Subgroups — C₃×#2368 #84232

Order 384.

548 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#391	C₂	1	2	yes	maximal, complemented
3	C₃×#387	C₂	1	2	yes	maximal, complemented
4	C₃×#387	C₂	1	2	yes	maximal, complemented
5	C₃×#385	C₂	1	2	yes	maximal
6	C₃×C₄×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₃×C₄×C₂²⋊C₄	C₂	1	2	yes	maximal
8	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂×C₃×#95	C₂	1	1	yes	maximal, characteristic
10	#2368	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	C₂²	1	2	yes	—
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₃×C₂²⋊Q₈	C₄	1	8	yes	complemented
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	2	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄²	C₂²	1	2	yes	—
23	C₂×C₃×C₄²	C₂²	1	2	yes	—
24	C₂×C₃×C₄²	C₂²	1	2	yes	—
25	C₃×#95	C₂²	1	1	yes	characteristic
26	C₃×#95	C₂²	1	1	yes	characteristic
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	2	yes	—
31	C₃×#95	C₂²	1	2	yes	—
32	C₃×#95	C₂²	1	2	yes	—
33	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
34	#391	C₂×C₃	1	2	yes	complemented
35	#387	C₂×C₃	1	2	yes	complemented
36	#387	C₂×C₃	1	2	yes	complemented
37	#385	C₂×C₃	1	2	yes	—
38	C₄×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
39	C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	—
40	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
41	C₂×#95	C₂×C₃	1	1	yes	characteristic
42	C₂⁴×C₃	C₂³	1	1	yes	characteristic
43	C₂×C₃×Q₈	—	4	2	no	—
44	C₂×C₃×Q₈	C₂×C₄	1	4	yes	complemented
45	C₂²×C₃×C₄	—	2	4	no	—
46	C₂²×C₃×C₄	—	2	4	no	—
47	C₂²×C₃×C₄	—	4	2	no	—
48	C₂²×C₃×C₄	—	4	2	no	—
49	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
50	C₂²×C₃×C₄	C₂³	1	2	yes	—
51	C₂²×C₃×C₄	C₂³	1	2	yes	—
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂³	1	2	yes	—
55	C₂²×C₃×C₄	C₂³	1	2	yes	—
56	C₂²×C₃×C₄	C₂³	1	2	yes	—
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	C₂×C₄	1	4	yes	complemented
61	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	complemented
62	C₃×C₂²⋊C₄	—	2	2	no	—
63	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
64	C₃×C₂²⋊C₄	Q₈	1	4	yes	complemented
65	C₃×C₄²	—	2	4	no	—
66	C₃×C₄²	—	4	2	no	—
67	C₃×C₄²	—	4	2	no	—
68	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
69	C₂³×C₄	C₂²×C₃	1	2	yes	—
70	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
71	C₂²⋊Q₈	C₃×C₄	1	8	yes	complemented
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×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₄²	C₂²×C₃	1	2	yes	—
80	C₂×C₄²	C₂²×C₃	1	2	yes	—
81	C₂×C₄²	C₂²×C₃	1	2	yes	—
82	#95	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	2	yes	—
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	C₂²×C₄	1	2	yes	—
93	C₂³×C₃	C₂×Q₈	1	2	yes	—
94	C₂³×C₃	#41	1	2	yes	—
95	C₂³×C₃	C₂⁴	1	1	yes	characteristic
96	C₃×Q₈	—	4	4	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	—	2	2	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	—	4	4	no	—
114	C₂×C₃×C₄	—	4	8	no	—
115	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
116	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
117	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
118	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
119	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
120	C₂×C₃×C₄	#41	1	4	yes	—
121	C₂⁴	C₂³×C₃	1	1	yes	characteristic
122	C₂×Q₈	—	4	2	no	—
123	C₂×Q₈	C₂×C₃×C₄	1	4	yes	complemented
124	C₂²×C₄	—	2	4	no	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	4	2	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
129	C₂²×C₄	C₂³×C₃	1	2	yes	—
130	C₂²×C₄	C₂³×C₃	1	2	yes	—
131	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
132	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
133	C₂²×C₄	C₂³×C₃	1	2	yes	—
134	C₂²×C₄	C₂³×C₃	1	2	yes	—
135	C₂²×C₄	C₂³×C₃	1	2	yes	—
136	C₂²×C₄	C₂³×C₃	1	2	yes	—
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
140	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
141	C₂²⋊C₄	—	2	2	no	—
142	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
143	C₂²⋊C₄	C₃×Q₈	1	4	yes	complemented
144	C₄²	—	2	4	no	—
145	C₄²	—	4	2	no	—
146	C₄²	—	4	2	no	—
147	C₂²×C₃	—	2	2	no	—
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	—	2	4	no	—
150	C₂²×C₃	—	2	4	no	—
151	C₂²×C₃	C₄×Q₈	1	4	yes	—
152	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
153	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
154	C₂²×C₃	C₂×#41	1	1	yes	characteristic
155	C₂²×C₃	C₂×#41	1	2	yes	—
156	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
157	C₃×C₄	—	2	4	no	—
158	C₃×C₄	—	2	2	no	—
159	C₃×C₄	—	2	2	no	—
160	C₃×C₄	—	2	4	no	—
161	C₃×C₄	—	2	4	no	—
162	C₃×C₄	—	2	4	no	—
163	C₃×C₄	—	4	2	no	—
164	C₃×C₄	—	4	2	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	C₂²×C₃×C₄	1	2	yes	—
168	C₂³	C₂×C₃×Q₈	1	2	yes	—
169	C₂³	C₃×#41	1	2	yes	—
170	C₂³	C₂⁴×C₃	1	1	yes	characteristic
171	Q₈	—	4	4	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	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	—	4	4	no	—
189	C₂×C₄	—	4	8	no	—
190	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
191	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
192	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
193	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
194	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
195	C₂×C₄	C₃×#41	1	4	yes	—
196	C₂×C₃	—	2	4	no	—
197	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
198	C₂×C₃	C₄×#41	1	1	yes	characteristic
199	C₂×C₃	#518	1	1	yes	characteristic
200	C₂×C₃	#551	1	2	yes	—
201	C₂×C₃	#554	1	2	yes	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	4	no	—
206	C₂²	C₃×C₄×Q₈	1	4	yes	—
207	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
208	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
209	C₂²	C₂×C₃×#41	1	1	yes	characteristic
210	C₂²	C₂×C₃×#41	1	2	yes	—
211	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
212	C₄	—	2	4	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	4	no	—
216	C₄	—	2	4	no	—
217	C₄	—	2	4	no	—
218	C₄	—	4	2	no	—
219	C₄	—	4	2	no	—
220	C₃	#2368	1	1	yes	characteristic, direct product factor, Sylow
221	C₂	—	2	4	no	—
222	C₂	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
223	C₂	C₃×C₄×#41	1	1	yes	characteristic
224	C₂	C₃×#518	1	1	yes	characteristic
225	C₂	C₃×#551	1	2	yes	—
226	C₂	C₃×#554	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃×#391	C₂	semidirect
3. C₃×#387	C₂	semidirect
4. C₃×#387	C₂	semidirect
6. C₃×C₄×C₄⋊C₄	C₂	semidirect
10. #2368	C₃	direct (N × H)
14. C₃×C₂²⋊Q₈	C₄	semidirect
34. #391	C₂×C₃	semidirect
35. #387	C₂×C₃	semidirect
36. #387	C₂×C₃	semidirect
38. C₄×C₄⋊C₄	C₂×C₃	semidirect
44. C₂×C₃×Q₈	C₂×C₄	semidirect
60. C₃×C₄⋊C₄	C₂×C₄	semidirect
61. C₃×C₄⋊C₄	C₂×C₄	semidirect
64. C₃×C₂²⋊C₄	Q₈	semidirect
71. C₂²⋊Q₈	C₃×C₄	semidirect
123. C₂×Q₈	C₂×C₃×C₄	semidirect
139. C₄⋊C₄	C₂×C₃×C₄	semidirect
140. C₄⋊C₄	C₂×C₃×C₄	semidirect
143. C₂²⋊C₄	C₃×Q₈	semidirect
220. C₃	#2368	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₂²⋊Q₈	2
C₃×#391	2
C₃×#387	2
C₃×#385	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
C₂×C₃×#95	2
#2368	3

Subgroups — C3×#2368 #84232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2368 #84232