Subgroups — C₃×#2784 #84648

Order 384.

472 conjugacy classes of subgroups.

The subgroup lattice has 238 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	1	yes	maximal, characteristic, complemented
2	C₃×#396	C₂	1	2	yes	maximal, complemented
3	C₃×#395	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#393	C₂	1	2	yes	maximal, complemented
5	C₃×#390	C₂	1	1	yes	maximal, characteristic
6	C₃×#387	C₂	1	2	yes	maximal, complemented
7	C₃×#386	C₂	1	2	yes	maximal
8	C₃×#385	C₂	1	2	yes	maximal, complemented
9	C₃×#380	C₂	1	2	yes	maximal, complemented
10	#2784	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
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×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₃×#95	C₂²	1	1	yes	characteristic
26	C₃×#95	C₂²	1	2	yes	—
27	C₃×#95	C₂²	1	1	yes	characteristic
28	C₃×#95	C₂²	1	2	yes	—
29	C₃×#95	C₂²	1	2	yes	—
30	C₃×#95	C₂²	1	2	yes	—
31	C₃×#95	C₂²	1	2	yes	—
32	#400	C₂×C₃	1	1	yes	characteristic, complemented
33	#396	C₂×C₃	1	2	yes	complemented
34	#395	C₂×C₃	1	1	yes	characteristic, complemented
35	#393	C₂×C₃	1	2	yes	complemented
36	#390	C₂×C₃	1	1	yes	characteristic
37	#387	C₂×C₃	1	2	yes	complemented
38	#386	C₂×C₃	1	2	yes	—
39	#385	C₂×C₃	1	2	yes	complemented
40	#380	C₂×C₃	1	2	yes	complemented
41	C₂⁴×C₃	C₂³	1	2	yes	—
42	C₂×C₃×Q₈	—	4	2	no	—
43	C₂×C₃×Q₈	—	4	1	no	—
44	C₂×C₃×D₄	—	4	1	no	—
45	C₂×C₃×D₄	—	4	1	no	—
46	C₂×C₃×D₄	Q₈	1	4	yes	complemented
47	C₂²×C₃×C₄	—	2	4	no	—
48	C₂²×C₃×C₄	—	4	2	no	—
49	C₂²×C₃×C₄	—	4	2	no	—
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	1	yes	characteristic
54	C₂²×C₃×C₄	C₂³	1	2	yes	—
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂³	1	2	yes	—
58	C₃×C₄⋊C₄	—	2	4	no	—
59	C₃×C₄⋊C₄	—	4	2	no	—
60	C₃×C₄⋊C₄	—	4	2	no	—
61	C₃×C₄⋊C₄	—	4	1	no	—
62	C₃×C₄⋊C₄	—	4	1	no	—
63	C₃×C₂²⋊C₄	—	2	4	no	—
64	C₃×C₂²⋊C₄	—	4	2	no	—
65	C₃×C₂²⋊C₄	—	4	2	no	—
66	C₃×C₂²⋊C₄	—	4	2	no	—
67	C₃×C₂²⋊C₄	—	4	2	no	—
68	C₃×C₄²	—	4	1	no	—
69	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
70	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
71	C₂³×C₄	C₂²×C₃	1	2	yes	—
72	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
73	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
74	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
75	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
81	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
82	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
83	#95	C₂²×C₃	1	1	yes	characteristic
84	#95	C₂²×C₃	1	2	yes	—
85	#95	C₂²×C₃	1	1	yes	characteristic
86	#95	C₂²×C₃	1	2	yes	—
87	#95	C₂²×C₃	1	2	yes	—
88	#95	C₂²×C₃	1	2	yes	—
89	#95	C₂²×C₃	1	2	yes	—
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	—	2	2	no	—
96	C₂³×C₃	C₂×Q₈	1	4	yes	—
97	C₂³×C₃	C₂⁴	1	1	yes	characteristic
98	C₃×Q₈	—	4	4	no	—
99	C₃×D₄	—	4	4	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	1	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	2	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	4	8	no	—
121	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
122	C₂×C₃×C₄	#41	1	4	yes	—
123	C₂⁴	C₂³×C₃	1	2	yes	—
124	C₂×Q₈	—	4	2	no	—
125	C₂×Q₈	—	4	1	no	—
126	C₂×D₄	—	4	1	no	—
127	C₂×D₄	—	4	1	no	—
128	C₂×D₄	C₃×Q₈	1	4	yes	complemented
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	C₂³×C₃	1	2	yes	—
133	C₂²×C₄	C₂³×C₃	1	2	yes	—
134	C₂²×C₄	C₂³×C₃	1	2	yes	—
135	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₃	1	2	yes	—
137	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
138	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
139	C₂²×C₄	C₂³×C₃	1	2	yes	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	1	no	—
144	C₄⋊C₄	—	4	1	no	—
145	C₂²⋊C₄	—	2	4	no	—
146	C₂²⋊C₄	—	4	2	no	—
147	C₂²⋊C₄	—	4	2	no	—
148	C₂²⋊C₄	—	4	2	no	—
149	C₂²⋊C₄	—	4	2	no	—
150	C₄²	—	4	1	no	—
151	C₂²×C₃	—	2	4	no	—
152	C₂²×C₃	—	2	4	no	—
153	C₂²×C₃	—	2	4	no	—
154	C₂²×C₃	—	4	2	no	—
155	C₂²×C₃	—	4	2	no	—
156	C₂²×C₃	—	4	2	no	—
157	C₂²×C₃	—	4	2	no	—
158	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
159	C₂²×C₃	C₂×#41	1	2	yes	—
160	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
162	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
163	C₃×C₄	—	2	4	no	—
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	—	4	2	no	—
166	C₃×C₄	—	4	2	no	—
167	C₃×C₄	—	4	2	no	—
168	C₃×C₄	—	4	1	no	—
169	C₃×C₄	—	4	1	no	—
170	C₃×C₄	—	4	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	C₂×C₃×Q₈	1	4	yes	—
178	C₂³	C₂⁴×C₃	1	1	yes	characteristic
179	Q₈	—	4	4	no	—
180	D₄	—	4	4	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	1	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	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	4	8	no	—
202	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
203	C₂×C₄	C₃×#41	1	4	yes	—
204	C₂×C₃	—	4	4	no	—
205	C₂×C₃	#538	1	2	yes	—
206	C₂×C₃	#543	1	1	yes	characteristic
207	C₂×C₃	#554	1	2	yes	—
208	C₂×C₃	#555	1	1	yes	characteristic
209	C₂×C₃	#560	1	1	yes	characteristic
210	C₂²	—	2	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
218	C₂²	C₂×C₃×#41	1	2	yes	—
219	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
220	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
221	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
222	C₄	—	2	4	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	1	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₃	#2784	1	1	yes	characteristic, direct product factor, Sylow
231	C₂	—	4	4	no	—
232	C₂	C₃×#538	1	2	yes	—
233	C₂	C₃×#543	1	1	yes	characteristic
234	C₂	C₃×#554	1	2	yes	—
235	C₂	C₃×#555	1	1	yes	characteristic
236	C₂	C₃×#560	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×#396	C₂	semidirect
3. C₃×#395	C₂	semidirect
4. C₃×#393	C₂	semidirect
6. C₃×#387	C₂	semidirect
8. C₃×#385	C₂	semidirect
9. C₃×#380	C₂	semidirect
10. #2784	C₃	direct (N × H)
32. #400	C₂×C₃	semidirect
33. #396	C₂×C₃	semidirect
34. #395	C₂×C₃	semidirect
35. #393	C₂×C₃	semidirect
37. #387	C₂×C₃	semidirect
39. #385	C₂×C₃	semidirect
40. #380	C₂×C₃	semidirect
46. C₂×C₃×D₄	Q₈	semidirect
128. C₂×D₄	C₃×Q₈	semidirect
230. C₃	#2784	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×#396	2
C₃×#395	2
C₃×#393	2
C₃×#390	2
C₃×#387	2
C₃×#386	2
C₃×#385	2
C₃×#380	2
#2784	3

Subgroups — C3×#2784 #84648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2784 #84648