Subgroups — C₂×#5835 #87276

Order 384.

1512 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#1009	C₂	1	1	yes	maximal, characteristic, complemented
3	#5835	C₂	1	16	yes	maximal, direct product factor
4	C₂×#950	C₂	1	3	yes	maximal, complemented
5	C₂×#949	C₂	1	3	yes	maximal, complemented
6	C₂×#948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#932	C₂	1	3	yes	maximal, complemented
8	C₂²×#213	C₂	1	3	yes	maximal, complemented
9	C₂×#576	—	3	1	no	maximal, Sylow
10	C₃×H(5,𝔽₂)	—	2	4	no	—
11	C₃×H(5,𝔽₂)	C₂²	1	8	yes	complemented
12	C₂×C₃×#41	—	2	1	no	—
13	C₂×C₃×#41	C₂²	1	1	yes	characteristic
14	C₂×C₃×#41	C₂²	1	3	yes	complemented
15	C₂²×C₃×D₄	—	2	3	no	—
16	C₂²×C₃×D₄	C₂²	1	3	yes	complemented
17	#1009	—	2	4	no	—
18	#1009	C₂²	1	8	yes	complemented
19	C₂×#239	—	2	1	no	—
20	C₂×D₃×D₄	—	2	3	no	—
21	C₂×#235	C₂²	1	3	yes	complemented
22	C₂²×D₁₂	C₂²	1	3	yes	complemented
23	#950	C₂²	1	24	yes	complemented
24	#949	C₂²	1	24	yes	complemented
25	#948	C₂²	1	8	yes	complemented
26	C₂×#216	C₂²	1	1	yes	characteristic, complemented
27	C₂×#215	C₂²	1	3	yes	complemented
28	C₂×#214	C₂²	1	3	yes	complemented
29	#932	C₂²	1	24	yes	complemented
30	C₂×#213	C₂²	1	3	yes	complemented
31	C₂×#213	C₂²	1	6	yes	complemented
32	C₂×#213	C₂²	1	24	yes	complemented
33	C₂×#208	C₂²	1	3	yes	complemented
34	C₂²×C₃⋊C₈	C₂²	1	3	yes	complemented
35	C₂×H(5,𝔽₂)	—	3	1	no	—
36	C₂×H(5,𝔽₂)	D₃	1	1	yes	characteristic, complemented
37	#576	—	3	16	no	—
38	C₂×AGL(1,ℤ/8)	—	3	3	no	—
39	C₂×AGL(1,ℤ/8)	—	3	3	no	—
40	C₂×#135	—	3	3	no	—
41	C₂²×D₈	—	3	3	no	—
42	C₂×#131	—	3	1	no	—
43	C₂⁴×C₃	—	2	3	no	—
44	C₂³×D₃	—	2	3	no	—
45	C₃×#41	—	2	8	no	—
46	C₃×#41	—	2	6	no	—
47	C₃×#41	D₄	1	4	yes	—
48	C₃×#41	C₂³	1	4	yes	—
49	C₃×#41	C₂³	1	12	yes	complemented
50	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
51	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
52	C₂×C₃×D₄	—	2	3	no	—
53	C₂×C₃×D₄	—	2	24	no	—
54	C₂×C₃×D₄	—	2	3	no	—
55	C₂×C₃×D₄	—	2	6	no	—
56	C₂×C₃×D₄	D₄	1	3	yes	—
57	C₂×C₃×D₄	C₂³	1	3	yes	—
58	C₂×C₃×D₄	C₂³	1	12	yes	complemented
59	C₂²×C₃×C₄	—	2	3	no	—
60	C₂²×C₃×C₄	C₂³	1	3	yes	—
61	C₂×#67	—	2	3	no	—
62	#239	—	2	8	no	—
63	D₃×D₄	—	2	24	no	—
64	#235	—	2	6	no	—
65	#235	C₂³	1	12	yes	complemented
66	C₂×D₁₂	—	2	3	no	—
67	C₂×D₁₂	—	2	6	no	—
68	C₂×D₁₂	C₂³	1	3	yes	—
69	C₂×D₁₂	C₂³	1	12	yes	complemented
70	C₂×C₄×D₃	—	2	3	no	—
71	C₂×Dic₆	C₂³	1	1	yes	characteristic
72	#216	C₂³	1	4	yes	complemented
73	#215	C₂³	1	12	yes	complemented
74	#214	C₂³	1	12	yes	complemented
75	#213	C₂³	1	12	yes	complemented
76	#213	C₂³	1	24	yes	complemented
77	#208	C₂³	1	12	yes	complemented
78	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₃⋊C₈	C₂³	1	3	yes	—
80	C₂×C₃⋊C₈	C₂³	1	12	yes	complemented
81	H(5,𝔽₂)	—	2	4	no	—
82	H(5,𝔽₂)	—	3	8	no	—
83	H(5,𝔽₂)	—	6	4	no	—
84	H(5,𝔽₂)	C₂×D₃	1	8	yes	complemented
85	C₂×#41	—	2	1	no	—
86	C₂×#41	—	3	3	no	—
87	C₂×#41	—	6	1	no	—
88	C₂×#41	C₂×D₃	1	1	yes	characteristic
89	C₂×#41	C₂×D₃	1	3	yes	complemented
90	C₂²×D₄	—	2	3	no	—
91	C₂²×D₄	—	3	3	no	—
92	C₂²×D₄	—	6	3	no	—
93	C₂²×D₄	C₂×D₃	1	3	yes	complemented
94	AGL(1,ℤ/8)	—	3	24	no	—
95	AGL(1,ℤ/8)	—	3	24	no	—
96	#135	—	3	24	no	—
97	C₂×Q₁₆	—	3	1	no	—
98	C₂×SD₁₆	—	3	3	no	—
99	C₂×SD₁₆	—	3	3	no	—
100	C₂×D₈	—	3	3	no	—
101	C₂×D₈	—	3	6	no	—
102	C₂×D₈	—	3	24	no	—
103	#131	—	3	8	no	—
104	C₂×M₁₆	—	3	3	no	—
105	C₂²×C₈	—	3	3	no	—
106	C₂³×C₃	—	2	3	no	—
107	C₂³×C₃	—	2	12	no	—
108	C₂³×C₃	—	8	6	no	—
109	C₂³×C₃	C₂×D₄	1	3	yes	—
110	C₂²×D₃	—	2	3	no	—
111	C₂²×D₃	—	2	12	no	—
112	C₂²×D₃	—	8	6	no	—
113	C₃×Q₈	—	2	1	no	—
114	C₃×Q₈	C₂×D₄	1	4	yes	—
115	C₃×Q₈	C₂⁴	1	2	yes	—
116	C₃×D₄	—	2	12	no	—
117	C₃×D₄	—	2	3	no	—
118	C₃×D₄	—	2	12	no	—
119	C₃×D₄	C₂×D₄	1	12	yes	—
120	C₃×D₄	C₂⁴	1	6	yes	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	12	no	—
123	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
124	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
125	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂⁴	1	6	yes	—
127	#67	—	2	12	no	—
128	C₂×Dic₃	—	2	1	no	—
129	D₁₂	—	2	3	no	—
130	D₁₂	—	2	12	no	—
131	D₁₂	C₂⁴	1	6	yes	—
132	C₄×D₃	—	2	12	no	—
133	Dic₆	—	2	1	no	—
134	Dic₆	C₂⁴	1	2	yes	—
135	C₃⋊C₈	C₂⁴	1	2	yes	—
136	C₃⋊C₈	C₂⁴	1	6	yes	—
137	C₂⁴	—	2	3	no	—
138	C₂⁴	—	6	3	no	—
139	#41	—	2	8	no	—
140	#41	—	2	6	no	—
141	#41	—	3	12	no	—
142	#41	—	6	8	no	—
143	#41	—	6	6	no	—
144	#41	#67	1	4	yes	—
145	#41	C₂²×D₃	1	4	yes	—
146	#41	C₂²×D₃	1	12	yes	complemented
147	C₂×Q₈	—	3	1	no	—
148	C₂×Q₈	#67	1	1	yes	characteristic
149	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
150	C₂×D₄	—	2	3	no	—
151	C₂×D₄	—	2	24	no	—
152	C₂×D₄	—	2	3	no	—
153	C₂×D₄	—	2	6	no	—
154	C₂×D₄	—	3	3	no	—
155	C₂×D₄	—	3	12	no	—
156	C₂×D₄	—	6	3	no	—
157	C₂×D₄	—	6	24	no	—
158	C₂×D₄	—	6	3	no	—
159	C₂×D₄	—	6	6	no	—
160	C₂×D₄	#67	1	3	yes	—
161	C₂×D₄	C₂²×D₃	1	3	yes	—
162	C₂×D₄	C₂²×D₃	1	12	yes	complemented
163	C₂²×C₄	—	2	3	no	—
164	C₂²×C₄	—	6	3	no	—
165	C₂²×C₄	C₂²×D₃	1	3	yes	—
166	Q₁₆	—	3	4	no	—
167	SD₁₆	—	3	12	no	—
168	SD₁₆	—	3	12	no	—
169	D₈	—	3	12	no	—
170	D₈	—	3	24	no	—
171	M₁₆	—	3	12	no	—
172	C₂×C₈	—	3	1	no	—
173	C₂×C₈	—	3	3	no	—
174	C₂×C₈	—	3	12	no	—
175	C₂²×C₃	—	2	6	no	—
176	C₂²×C₃	—	2	6	no	—
177	C₂²×C₃	—	4	6	no	—
178	C₂²×C₃	—	8	12	no	—
179	C₂²×C₃	C₂²×D₄	1	6	yes	—
180	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
181	C₂×D₃	—	2	6	no	—
182	C₂×D₃	—	4	6	no	—
183	C₂×D₃	—	8	12	no	—
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
186	C₃×C₄	C₂²×D₄	1	6	yes	—
187	C₃×C₄	C₂⁵	1	1	yes	characteristic
188	Dic₃	—	2	2	no	—
189	C₂³	—	2	3	no	—
190	C₂³	—	2	12	no	—
191	C₂³	—	6	3	no	—
192	C₂³	—	6	12	no	—
193	C₂³	—	8	6	no	—
194	C₂³	—	24	6	no	—
195	C₂³	C₂×#67	1	3	yes	—
196	Q₈	—	2	1	no	—
197	Q₈	—	3	2	no	—
198	Q₈	—	6	1	no	—
199	Q₈	C₂×#67	1	4	yes	—
200	Q₈	C₂³×D₃	1	2	yes	—
201	D₄	—	2	12	no	—
202	D₄	—	2	3	no	—
203	D₄	—	2	12	no	—
204	D₄	—	3	6	no	—
205	D₄	—	6	12	no	—
206	D₄	—	6	12	no	—
207	D₄	—	6	3	no	—
208	D₄	C₂×#67	1	12	yes	—
209	D₄	C₂³×D₃	1	6	yes	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	12	no	—
212	C₂×C₄	—	6	1	no	—
213	C₂×C₄	—	6	12	no	—
214	C₂×C₄	C₂×#67	1	6	yes	—
215	C₂×C₄	C₂×#67	1	3	yes	—
216	C₂×C₄	C₂³×D₃	1	6	yes	—
217	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
218	C₈	—	3	2	no	—
219	C₈	—	3	6	no	—
220	C₂×C₃	—	2	6	no	—
221	C₂×C₃	—	4	6	no	—
222	C₂×C₃	#576	1	2	yes	complemented
223	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
224	D₃	—	4	6	no	—
225	C₂²	—	2	6	no	—
226	C₂²	—	2	6	no	—
227	C₂²	—	4	6	no	—
228	C₂²	—	6	6	no	—
229	C₂²	—	8	12	no	—
230	C₂²	—	12	6	no	—
231	C₂²	—	24	12	no	—
232	C₂²	C₂²×#67	1	6	yes	—
233	C₂²	C₂²×#67	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	6	2	no	—
236	C₄	C₂²×#67	1	6	yes	—
237	C₄	C₂²×#67	1	1	yes	characteristic
238	C₄	C₂⁴×D₃	1	1	yes	characteristic
239	C₃	C₂×#576	1	1	yes	characteristic, complemented, Sylow
240	C₂	—	2	6	no	—
241	C₂	—	4	6	no	—
242	C₂	—	12	6	no	—
243	C₂	#5835	1	2	yes	direct product factor
244	C₂	C₂³×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#67
commutator, cosocle	C₃×C₄	C₂⁵
Frattini	C₄	C₂⁴×D₃
Fitting	C₂×C₃×H(5,𝔽₂)	C₂
socle	C₂²×C₃	C₂²×D₄
hypercenter	C₂×H(5,𝔽₂)	D₃
hypocenter	C₃	C₂×#576
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×H(5,𝔽₂)	C₂	semidirect
2. C₂×#1009	C₂	semidirect
3. #5835	C₂	direct (N × H)
4. C₂×#950	C₂	semidirect
5. C₂×#949	C₂	semidirect
6. C₂×#948	C₂	semidirect
7. C₂×#932	C₂	semidirect
8. C₂²×#213	C₂	semidirect
11. C₃×H(5,𝔽₂)	C₂²	semidirect
14. C₂×C₃×#41	C₂²	semidirect
16. C₂²×C₃×D₄	C₂²	semidirect
18. #1009	C₂²	semidirect
21. C₂×#235	C₂²	semidirect
22. C₂²×D₁₂	C₂²	semidirect
23. #950	C₂²	semidirect
24. #949	C₂²	semidirect
25. #948	C₂²	semidirect
26. C₂×#216	C₂²	semidirect
27. C₂×#215	C₂²	semidirect
28. C₂×#214	C₂²	semidirect
29. #932	C₂²	semidirect
30. C₂×#213	C₂²	semidirect
31. C₂×#213	C₂²	semidirect
32. C₂×#213	C₂²	semidirect
33. C₂×#208	C₂²	semidirect
34. C₂²×C₃⋊C₈	C₂²	semidirect
36. C₂×H(5,𝔽₂)	D₃	semidirect
49. C₃×#41	C₂³	semidirect
58. C₂×C₃×D₄	C₂³	semidirect
65. #235	C₂³	semidirect
69. C₂×D₁₂	C₂³	semidirect
72. #216	C₂³	semidirect
73. #215	C₂³	semidirect
74. #214	C₂³	semidirect
75. #213	C₂³	semidirect
76. #213	C₂³	semidirect
77. #208	C₂³	semidirect
80. C₂×C₃⋊C₈	C₂³	semidirect
84. H(5,𝔽₂)	C₂×D₃	semidirect
89. C₂×#41	C₂×D₃	semidirect
93. C₂²×D₄	C₂×D₃	semidirect
146. #41	C₂²×D₃	semidirect
162. C₂×D₄	C₂²×D₃	semidirect
222. C₂×C₃	#576	semidirect
239. C₃	C₂×#576	semidirect
243. C₂	#5835	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×H(5,𝔽₂)	2
C₂×#1009	2
#5835	2
C₂×#950	2
C₂×#949	2
C₂×#948	2
C₂×#932	2
C₂²×#213	2
C₂×#576	3

Subgroups — C2×#5835 #87276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#5835 #87276