Subgroups — C₃×#2298 #84162

Order 384.

524 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₃×#388	C₂	1	4	yes	maximal
3	C₃×#387	C₂	1	4	yes	maximal, complemented
4	C₃×C₄×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₂²⋊C₄	C₂	1	2	yes	maximal
6	C₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄³	C₂	1	1	yes	maximal, characteristic, complemented
8	#2298	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₃×#117	C₄	1	8	yes	complemented
11	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄²	C₂²	1	4	yes	—
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	1	yes	characteristic
20	C₃×#95	C₂²	1	4	yes	—
21	C₃×#95	C₂²	1	4	yes	—
22	C₃×#95	C₂²	1	4	yes	—
23	C₂×#117	C₂×C₃	1	1	yes	characteristic
24	#388	C₂×C₃	1	4	yes	—
25	#387	C₂×C₃	1	4	yes	complemented
26	C₄×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
27	C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	—
28	#376	C₂×C₃	1	1	yes	characteristic, complemented
29	C₄³	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂⁴×C₃	C₂³	1	1	yes	characteristic
31	C₂²×C₃×C₄	—	2	4	no	—
32	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
33	C₂²×C₃×C₄	C₂³	1	4	yes	—
34	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂²×C₃×C₄	C₂³	1	2	yes	—
36	C₂²×C₃×C₄	C₂³	1	2	yes	—
37	C₂²×C₃×C₄	C₂³	1	4	yes	—
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₃×C₄⋊C₄	—	2	8	no	—
40	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
41	C₃×C₂²⋊C₄	—	2	8	no	—
42	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
43	C₃×C₄²	—	2	8	no	—
44	C₃×C₄²	—	2	4	no	—
45	C₃×C₄²	—	2	4	no	—
46	C₃×C₄²	C₂×C₄	1	4	yes	—
47	C₃×C₄²	C₂×C₄	1	4	yes	—
48	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
49	#117	C₃×C₄	1	8	yes	complemented
50	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
51	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
52	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
53	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
54	C₂×C₄²	C₂²×C₃	1	2	yes	—
55	C₂×C₄²	C₂²×C₃	1	4	yes	—
56	C₂×C₄²	C₂²×C₃	1	2	yes	—
57	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
59	#95	C₂²×C₃	1	4	yes	—
60	#95	C₂²×C₃	1	4	yes	—
61	#95	C₂²×C₃	1	4	yes	—
62	C₂³×C₃	—	2	4	no	—
63	C₂³×C₃	—	2	2	no	—
64	C₂³×C₃	C₂²×C₄	1	2	yes	—
65	C₂³×C₃	C₂⁴	1	1	yes	characteristic
66	C₂×C₃×C₄	—	2	4	no	—
67	C₂×C₃×C₄	—	2	4	no	—
68	C₂×C₃×C₄	—	2	4	no	—
69	C₂×C₃×C₄	—	2	4	no	—
70	C₂×C₃×C₄	—	2	2	no	—
71	C₂×C₃×C₄	—	2	8	no	—
72	C₂×C₃×C₄	—	2	4	no	—
73	C₂×C₃×C₄	—	2	4	no	—
74	C₂×C₃×C₄	—	2	4	no	—
75	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
76	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
77	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
78	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
79	C₂×C₃×C₄	#41	1	8	yes	—
80	C₂×C₃×C₄	#41	1	4	yes	—
81	C₂⁴	C₂³×C₃	1	1	yes	characteristic
82	C₂²×C₄	—	2	4	no	—
83	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
84	C₂²×C₄	C₂³×C₃	1	4	yes	—
85	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
86	C₂²×C₄	C₂³×C₃	1	2	yes	—
87	C₂²×C₄	C₂³×C₃	1	2	yes	—
88	C₂²×C₄	C₂³×C₃	1	4	yes	—
89	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
90	C₄⋊C₄	—	2	8	no	—
91	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
92	C₂²⋊C₄	—	2	8	no	—
93	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
94	C₄²	—	2	8	no	—
95	C₄²	—	2	4	no	—
96	C₄²	—	2	4	no	—
97	C₄²	C₂×C₃×C₄	1	4	yes	—
98	C₄²	C₂×C₃×C₄	1	4	yes	—
99	C₂²×C₃	—	2	2	no	—
100	C₂²×C₃	—	2	4	no	—
101	C₂²×C₃	—	4	4	no	—
102	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
103	C₂²×C₃	C₂×#41	1	2	yes	—
104	C₂²×C₃	C₂×#41	1	4	yes	—
105	C₃×C₄	—	2	2	no	—
106	C₃×C₄	—	2	4	no	—
107	C₃×C₄	—	2	4	no	—
108	C₃×C₄	—	2	8	no	—
109	C₃×C₄	—	4	4	no	—
110	C₃×C₄	#117	1	4	yes	—
111	C₂³	—	2	4	no	—
112	C₂³	—	2	2	no	—
113	C₂³	C₂²×C₃×C₄	1	2	yes	—
114	C₂³	C₂⁴×C₃	1	1	yes	characteristic
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
125	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
126	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
127	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
128	C₂×C₄	C₃×#41	1	8	yes	—
129	C₂×C₄	C₃×#41	1	4	yes	—
130	C₂×C₃	—	4	2	no	—
131	C₂×C₃	C₂×#117	1	1	yes	characteristic
132	C₂×C₃	C₄×#41	1	2	yes	—
133	C₂×C₃	#529	1	4	yes	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	4	no	—
136	C₂²	—	4	4	no	—
137	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
138	C₂²	C₂×C₃×#41	1	2	yes	—
139	C₂²	C₂×C₃×#41	1	4	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	2	4	no	—
142	C₄	—	2	4	no	—
143	C₄	—	2	8	no	—
144	C₄	—	4	4	no	—
145	C₄	C₃×#117	1	4	yes	—
146	C₃	#2298	1	1	yes	characteristic, direct product factor, Sylow
147	C₂	—	4	2	no	—
148	C₂	C₂×C₃×#117	1	1	yes	characteristic
149	C₂	C₃×C₄×#41	1	2	yes	—
150	C₂	C₃×#529	1	4	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
socle	C₂³×C₃	C₂⁴
prime radical	C₂⁴×C₃	C₂³
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
3. C₃×#387	C₂	semidirect
4. C₃×C₄×C₄⋊C₄	C₂	semidirect
6. C₃×#376	C₂	semidirect
7. C₃×C₄³	C₂	semidirect
8. #2298	C₃	direct (N × H)
10. C₃×#117	C₄	semidirect
25. #387	C₂×C₃	semidirect
26. C₄×C₄⋊C₄	C₂×C₃	semidirect
28. #376	C₂×C₃	semidirect
29. C₄³	C₂×C₃	semidirect
49. #117	C₃×C₄	semidirect
146. C₃	#2298	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×#117	2
C₃×#388	2
C₃×#387	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
C₃×#376	2
C₃×C₄³	2
#2298	3

Subgroups — C3×#2298 #84162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2298 #84162