Subgroups — C₂×C₃×#400 #84258

Order 384.

604 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	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#400	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×#95	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃×#95	C₂	1	2	yes	maximal
6	C₂×C₃×#95	C₂	1	2	yes	maximal
7	C₂×#400	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₃×C₄	C₂²	1	2	yes	—
9	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂³×C₃×C₄	C₂²	1	2	yes	—
11	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊C₄	—	2	4	no	—
14	C₂×C₃×C₄⋊C₄	—	2	4	no	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
17	C₃×#95	C₂²	1	4	yes	—
18	C₃×#95	C₂²	1	8	yes	—
19	C₃×#95	C₂²	1	8	yes	—
20	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
21	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
22	#400	C₂×C₃	1	8	yes	direct product factor
23	C₂×#95	C₂×C₃	1	2	yes	—
24	C₂×#95	C₂×C₃	1	2	yes	—
25	C₂×#95	C₂×C₃	1	1	yes	characteristic
26	C₂⁴×C₃	C₂³	1	1	yes	characteristic
27	C₂²×C₃×C₄	—	2	2	no	—
28	C₂²×C₃×C₄	—	2	2	no	—
29	C₂²×C₃×C₄	—	2	4	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	8	no	—
34	C₂²×C₃×C₄	—	2	4	no	—
35	C₂²×C₃×C₄	—	2	4	no	—
36	C₂²×C₃×C₄	—	2	4	no	—
37	C₂²×C₃×C₄	D₄	1	4	yes	—
38	C₂²×C₃×C₄	Q₈	1	4	yes	—
39	C₂²×C₃×C₄	C₂³	1	4	yes	—
40	C₂²×C₃×C₄	C₂³	1	2	yes	—
41	C₂²×C₃×C₄	C₂³	1	4	yes	—
42	C₂²×C₃×C₄	C₂³	1	2	yes	—
43	C₂²×C₃×C₄	C₂³	1	2	yes	—
44	C₃×C₄⋊C₄	—	2	8	no	—
45	C₃×C₄⋊C₄	—	2	8	no	—
46	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
47	C₂³×C₄	C₂²×C₃	1	2	yes	—
48	C₂³×C₄	C₂²×C₃	1	2	yes	—
49	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
50	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₄⋊C₄	—	2	4	no	—
52	C₂×C₄⋊C₄	—	2	4	no	—
53	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
54	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
55	#95	C₂²×C₃	1	4	yes	—
56	#95	C₂²×C₃	1	8	yes	—
57	#95	C₂²×C₃	1	8	yes	—
58	C₂³×C₃	C₂×D₄	1	2	yes	—
59	C₂³×C₃	C₂×Q₈	1	2	yes	—
60	C₂³×C₃	#41	1	2	yes	—
61	C₂³×C₃	#41	1	4	yes	—
62	C₂³×C₃	#41	1	4	yes	—
63	C₂³×C₃	C₂⁴	1	1	yes	characteristic
64	C₂×C₃×C₄	—	2	8	no	—
65	C₂×C₃×C₄	—	2	8	no	—
66	C₂×C₃×C₄	—	2	2	no	—
67	C₂×C₃×C₄	—	2	2	no	—
68	C₂×C₃×C₄	—	2	4	no	—
69	C₂×C₃×C₄	—	2	2	no	—
70	C₂×C₃×C₄	—	2	4	no	—
71	C₂×C₃×C₄	—	2	4	no	—
72	C₂×C₃×C₄	—	2	8	no	—
73	C₂×C₃×C₄	—	2	4	no	—
74	C₂×C₃×C₄	—	2	4	no	—
75	C₂×C₃×C₄	—	2	4	no	—
76	C₂×C₃×C₄	—	4	4	no	—
77	C₂×C₃×C₄	—	4	8	no	—
78	C₂×C₃×C₄	—	4	8	no	—
79	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
80	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
81	C₂⁴	C₂³×C₃	1	1	yes	characteristic
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	2	4	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	2	4	no	—
87	C₂²×C₄	—	2	4	no	—
88	C₂²×C₄	—	2	8	no	—
89	C₂²×C₄	—	2	4	no	—
90	C₂²×C₄	—	2	4	no	—
91	C₂²×C₄	—	2	4	no	—
92	C₂²×C₄	C₃×D₄	1	4	yes	—
93	C₂²×C₄	C₃×Q₈	1	4	yes	—
94	C₂²×C₄	C₂³×C₃	1	4	yes	—
95	C₂²×C₄	C₂³×C₃	1	2	yes	—
96	C₂²×C₄	C₂³×C₃	1	4	yes	—
97	C₂²×C₄	C₂³×C₃	1	2	yes	—
98	C₂²×C₄	C₂³×C₃	1	2	yes	—
99	C₄⋊C₄	—	2	8	no	—
100	C₄⋊C₄	—	2	8	no	—
101	C₂²×C₃	C₂²⋊Q₈	1	4	yes	—
102	C₂²×C₃	#123	1	4	yes	—
103	C₂²×C₃	#124	1	4	yes	—
104	C₂²×C₃	#125	1	8	yes	—
105	C₂²×C₃	#126	1	8	yes	—
106	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
107	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
108	C₂²×C₃	C₂×#41	1	1	yes	characteristic
109	C₂²×C₃	C₂×#41	1	2	yes	—
110	C₂²×C₃	C₂×#41	1	2	yes	—
111	C₃×C₄	—	2	4	no	—
112	C₃×C₄	—	2	4	no	—
113	C₃×C₄	—	4	2	no	—
114	C₃×C₄	—	4	4	no	—
115	C₃×C₄	—	4	4	no	—
116	C₂³	C₂×C₃×D₄	1	2	yes	—
117	C₂³	C₂×C₃×Q₈	1	2	yes	—
118	C₂³	C₃×#41	1	2	yes	—
119	C₂³	C₃×#41	1	4	yes	—
120	C₂³	C₃×#41	1	4	yes	—
121	C₂³	C₂⁴×C₃	1	1	yes	characteristic
122	C₂×C₄	—	2	8	no	—
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	4	8	no	—
136	C₂×C₄	—	4	8	no	—
137	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
138	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
139	C₂×C₃	#400	1	8	yes	direct product factor
140	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₃	C₂×#123	1	1	yes	characteristic
142	C₂×C₃	C₂×#124	1	1	yes	characteristic
143	C₂×C₃	C₂×#125	1	2	yes	—
144	C₂×C₃	C₂×#126	1	2	yes	—
145	C₂²	C₃×C₂²⋊Q₈	1	4	yes	—
146	C₂²	C₃×#123	1	4	yes	—
147	C₂²	C₃×#124	1	4	yes	—
148	C₂²	C₃×#125	1	8	yes	—
149	C₂²	C₃×#126	1	8	yes	—
150	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
151	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
152	C₂²	C₂×C₃×#41	1	1	yes	characteristic
153	C₂²	C₂×C₃×#41	1	2	yes	—
154	C₂²	C₂×C₃×#41	1	2	yes	—
155	C₄	—	2	4	no	—
156	C₄	—	2	4	no	—
157	C₄	—	4	2	no	—
158	C₄	—	4	4	no	—
159	C₄	—	4	4	no	—
160	C₃	C₂×#400	1	1	yes	characteristic, direct product factor, Sylow
161	C₂	C₃×#400	1	8	yes	direct product factor
162	C₂	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
163	C₂	C₂×C₃×#123	1	1	yes	characteristic
164	C₂	C₂×C₃×#124	1	1	yes	characteristic
165	C₂	C₂×C₃×#125	1	2	yes	—
166	C₂	C₂×C₃×#126	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃×#400	C₂	direct (N × H)
7. C₂×#400	C₃	direct (N × H)
22. #400	C₂×C₃	direct (N × H)
139. C₂×C₃	#400	direct (N × H)
160. C₃	C₂×#400	direct (N × H)
161. C₂	C₃×#400	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
C₃×#400	2
C₂×C₃×#95	2
C₂×#400	3

Subgroups — C2×C3×#400 #84258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#400 #84258