—
68	C₃×SD₁₆	—	2	2	no	—
69	C₃×SD₁₆	—	4	2	no	—
70	C₃×M₁₆	—	4	2	no	—
71	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₃×C₈	C₂³	1	2	yes	—
74	C₃×C₄⋊C₄	—	2	2	no	—
75	C₃×C₄⋊C₄	—	2	1	no	—
76	C₃×C₄⋊C₄	—	2	1	no	—
77	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
78	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₃×C₄⋊C₄	C₂³	1	2	yes	—
80	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
81	C₃×C₂²⋊C₄	—	2	2	no	—
82	C₃×C₂²⋊C₄	—	2	2	no	—
83	C₃×C₂²⋊C₄	—	2	1	no	—
84	C₃×C₂²⋊C₄	—	2	2	no	—
85	C₃×C₂²⋊C₄	D₄	1	2	yes	—
86	C₃×C₄²	—	2	2	no	—
87	C₃×C₄²	—	2	1	no	—
88	C₃×C₄²	C₂³	1	1	yes	characteristic
89	C₂²×D₄	—	2	1	no	—
90	C₂²×D₄	C₂²×C₃	1	2	yes	—
91	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic
92	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
93	C₂×M₁₆	C₂²×C₃	1	2	yes	complemented
94	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
95	#127	C₂²×C₃	1	1	yes	characteristic
96	#126	—	2	2	no	—
97	#124	—	2	1	no	—
98	C₂²⋊Q₈	C₂²×C₃	1	2	yes	complemented
99	#121	—	2	1	no	—
100	#121	C₂²×C₃	1	2	yes	—
101	#120	—	2	2	no	—
102	C₄×Q₈	—	2	1	no	—
103	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
104	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
105	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
106	#117	C₂²×C₃	1	2	yes	complemented
107	#107	C₂²×C₃	1	2	yes	complemented
108	#106	C₂²×C₃	1	1	yes	characteristic, complemented
109	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
110	#103	C₂²×C₃	1	1	yes	characteristic, complemented
111	#102	C₂²×C₃	1	1	yes	characteristic
112	#102	C₂²×C₃	1	2	yes	—
113	#102	C₂²×C₃	1	2	yes	complemented
114	#102	C₂²×C₃	1	2	yes	complemented
115	#102	C₂²×C₃	1	2	yes	complemented
116	C₂²⋊C₈	C₂²×C₃	1	2	yes	complemented
117	C₄×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
118	C₂³×C₃	—	2	1	no	—
119	C₂³×C₃	—	2	2	no	—
120	C₂³×C₃	—	4	4	no	—
121	C₂³×C₃	—	4	4	no	—
122	C₂³×C₃	—	8	2	no	—
123	C₂³×C₃	C₂×D₄	1	2	yes	—
124	C₃×Q₈	—	2	1	no	—
125	C₃×Q₈	—	2	1	no	—
126	C₃×Q₈	—	4	2	no	—
127	C₃×D₄	—	2	2	no	—
128	C₃×D₄	—	2	2	no	—
129	C₃×D₄	—	2	1	no	—
130	C₃×D₄	—	2	2	no	—
131	C₃×D₄	—	4	4	no	—
132	C₃×D₄	—	4	4	no	—
133	C₃×D₄	—	4	2	no	—
134	C₃×D₄	—	8	1	no	—
135	C₃×D₄	#41	1	2	yes	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	4	2	no	—
143	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
144	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
147	C₃×C₈	—	2	1	no	—
148	C₃×C₈	—	2	1	no	—
149	C₃×C₈	—	2	2	no	—
150	C₂⁴	—	2	2	no	—
151	C₂×Q₈	—	2	1	no	—
152	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
153	C₂×D₄	—	2	4	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	—	2	1	no	—
158	C₂×D₄	—	4	2	no	—
159	C₂×D₄	—	4	4	no	—
160	C₂×D₄	C₃×D₄	1	1	yes	characteristic
161	C₂×D₄	C₂³×C₃	1	2	yes	—
162	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	C₂³×C₃	1	2	yes	—
165	SD₁₆	—	2	2	no	—
166	SD₁₆	—	4	2	no	—
167	M₁₆	—	4	2	no	—
168	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
169	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
170	C₂×C₈	C₂³×C₃	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
175	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
176	C₄⋊C₄	C₂³×C₃	1	2	yes	—
177	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
178	C₂²⋊C₄	—	2	2	no	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	—	2	1	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	C₃×D₄	1	2	yes	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	1	no	—
185	C₄²	C₂³×C₃	1	1	yes	characteristic
186	C₂²×C₃	—	2	2	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	4	1	no	—
191	C₂²×C₃	—	4	1	no	—
192	C₂²×C₃	—	4	2	no	—
193	C₂²×C₃	—	4	2	no	—
194	C₂²×C₃	—	4	2	no	—
195	C₂²×C₃	—	8	8	no	—
196	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	4	1	no	—
200	C₃×C₄	—	4	1	no	—
201	C₃×C₄	—	4	2	no	—
202	C₃×C₄	—	4	1	no	—
203	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
204	C₃×C₄	C₂×#41	1	1	yes	characteristic
205	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
206	C₂³	—	2	1	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	4	4	no	—
209	C₂³	—	4	4	no	—
210	C₂³	—	8	2	no	—
211	C₂³	C₂×C₃×D₄	1	2	yes	—
212	Q₈	—	2	1	no	—
213	Q₈	—	2	1	no	—
214	Q₈	—	4	2	no	—
215	D₄	—	2	2	no	—
216	D₄	—	2	2	no	—
217	D₄	—	2	1	no	—
218	D₄	—	2	2	no	—
219	D₄	—	4	4	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	2	no	—
222	D₄	—	8	1	no	—
223	D₄	C₃×#41	1	2	yes	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
232	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
233	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
234	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₂×C₃	—	4	2	no	—
239	C₂×C₃	—	4	2	no	—
240	C₂×C₃	—	8	2	no	—
241	C₂×C₃	#546	1	1	yes	characteristic
242	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
243	C₂×C₃	#577	1	1	yes	characteristic
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	8	8	no	—
254	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	2	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	1	no	—
261	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
262	C₄	C₂×C₃×#41	1	1	yes	characteristic
263	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
264	C₃	#3324	1	1	yes	characteristic, direct product factor, Sylow
265	C₂	—	4	2	no	—
266	C₂	—	4	2	no	—
267	C₂	—	8	2	no	—
268	C₂	C₃×#546	1	1	yes	characteristic
269	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
270	C₂	C₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×D₄²	C₂
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. C₃×#556	C₂	semidirect
3. C₃×#486	C₂	semidirect
4. C₃×#482	C₂	semidirect
5. C₃×#474	C₂	semidirect
6. C₃×#468	C₂	semidirect
7. C₃×#460	C₂	semidirect
8. C₃×#450	C₂	semidirect
9. C₃×C₄×SD₁₆	C₂	semidirect
10. C₃×#436	C₂	semidirect
11. C₃×#418	C₂	semidirect
12. #3324	C₃	direct (N × H)
16. C₂×C₃×SD₁₆	C₂²	semidirect
17. C₂×C₃×M₁₆	C₂²	semidirect
18. C₃×C₄⋊Q₈	C₂²	semidirect
22. C₃×C₂²⋊Q₈	C₂²	semidirect
27. C₃×C₄×Q₈	C₂²	semidirect
30. C₃×#117	C₂²	semidirect
31. C₃×#107	C₂²	semidirect
32. C₃×#106	C₂²	semidirect
33. C₃×C₄⋊C₈	C₂²	semidirect
34. C₃×#103	C₂²	semidirect
37. C₃×#102	C₂²	semidirect
38. C₃×#102	C₂²	semidirect
39. C₃×#102	C₂²	semidirect
40. C₃×C₂²⋊C₈	C₂²	semidirect
41. C₃×C₄×C₈	C₂²	semidirect
42. #556	C₂×C₃	semidirect
44. #486	C₂×C₃	semidirect
45. #482	C₂×C₃	semidirect
46. #474	C₂×C₃	semidirect
47. #468	C₂×C₃	semidirect
48. #460	C₂×C₃	semidirect
49. #450	C₂×C₃	semidirect
50. C₄×SD₁₆	C₂×C₃	semidirect
51. #436	C₂×C₃	semidirect
52. #418	C₂×C₃	semidirect
92. C₂×SD₁₆	C₂²×C₃	semidirect
93. C₂×M₁₆	C₂²×C₃	semidirect
94. C₄⋊Q₈	C₂²×C₃	semidirect
98. C₂²⋊Q₈	C₂²×C₃	semidirect
103. C₄×Q₈	C₂²×C₃	semidirect
106. #117	C₂²×C₃	semidirect
107. #107	C₂²×C₃	semidirect
108. #106	C₂²×C₃	semidirect
109. C₄⋊C₈	C₂²×C₃	semidirect
110. #103	C₂²×C₃	semidirect
113. #102	C₂²×C₃	semidirect
114. #102	C₂²×C₃	semidirect
115. #102	C₂²×C₃	semidirect
116. C₂²⋊C₈	C₂²×C₃	semidirect
117. C₄×C₈	C₂²×C₃	semidirect
264. C₃	#3324	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×#556	2
C₃×D₄²	2
C₃×#486	2
C₃×#482	2
C₃×#474	2
C₃×#468	2
C₃×#460	2
C₃×#450	2
C₃×C₄×SD₁₆	2
C₃×#436	2
C₃×#418	2
#3324	3

Subgroups — C3×#3324 #85188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3324 #85188