Subgroups — C₃×#2732 #84596

Order 384.

528 conjugacy classes of subgroups.

The subgroup lattice has 260 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	2	yes	maximal, complemented
2	C₃×#399	C₂	1	2	yes	maximal, complemented
3	C₃×#398	C₂	1	2	yes	maximal, complemented
4	C₃×#396	C₂	1	2	yes	maximal, complemented
5	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#390	C₂	1	1	yes	maximal, characteristic
7	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#386	C₂	1	2	yes	maximal
9	C₃×#385	C₂	1	2	yes	maximal, complemented
10	#2732	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	C₃×C₂²⋊Q₈	—	2	8	no	—
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	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
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	2	yes	complemented
34	#399	C₂×C₃	1	2	yes	complemented
35	#398	C₂×C₃	1	2	yes	complemented
36	#396	C₂×C₃	1	2	yes	complemented
37	#391	C₂×C₃	1	1	yes	characteristic, complemented
38	#390	C₂×C₃	1	1	yes	characteristic
39	#389	C₂×C₃	1	1	yes	characteristic, complemented
40	#386	C₂×C₃	1	2	yes	—
41	#385	C₂×C₃	1	2	yes	complemented
42	C₂⁴×C₃	C₂³	1	2	yes	—
43	C₂×C₃×Q₈	—	4	1	no	—
44	C₂×C₃×Q₈	—	4	1	no	—
45	C₂×C₃×Q₈	D₄	1	4	yes	complemented
46	C₂×C₃×D₄	—	4	1	no	—
47	C₂×C₃×D₄	—	4	1	no	—
48	C₂×C₃×D₄	Q₈	1	4	yes	complemented
49	C₂²×C₃×C₄	—	2	4	no	—
50	C₂²×C₃×C₄	—	4	2	no	—
51	C₂²×C₃×C₄	—	4	2	no	—
52	C₂²×C₃×C₄	C₂³	1	2	yes	—
53	C₂²×C₃×C₄	C₂³	1	2	yes	—
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	1	yes	characteristic
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₃×C₄⋊C₄	—	2	4	no	—
61	C₃×C₄⋊C₄	—	2	4	no	—
62	C₃×C₄⋊C₄	—	2	4	no	—
63	C₃×C₄⋊C₄	—	2	2	no	—
64	C₃×C₄⋊C₄	—	4	1	no	—
65	C₃×C₄⋊C₄	—	4	1	no	—
66	C₃×C₄⋊C₄	—	4	1	no	—
67	C₃×C₄⋊C₄	—	4	2	no	—
68	C₃×C₂²⋊C₄	—	2	4	no	—
69	C₃×C₂²⋊C₄	—	2	4	no	—
70	C₃×C₂²⋊C₄	—	2	4	no	—
71	C₃×C₂²⋊C₄	—	4	2	no	—
72	C₃×C₂²⋊C₄	—	4	2	no	—
73	C₃×C₄²	—	4	1	no	—
74	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
75	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
76	C₂³×C₄	C₂²×C₃	1	2	yes	—
77	C₂²⋊Q₈	—	2	8	no	—
78	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
81	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
86	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
88	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
89	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
90	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
91	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
92	#95	C₂²×C₃	1	2	yes	—
93	#95	C₂²×C₃	1	2	yes	—
94	#95	C₂²×C₃	1	2	yes	—
95	#95	C₂²×C₃	1	2	yes	—
96	C₂³×C₃	—	2	2	no	—
97	C₂³×C₃	—	2	2	no	—
98	C₂³×C₃	—	2	2	no	—
99	C₂³×C₃	—	2	2	no	—
100	C₂³×C₃	—	2	2	no	—
101	C₂³×C₃	—	2	2	no	—
102	C₂³×C₃	C₂×Q₈	1	4	yes	—
103	C₂³×C₃	C₂⁴	1	1	yes	characteristic
104	C₃×Q₈	—	4	4	no	—
105	C₃×D₄	—	4	4	no	—
106	C₂×C₃×C₄	—	2	1	no	—
107	C₂×C₃×C₄	—	2	1	no	—
108	C₂×C₃×C₄	—	2	1	no	—
109	C₂×C₃×C₄	—	2	1	no	—
110	C₂×C₃×C₄	—	2	1	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	2	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	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	4	8	no	—
127	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
128	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
129	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
130	C₂×C₃×C₄	#41	1	2	yes	—
131	C₂⁴	C₂³×C₃	1	2	yes	—
132	C₂×Q₈	—	4	1	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	C₃×D₄	1	4	yes	complemented
135	C₂×D₄	—	4	1	no	—
136	C₂×D₄	—	4	1	no	—
137	C₂×D₄	C₃×Q₈	1	4	yes	complemented
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	C₂³×C₃	1	2	yes	—
142	C₂²×C₄	C₂³×C₃	1	2	yes	—
143	C₂²×C₄	C₂³×C₃	1	2	yes	—
144	C₂²×C₄	C₂³×C₃	1	2	yes	—
145	C₂²×C₄	C₂³×C₃	1	2	yes	—
146	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
147	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	4	1	no	—
154	C₄⋊C₄	—	4	1	no	—
155	C₄⋊C₄	—	4	1	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₂²⋊C₄	—	2	4	no	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₂²⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	4	2	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₄²	—	4	1	no	—
163	C₂²×C₃	—	2	4	no	—
164	C₂²×C₃	—	2	4	no	—
165	C₂²×C₃	—	2	4	no	—
166	C₂²×C₃	—	4	2	no	—
167	C₂²×C₃	—	4	2	no	—
168	C₂²×C₃	—	4	2	no	—
169	C₂²×C₃	—	4	2	no	—
170	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
171	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
172	C₂²×C₃	C₂×#41	1	1	yes	characteristic
173	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
174	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
175	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
176	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
177	C₃×C₄	—	2	4	no	—
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	—	2	2	no	—
180	C₃×C₄	—	2	2	no	—
181	C₃×C₄	—	4	2	no	—
182	C₃×C₄	—	4	2	no	—
183	C₃×C₄	—	4	2	no	—
184	C₃×C₄	—	4	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	C₂×C₃×Q₈	1	4	yes	—
192	C₂³	C₂⁴×C₃	1	1	yes	characteristic
193	Q₈	—	4	4	no	—
194	D₄	—	4	4	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	8	no	—
216	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
217	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
218	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
219	C₂×C₄	C₃×#41	1	2	yes	—
220	C₂×C₃	—	4	4	no	—
221	C₂×C₃	#534	1	1	yes	characteristic
222	C₂×C₃	#539	1	1	yes	characteristic
223	C₂×C₃	#543	1	1	yes	characteristic
224	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
225	C₂×C₃	#550	1	1	yes	characteristic
226	C₂×C₃	#554	1	1	yes	characteristic
227	C₂×C₃	#562	1	1	yes	characteristic
228	C₂²	—	2	4	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
236	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
237	C₂²	C₂×C₃×#41	1	1	yes	characteristic
238	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
239	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
240	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
241	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
242	C₄	—	2	4	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₃	#2732	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	4	4	no	—
252	C₂	C₃×#534	1	1	yes	characteristic
253	C₂	C₃×#539	1	1	yes	characteristic
254	C₂	C₃×#543	1	1	yes	characteristic
255	C₂	C₃×D₄×Q₈	1	1	yes	characteristic
256	C₂	C₃×#550	1	1	yes	characteristic
257	C₂	C₃×#554	1	1	yes	characteristic
258	C₂	C₃×#562	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. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
2. C₃×#399	C₂	semidirect
3. C₃×#398	C₂	semidirect
4. C₃×#396	C₂	semidirect
5. C₃×#391	C₂	semidirect
7. C₃×#389	C₂	semidirect
9. C₃×#385	C₂	semidirect
10. #2732	C₃	direct (N × H)
33. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
34. #399	C₂×C₃	semidirect
35. #398	C₂×C₃	semidirect
36. #396	C₂×C₃	semidirect
37. #391	C₂×C₃	semidirect
39. #389	C₂×C₃	semidirect
41. #385	C₂×C₃	semidirect
45. C₂×C₃×Q₈	D₄	semidirect
48. C₂×C₃×D₄	Q₈	semidirect
134. C₂×Q₈	C₃×D₄	semidirect
137. C₂×D₄	C₃×Q₈	semidirect
250. C₃	#2732	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×#399	2
C₃×#398	2
C₃×#396	2
C₃×#391	2
C₃×#390	2
C₃×#389	2
C₃×#386	2
C₃×#385	2
#2732	3

Subgroups — C3×#2732 #84596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2732 #84596