Subgroups — C₃×#2832 #84696

Order 384.

416 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#400	C₂	1	2	yes	maximal, complemented
2	C₃×#397	C₂	1	2	yes	maximal, complemented
3	C₃×#395	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#394	C₂	1	2	yes	maximal
5	C₃×#394	C₂	1	1	yes	maximal, characteristic
6	C₃×#393	C₂	1	2	yes	maximal, complemented
7	C₃×#393	C₂	1	2	yes	maximal, complemented
8	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#388	C₂	1	2	yes	maximal, complemented
10	#2832	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
24	C₃×#95	C₂²	1	2	yes	—
25	C₃×#95	C₂²	1	2	yes	—
26	C₃×#95	C₂²	1	2	yes	—
27	C₃×#95	C₂²	1	2	yes	—
28	C₃×#95	C₂²	1	2	yes	—
29	C₃×#95	C₂²	1	2	yes	—
30	C₃×#95	C₂²	1	1	yes	characteristic
31	C₃×#95	C₂²	1	1	yes	characteristic
32	#400	C₂×C₃	1	2	yes	complemented
33	#397	C₂×C₃	1	2	yes	complemented
34	#395	C₂×C₃	1	1	yes	characteristic, complemented
35	#394	C₂×C₃	1	2	yes	—
36	#394	C₂×C₃	1	1	yes	characteristic
37	#393	C₂×C₃	1	2	yes	complemented
38	#393	C₂×C₃	1	2	yes	complemented
39	#391	C₂×C₃	1	1	yes	characteristic, complemented
40	#388	C₂×C₃	1	2	yes	complemented
41	C₂⁴×C₃	C₂³	1	2	yes	—
42	C₂×C₃×Q₈	—	2	2	no	—
43	C₂×C₃×Q₈	—	4	1	no	—
44	C₂×C₃×Q₈	—	4	1	no	—
45	C₂×C₃×Q₈	—	4	1	no	—
46	C₂×C₃×Q₈	—	4	2	no	—
47	C₂×C₃×D₄	—	4	1	no	—
48	C₂×C₃×D₄	—	4	2	no	—
49	C₂²×C₃×C₄	C₂³	1	2	yes	—
50	C₂²×C₃×C₄	C₂³	1	2	yes	—
51	C₂²×C₃×C₄	C₂³	1	2	yes	—
52	C₂²×C₃×C₄	C₂³	1	2	yes	—
53	C₂²×C₃×C₄	C₂³	1	2	yes	—
54	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₃×C₄⋊C₄	—	4	1	no	—
58	C₃×C₄⋊C₄	—	4	2	no	—
59	C₃×C₄⋊C₄	—	4	2	no	—
60	C₃×C₂²⋊C₄	—	2	4	no	—
61	C₃×C₂²⋊C₄	—	4	2	no	—
62	C₃×C₂²⋊C₄	—	4	2	no	—
63	C₃×C₂²⋊C₄	—	4	2	no	—
64	C₃×C₂²⋊C₄	—	4	2	no	—
65	C₃×C₂²⋊C₄	—	4	2	no	—
66	C₃×C₄²	—	4	1	no	—
67	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
68	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
69	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
73	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
74	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
75	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
76	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
77	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
80	#95	C₂²×C₃	1	2	yes	—
81	#95	C₂²×C₃	1	2	yes	—
82	#95	C₂²×C₃	1	2	yes	—
83	#95	C₂²×C₃	1	2	yes	—
84	#95	C₂²×C₃	1	2	yes	—
85	#95	C₂²×C₃	1	2	yes	—
86	#95	C₂²×C₃	1	1	yes	characteristic
87	#95	C₂²×C₃	1	1	yes	characteristic
88	C₂³×C₃	—	2	2	no	—
89	C₂³×C₃	—	2	2	no	—
90	C₂³×C₃	—	2	2	no	—
91	C₂³×C₃	—	2	2	no	—
92	C₂³×C₃	—	2	2	no	—
93	C₂³×C₃	—	2	2	no	—
94	C₂³×C₃	—	2	2	no	—
95	C₂³×C₃	C₂⁴	1	1	yes	characteristic
96	C₃×Q₈	—	4	4	no	—
97	C₃×Q₈	—	4	4	no	—
98	C₃×D₄	—	4	4	no	—
99	C₂×C₃×C₄	—	2	1	no	—
100	C₂×C₃×C₄	—	2	1	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	#41	1	2	yes	—
121	C₂⁴	C₂³×C₃	1	2	yes	—
122	C₂×Q₈	—	2	2	no	—
123	C₂×Q₈	—	4	1	no	—
124	C₂×Q₈	—	4	1	no	—
125	C₂×Q₈	—	4	2	no	—
126	C₂×Q₈	—	4	1	no	—
127	C₂×D₄	—	4	2	no	—
128	C₂×D₄	—	4	1	no	—
129	C₂²×C₄	C₂³×C₃	1	2	yes	—
130	C₂²×C₄	C₂³×C₃	1	2	yes	—
131	C₂²×C₄	C₂³×C₃	1	2	yes	—
132	C₂²×C₄	C₂³×C₃	1	2	yes	—
133	C₂²×C₄	C₂³×C₃	1	2	yes	—
134	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₄⋊C₄	—	4	1	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₂²⋊C₄	—	2	4	no	—
141	C₂²⋊C₄	—	4	2	no	—
142	C₂²⋊C₄	—	4	2	no	—
143	C₂²⋊C₄	—	4	2	no	—
144	C₂²⋊C₄	—	4	2	no	—
145	C₂²⋊C₄	—	4	2	no	—
146	C₄²	—	4	1	no	—
147	C₂²×C₃	—	4	2	no	—
148	C₂²×C₃	—	4	2	no	—
149	C₂²×C₃	—	4	2	no	—
150	C₂²×C₃	—	4	2	no	—
151	C₂²×C₃	—	4	2	no	—
152	C₂²×C₃	—	4	2	no	—
153	C₂²×C₃	—	4	2	no	—
154	C₂²×C₃	C₂×#41	1	1	yes	characteristic
155	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
158	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
159	C₃×C₄	—	2	2	no	—
160	C₃×C₄	—	4	2	no	—
161	C₃×C₄	—	4	2	no	—
162	C₃×C₄	—	4	1	no	—
163	C₃×C₄	—	4	2	no	—
164	C₃×C₄	—	4	1	no	—
165	C₃×C₄	—	4	2	no	—
166	C₃×C₄	—	4	2	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	C₂⁴×C₃	1	1	yes	characteristic
175	Q₈	—	4	4	no	—
176	Q₈	—	4	4	no	—
177	D₄	—	4	4	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	C₃×#41	1	2	yes	—
200	C₂×C₃	—	8	2	no	—
201	C₂×C₃	#538	1	1	yes	characteristic
202	C₂×C₃	#542	1	2	yes	—
203	C₂×C₃	H(3,𝔽₄)	1	1	yes	characteristic
204	C₂×C₃	#562	1	2	yes	—
205	C₂×C₃	#563	1	1	yes	characteristic
206	C₂²	—	4	2	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	2	no	—
213	C₂²	C₂×C₃×#41	1	1	yes	characteristic
214	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
215	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
216	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
217	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
218	C₄	—	2	2	no	—
219	C₄	—	4	1	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₃	#2832	1	1	yes	characteristic, direct product factor, Sylow
227	C₂	—	8	2	no	—
228	C₂	C₃×#538	1	1	yes	characteristic
229	C₂	C₃×#542	1	2	yes	—
230	C₂	C₃×H(3,𝔽₄)	1	1	yes	characteristic
231	C₂	C₃×#562	1	2	yes	—
232	C₂	C₃×#563	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃×#400	C₂	semidirect
2. C₃×#397	C₂	semidirect
3. C₃×#395	C₂	semidirect
6. C₃×#393	C₂	semidirect
7. C₃×#393	C₂	semidirect
8. C₃×#391	C₂	semidirect
9. C₃×#388	C₂	semidirect
10. #2832	C₃	direct (N × H)
32. #400	C₂×C₃	semidirect
33. #397	C₂×C₃	semidirect
34. #395	C₂×C₃	semidirect
37. #393	C₂×C₃	semidirect
38. #393	C₂×C₃	semidirect
39. #391	C₂×C₃	semidirect
40. #388	C₂×C₃	semidirect
226. C₃	#2832	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×#400	2
C₃×#397	2
C₃×#395	2
C₃×#394	2
C₃×#393	2
C₃×#391	2
C₃×#388	2
#2832	3

Subgroups — C3×#2832 #84696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2832 #84696