Subgroups — C₂×C₃×#389 #84155

Order 384.

800 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
5	C₃×#389	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×#95	C₂	1	2	yes	maximal
7	C₂×#389	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₃×C₄	C₂²	1	2	yes	—
9	C₂³×C₃×C₄	C₂²	1	2	yes	—
10	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊C₄	—	2	8	no	—
14	C₂×C₃×C₄⋊C₄	—	2	4	no	—
15	C₂×C₃×C₄⋊C₄	C₄	1	8	yes	complemented
16	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	8	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
19	C₂×C₃×C₄²	—	2	4	no	—
20	C₂×C₃×C₄²	C₂²	1	4	yes	—
21	C₃×#95	C₂²	1	8	yes	—
22	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₄⋊C₄	C₂×C₃	1	2	yes	—
24	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
25	C₂²×C₄²	C₂×C₃	1	1	yes	characteristic
26	#389	C₂×C₃	1	8	yes	direct product factor
27	C₂×#95	C₂×C₃	1	2	yes	—
28	C₂⁴×C₃	C₂³	1	1	yes	characteristic
29	C₂²×C₃×C₄	—	2	2	no	—
30	C₂²×C₃×C₄	—	2	2	no	—
31	C₂²×C₃×C₄	—	2	4	no	—
32	C₂²×C₃×C₄	—	2	4	no	—
33	C₂²×C₃×C₄	—	2	2	no	—
34	C₂²×C₃×C₄	—	2	2	no	—
35	C₂²×C₃×C₄	—	2	4	no	—
36	C₂²×C₃×C₄	—	2	4	no	—
37	C₂²×C₃×C₄	—	2	4	no	—
38	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
39	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
40	C₂²×C₃×C₄	D₄	1	4	yes	—
41	C₂²×C₃×C₄	D₄	1	4	yes	—
42	C₂²×C₃×C₄	Q₈	1	4	yes	—
43	C₂²×C₃×C₄	Q₈	1	4	yes	—
44	C₂²×C₃×C₄	C₂³	1	4	yes	—
45	C₂²×C₃×C₄	C₂³	1	4	yes	—
46	C₂²×C₃×C₄	C₂³	1	2	yes	—
47	C₂²×C₃×C₄	C₂³	1	2	yes	—
48	C₂²×C₃×C₄	C₂³	1	2	yes	—
49	C₃×C₄⋊C₄	—	2	16	no	—
50	C₃×C₄⋊C₄	—	2	8	no	—
51	C₃×C₄⋊C₄	C₂×C₄	1	16	yes	complemented
52	C₃×C₄²	—	2	8	no	—
53	C₂³×C₄	C₂²×C₃	1	2	yes	—
54	C₂³×C₄	C₂²×C₃	1	2	yes	—
55	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
56	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
57	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₄⋊C₄	—	2	8	no	—
59	C₂×C₄⋊C₄	—	2	4	no	—
60	C₂×C₄⋊C₄	C₃×C₄	1	8	yes	complemented
61	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
62	C₂×C₄⋊C₄	C₂²×C₃	1	8	yes	—
63	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
64	C₂×C₄²	—	2	4	no	—
65	C₂×C₄²	C₂²×C₃	1	4	yes	—
66	#95	C₂²×C₃	1	8	yes	—
67	C₂³×C₃	C₂²×C₄	1	2	yes	—
68	C₂³×C₃	C₂×D₄	1	2	yes	—
69	C₂³×C₃	C₂×D₄	1	2	yes	—
70	C₂³×C₃	C₂×Q₈	1	2	yes	—
71	C₂³×C₃	C₂×Q₈	1	2	yes	—
72	C₂³×C₃	#41	1	2	yes	—
73	C₂³×C₃	#41	1	2	yes	—
74	C₂³×C₃	C₂⁴	1	1	yes	characteristic
75	C₂×C₃×C₄	—	2	16	no	—
76	C₂×C₃×C₄	—	2	8	no	—
77	C₂×C₃×C₄	—	2	8	no	—
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	—	2	2	no	—
80	C₂×C₃×C₄	—	2	4	no	—
81	C₂×C₃×C₄	—	2	4	no	—
82	C₂×C₃×C₄	—	2	2	no	—
83	C₂×C₃×C₄	—	2	2	no	—
84	C₂×C₃×C₄	—	2	4	no	—
85	C₂×C₃×C₄	—	2	4	no	—
86	C₂×C₃×C₄	—	2	4	no	—
87	C₂×C₃×C₄	—	4	8	no	—
88	C₂×C₃×C₄	C₄⋊C₄	1	16	yes	complemented
89	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
90	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
91	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
92	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
93	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
94	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
95	C₂⁴	C₂³×C₃	1	1	yes	characteristic
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	—	2	4	no	—
99	C₂²×C₄	—	2	4	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	2	4	no	—
103	C₂²×C₄	—	2	4	no	—
104	C₂²×C₄	—	2	4	no	—
105	C₂²×C₄	C₂×C₃×C₄	1	8	yes	—
106	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
107	C₂²×C₄	C₃×D₄	1	4	yes	—
108	C₂²×C₄	C₃×D₄	1	4	yes	—
109	C₂²×C₄	C₃×Q₈	1	4	yes	—
110	C₂²×C₄	C₃×Q₈	1	4	yes	—
111	C₂²×C₄	C₂³×C₃	1	4	yes	—
112	C₂²×C₄	C₂³×C₃	1	4	yes	—
113	C₂²×C₄	C₂³×C₃	1	2	yes	—
114	C₂²×C₄	C₂³×C₃	1	2	yes	—
115	C₂²×C₄	C₂³×C₃	1	2	yes	—
116	C₄⋊C₄	—	2	16	no	—
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	C₂×C₃×C₄	1	16	yes	complemented
119	C₄²	—	2	8	no	—
120	C₂²×C₃	C₂×C₄⋊C₄	1	4	yes	—
121	C₂²×C₃	C₄×D₄	1	4	yes	—
122	C₂²×C₃	C₄×Q₈	1	4	yes	—
123	C₂²×C₃	#121	1	4	yes	—
124	C₂²×C₃	C₂²⋊Q₈	1	4	yes	—
125	C₂²×C₃	#125	1	4	yes	—
126	C₂²×C₃	C₄⋊Q₈	1	4	yes	—
127	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
128	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
129	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
130	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
131	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
132	C₂²×C₃	C₂×#41	1	1	yes	characteristic
133	C₂²×C₃	C₂×#41	1	1	yes	characteristic
134	C₃×C₄	—	2	8	no	—
135	C₃×C₄	—	2	4	no	—
136	C₃×C₄	—	2	4	no	—
137	C₃×C₄	—	4	4	no	—
138	C₃×C₄	C₂×C₄⋊C₄	1	8	yes	complemented
139	C₂³	C₂²×C₃×C₄	1	2	yes	—
140	C₂³	C₂×C₃×D₄	1	2	yes	—
141	C₂³	C₂×C₃×D₄	1	2	yes	—
142	C₂³	C₂×C₃×Q₈	1	2	yes	—
143	C₂³	C₂×C₃×Q₈	1	2	yes	—
144	C₂³	C₃×#41	1	2	yes	—
145	C₂³	C₃×#41	1	2	yes	—
146	C₂³	C₂⁴×C₃	1	1	yes	characteristic
147	C₂×C₄	—	2	16	no	—
148	C₂×C₄	—	2	8	no	—
149	C₂×C₄	—	2	8	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	8	no	—
160	C₂×C₄	C₃×C₄⋊C₄	1	16	yes	complemented
161	C₂×C₄	C₂²×C₃×C₄	1	8	yes	—
162	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
163	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
164	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
165	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
166	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
167	C₂×C₃	#389	1	8	yes	direct product factor
168	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
171	C₂×C₃	C₂×#121	1	1	yes	characteristic
172	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
173	C₂×C₃	C₂×#125	1	1	yes	characteristic
174	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
175	C₂²	C₂×C₃×C₄⋊C₄	1	4	yes	—
176	C₂²	C₃×C₄×D₄	1	4	yes	—
177	C₂²	C₃×C₄×Q₈	1	4	yes	—
178	C₂²	C₃×#121	1	4	yes	—
179	C₂²	C₃×C₂²⋊Q₈	1	4	yes	—
180	C₂²	C₃×#125	1	4	yes	—
181	C₂²	C₃×C₄⋊Q₈	1	4	yes	—
182	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
183	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
184	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
185	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
186	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
187	C₂²	C₂×C₃×#41	1	1	yes	characteristic
188	C₂²	C₂×C₃×#41	1	1	yes	characteristic
189	C₄	—	2	8	no	—
190	C₄	—	2	4	no	—
191	C₄	—	2	4	no	—
192	C₄	—	4	4	no	—
193	C₄	C₂×C₃×C₄⋊C₄	1	8	yes	complemented
194	C₃	C₂×#389	1	1	yes	characteristic, direct product factor, Sylow
195	C₂	C₃×#389	1	8	yes	direct product factor
196	C₂	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
197	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
198	C₂	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
199	C₂	C₂×C₃×#121	1	1	yes	characteristic
200	C₂	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
201	C₂	C₂×C₃×#125	1	1	yes	characteristic
202	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
5. C₃×#389	C₂	direct (N × H)
7. C₂×#389	C₃	direct (N × H)
15. C₂×C₃×C₄⋊C₄	C₄	semidirect
26. #389	C₂×C₃	direct (N × H)
51. C₃×C₄⋊C₄	C₂×C₄	semidirect
60. C₂×C₄⋊C₄	C₃×C₄	semidirect
88. C₂×C₃×C₄	C₄⋊C₄	semidirect
118. C₄⋊C₄	C₂×C₃×C₄	semidirect
138. C₃×C₄	C₂×C₄⋊C₄	semidirect
160. C₂×C₄	C₃×C₄⋊C₄	semidirect
167. C₂×C₃	#389	direct (N × H)
193. C₄	C₂×C₃×C₄⋊C₄	semidirect
194. C₃	C₂×#389	direct (N × H)
195. C₂	C₃×#389	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄⋊C₄	2
C₂²×C₃×C₄²	2
C₃×#389	2
C₂×C₃×#95	2
C₂×#389	3

Subgroups — C2×C3×#389 #84155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#389 #84155