Subgroups — C₃×#2857 #84721

Order 384.

644 conjugacy classes of subgroups.

The subgroup lattice has 192 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×C₂²⋊Q₈	C₂	1	2	yes	maximal
3	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
4	C₃×#391	C₂	1	2	yes	maximal, complemented
5	C₃×#389	C₂	1	4	yes	maximal, complemented
6	C₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#380	C₂	1	4	yes	maximal
8	#2857	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×Q₈	C₂²	1	2	yes	—
10	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	2	yes	—
12	C₃×C₄⋊Q₈	—	2	4	no	—
13	C₃×C₂²⋊Q₈	—	2	8	no	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
18	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
19	C₂×C₃×C₄²	—	2	4	no	—
20	C₂×C₃×C₄²	C₂²	1	2	yes	—
21	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
23	C₃×#95	C₂²	1	8	yes	—
24	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
26	C₂²×C₄²	C₂×C₃	1	1	yes	characteristic
27	#391	C₂×C₃	1	2	yes	complemented
28	#389	C₂×C₃	1	4	yes	complemented
29	#384	C₂×C₃	1	1	yes	characteristic, complemented
30	#380	C₂×C₃	1	4	yes	—
31	C₂⁴×C₃	C₂³	1	1	yes	characteristic
32	C₂×C₃×Q₈	—	2	4	no	—
33	C₂×C₃×Q₈	—	2	4	no	—
34	C₂×C₃×Q₈	—	2	4	no	—
35	C₂²×C₃×C₄	—	2	2	no	—
36	C₂²×C₃×C₄	—	2	4	no	—
37	C₂²×C₃×C₄	—	2	2	no	—
38	C₂²×C₃×C₄	—	2	4	no	—
39	C₂²×C₃×C₄	D₄	1	4	yes	—
40	C₂²×C₃×C₄	Q₈	1	8	yes	complemented
41	C₂²×C₃×C₄	C₂³	1	4	yes	—
42	C₂²×C₃×C₄	C₂³	1	4	yes	—
43	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
44	C₂²×C₃×C₄	C₂³	1	2	yes	—
45	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
46	C₂²×C₃×C₄	C₂³	1	2	yes	—
47	C₃×C₄⋊C₄	—	2	4	no	—
48	C₃×C₄⋊C₄	—	2	8	no	—
49	C₃×C₄⋊C₄	—	2	8	no	—
50	C₃×C₄⋊C₄	—	2	8	no	—
51	C₃×C₂²⋊C₄	—	2	8	no	—
52	C₃×C₄²	—	2	2	no	—
53	C₃×C₄²	—	2	4	no	—
54	C₃×C₄²	D₄	1	4	yes	—
55	C₂²×Q₈	C₂²×C₃	1	2	yes	—
56	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
57	C₂³×C₄	C₂²×C₃	1	2	yes	—
58	C₄⋊Q₈	—	2	4	no	—
59	C₂²⋊Q₈	—	2	8	no	—
60	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
61	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
62	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
63	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
64	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
65	C₂×C₄²	—	2	4	no	—
66	C₂×C₄²	C₂²×C₃	1	2	yes	—
67	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
69	#95	C₂²×C₃	1	8	yes	—
70	C₂³×C₃	—	2	1	no	—
71	C₂³×C₃	—	2	1	no	—
72	C₂³×C₃	—	2	2	no	—
73	C₂³×C₃	C₂×D₄	1	2	yes	—
74	C₂³×C₃	C₂×Q₈	1	4	yes	—
75	C₂³×C₃	C₂⁴	1	1	yes	characteristic
76	C₃×Q₈	—	4	8	no	—
77	C₂×C₃×C₄	—	2	1	no	—
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	—	2	1	no	—
80	C₂×C₃×C₄	—	2	2	no	—
81	C₂×C₃×C₄	—	2	8	no	—
82	C₂×C₃×C₄	—	2	16	no	—
83	C₂×C₃×C₄	—	2	4	no	—
84	C₂×C₃×C₄	—	2	4	no	—
85	C₂×C₃×C₄	—	2	4	no	—
86	C₂×C₃×C₄	—	2	8	no	—
87	C₂×C₃×C₄	—	2	4	no	—
88	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
89	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
90	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
91	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
92	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
93	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
94	C₂×C₃×C₄	#41	1	2	yes	—
95	C₂×C₃×C₄	#41	1	4	yes	—
96	C₂⁴	C₂³×C₃	1	1	yes	characteristic
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	—	2	4	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	2	4	no	—
104	C₂²×C₄	C₃×D₄	1	4	yes	—
105	C₂²×C₄	C₃×Q₈	1	8	yes	complemented
106	C₂²×C₄	C₂³×C₃	1	4	yes	—
107	C₂²×C₄	C₂³×C₃	1	4	yes	—
108	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
109	C₂²×C₄	C₂³×C₃	1	2	yes	—
110	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
111	C₂²×C₄	C₂³×C₃	1	2	yes	—
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	2	8	no	—
114	C₄⋊C₄	—	2	8	no	—
115	C₄⋊C₄	—	2	8	no	—
116	C₂²⋊C₄	—	2	8	no	—
117	C₄²	—	2	2	no	—
118	C₄²	—	2	4	no	—
119	C₄²	C₃×D₄	1	4	yes	—
120	C₂²×C₃	—	2	2	no	—
121	C₂²×C₃	—	2	4	no	—
122	C₂²×C₃	—	2	2	no	—
123	C₂²×C₃	—	2	4	no	—
124	C₂²×C₃	C₄⋊Q₈	1	4	yes	complemented
125	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₃	C₂²×Q₈	1	2	yes	—
128	C₂²×C₃	C₂×#41	1	1	yes	characteristic
129	C₂²×C₃	C₂×#41	1	2	yes	—
130	C₃×C₄	—	2	2	no	—
131	C₃×C₄	—	2	4	no	—
132	C₃×C₄	—	4	4	no	—
133	C₃×C₄	—	4	4	no	—
134	C₃×C₄	#121	1	4	yes	—
135	C₃×C₄	C₂²⋊Q₈	1	8	yes	complemented
136	C₂³	—	2	1	no	—
137	C₂³	—	2	1	no	—
138	C₂³	—	2	2	no	—
139	C₂³	C₂×C₃×D₄	1	2	yes	—
140	C₂³	C₂×C₃×Q₈	1	4	yes	—
141	C₂³	C₂⁴×C₃	1	1	yes	characteristic
142	Q₈	—	4	8	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	2	16	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	8	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
155	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
156	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
157	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
158	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
159	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
160	C₂×C₄	C₃×#41	1	2	yes	—
161	C₂×C₄	C₃×#41	1	4	yes	—
162	C₂×C₃	—	2	4	no	—
163	C₂×C₃	C₂×#121	1	1	yes	characteristic
164	C₂×C₃	C₂×C₂²⋊Q₈	1	2	yes	—
165	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
166	C₂×C₃	#532	1	1	yes	characteristic
167	C₂×C₃	#533	1	2	yes	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	4	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	4	no	—
172	C₂²	C₃×C₄⋊Q₈	1	4	yes	complemented
173	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
174	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
175	C₂²	C₂²×C₃×Q₈	1	2	yes	—
176	C₂²	C₂×C₃×#41	1	1	yes	characteristic
177	C₂²	C₂×C₃×#41	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	2	4	no	—
180	C₄	—	4	4	no	—
181	C₄	—	4	4	no	—
182	C₄	C₃×#121	1	4	yes	—
183	C₄	C₃×C₂²⋊Q₈	1	8	yes	complemented
184	C₃	#2857	1	1	yes	characteristic, direct product factor, Sylow
185	C₂	—	2	4	no	—
186	C₂	C₂×C₃×#121	1	1	yes	characteristic
187	C₂	C₂×C₃×C₂²⋊Q₈	1	2	yes	—
188	C₂	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
189	C₂	C₃×#532	1	1	yes	characteristic
190	C₂	C₃×#533	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
4. C₃×#391	C₂	semidirect
5. C₃×#389	C₂	semidirect
6. C₃×#384	C₂	semidirect
8. #2857	C₃	direct (N × H)
24. C₂×C₄⋊Q₈	C₂×C₃	semidirect
27. #391	C₂×C₃	semidirect
28. #389	C₂×C₃	semidirect
29. #384	C₂×C₃	semidirect
40. C₂²×C₃×C₄	Q₈	semidirect
105. C₂²×C₄	C₃×Q₈	semidirect
124. C₂²×C₃	C₄⋊Q₈	semidirect
135. C₃×C₄	C₂²⋊Q₈	semidirect
172. C₂²	C₃×C₄⋊Q₈	semidirect
183. C₄	C₃×C₂²⋊Q₈	semidirect
184. C₃	#2857	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×C₂²⋊Q₈	2
C₂²×C₃×C₄²	2
C₃×#391	2
C₃×#389	2
C₃×#384	2
C₃×#380	2
#2857	3

Subgroups — C3×#2857 #84721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2857 #84721