Subgroups — C₃×#3398 #85262

Order 384.

350 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#556	C₂	1	1	yes	maximal, characteristic
3	C₃×#497	C₂	1	2	yes	maximal
4	C₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#464	C₂	1	2	yes	maximal
6	C₃×#462	C₂	1	2	yes	maximal, complemented
7	C₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#461	C₂	1	1	yes	maximal, characteristic
9	C₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#441	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic
12	#3398	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
15	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
16	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
18	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
21	C₃×#126	—	2	2	no	—
22	C₃×#125	—	2	1	no	—
23	C₃×#125	—	2	2	no	—
24	C₃×#124	C₂²	1	2	yes	—
25	C₃×C₂²⋊Q₈	—	2	1	no	—
26	C₃×C₄×Q₈	—	2	1	no	—
27	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
28	C₃×C₄×Q₈	C₂²	1	2	yes	—
29	C₃×C₄×Q₈	C₂²	1	2	yes	—
30	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
31	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
32	C₃×#117	—	2	1	no	—
33	C₃×#106	C₂²	1	1	yes	characteristic
34	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₃×C₄⋊C₈	C₂²	1	2	yes	—
36	C₃×#103	C₂²	1	1	yes	characteristic
37	C₃×#103	C₂²	1	2	yes	—
38	C₃×#103	C₂²	1	2	yes	—
39	C₃×#102	C₂²	1	1	yes	characteristic
40	C₃×#97	C₂²	1	2	yes	—
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
42	#557	C₂×C₃	1	1	yes	characteristic, complemented
43	#556	C₂×C₃	1	1	yes	characteristic
44	#497	C₂×C₃	1	2	yes	—
45	#494	C₂×C₃	1	1	yes	characteristic, complemented
46	#464	C₂×C₃	1	2	yes	—
47	#462	C₂×C₃	1	2	yes	complemented
48	#462	C₂×C₃	1	1	yes	characteristic, complemented
49	#461	C₂×C₃	1	1	yes	characteristic
50	#446	C₂×C₃	1	1	yes	characteristic, complemented
51	#441	C₂×C₃	1	2	yes	complemented
52	C₄×SD₁₆	C₂×C₃	1	1	yes	characteristic
53	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
54	C₂×C₃×Q₈	C₂³	1	2	yes	—
55	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
56	C₂×C₃×Q₈	C₂³	1	2	yes	—
57	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄	—	2	1	no	—
59	C₃×Q₁₆	—	2	4	no	—
60	C₃×Q₁₆	—	4	2	no	—
61	C₃×Q₁₆	—	4	2	no	—
62	C₃×SD₁₆	—	2	2	no	—
63	C₃×SD₁₆	—	4	2	no	—
64	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂³	1	2	yes	—
66	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
67	C₃×C₄⋊C₄	—	2	1	no	—
68	C₃×C₄⋊C₄	—	2	2	no	—
69	C₃×C₄⋊C₄	—	2	1	no	—
70	C₃×C₄⋊C₄	—	2	2	no	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	—	2	1	no	—
73	C₃×C₄⋊C₄	—	2	2	no	—
74	C₃×C₄⋊C₄	—	2	1	no	—
75	C₃×C₄⋊C₄	—	2	1	no	—
76	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
77	C₃×C₄⋊C₄	D₄	1	2	yes	—
78	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₃×C₂²⋊C₄	—	2	2	no	—
81	C₃×C₂²⋊C₄	—	2	1	no	—
82	C₃×C₂²⋊C₄	—	2	2	no	—
83	C₃×C₄²	—	2	2	no	—
84	C₃×C₄²	—	2	2	no	—
85	C₃×C₄²	—	2	1	no	—
86	C₃×C₄²	C₂³	1	1	yes	characteristic
87	C₃×C₄²	C₂³	1	2	yes	—
88	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
89	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
90	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
91	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic
92	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
93	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
94	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
95	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
96	#126	—	2	2	no	—
97	#125	—	2	1	no	—
98	#125	—	2	2	no	—
99	#124	C₂²×C₃	1	2	yes	—
100	C₂²⋊Q₈	—	2	1	no	—
101	C₄×Q₈	—	2	1	no	—
102	C₄×Q₈	C₂²×C₃	1	2	yes	—
103	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
104	C₄×Q₈	C₂²×C₃	1	2	yes	—
105	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
106	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
107	#117	—	2	1	no	—
108	#106	C₂²×C₃	1	1	yes	characteristic
109	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
110	C₄⋊C₈	C₂²×C₃	1	2	yes	—
111	#103	C₂²×C₃	1	1	yes	characteristic
112	#103	C₂²×C₃	1	2	yes	—
113	#103	C₂²×C₃	1	2	yes	—
114	#102	C₂²×C₃	1	1	yes	characteristic
115	#97	C₂²×C₃	1	2	yes	—
116	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
117	C₂³×C₃	—	2	1	no	—
118	C₃×Q₈	—	2	2	no	—
119	C₃×Q₈	—	2	1	no	—
120	C₃×Q₈	—	2	2	no	—
121	C₃×Q₈	—	2	2	no	—
122	C₃×Q₈	—	2	2	no	—
123	C₃×Q₈	—	2	2	no	—
124	C₃×Q₈	—	2	1	no	—
125	C₃×D₄	—	2	1	no	—
126	C₃×D₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	4	2	no	—
133	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
134	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
135	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
138	C₃×C₈	—	2	2	no	—
139	C₃×C₈	—	2	1	no	—
140	C₃×C₈	#41	1	2	yes	—
141	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
142	C₂×Q₈	C₂³×C₃	1	2	yes	—
143	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
144	C₂×Q₈	C₂³×C₃	1	2	yes	—
145	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
146	C₂²×C₄	—	2	1	no	—
147	Q₁₆	—	2	4	no	—
148	Q₁₆	—	4	2	no	—
149	Q₁₆	—	4	2	no	—
150	SD₁₆	—	2	2	no	—
151	SD₁₆	—	4	2	no	—
152	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
153	C₂×C₈	C₂³×C₃	1	2	yes	—
154	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
155	C₄⋊C₄	—	2	1	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
165	C₄⋊C₄	C₃×D₄	1	2	yes	—
166	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
167	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	2	1	no	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₄²	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	1	no	—
174	C₄²	C₂³×C₃	1	1	yes	characteristic
175	C₄²	C₂³×C₃	1	2	yes	—
176	C₂²×C₃	—	4	1	no	—
177	C₂²×C₃	—	4	1	no	—
178	C₂²×C₃	—	4	1	no	—
179	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
180	C₃×C₄	—	2	2	no	—
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	4	2	no	—
184	C₃×C₄	—	4	1	no	—
185	C₃×C₄	—	4	2	no	—
186	C₃×C₄	—	4	1	no	—
187	C₃×C₄	—	4	1	no	—
188	C₃×C₄	#137	1	2	yes	—
189	C₃×C₄	C₂×#41	1	1	yes	characteristic
190	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
191	C₂³	—	2	1	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	2	1	no	—
194	Q₈	—	2	2	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	2	2	no	—
197	Q₈	—	2	2	no	—
198	Q₈	—	2	1	no	—
199	D₄	—	2	1	no	—
200	D₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
208	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
209	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
210	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
211	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	2	1	no	—
214	C₈	C₃×#41	1	2	yes	—
215	C₂×C₃	—	8	1	no	—
216	C₂×C₃	#550	1	1	yes	characteristic
217	C₂×C₃	C₂×#137	1	1	yes	characteristic
218	C₂×C₃	#578	1	1	yes	characteristic
219	C₂²	—	4	1	no	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	1	no	—
222	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	1	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	1	no	—
230	C₄	—	4	2	no	—
231	C₄	C₃×#137	1	2	yes	—
232	C₄	C₂×C₃×#41	1	1	yes	characteristic
233	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
234	C₃	#3398	1	1	yes	characteristic, direct product factor, Sylow
235	C₂	—	8	1	no	—
236	C₂	C₃×#550	1	1	yes	characteristic
237	C₂	C₂×C₃×#137	1	1	yes	characteristic
238	C₂	C₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂×C₃×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#557	C₂	semidirect
4. C₃×#494	C₂	semidirect
6. C₃×#462	C₂	semidirect
7. C₃×#462	C₂	semidirect
9. C₃×#446	C₂	semidirect
10. C₃×#441	C₂	semidirect
12. #3398	C₃	direct (N × H)
42. #557	C₂×C₃	semidirect
45. #494	C₂×C₃	semidirect
47. #462	C₂×C₃	semidirect
48. #462	C₂×C₃	semidirect
50. #446	C₂×C₃	semidirect
51. #441	C₂×C₃	semidirect
234. C₃	#3398	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×#557	2
C₃×#556	2
C₃×#497	2
C₃×#494	2
C₃×#464	2
C₃×#462	2
C₃×#461	2
C₃×#446	2
C₃×#441	2
C₃×C₄×SD₁₆	2
#3398	3

Subgroups — C3×#3398 #85262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3398 #85262