Subgroups — C₃×#3400 #85264

Order 384.

362 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#559	C₂	1	1	yes	maximal, characteristic
2	C₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#497	C₂	1	2	yes	maximal, complemented
4	C₃×#492	C₂	1	1	yes	maximal, characteristic
5	C₃×#464	C₂	1	2	yes	maximal, complemented
6	C₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#461	C₂	1	2	yes	maximal, complemented
8	C₃×#460	C₂	1	1	yes	maximal, characteristic
9	C₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#440	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#3400	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
15	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
16	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
17	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
18	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
19	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₃×#127	C₂²	1	1	yes	characteristic
21	C₃×#125	—	2	1	no	—
22	C₃×#125	—	2	2	no	—
23	C₃×#124	—	2	1	no	—
24	C₃×#124	C₂²	1	2	yes	—
25	C₃×#123	—	2	2	no	—
26	C₃×C₄×Q₈	—	2	1	no	—
27	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
28	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
29	C₃×C₄×Q₈	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	2	yes	—
33	C₃×#107	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	1	yes	characteristic
38	C₃×#103	C₂²	1	2	yes	—
39	C₃×#102	C₂²	1	2	yes	—
40	C₃×#97	C₂²	1	2	yes	—
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
42	#559	C₂×C₃	1	1	yes	characteristic
43	#557	C₂×C₃	1	1	yes	characteristic, complemented
44	#497	C₂×C₃	1	2	yes	complemented
45	#492	C₂×C₃	1	1	yes	characteristic
46	#464	C₂×C₃	1	2	yes	complemented
47	#462	C₂×C₃	1	1	yes	characteristic, complemented
48	#461	C₂×C₃	1	2	yes	complemented
49	#460	C₂×C₃	1	1	yes	characteristic
50	#446	C₂×C₃	1	1	yes	characteristic, complemented
51	#440	C₂×C₃	1	2	yes	complemented
52	C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
53	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
54	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
55	C₂×C₃×Q₈	C₂³	1	2	yes	—
56	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
57	C₂×C₃×D₄	—	2	1	no	—
58	C₂×C₃×D₄	—	2	1	no	—
59	C₂×C₃×D₄	C₂³	1	2	yes	—
60	C₂²×C₃×C₄	—	2	2	no	—
61	C₃×Q₁₆	—	2	2	no	—
62	C₃×Q₁₆	—	4	2	no	—
63	C₃×SD₁₆	—	2	4	no	—
64	C₃×SD₁₆	—	4	2	no	—
65	C₃×SD₁₆	—	4	2	no	—
66	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂³	1	2	yes	—
68	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
69	C₃×C₄⋊C₄	—	2	2	no	—
70	C₃×C₄⋊C₄	—	2	1	no	—
71	C₃×C₄⋊C₄	—	2	1	no	—
72	C₃×C₄⋊C₄	—	2	2	no	—
73	C₃×C₄⋊C₄	—	2	1	no	—
74	C₃×C₄⋊C₄	—	2	2	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	2	no	—
82	C₃×C₂²⋊C₄	—	2	2	no	—
83	C₃×C₄²	—	2	1	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	1	yes	characteristic
89	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
90	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
91	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
92	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
93	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
94	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
95	#127	C₂²×C₃	1	1	yes	characteristic
96	#125	—	2	1	no	—
97	#125	—	2	2	no	—
98	#124	—	2	1	no	—
99	#124	C₂²×C₃	1	2	yes	—
100	#123	—	2	2	no	—
101	C₄×Q₈	—	2	1	no	—
102	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
103	C₄×Q₈	C₂²×C₃	1	2	yes	—
104	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
105	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
106	C₄×D₄	—	2	1	no	—
107	C₄×D₄	C₂²×C₃	1	2	yes	—
108	#107	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	1	yes	characteristic
113	#103	C₂²×C₃	1	2	yes	—
114	#102	C₂²×C₃	1	2	yes	—
115	#97	C₂²×C₃	1	2	yes	—
116	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
117	C₂³×C₃	—	2	2	no	—
118	C₃×Q₈	—	2	2	no	—
119	C₃×Q₈	—	2	2	no	—
120	C₃×Q₈	—	2	1	no	—
121	C₃×Q₈	—	2	1	no	—
122	C₃×Q₈	—	2	1	no	—
123	C₃×Q₈	—	2	2	no	—
124	C₃×Q₈	—	2	1	no	—
125	C₃×D₄	—	2	2	no	—
126	C₃×D₄	—	2	2	no	—
127	C₃×D₄	—	4	2	no	—
128	C₃×D₄	—	8	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	8	2	no	—
135	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
136	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
137	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
140	C₃×C₈	—	2	1	no	—
141	C₃×C₈	—	2	2	no	—
142	C₃×C₈	#41	1	2	yes	—
143	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
144	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
145	C₂×Q₈	C₂³×C₃	1	2	yes	—
146	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
147	C₂×D₄	—	2	1	no	—
148	C₂×D₄	—	2	1	no	—
149	C₂×D₄	C₂³×C₃	1	2	yes	—
150	C₂²×C₄	—	2	2	no	—
151	Q₁₆	—	2	2	no	—
152	Q₁₆	—	4	2	no	—
153	SD₁₆	—	2	4	no	—
154	SD₁₆	—	4	2	no	—
155	SD₁₆	—	4	2	no	—
156	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
157	C₂×C₈	C₂³×C₃	1	2	yes	—
158	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
167	C₄⋊C₄	C₃×D₄	1	2	yes	—
168	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
169	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
170	C₂²⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	1	no	—
176	C₄²	C₂³×C₃	1	1	yes	characteristic
177	C₄²	C₂³×C₃	1	2	yes	—
178	C₂²×C₃	—	4	2	no	—
179	C₂²×C₃	—	4	2	no	—
180	C₂²×C₃	—	4	2	no	—
181	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	4	1	no	—
186	C₃×C₄	—	4	2	no	—
187	C₃×C₄	—	4	1	no	—
188	C₃×C₄	—	4	1	no	—
189	C₃×C₄	—	4	1	no	—
190	C₃×C₄	#137	1	2	yes	—
191	C₃×C₄	C₂×#41	1	1	yes	characteristic
192	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
193	C₂³	—	2	2	no	—
194	Q₈	—	2	2	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	2	1	no	—
197	Q₈	—	2	1	no	—
198	Q₈	—	2	1	no	—
199	Q₈	—	2	2	no	—
200	Q₈	—	2	1	no	—
201	D₄	—	2	2	no	—
202	D₄	—	2	2	no	—
203	D₄	—	4	2	no	—
204	D₄	—	8	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
212	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
213	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
214	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
215	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	2	2	no	—
218	C₈	C₃×#41	1	2	yes	—
219	C₂×C₃	—	8	2	no	—
220	C₂×C₃	#550	1	1	yes	characteristic
221	C₂×C₃	C₂×#137	1	1	yes	characteristic
222	C₂×C₃	#577	1	1	yes	characteristic
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	1	no	—
235	C₄	C₃×#137	1	2	yes	—
236	C₄	C₂×C₃×#41	1	1	yes	characteristic
237	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
238	C₃	#3400	1	1	yes	characteristic, direct product factor, Sylow
239	C₂	—	8	2	no	—
240	C₂	C₃×#550	1	1	yes	characteristic
241	C₂	C₂×C₃×#137	1	1	yes	characteristic
242	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×#127	C₂²
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
2. C₃×#557	C₂	semidirect
3. C₃×#497	C₂	semidirect
5. C₃×#464	C₂	semidirect
6. C₃×#462	C₂	semidirect
7. C₃×#461	C₂	semidirect
9. C₃×#446	C₂	semidirect
10. C₃×#440	C₂	semidirect
11. C₃×C₄×Q₁₆	C₂	semidirect
12. #3400	C₃	direct (N × H)
43. #557	C₂×C₃	semidirect
44. #497	C₂×C₃	semidirect
46. #464	C₂×C₃	semidirect
47. #462	C₂×C₃	semidirect
48. #461	C₂×C₃	semidirect
50. #446	C₂×C₃	semidirect
51. #440	C₂×C₃	semidirect
52. C₄×Q₁₆	C₂×C₃	semidirect
238. C₃	#3400	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×#559	2
C₃×#557	2
C₃×#497	2
C₃×#492	2
C₃×#464	2
C₃×#462	2
C₃×#461	2
C₃×#460	2
C₃×#446	2
C₃×#440	2
C₃×C₄×Q₁₆	2
#3400	3

Subgroups — C3×#3400 #85264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3400 #85264