Subgroups — C₃×#2628 #84492

Order 384.

488 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₃×#400	C₂	1	2	yes	maximal, complemented
4	C₃×#399	C₂	1	2	yes	maximal
5	C₃×#395	C₂	1	2	yes	maximal, complemented
6	C₃×#393	C₂	1	2	yes	maximal
7	C₃×#389	C₂	1	2	yes	maximal, complemented
8	C₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#380	C₂	1	2	yes	maximal
10	#2628	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	2	yes	—
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₃×C₄⋊Q₈	—	2	4	no	—
14	C₃×#117	—	2	4	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	4	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
24	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
26	C₃×#95	C₂²	1	4	yes	—
27	C₃×#95	C₂²	1	2	yes	—
28	C₃×#95	C₂²	1	2	yes	—
29	C₃×#95	C₂²	1	4	yes	—
30	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#117	C₂×C₃	1	1	yes	characteristic
32	#400	C₂×C₃	1	2	yes	complemented
33	#399	C₂×C₃	1	2	yes	—
34	#395	C₂×C₃	1	2	yes	complemented
35	#393	C₂×C₃	1	2	yes	—
36	#389	C₂×C₃	1	2	yes	complemented
37	#382	C₂×C₃	1	1	yes	characteristic, complemented
38	#380	C₂×C₃	1	2	yes	—
39	C₂⁴×C₃	C₂³	1	1	yes	characteristic
40	C₂×C₃×Q₈	—	2	4	no	—
41	C₂×C₃×Q₈	—	4	4	no	—
42	C₂²×C₃×C₄	—	2	2	no	—
43	C₂²×C₃×C₄	—	2	2	no	—
44	C₂²×C₃×C₄	Q₈	1	4	yes	complemented
45	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
46	C₂²×C₃×C₄	C₂³	1	4	yes	—
47	C₂²×C₃×C₄	C₂³	1	4	yes	—
48	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
49	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
50	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
51	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₃×C₄⋊C₄	—	2	2	no	—
54	C₃×C₄⋊C₄	—	2	2	no	—
55	C₃×C₄⋊C₄	—	2	4	no	—
56	C₃×C₄⋊C₄	—	2	4	no	—
57	C₃×C₄⋊C₄	—	2	4	no	—
58	C₃×C₄⋊C₄	—	4	4	no	—
59	C₃×C₂²⋊C₄	—	2	2	no	—
60	C₃×C₂²⋊C₄	—	2	2	no	—
61	C₃×C₂²⋊C₄	—	4	4	no	—
62	C₃×C₄²	—	2	2	no	—
63	C₃×C₄²	D₄	1	4	yes	—
64	C₂²×Q₈	C₂²×C₃	1	2	yes	—
65	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
66	C₄⋊Q₈	—	2	4	no	—
67	#117	—	2	4	no	—
68	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
72	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
73	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
74	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
77	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
79	#95	C₂²×C₃	1	4	yes	—
80	#95	C₂²×C₃	1	2	yes	—
81	#95	C₂²×C₃	1	2	yes	—
82	#95	C₂²×C₃	1	4	yes	—
83	C₂³×C₃	—	2	1	no	—
84	C₂³×C₃	—	2	1	no	—
85	C₂³×C₃	—	2	2	no	—
86	C₂³×C₃	—	2	2	no	—
87	C₂³×C₃	C₂×Q₈	1	2	yes	—
88	C₂³×C₃	C₂⁴	1	1	yes	characteristic
89	C₃×Q₈	—	4	8	no	—
90	C₂×C₃×C₄	—	2	1	no	—
91	C₂×C₃×C₄	—	2	1	no	—
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	—	2	8	no	—
95	C₂×C₃×C₄	—	2	1	no	—
96	C₂×C₃×C₄	—	2	1	no	—
97	C₂×C₃×C₄	—	2	4	no	—
98	C₂×C₃×C₄	—	2	4	no	—
99	C₂×C₃×C₄	—	2	4	no	—
100	C₂×C₃×C₄	—	2	4	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	4	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	4	no	—
105	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
106	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
107	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
108	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
109	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
110	C₂×C₃×C₄	#41	1	2	yes	—
111	C₂⁴	C₂³×C₃	1	1	yes	characteristic
112	C₂×Q₈	—	2	4	no	—
113	C₂×Q₈	—	4	4	no	—
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	C₃×Q₈	1	4	yes	complemented
117	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₂²×C₄	C₂³×C₃	1	4	yes	—
119	C₂²×C₄	C₂³×C₃	1	4	yes	—
120	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
121	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
122	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
123	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
124	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	4	4	no	—
131	C₂²⋊C₄	—	2	2	no	—
132	C₂²⋊C₄	—	2	2	no	—
133	C₂²⋊C₄	—	4	4	no	—
134	C₄²	—	2	2	no	—
135	C₄²	C₃×D₄	1	4	yes	—
136	C₂²×C₃	—	2	2	no	—
137	C₂²×C₃	—	2	4	no	—
138	C₂²×C₃	—	4	1	no	—
139	C₂²×C₃	—	4	1	no	—
140	C₂²×C₃	—	4	2	no	—
141	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
142	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
143	C₂²×C₃	C₂×#41	1	1	yes	characteristic
144	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
145	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
146	C₃×C₄	—	2	2	no	—
147	C₃×C₄	—	2	2	no	—
148	C₃×C₄	—	2	2	no	—
149	C₃×C₄	—	4	1	no	—
150	C₃×C₄	—	4	4	no	—
151	C₃×C₄	—	4	1	no	—
152	C₃×C₄	—	4	4	no	—
153	C₃×C₄	C₂²⋊Q₈	1	4	yes	—
154	C₂³	—	2	1	no	—
155	C₂³	—	2	1	no	—
156	C₂³	—	2	2	no	—
157	C₂³	—	2	2	no	—
158	C₂³	C₂×C₃×Q₈	1	2	yes	—
159	C₂³	C₂⁴×C₃	1	1	yes	characteristic
160	Q₈	—	4	8	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
177	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
178	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
179	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
180	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
181	C₂×C₄	C₃×#41	1	2	yes	—
182	C₂×C₃	—	4	2	no	—
183	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
184	C₂×C₃	#535	1	1	yes	characteristic
185	C₂×C₃	#536	1	1	yes	characteristic
186	C₂×C₃	#542	1	2	yes	—
187	C₂×C₃	#544	1	2	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	4	1	no	—
191	C₂²	—	4	1	no	—
192	C₂²	—	4	2	no	—
193	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
194	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
195	C₂²	C₂×C₃×#41	1	1	yes	characteristic
196	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
197	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	4	no	—
202	C₄	—	4	1	no	—
203	C₄	—	4	1	no	—
204	C₄	—	4	4	no	—
205	C₄	C₃×C₂²⋊Q₈	1	4	yes	—
206	C₃	#2628	1	1	yes	characteristic, direct product factor, Sylow
207	C₂	—	4	2	no	—
208	C₂	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
209	C₂	C₃×#535	1	1	yes	characteristic
210	C₂	C₃×#536	1	1	yes	characteristic
211	C₂	C₃×#542	1	2	yes	—
212	C₂	C₃×#544	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³
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×C₄⋊Q₈	C₂	semidirect
3. C₃×#400	C₂	semidirect
5. C₃×#395	C₂	semidirect
7. C₃×#389	C₂	semidirect
8. C₃×#382	C₂	semidirect
10. #2628	C₃	direct (N × H)
30. C₂×C₄⋊Q₈	C₂×C₃	semidirect
32. #400	C₂×C₃	semidirect
34. #395	C₂×C₃	semidirect
36. #389	C₂×C₃	semidirect
37. #382	C₂×C₃	semidirect
44. C₂²×C₃×C₄	Q₈	semidirect
116. C₂²×C₄	C₃×Q₈	semidirect
206. C₃	#2628	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×C₄⋊Q₈	2
C₂×C₃×#117	2
C₃×#400	2
C₃×#399	2
C₃×#395	2
C₃×#393	2
C₃×#389	2
C₃×#382	2
C₃×#380	2
#2628	3

Subgroups — C3×#2628 #84492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2628 #84492