Subgroups — C₃×#2626 #84490

Order 384.

668 conjugacy classes of subgroups.

The subgroup lattice has 236 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, complemented
2	C₂²×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#399	C₂	1	2	yes	maximal, complemented
4	C₃×#396	C₂	1	4	yes	maximal, complemented
5	C₃×#393	C₂	1	2	yes	maximal, complemented
6	C₃×#385	C₂	1	2	yes	maximal, complemented
7	C₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#379	C₂	1	2	yes	maximal
9	#2626	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂⁵×C₃	C₂²	1	1	yes	characteristic
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₃×C₂²⋊Q₈	—	2	4	no	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
18	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	complemented
19	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
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×C₂²⋊C₄	C₂²	1	4	yes	—
25	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₃×#95	C₂²	1	2	yes	complemented
27	C₃×#95	C₂²	1	4	yes	complemented
28	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂²×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
30	#399	C₂×C₃	1	2	yes	complemented
31	#396	C₂×C₃	1	4	yes	complemented
32	#393	C₂×C₃	1	2	yes	complemented
33	#385	C₂×C₃	1	2	yes	complemented
34	#380	C₂×C₃	1	1	yes	characteristic, complemented
35	#379	C₂×C₃	1	2	yes	—
36	C₂⁴×C₃	—	2	2	no	—
37	C₂⁴×C₃	—	2	2	no	—
38	C₂⁴×C₃	—	4	2	no	—
39	C₂⁴×C₃	—	4	2	no	—
40	C₂⁴×C₃	Q₈	1	4	yes	complemented
41	C₂⁴×C₃	C₂³	1	2	yes	—
42	C₂⁴×C₃	C₂³	1	1	yes	characteristic
43	C₂×C₃×Q₈	—	2	2	no	—
44	C₂×C₃×Q₈	—	4	2	no	—
45	C₂²×C₃×C₄	—	2	2	no	—
46	C₂²×C₃×C₄	—	4	2	no	—
47	C₂²×C₃×C₄	—	4	2	no	—
48	C₂²×C₃×C₄	D₄	1	4	yes	—
49	C₂²×C₃×C₄	C₂³	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	2	yes	—
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
55	C₃×C₄⋊C₄	—	2	2	no	—
56	C₃×C₄⋊C₄	—	2	4	no	—
57	C₃×C₄⋊C₄	—	4	2	no	—
58	C₃×C₄⋊C₄	—	4	4	no	—
59	C₃×C₂²⋊C₄	—	2	4	no	—
60	C₃×C₂²⋊C₄	—	2	4	no	—
61	C₃×C₂²⋊C₄	—	2	4	no	—
62	C₃×C₂²⋊C₄	—	2	4	no	—
63	C₃×C₂²⋊C₄	—	4	4	no	—
64	C₃×C₂²⋊C₄	—	4	4	no	—
65	C₂⁵	C₂²×C₃	1	1	yes	characteristic
66	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
67	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
68	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
69	C₂²⋊Q₈	—	2	4	no	—
70	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
71	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
72	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
73	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
74	C₂×C₂²⋊C₄	—	2	4	no	—
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×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
80	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
81	#95	C₂²×C₃	1	2	yes	complemented
82	#95	C₂²×C₃	1	4	yes	complemented
83	C₂³×C₃	—	2	2	no	—
84	C₂³×C₃	—	2	2	no	—
85	C₂³×C₃	—	2	8	no	—
86	C₂³×C₃	—	2	2	no	—
87	C₂³×C₃	—	2	2	no	—
88	C₂³×C₃	—	2	4	no	—
89	C₂³×C₃	—	2	4	no	—
90	C₂³×C₃	—	4	8	no	—
91	C₂³×C₃	—	4	4	no	—
92	C₂³×C₃	—	4	4	no	—
93	C₂³×C₃	—	4	8	no	—
94	C₂³×C₃	C₂×D₄	1	2	yes	—
95	C₂³×C₃	C₂×Q₈	1	4	yes	—
96	C₂³×C₃	C₂×Q₈	1	2	yes	—
97	C₂³×C₃	#41	1	2	yes	—
98	C₂³×C₃	C₂⁴	1	1	yes	characteristic
99	C₃×Q₈	—	4	4	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	4	no	—
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	—	2	4	no	—
108	C₂×C₃×C₄	—	2	4	no	—
109	C₂×C₃×C₄	—	2	4	no	—
110	C₂×C₃×C₄	—	2	4	no	—
111	C₂×C₃×C₄	—	4	4	no	—
112	C₂×C₃×C₄	—	4	4	no	—
113	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
114	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
115	C₂⁴	—	2	2	no	—
116	C₂⁴	—	2	2	no	—
117	C₂⁴	—	4	2	no	—
118	C₂⁴	—	4	2	no	—
119	C₂⁴	C₃×Q₈	1	4	yes	complemented
120	C₂⁴	C₂³×C₃	1	2	yes	—
121	C₂⁴	C₂³×C₃	1	1	yes	characteristic
122	C₂×Q₈	—	2	2	no	—
123	C₂×Q₈	—	4	2	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	4	2	no	—
126	C₂²×C₄	—	4	2	no	—
127	C₂²×C₄	C₃×D₄	1	4	yes	—
128	C₂²×C₄	C₂³×C₃	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	2	yes	—
132	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
133	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₂²⋊C₄	—	2	4	no	—
139	C₂²⋊C₄	—	2	4	no	—
140	C₂²⋊C₄	—	2	4	no	—
141	C₂²⋊C₄	—	2	4	no	—
142	C₂²⋊C₄	—	4	4	no	—
143	C₂²⋊C₄	—	4	4	no	—
144	C₂²×C₃	—	2	2	no	—
145	C₂²×C₃	—	2	2	no	—
146	C₂²×C₃	—	2	4	no	—
147	C₂²×C₃	—	2	4	no	—
148	C₂²×C₃	—	2	4	no	—
149	C₂²×C₃	—	2	8	no	—
150	C₂²×C₃	—	4	16	no	—
151	C₂²×C₃	—	4	2	no	—
152	C₂²×C₃	—	4	2	no	—
153	C₂²×C₃	—	4	4	no	—
154	C₂²×C₃	C₂²⋊Q₈	1	4	yes	—
155	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
156	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
157	C₂²×C₃	C₂×#41	1	1	yes	characteristic
158	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
159	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
160	C₃×C₄	—	2	2	no	—
161	C₃×C₄	—	2	2	no	—
162	C₃×C₄	—	4	2	no	—
163	C₃×C₄	—	4	2	no	—
164	C₃×C₄	—	4	2	no	—
165	C₃×C₄	—	4	4	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	8	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	4	no	—
172	C₂³	—	2	4	no	—
173	C₂³	—	4	8	no	—
174	C₂³	—	4	4	no	—
175	C₂³	—	4	4	no	—
176	C₂³	—	4	8	no	—
177	C₂³	C₂×C₃×D₄	1	2	yes	—
178	C₂³	C₂×C₃×Q₈	1	4	yes	—
179	C₂³	C₂×C₃×Q₈	1	2	yes	—
180	C₂³	C₃×#41	1	2	yes	—
181	C₂³	C₂⁴×C₃	1	1	yes	characteristic
182	Q₈	—	4	4	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	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
197	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
198	C₂×C₃	—	2	4	no	—
199	C₂×C₃	—	4	4	no	—
200	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
201	C₂×C₃	#534	1	1	yes	characteristic
202	C₂×C₃	#535	1	1	yes	characteristic
203	C₂×C₃	#538	1	2	yes	—
204	C₂×C₃	#543	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	8	no	—
211	C₂²	—	4	16	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	4	no	—
215	C₂²	C₃×C₂²⋊Q₈	1	4	yes	—
216	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
217	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
218	C₂²	C₂×C₃×#41	1	1	yes	characteristic
219	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
220	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	4	no	—
227	C₃	#2626	1	1	yes	characteristic, direct product factor, Sylow
228	C₂	—	2	4	no	—
229	C₂	—	4	4	no	—
230	C₂	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
231	C₂	C₃×#534	1	1	yes	characteristic
232	C₂	C₃×#535	1	1	yes	characteristic
233	C₂	C₃×#538	1	2	yes	—
234	C₂	C₃×#543	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁵×C₃	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
3. C₃×#399	C₂	semidirect
4. C₃×#396	C₂	semidirect
5. C₃×#393	C₂	semidirect
6. C₃×#385	C₂	semidirect
7. C₃×#380	C₂	semidirect
9. #2626	C₃	direct (N × H)
11. C₂²×C₃×Q₈	C₂²	semidirect
15. C₂×C₃×C₄⋊C₄	C₂²	semidirect
16. C₂×C₃×C₄⋊C₄	C₂²	semidirect
17. C₂×C₃×C₄⋊C₄	C₂²	semidirect
18. C₂×C₃×C₄⋊C₄	C₂²	semidirect
26. C₃×#95	C₂²	semidirect
27. C₃×#95	C₂²	semidirect
28. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
30. #399	C₂×C₃	semidirect
31. #396	C₂×C₃	semidirect
32. #393	C₂×C₃	semidirect
33. #385	C₂×C₃	semidirect
34. #380	C₂×C₃	semidirect
40. C₂⁴×C₃	Q₈	semidirect
66. C₂²×Q₈	C₂²×C₃	semidirect
70. C₂×C₄⋊C₄	C₂²×C₃	semidirect
71. C₂×C₄⋊C₄	C₂²×C₃	semidirect
72. C₂×C₄⋊C₄	C₂²×C₃	semidirect
73. C₂×C₄⋊C₄	C₂²×C₃	semidirect
81. #95	C₂²×C₃	semidirect
82. #95	C₂²×C₃	semidirect
119. C₂⁴	C₃×Q₈	semidirect
227. C₃	#2626	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
C₂²×C₃×C₂²⋊C₄	2
C₃×#399	2
C₃×#396	2
C₃×#393	2
C₃×#385	2
C₃×#380	2
C₃×#379	2
#2626	3

Subgroups — C3×#2626 #84490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2626 #84490