Subgroups — C₃×#2295 #84159

Order 384.

500 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	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#391	C₂	1	2	yes	maximal
3	C₃×#389	C₂	1	2	yes	maximal
4	C₃×#387	C₂	1	4	yes	maximal
5	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₃×C₄×C₄⋊C₄	C₂	1	2	yes	maximal
7	C₃×#376	C₂	1	2	yes	maximal
8	C₃×C₄³	C₂	1	1	yes	maximal, characteristic
9	#2295	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×Q₈	C₄	1	8	yes	complemented
12	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄²	C₂²	1	2	yes	—
17	C₂×C₃×C₄²	C₂²	1	2	yes	—
18	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄²	C₂²	1	2	yes	—
20	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄²	C₂²	1	2	yes	—
22	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
23	C₃×#95	C₂²	1	4	yes	—
24	C₃×#95	C₂²	1	4	yes	—
25	C₃×#95	C₂²	1	2	yes	—
26	C₃×#95	C₂²	1	4	yes	—
27	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
28	#391	C₂×C₃	1	2	yes	—
29	#389	C₂×C₃	1	2	yes	—
30	#387	C₂×C₃	1	4	yes	—
31	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
32	C₄×C₄⋊C₄	C₂×C₃	1	2	yes	—
33	#376	C₂×C₃	1	2	yes	—
34	C₄³	C₂×C₃	1	1	yes	characteristic
35	C₂×C₃×Q₈	—	2	4	no	—
36	C₂×C₃×Q₈	C₂×C₄	1	4	yes	—
37	C₂²×C₃×C₄	C₂³	1	4	yes	—
38	C₂²×C₃×C₄	C₂³	1	2	yes	—
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₂²×C₃×C₄	C₂³	1	2	yes	—
41	C₂²×C₃×C₄	C₂³	1	2	yes	—
42	C₂²×C₃×C₄	C₂³	1	2	yes	—
43	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
44	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₃×C₄⋊C₄	—	2	8	no	—
46	C₃×C₄⋊C₄	—	2	4	no	—
47	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
48	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
49	C₃×C₄²	—	2	4	no	—
50	C₃×C₄²	—	2	4	no	—
51	C₃×C₄²	—	2	2	no	—
52	C₃×C₄²	—	2	4	no	—
53	C₃×C₄²	C₂×C₄	1	8	yes	—
54	C₃×C₄²	C₂×C₄	1	4	yes	—
55	C₃×C₄²	Q₈	1	4	yes	complemented
56	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
57	C₄×Q₈	C₃×C₄	1	8	yes	complemented
58	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
59	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
60	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
61	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₄²	C₂²×C₃	1	2	yes	—
64	C₂×C₄²	C₂²×C₃	1	2	yes	—
65	C₂×C₄²	C₂²×C₃	1	2	yes	—
66	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₄²	C₂²×C₃	1	2	yes	—
68	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
69	#95	C₂²×C₃	1	4	yes	—
70	#95	C₂²×C₃	1	4	yes	—
71	#95	C₂²×C₃	1	2	yes	—
72	#95	C₂²×C₃	1	4	yes	—
73	C₂³×C₃	C₂⁴	1	1	yes	characteristic
74	C₃×Q₈	—	2	8	no	—
75	C₂×C₃×C₄	—	2	2	no	—
76	C₂×C₃×C₄	—	2	2	no	—
77	C₂×C₃×C₄	—	2	2	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	4	no	—
83	C₂×C₃×C₄	—	2	4	no	—
84	C₂×C₃×C₄	—	2	4	no	—
85	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
86	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
87	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
88	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
89	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
90	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
91	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
92	C₂×C₃×C₄	#41	1	4	yes	—
93	C₂×C₃×C₄	#41	1	4	yes	—
94	C₂×C₃×C₄	#41	1	2	yes	—
95	C₂×Q₈	—	2	4	no	—
96	C₂×Q₈	C₂×C₃×C₄	1	4	yes	—
97	C₂²×C₄	C₂³×C₃	1	4	yes	—
98	C₂²×C₄	C₂³×C₃	1	2	yes	—
99	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
100	C₂²×C₄	C₂³×C₃	1	2	yes	—
101	C₂²×C₄	C₂³×C₃	1	2	yes	—
102	C₂²×C₄	C₂³×C₃	1	2	yes	—
103	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
104	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
105	C₄⋊C₄	—	2	8	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
108	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
109	C₄²	—	2	4	no	—
110	C₄²	—	2	4	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	2	4	no	—
113	C₄²	C₂×C₃×C₄	1	8	yes	—
114	C₄²	C₂×C₃×C₄	1	4	yes	—
115	C₄²	C₃×Q₈	1	4	yes	complemented
116	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
117	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂²×C₃	C₂×#41	1	2	yes	—
119	C₂²×C₃	C₂×#41	1	1	yes	characteristic
120	C₂²×C₃	C₂×#41	1	2	yes	—
121	C₃×C₄	—	2	4	no	—
122	C₃×C₄	—	2	2	no	—
123	C₃×C₄	—	2	4	no	—
124	C₃×C₄	—	2	4	no	—
125	C₃×C₄	—	2	4	no	—
126	C₃×C₄	—	4	4	no	—
127	C₃×C₄	#117	1	4	yes	—
128	C₃×C₄	C₄×Q₈	1	4	yes	—
129	C₂³	C₂⁴×C₃	1	1	yes	characteristic
130	Q₈	—	2	8	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
142	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
143	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
144	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
145	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
146	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
147	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
148	C₂×C₄	C₃×#41	1	4	yes	—
149	C₂×C₄	C₃×#41	1	4	yes	—
150	C₂×C₄	C₃×#41	1	2	yes	—
151	C₂×C₃	C₂×#117	1	1	yes	characteristic
152	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
153	C₂×C₃	C₄×#41	1	1	yes	characteristic
154	C₂×C₃	#529	1	2	yes	—
155	C₂×C₃	#533	1	2	yes	—
156	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
157	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
158	C₂²	C₂×C₃×#41	1	2	yes	—
159	C₂²	C₂×C₃×#41	1	1	yes	characteristic
160	C₂²	C₂×C₃×#41	1	2	yes	—
161	C₄	—	2	4	no	—
162	C₄	—	2	2	no	—
163	C₄	—	2	4	no	—
164	C₄	—	2	4	no	—
165	C₄	—	2	4	no	—
166	C₄	—	4	4	no	—
167	C₄	C₃×#117	1	4	yes	—
168	C₄	C₃×C₄×Q₈	1	4	yes	—
169	C₃	#2295	1	1	yes	characteristic, direct product factor, Sylow
170	C₂	C₂×C₃×#117	1	1	yes	characteristic
171	C₂	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
172	C₂	C₃×C₄×#41	1	1	yes	characteristic
173	C₂	C₃×#529	1	2	yes	—
174	C₂	C₃×#533	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
9. #2295	C₃	direct (N × H)
11. C₃×C₄×Q₈	C₄	semidirect
55. C₃×C₄²	Q₈	semidirect
57. C₄×Q₈	C₃×C₄	semidirect
115. C₄²	C₃×Q₈	semidirect
169. C₃	#2295	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
C₃×#391	2
C₃×#389	2
C₃×#387	2
C₃×C₄×C₄⋊C₄	2
C₃×#376	2
C₃×C₄³	2
#2295	3

Subgroups — C3×#2295 #84159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2295 #84159