Subgroups — C₂×C₃×#554 #87356

Order 384.

1200 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#554	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#125	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×C₄×D₄	C₂	1	2	yes	maximal
8	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₃×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
10	C₂×#554	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×Q₈	C₂²	1	2	yes	—
13	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₄	C₂²	1	2	yes	—
15	C₂³×C₃×C₄	C₂²	1	4	yes	—
16	C₃×C₄⋊Q₈	C₂²	1	8	yes	complemented
17	C₃×#125	C₂²	1	16	yes	complemented
18	C₃×C₂²⋊Q₈	C₂²	1	32	yes	complemented
19	C₃×C₂²⋊Q₈	C₂²	1	16	yes	complemented
20	C₃×C₄×Q₈	C₂²	1	8	yes	complemented
21	C₃×C₄×D₄	C₂²	1	16	yes	—
22	C₃×C₄×D₄	C₂²	1	8	yes	—
23	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
25	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
28	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
29	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
30	C₂×C₃×C₄⋊C₄	C₂²	1	16	yes	complemented
31	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
32	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
33	C₂×C₃×C₄²	C₂²	1	2	yes	—
34	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
35	#554	C₂×C₃	1	16	yes	direct product factor
36	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂×#125	C₂×C₃	1	2	yes	complemented
38	C₂×C₂²⋊Q₈	C₂×C₃	1	4	yes	complemented
39	C₂×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
40	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×C₄×D₄	C₂×C₃	1	2	yes	—
42	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
43	C₂²×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
44	C₂⁴×C₃	C₂³	1	2	yes	—
45	C₂×C₃×Q₈	—	2	4	no	—
46	C₂×C₃×Q₈	—	4	4	no	—
47	C₂×C₃×Q₈	C₂³	1	4	yes	—
48	C₂×C₃×Q₈	C₂³	1	8	yes	—
49	C₂×C₃×D₄	Q₈	1	8	yes	complemented
50	C₂×C₃×D₄	C₂³	1	4	yes	—
51	C₂²×C₃×C₄	—	2	8	no	—
52	C₂²×C₃×C₄	—	4	8	no	—
53	C₂²×C₃×C₄	C₂³	1	8	yes	—
54	C₂²×C₃×C₄	C₂³	1	2	yes	—
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂³	1	2	yes	—
57	C₂²×C₃×C₄	C₂³	1	2	yes	—
58	C₂²×C₃×C₄	C₂³	1	4	yes	—
59	C₂²×C₃×C₄	C₂³	1	16	yes	—
60	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
62	C₃×C₄⋊C₄	C₂³	1	4	yes	—
63	C₃×C₄⋊C₄	C₂³	1	16	yes	—
64	C₃×C₄⋊C₄	C₂³	1	8	yes	—
65	C₃×C₄⋊C₄	C₂³	1	4	yes	—
66	C₃×C₄⋊C₄	C₂³	1	8	yes	—
67	C₃×C₄⋊C₄	C₂³	1	8	yes	—
68	C₃×C₄⋊C₄	C₂³	1	16	yes	—
69	C₃×C₂²⋊C₄	C₂³	1	16	yes	—
70	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
71	C₃×C₄²	C₂³	1	8	yes	—
72	C₃×C₄²	C₂³	1	4	yes	—
73	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
74	C₂²×Q₈	C₂²×C₃	1	2	yes	—
75	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
76	C₂³×C₄	C₂²×C₃	1	2	yes	—
77	C₂³×C₄	C₂²×C₃	1	4	yes	—
78	C₄⋊Q₈	C₂²×C₃	1	8	yes	complemented
79	#125	C₂²×C₃	1	16	yes	complemented
80	C₂²⋊Q₈	C₂²×C₃	1	32	yes	complemented
81	C₂²⋊Q₈	C₂²×C₃	1	16	yes	complemented
82	C₄×Q₈	C₂²×C₃	1	8	yes	complemented
83	C₄×D₄	C₂²×C₃	1	16	yes	—
84	C₄×D₄	C₂²×C₃	1	8	yes	—
85	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
87	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
88	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
90	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
91	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
92	C₂×C₄⋊C₄	C₂²×C₃	1	16	yes	complemented
93	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
94	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
95	C₂×C₄²	C₂²×C₃	1	2	yes	—
96	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
97	C₂³×C₃	—	2	4	no	—
98	C₂³×C₃	—	2	4	no	—
99	C₂³×C₃	C₂×Q₈	1	8	yes	—
100	C₂³×C₃	C₂⁴	1	4	yes	—
101	C₂³×C₃	C₂⁴	1	1	yes	characteristic
102	C₃×Q₈	—	2	8	no	—
103	C₃×Q₈	—	4	8	no	—
104	C₃×D₄	C₂×Q₈	1	16	yes	complemented
105	C₂×C₃×C₄	—	2	4	no	—
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	—	2	4	no	—
108	C₂×C₃×C₄	—	2	8	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	16	no	—
111	C₂×C₃×C₄	—	4	16	no	—
112	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
113	C₂×C₃×C₄	#41	1	4	yes	—
114	C₂×C₃×C₄	C₂⁴	1	4	yes	—
115	C₂×C₃×C₄	C₂⁴	1	2	yes	—
116	C₂×C₃×C₄	C₂⁴	1	4	yes	—
117	C₂×C₃×C₄	C₂⁴	1	4	yes	—
118	C₂×C₃×C₄	C₂⁴	1	8	yes	—
119	C₂×C₃×C₄	C₂⁴	1	2	yes	—
120	C₂×C₃×C₄	C₂⁴	1	2	yes	—
121	C₂⁴	C₂³×C₃	1	2	yes	—
122	C₂×Q₈	—	2	4	no	—
123	C₂×Q₈	—	4	4	no	—
124	C₂×Q₈	C₂³×C₃	1	4	yes	—
125	C₂×Q₈	C₂³×C₃	1	8	yes	—
126	C₂×D₄	C₃×Q₈	1	8	yes	complemented
127	C₂×D₄	C₂³×C₃	1	4	yes	—
128	C₂²×C₄	—	2	8	no	—
129	C₂²×C₄	—	4	8	no	—
130	C₂²×C₄	C₂³×C₃	1	2	yes	—
131	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
132	C₂²×C₄	C₂³×C₃	1	2	yes	—
133	C₂²×C₄	C₂³×C₃	1	2	yes	—
134	C₂²×C₄	C₂³×C₃	1	4	yes	—
135	C₂²×C₄	C₂³×C₃	1	8	yes	—
136	C₂²×C₄	C₂³×C₃	1	16	yes	—
137	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
138	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
139	C₄⋊C₄	C₂³×C₃	1	4	yes	—
140	C₄⋊C₄	C₂³×C₃	1	16	yes	—
141	C₄⋊C₄	C₂³×C₃	1	8	yes	—
142	C₄⋊C₄	C₂³×C₃	1	4	yes	—
143	C₄⋊C₄	C₂³×C₃	1	8	yes	—
144	C₄⋊C₄	C₂³×C₃	1	8	yes	—
145	C₄⋊C₄	C₂³×C₃	1	16	yes	—
146	C₂²⋊C₄	C₂³×C₃	1	16	yes	—
147	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
148	C₄²	C₂³×C₃	1	8	yes	—
149	C₄²	C₂³×C₃	1	4	yes	—
150	C₂²×C₃	—	2	16	no	—
151	C₂²×C₃	—	2	4	no	—
152	C₂²×C₃	—	2	4	no	—
153	C₂²×C₃	C₂²×Q₈	1	8	yes	—
154	C₂²×C₃	C₂²×Q₈	1	2	yes	—
155	C₂²×C₃	C₂×#41	1	2	yes	—
156	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
157	C₂²×C₃	C₂⁵	1	1	yes	characteristic
158	C₃×C₄	—	2	4	no	—
159	C₃×C₄	—	2	4	no	—
160	C₃×C₄	—	2	4	no	—
161	C₃×C₄	—	2	8	no	—
162	C₃×C₄	—	2	2	no	—
163	C₃×C₄	C₂²×Q₈	1	4	yes	—
164	C₃×C₄	C₂×#41	1	4	yes	—
165	C₂³	—	2	4	no	—
166	C₂³	—	2	4	no	—
167	C₂³	C₂×C₃×Q₈	1	8	yes	—
168	C₂³	C₂⁴×C₃	1	4	yes	—
169	C₂³	C₂⁴×C₃	1	1	yes	characteristic
170	Q₈	—	2	8	no	—
171	Q₈	—	4	8	no	—
172	D₄	C₂×C₃×Q₈	1	16	yes	complemented
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	16	no	—
179	C₂×C₄	—	4	16	no	—
180	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
181	C₂×C₄	C₃×#41	1	4	yes	—
182	C₂×C₄	C₂⁴×C₃	1	4	yes	—
183	C₂×C₄	C₂⁴×C₃	1	2	yes	—
184	C₂×C₄	C₂⁴×C₃	1	4	yes	—
185	C₂×C₄	C₂⁴×C₃	1	4	yes	—
186	C₂×C₄	C₂⁴×C₃	1	8	yes	—
187	C₂×C₄	C₂⁴×C₃	1	2	yes	—
188	C₂×C₄	C₂⁴×C₃	1	2	yes	—
189	C₂×C₃	—	2	8	no	—
190	C₂×C₃	#554	1	4	yes	direct product factor
191	C₂×C₃	C₂³×Q₈	1	1	yes	characteristic
192	C₂×C₃	C₂²×#41	1	1	yes	characteristic
193	C₂×C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
194	C₂²	—	2	4	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	16	no	—
197	C₂²	C₂²×C₃×Q₈	1	8	yes	—
198	C₂²	C₂²×C₃×Q₈	1	2	yes	—
199	C₂²	C₂×C₃×#41	1	2	yes	—
200	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
201	C₂²	C₂⁵×C₃	1	1	yes	characteristic
202	C₄	—	2	4	no	—
203	C₄	—	2	4	no	—
204	C₄	—	2	4	no	—
205	C₄	—	2	8	no	—
206	C₄	—	2	2	no	—
207	C₄	C₂²×C₃×Q₈	1	4	yes	—
208	C₄	C₂×C₃×#41	1	4	yes	—
209	C₃	C₂×#554	1	1	yes	characteristic, direct product factor, Sylow
210	C₂	—	2	8	no	—
211	C₂	C₃×#554	1	4	yes	direct product factor
212	C₂	C₂³×C₃×Q₈	1	1	yes	characteristic
213	C₂	C₂²×C₃×#41	1	1	yes	characteristic
214	C₂	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
prime radical	C₂²×C₃×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#554	C₂	direct (N × H)
2. C₂×C₃×C₄⋊Q₈	C₂	semidirect
3. C₂×C₃×#125	C₂	semidirect
4. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
5. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
6. C₂×C₃×C₄×Q₈	C₂	semidirect
9. C₂²×C₃×C₄⋊C₄	C₂	semidirect
10. C₂×#554	C₃	direct (N × H)
16. C₃×C₄⋊Q₈	C₂²	semidirect
17. C₃×#125	C₂²	semidirect
18. C₃×C₂²⋊Q₈	C₂²	semidirect
19. C₃×C₂²⋊Q₈	C₂²	semidirect
20. C₃×C₄×Q₈	C₂²	semidirect
30. C₂×C₃×C₄⋊C₄	C₂²	semidirect
35. #554	C₂×C₃	direct (N × H)
36. C₂×C₄⋊Q₈	C₂×C₃	semidirect
37. C₂×#125	C₂×C₃	semidirect
38. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
39. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
40. C₂×C₄×Q₈	C₂×C₃	semidirect
43. C₂²×C₄⋊C₄	C₂×C₃	semidirect
49. C₂×C₃×D₄	Q₈	semidirect
78. C₄⋊Q₈	C₂²×C₃	semidirect
79. #125	C₂²×C₃	semidirect
80. C₂²⋊Q₈	C₂²×C₃	semidirect
81. C₂²⋊Q₈	C₂²×C₃	semidirect
82. C₄×Q₈	C₂²×C₃	semidirect
92. C₂×C₄⋊C₄	C₂²×C₃	semidirect
104. C₃×D₄	C₂×Q₈	semidirect
126. C₂×D₄	C₃×Q₈	semidirect
172. D₄	C₂×C₃×Q₈	semidirect
190. C₂×C₃	#554	direct (N × H)
209. C₃	C₂×#554	direct (N × H)
211. C₂	C₃×#554	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#554	2
C₂×C₃×C₄⋊Q₈	2
C₂×C₃×#125	2
C₂×C₃×C₂²⋊Q₈	2
C₂×C₃×C₄×Q₈	2
C₂×C₃×C₄×D₄	2
C₂²×C₃×C₄⋊C₄	2
C₂×#554	3

Subgroups — C2×C3×#554 #87356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#554 #87356