Subgroups — C₂×C₃×#391 #84158

Order 384.

928 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
4	C₃×#391	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#95	C₂	1	4	yes	maximal
6	C₂×#391	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×Q₈	—	2	4	no	—
8	C₂²×C₃×Q₈	—	2	4	no	—
9	C₂²×C₃×Q₈	C₄	1	8	yes	complemented
10	C₂²×C₃×Q₈	C₂²	1	4	yes	—
11	C₂³×C₃×C₄	C₂²	1	2	yes	—
12	C₂³×C₃×C₄	C₂²	1	2	yes	—
13	C₂³×C₃×C₄	C₂²	1	2	yes	—
14	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊C₄	—	2	4	no	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
17	C₂×C₃×C₄²	—	2	4	no	—
18	C₂×C₃×C₄²	C₂²	1	4	yes	—
19	C₃×#95	C₂²	1	16	yes	—
20	C₂³×Q₈	C₂×C₃	1	1	yes	characteristic
21	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
22	C₂²×C₄²	C₂×C₃	1	1	yes	characteristic
23	#391	C₂×C₃	1	8	yes	direct product factor
24	C₂×#95	C₂×C₃	1	4	yes	—
25	C₂⁴×C₃	C₂³	1	1	yes	characteristic
26	C₂×C₃×Q₈	—	2	32	no	—
27	C₂×C₃×Q₈	—	2	8	no	—
28	C₂×C₃×Q₈	—	2	8	no	—
29	C₂×C₃×Q₈	C₂×C₄	1	16	yes	complemented
30	C₂²×C₃×C₄	—	2	4	no	—
31	C₂²×C₃×C₄	—	2	4	no	—
32	C₂²×C₃×C₄	—	2	4	no	—
33	C₂²×C₃×C₄	—	2	4	no	—
34	C₂²×C₃×C₄	—	2	4	no	—
35	C₂²×C₃×C₄	—	2	8	no	—
36	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
37	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
38	C₂²×C₃×C₄	D₄	1	4	yes	—
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	4	yes	—
44	C₂²×C₃×C₄	C₂³	1	2	yes	—
45	C₃×C₄⋊C₄	—	2	8	no	—
46	C₃×C₄²	—	2	8	no	—
47	C₂²×Q₈	—	2	4	no	—
48	C₂²×Q₈	—	2	4	no	—
49	C₂²×Q₈	C₃×C₄	1	8	yes	complemented
50	C₂²×Q₈	C₂²×C₃	1	4	yes	—
51	C₂³×C₄	C₂²×C₃	1	2	yes	—
52	C₂³×C₄	C₂²×C₃	1	2	yes	—
53	C₂³×C₄	C₂²×C₃	1	2	yes	—
54	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₄⋊C₄	—	2	4	no	—
56	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
57	C₂×C₄²	—	2	4	no	—
58	C₂×C₄²	C₂²×C₃	1	4	yes	—
59	#95	C₂²×C₃	1	16	yes	—
60	C₂³×C₃	C₂²×C₄	1	2	yes	—
61	C₂³×C₃	C₂×D₄	1	2	yes	—
62	C₂³×C₃	C₂×D₄	1	2	yes	—
63	C₂³×C₃	C₂×Q₈	1	4	yes	—
64	C₂³×C₃	#41	1	4	yes	—
65	C₂³×C₃	C₂⁴	1	1	yes	characteristic
66	C₃×Q₈	—	2	32	no	—
67	C₂×C₃×C₄	—	2	16	no	—
68	C₂×C₃×C₄	—	2	16	no	—
69	C₂×C₃×C₄	—	2	4	no	—
70	C₂×C₃×C₄	—	2	4	no	—
71	C₂×C₃×C₄	—	2	4	no	—
72	C₂×C₃×C₄	—	2	4	no	—
73	C₂×C₃×C₄	—	2	4	no	—
74	C₂×C₃×C₄	—	2	8	no	—
75	C₂×C₃×C₄	—	4	8	no	—
76	C₂×C₃×C₄	C₂²⋊C₄	1	16	yes	—
77	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
78	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
79	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
80	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
81	C₂×C₃×C₄	C₂×Q₈	1	8	yes	complemented
82	C₂⁴	C₂³×C₃	1	1	yes	characteristic
83	C₂×Q₈	—	2	32	no	—
84	C₂×Q₈	—	2	8	no	—
85	C₂×Q₈	—	2	8	no	—
86	C₂×Q₈	C₂×C₃×C₄	1	16	yes	complemented
87	C₂²×C₄	—	2	4	no	—
88	C₂²×C₄	—	2	4	no	—
89	C₂²×C₄	—	2	4	no	—
90	C₂²×C₄	—	2	4	no	—
91	C₂²×C₄	—	2	4	no	—
92	C₂²×C₄	—	2	8	no	—
93	C₂²×C₄	C₂×C₃×C₄	1	8	yes	—
94	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
95	C₂²×C₄	C₃×D₄	1	4	yes	—
96	C₂²×C₄	C₃×D₄	1	4	yes	—
97	C₂²×C₄	C₃×Q₈	1	8	yes	complemented
98	C₂²×C₄	C₂³×C₃	1	4	yes	—
99	C₂²×C₄	C₂³×C₃	1	4	yes	—
100	C₂²×C₄	C₂³×C₃	1	4	yes	—
101	C₂²×C₄	C₂³×C₃	1	2	yes	—
102	C₄⋊C₄	—	2	8	no	—
103	C₄²	—	2	8	no	—
104	C₂²×C₃	C₂×C₂²⋊C₄	1	4	yes	—
105	C₂²×C₃	C₄×Q₈	1	8	yes	—
106	C₂²×C₃	C₂²⋊Q₈	1	8	yes	—
107	C₂²×C₃	#124	1	4	yes	—
108	C₂²×C₃	C₄⋊Q₈	1	4	yes	—
109	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
110	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
111	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
112	C₂²×C₃	C₂²×Q₈	1	2	yes	—
113	C₂²×C₃	C₂×#41	1	2	yes	—
114	C₃×C₄	—	2	8	no	—
115	C₃×C₄	—	2	8	no	—
116	C₃×C₄	—	4	4	no	—
117	C₃×C₄	C₂×C₂²⋊C₄	1	8	yes	—
118	C₂³	C₂²×C₃×C₄	1	2	yes	—
119	C₂³	C₂×C₃×D₄	1	2	yes	—
120	C₂³	C₂×C₃×D₄	1	2	yes	—
121	C₂³	C₂×C₃×Q₈	1	4	yes	—
122	C₂³	C₃×#41	1	4	yes	—
123	C₂³	C₂⁴×C₃	1	1	yes	characteristic
124	Q₈	—	2	32	no	—
125	C₂×C₄	—	2	16	no	—
126	C₂×C₄	—	2	16	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	8	no	—
133	C₂×C₄	—	4	8	no	—
134	C₂×C₄	C₃×C₂²⋊C₄	1	16	yes	—
135	C₂×C₄	C₂²×C₃×C₄	1	8	yes	—
136	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
137	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
138	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
139	C₂×C₄	C₂×C₃×Q₈	1	8	yes	complemented
140	C₂×C₃	#391	1	8	yes	direct product factor
141	C₂×C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₃	C₂×C₄×Q₈	1	2	yes	—
143	C₂×C₃	C₂×C₂²⋊Q₈	1	2	yes	—
144	C₂×C₃	C₂×#124	1	1	yes	characteristic
145	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	C₂²	C₂×C₃×C₂²⋊C₄	1	4	yes	—
147	C₂²	C₃×C₄×Q₈	1	8	yes	—
148	C₂²	C₃×C₂²⋊Q₈	1	8	yes	—
149	C₂²	C₃×#124	1	4	yes	—
150	C₂²	C₃×C₄⋊Q₈	1	4	yes	—
151	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
152	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
153	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
154	C₂²	C₂²×C₃×Q₈	1	2	yes	—
155	C₂²	C₂×C₃×#41	1	2	yes	—
156	C₄	—	2	8	no	—
157	C₄	—	2	8	no	—
158	C₄	—	4	4	no	—
159	C₄	C₂×C₃×C₂²⋊C₄	1	8	yes	—
160	C₃	C₂×#391	1	1	yes	characteristic, direct product factor, Sylow
161	C₂	C₃×#391	1	8	yes	direct product factor
162	C₂	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
163	C₂	C₂×C₃×C₄×Q₈	1	2	yes	—
164	C₂	C₂×C₃×C₂²⋊Q₈	1	2	yes	—
165	C₂	C₂×C₃×#124	1	1	yes	characteristic
166	C₂	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃	C₂³
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₃×#391	C₂	direct (N × H)
6. C₂×#391	C₃	direct (N × H)
9. C₂²×C₃×Q₈	C₄	semidirect
23. #391	C₂×C₃	direct (N × H)
29. C₂×C₃×Q₈	C₂×C₄	semidirect
40. C₂²×C₃×C₄	Q₈	semidirect
49. C₂²×Q₈	C₃×C₄	semidirect
81. C₂×C₃×C₄	C₂×Q₈	semidirect
86. C₂×Q₈	C₂×C₃×C₄	semidirect
97. C₂²×C₄	C₃×Q₈	semidirect
139. C₂×C₄	C₂×C₃×Q₈	semidirect
140. C₂×C₃	#391	direct (N × H)
160. C₃	C₂×#391	direct (N × H)
161. C₂	C₃×#391	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₃×Q₈	2
C₂²×C₃×C₄⋊C₄	2
C₂²×C₃×C₄²	2
C₃×#391	2
C₂×C₃×#95	2
C₂×#391	3

Subgroups — C2×C3×#391 #84158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#391 #84158