Subgroups — C₃×#2332 #84196

Order 384.

484 conjugacy classes of subgroups.

The subgroup lattice has 232 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×#388	C₂	1	2	yes	maximal
4	C₃×#388	C₂	1	2	yes	maximal
5	C₃×#387	C₂	1	2	yes	maximal, complemented
6	C₃×#387	C₂	1	2	yes	maximal, complemented
7	C₃×#381	C₂	1	1	yes	maximal, characteristic
8	C₃×C₄×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
10	#2332	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	1	yes	characteristic
13	C₃×C₂²⋊Q₈	C₄	1	8	yes	complemented
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×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₄²	C₂²	1	2	yes	—
23	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₄²	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	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	#388	C₂×C₃	1	2	yes	—
36	#388	C₂×C₃	1	2	yes	—
37	#387	C₂×C₃	1	2	yes	complemented
38	#387	C₂×C₃	1	2	yes	complemented
39	#381	C₂×C₃	1	1	yes	characteristic
40	C₄×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
41	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
42	C₂⁴×C₃	C₂³	1	1	yes	characteristic
43	C₂×C₃×Q₈	—	4	1	no	—
44	C₂×C₃×Q₈	—	4	1	no	—
45	C₂×C₃×Q₈	C₂×C₄	1	4	yes	—
46	C₂²×C₃×C₄	—	4	1	no	—
47	C₂²×C₃×C₄	—	4	1	no	—
48	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
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	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	C₃×C₄⋊C₄	—	2	4	no	—
58	C₃×C₄⋊C₄	—	4	2	no	—
59	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
60	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
61	C₃×C₂²⋊C₄	—	2	4	no	—
62	C₃×C₂²⋊C₄	—	4	1	no	—
63	C₃×C₂²⋊C₄	—	4	1	no	—
64	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
65	C₃×C₄²	—	2	4	no	—
66	C₃×C₄²	—	4	1	no	—
67	C₃×C₄²	—	4	2	no	—
68	C₃×C₄²	—	4	1	no	—
69	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
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	2	yes	—
77	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₄²	C₂²×C₃	1	2	yes	—
81	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₄²	C₂²×C₃	1	2	yes	—
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	#95	C₂²×C₃	1	2	yes	—
91	C₂³×C₃	—	2	2	no	—
92	C₂³×C₃	—	2	1	no	—
93	C₂³×C₃	—	2	1	no	—
94	C₂³×C₃	—	2	2	no	—
95	C₂³×C₃	C₂²×C₄	1	2	yes	—
96	C₂³×C₃	C₂⁴	1	1	yes	characteristic
97	C₃×Q₈	—	4	4	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	1	no	—
102	C₂×C₃×C₄	—	2	1	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	1	no	—
106	C₂×C₃×C₄	—	2	1	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	2	no	—
117	C₂×C₃×C₄	—	4	4	no	—
118	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
119	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
120	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
121	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
122	C₂×C₃×C₄	#41	1	4	yes	—
123	C₂⁴	C₂³×C₃	1	1	yes	characteristic
124	C₂×Q₈	—	4	1	no	—
125	C₂×Q₈	—	4	1	no	—
126	C₂×Q₈	C₂×C₃×C₄	1	4	yes	—
127	C₂²×C₄	—	4	1	no	—
128	C₂²×C₄	—	4	1	no	—
129	C₂²×C₄	C₂×C₃×C₄	1	4	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	1	yes	characteristic
134	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₃	1	2	yes	—
136	C₂²×C₄	C₂³×C₃	1	2	yes	—
137	C₂²×C₄	C₂³×C₃	1	2	yes	—
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
141	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
142	C₂²⋊C₄	—	2	4	no	—
143	C₂²⋊C₄	—	4	1	no	—
144	C₂²⋊C₄	—	4	1	no	—
145	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
146	C₄²	—	2	4	no	—
147	C₄²	—	4	1	no	—
148	C₄²	—	4	2	no	—
149	C₄²	—	4	1	no	—
150	C₂²×C₃	—	2	2	no	—
151	C₂²×C₃	—	2	2	no	—
152	C₂²×C₃	—	2	2	no	—
153	C₂²×C₃	—	4	2	no	—
154	C₂²×C₃	—	4	2	no	—
155	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
156	C₂²×C₃	C₂×#41	1	2	yes	—
157	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
160	C₃×C₄	—	2	4	no	—
161	C₃×C₄	—	2	2	no	—
162	C₃×C₄	—	2	2	no	—
163	C₃×C₄	—	2	4	no	—
164	C₃×C₄	—	2	4	no	—
165	C₃×C₄	—	4	2	no	—
166	C₃×C₄	—	4	2	no	—
167	C₃×C₄	—	4	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	1	no	—
170	C₂³	—	2	1	no	—
171	C₂³	—	2	2	no	—
172	C₂³	C₂²×C₃×C₄	1	2	yes	—
173	C₂³	C₂⁴×C₃	1	1	yes	characteristic
174	Q₈	—	4	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	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	1	no	—
183	C₂×C₄	—	2	1	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	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
196	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
197	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
198	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
199	C₂×C₄	C₃×#41	1	4	yes	—
200	C₂×C₃	—	4	2	no	—
201	C₂×C₃	C₄×#41	1	1	yes	characteristic
202	C₂×C₃	#518	1	1	yes	characteristic
203	C₂×C₃	#519	1	1	yes	characteristic
204	C₂×C₃	#539	1	2	yes	—
205	C₂×C₃	#542	1	2	yes	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
212	C₂²	C₂×C₃×#41	1	2	yes	—
213	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
214	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
215	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
216	C₄	—	2	4	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	4	no	—
220	C₄	—	2	4	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	2	no	—
224	C₃	#2332	1	1	yes	characteristic, direct product factor, Sylow
225	C₂	—	4	2	no	—
226	C₂	C₃×C₄×#41	1	1	yes	characteristic
227	C₂	C₃×#518	1	1	yes	characteristic
228	C₂	C₃×#519	1	1	yes	characteristic
229	C₂	C₃×#539	1	2	yes	—
230	C₂	C₃×#542	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
5. C₃×#387	C₂	semidirect
6. C₃×#387	C₂	semidirect
8. C₃×C₄×C₄⋊C₄	C₂	semidirect
10. #2332	C₃	direct (N × H)
13. C₃×C₂²⋊Q₈	C₄	semidirect
34. #391	C₂×C₃	semidirect
37. #387	C₂×C₃	semidirect
38. #387	C₂×C₃	semidirect
40. C₄×C₄⋊C₄	C₂×C₃	semidirect
71. C₂²⋊Q₈	C₃×C₄	semidirect
224. C₃	#2332	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×#388	2
C₃×#387	2
C₃×#381	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
#2332	3

Subgroups — C3×#2332 #84196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2332 #84196