Subgroups — C₂×C₃×#398 #84255

Order 384.

692 conjugacy classes of subgroups.

The subgroup lattice has 176 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	C₂²×C₃×C₄⋊C₄	C₂	1	2	yes	maximal
4	C₃×#398	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#95	C₂	1	2	yes	maximal
6	C₂×C₃×#95	C₂	1	1	yes	maximal, characteristic
7	C₂×#398	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
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	2	yes	—
13	C₂×C₃×C₄⋊C₄	—	2	4	no	—
14	C₂×C₃×C₄⋊C₄	—	2	8	no	—
15	C₂×C₃×C₄⋊C₄	—	2	4	no	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	8	yes	—
19	C₃×#95	C₂²	1	8	yes	—
20	C₃×#95	C₂²	1	4	yes	—
21	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
22	C₂²×C₄⋊C₄	C₂×C₃	1	2	yes	—
23	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
24	#398	C₂×C₃	1	8	yes	direct product factor
25	C₂×#95	C₂×C₃	1	2	yes	—
26	C₂×#95	C₂×C₃	1	1	yes	characteristic
27	C₂⁴×C₃	C₂³	1	1	yes	characteristic
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	4	no	—
34	C₂²×C₃×C₄	—	2	4	no	—
35	C₂²×C₃×C₄	—	2	2	no	—
36	C₂²×C₃×C₄	—	2	4	no	—
37	C₂²×C₃×C₄	—	2	4	no	—
38	C₂²×C₃×C₄	D₄	1	4	yes	—
39	C₂²×C₃×C₄	D₄	1	4	yes	—
40	C₂²×C₃×C₄	Q₈	1	8	yes	—
41	C₂²×C₃×C₄	C₂³	1	2	yes	—
42	C₂²×C₃×C₄	C₂³	1	4	yes	—
43	C₂²×C₃×C₄	C₂³	1	2	yes	—
44	C₂²×C₃×C₄	C₂³	1	2	yes	—
45	C₂²×C₃×C₄	C₂³	1	4	yes	—
46	C₃×C₄⋊C₄	—	2	8	no	—
47	C₃×C₄⋊C₄	—	2	16	no	—
48	C₃×C₄⋊C₄	—	2	8	no	—
49	C₂³×C₄	C₂²×C₃	1	2	yes	—
50	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
51	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
52	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
53	C₂³×C₄	C₂²×C₃	1	2	yes	—
54	C₂×C₄⋊C₄	—	2	4	no	—
55	C₂×C₄⋊C₄	—	2	8	no	—
56	C₂×C₄⋊C₄	—	2	4	no	—
57	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
58	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
59	C₂×C₄⋊C₄	C₂²×C₃	1	8	yes	—
60	#95	C₂²×C₃	1	8	yes	—
61	#95	C₂²×C₃	1	4	yes	—
62	C₂³×C₃	C₂×D₄	1	2	yes	—
63	C₂³×C₃	C₂×D₄	1	2	yes	—
64	C₂³×C₃	C₂×Q₈	1	4	yes	—
65	C₂³×C₃	#41	1	2	yes	—
66	C₂³×C₃	#41	1	4	yes	—
67	C₂³×C₃	C₂⁴	1	1	yes	characteristic
68	C₂×C₃×C₄	—	2	8	no	—
69	C₂×C₃×C₄	—	2	16	no	—
70	C₂×C₃×C₄	—	2	8	no	—
71	C₂×C₃×C₄	—	2	2	no	—
72	C₂×C₃×C₄	—	2	4	no	—
73	C₂×C₃×C₄	—	2	2	no	—
74	C₂×C₃×C₄	—	2	4	no	—
75	C₂×C₃×C₄	—	2	4	no	—
76	C₂×C₃×C₄	—	2	4	no	—
77	C₂×C₃×C₄	—	2	4	no	—
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	—	2	4	no	—
80	C₂×C₃×C₄	—	2	4	no	—
81	C₂×C₃×C₄	—	4	8	no	—
82	C₂×C₃×C₄	—	4	4	no	—
83	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
84	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
85	C₂×C₃×C₄	C₂×Q₈	1	8	yes	—
86	C₂⁴	C₂³×C₃	1	1	yes	characteristic
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	2	4	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	2	4	no	—
91	C₂²×C₄	—	2	4	no	—
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	2	4	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	2	4	no	—
96	C₂²×C₄	—	2	4	no	—
97	C₂²×C₄	C₃×D₄	1	4	yes	—
98	C₂²×C₄	C₃×D₄	1	4	yes	—
99	C₂²×C₄	C₃×Q₈	1	8	yes	—
100	C₂²×C₄	C₂³×C₃	1	2	yes	—
101	C₂²×C₄	C₂³×C₃	1	4	yes	—
102	C₂²×C₄	C₂³×C₃	1	2	yes	—
103	C₂²×C₄	C₂³×C₃	1	2	yes	—
104	C₂²×C₄	C₂³×C₃	1	4	yes	—
105	C₄⋊C₄	—	2	8	no	—
106	C₄⋊C₄	—	2	16	no	—
107	C₄⋊C₄	—	2	8	no	—
108	C₂²×C₃	#121	1	4	yes	—
109	C₂²×C₃	C₂²⋊Q₈	1	8	yes	—
110	C₂²×C₃	#123	1	4	yes	—
111	C₂²×C₃	#125	1	8	yes	—
112	C₂²×C₃	C₄⋊Q₈	1	4	yes	—
113	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
114	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
115	C₂²×C₃	C₂²×Q₈	1	2	yes	—
116	C₂²×C₃	C₂×#41	1	1	yes	characteristic
117	C₂²×C₃	C₂×#41	1	2	yes	—
118	C₃×C₄	—	2	4	no	—
119	C₃×C₄	—	2	8	no	—
120	C₃×C₄	—	2	4	no	—
121	C₃×C₄	—	4	4	no	—
122	C₃×C₄	—	4	2	no	—
123	C₂³	C₂×C₃×D₄	1	2	yes	—
124	C₂³	C₂×C₃×D₄	1	2	yes	—
125	C₂³	C₂×C₃×Q₈	1	4	yes	—
126	C₂³	C₃×#41	1	2	yes	—
127	C₂³	C₃×#41	1	4	yes	—
128	C₂³	C₂⁴×C₃	1	1	yes	characteristic
129	C₂×C₄	—	2	8	no	—
130	C₂×C₄	—	2	16	no	—
131	C₂×C₄	—	2	8	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	8	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
145	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
146	C₂×C₄	C₂×C₃×Q₈	1	8	yes	—
147	C₂×C₃	#398	1	8	yes	direct product factor
148	C₂×C₃	C₂×#121	1	1	yes	characteristic
149	C₂×C₃	C₂×C₂²⋊Q₈	1	2	yes	—
150	C₂×C₃	C₂×#123	1	1	yes	characteristic
151	C₂×C₃	C₂×#125	1	2	yes	—
152	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	C₂²	C₃×#121	1	4	yes	—
154	C₂²	C₃×C₂²⋊Q₈	1	8	yes	—
155	C₂²	C₃×#123	1	4	yes	—
156	C₂²	C₃×#125	1	8	yes	—
157	C₂²	C₃×C₄⋊Q₈	1	4	yes	—
158	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
159	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
160	C₂²	C₂²×C₃×Q₈	1	2	yes	—
161	C₂²	C₂×C₃×#41	1	1	yes	characteristic
162	C₂²	C₂×C₃×#41	1	2	yes	—
163	C₄	—	2	4	no	—
164	C₄	—	2	8	no	—
165	C₄	—	2	4	no	—
166	C₄	—	4	4	no	—
167	C₄	—	4	2	no	—
168	C₃	C₂×#398	1	1	yes	characteristic, direct product factor, Sylow
169	C₂	C₃×#398	1	8	yes	direct product factor
170	C₂	C₂×C₃×#121	1	1	yes	characteristic
171	C₂	C₂×C₃×C₂²⋊Q₈	1	2	yes	—
172	C₂	C₂×C₃×#123	1	1	yes	characteristic
173	C₂	C₂×C₃×#125	1	2	yes	—
174	C₂	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic

	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
4. C₃×#398	C₂	direct (N × H)
7. C₂×#398	C₃	direct (N × H)
24. #398	C₂×C₃	direct (N × H)
147. C₂×C₃	#398	direct (N × H)
168. C₃	C₂×#398	direct (N × H)
169. C₂	C₃×#398	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×#398	2
C₂×C₃×#95	2
C₂×#398	3

Subgroups — C2×C3×#398 #84255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#398 #84255