Subgroups — C₂×#5721 #86202

Order 384.

1048 conjugacy classes of subgroups.

The subgroup lattice has 196 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	C₂×Q₈×Dic₃	C₂	1	2	yes	maximal
3	C₂×#944	C₂	1	2	yes	maximal
4	#5721	C₂	1	16	yes	maximal, direct product factor
5	C₂×#890	C₂	1	2	yes	maximal
6	C₂×#889	C₂	1	4	yes	maximal
7	C₂×#887	C₂	1	2	yes	maximal
8	C₂×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
10	C₂×#557	—	3	1	no	maximal, Sylow
11	C₂²×C₃×Q₈	C₂²	1	2	yes	—
12	C₂²×Dic₆	C₂²	1	1	yes	characteristic
13	C₂²×Dic₆	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊Q₈	C₂²	1	8	yes	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
18	Q₈×Dic₃	C₂²	1	16	yes	—
19	#944	C₂²	1	16	yes	—
20	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
23	C₂×Dic₃⋊C₄	C₂²	1	2	yes	—
24	C₂×Dic₃⋊C₄	C₂²	1	2	yes	—
25	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
26	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
27	C₂×C₄×Dic₃	C₂²	1	2	yes	—
28	C₂×C₄×Dic₃	C₂²	1	2	yes	—
29	C₂×C₄×Dic₃	C₂²	1	4	yes	—
30	#890	C₂²	1	16	yes	—
31	#889	C₂²	1	32	yes	—
32	#887	C₂²	1	16	yes	—
33	C₄×Dic₆	C₂²	1	8	yes	—
34	C₄×Dic₆	C₂²	1	8	yes	—
35	#557	—	3	16	no	—
36	C₂×C₄⋊Q₈	—	3	2	no	—
37	C₂×C₄⋊Q₈	D₃	1	1	yes	characteristic
38	C₂×#125	—	3	4	no	—
39	C₂×#125	—	3	2	no	—
40	C₂×C₄×Q₈	—	3	2	no	—
41	C₂×C₄×Q₈	—	3	2	no	—
42	C₂×C₄×Q₈	—	3	1	no	—
43	C₂×C₄×Q₈	—	3	1	no	—
44	C₂×C₃×Q₈	—	2	8	no	—
45	C₂×C₃×Q₈	C₂³	1	8	yes	—
46	C₂²×C₃×C₄	C₂³	1	4	yes	—
47	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
49	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
50	C₂²×Dic₃	C₂³	1	2	yes	—
51	C₂²×Dic₃	C₂³	1	2	yes	—
52	C₂²×Dic₃	C₂³	1	2	yes	—
53	C₂²×Dic₃	C₂³	1	2	yes	—
54	C₂×Dic₆	—	2	4	no	—
55	C₂×Dic₆	Q₈	1	8	yes	complemented
56	C₂×Dic₆	C₂³	1	4	yes	—
57	C₂×Dic₆	C₂³	1	4	yes	—
58	C₃×C₄⋊C₄	C₂³	1	8	yes	—
59	C₃×C₄⋊C₄	C₂³	1	8	yes	—
60	C₃×C₄²	C₂³	1	4	yes	—
61	C₄⋊Dic₃	C₂³	1	4	yes	—
62	C₄⋊Dic₃	C₂³	1	4	yes	—
63	C₄⋊Dic₃	C₂³	1	16	yes	—
64	Dic₃⋊C₄	C₂³	1	8	yes	—
65	Dic₃⋊C₄	C₂³	1	8	yes	—
66	Dic₃⋊C₄	C₂³	1	16	yes	—
67	Dic₃⋊C₄	C₂³	1	16	yes	—
68	C₄×Dic₃	C₂³	1	8	yes	—
69	C₄×Dic₃	C₂³	1	8	yes	—
70	C₄×Dic₃	C₂³	1	16	yes	—
71	C₂²×Q₈	—	3	1	no	—
72	C₂²×Q₈	—	3	1	no	—
73	C₂²×Q₈	C₂×D₃	1	2	yes	—
74	C₄⋊Q₈	—	3	16	no	—
75	C₄⋊Q₈	C₂×D₃	1	8	yes	—
76	#125	—	3	32	no	—
77	#125	—	3	16	no	—
78	C₄×Q₈	—	3	16	no	—
79	C₄×Q₈	—	3	16	no	—
80	C₄×Q₈	—	3	8	no	—
81	C₄×Q₈	—	3	8	no	—
82	C₂×C₄⋊C₄	—	3	1	no	—
83	C₂×C₄⋊C₄	—	3	2	no	—
84	C₂×C₄⋊C₄	—	3	2	no	—
85	C₂×C₄⋊C₄	—	3	4	no	—
86	C₂×C₄⋊C₄	—	3	1	no	—
87	C₂×C₄⋊C₄	—	3	4	no	—
88	C₂×C₄⋊C₄	—	3	4	no	—
89	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
90	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
91	C₂×C₄²	—	3	4	no	—
92	C₂×C₄²	—	3	2	no	—
93	C₂×C₄²	—	3	2	no	—
94	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
95	C₂³×C₃	C₂⁴	1	1	yes	characteristic
96	C₃×Q₈	—	2	16	no	—
97	C₂×C₃×C₄	—	2	8	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
100	C₂×C₃×C₄	#41	1	4	yes	—
101	C₂×C₃×C₄	C₂⁴	1	8	yes	—
102	C₂×C₃×C₄	C₂⁴	1	2	yes	—
103	C₂×C₃×C₄	C₂⁴	1	2	yes	—
104	C₂×C₃×C₄	C₂⁴	1	2	yes	—
105	C₂×Dic₃	—	2	4	no	—
106	C₂×Dic₃	—	2	4	no	—
107	C₂×Dic₃	—	2	4	no	—
108	C₂×Dic₃	C₂×Q₈	1	8	yes	—
109	C₂×Dic₃	C₂⁴	1	4	yes	—
110	C₂×Dic₃	C₂⁴	1	4	yes	—
111	C₂×Dic₃	C₂⁴	1	4	yes	—
112	C₂×Dic₃	C₂⁴	1	4	yes	—
113	Dic₆	—	2	8	no	—
114	Dic₆	C₂×Q₈	1	16	yes	complemented
115	C₂×Q₈	—	2	8	no	—
116	C₂×Q₈	—	3	8	no	—
117	C₂×Q₈	—	3	4	no	—
118	C₂×Q₈	—	3	4	no	—
119	C₂×Q₈	—	6	4	no	—
120	C₂×Q₈	C₂²×D₃	1	8	yes	—
121	C₂²×C₄	—	3	2	no	—
122	C₂²×C₄	—	3	2	no	—
123	C₂²×C₄	—	3	2	no	—
124	C₂²×C₄	—	3	2	no	—
125	C₂²×C₄	C₂²×D₃	1	4	yes	—
126	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
129	C₄⋊C₄	—	3	4	no	—
130	C₄⋊C₄	—	3	8	no	—
131	C₄⋊C₄	—	3	8	no	—
132	C₄⋊C₄	—	3	16	no	—
133	C₄⋊C₄	—	3	4	no	—
134	C₄⋊C₄	—	3	16	no	—
135	C₄⋊C₄	—	3	16	no	—
136	C₄⋊C₄	C₂²×D₃	1	8	yes	—
137	C₄⋊C₄	C₂²×D₃	1	8	yes	—
138	C₄²	—	3	16	no	—
139	C₄²	—	3	8	no	—
140	C₄²	—	3	8	no	—
141	C₄²	C₂²×D₃	1	4	yes	—
142	C₂²×C₃	C₂²×Q₈	1	2	yes	—
143	C₂²×C₃	C₂×#41	1	2	yes	—
144	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
145	C₂²×C₃	C₂⁵	1	1	yes	characteristic
146	C₃×C₄	—	2	8	no	—
147	C₃×C₄	—	2	2	no	—
148	C₃×C₄	C₂²×Q₈	1	4	yes	—
149	C₃×C₄	C₂×#41	1	4	yes	—
150	Dic₃	—	2	4	no	—
151	Dic₃	—	2	4	no	—
152	Dic₃	—	2	4	no	—
153	Dic₃	C₂²×Q₈	1	8	yes	—
154	C₂³	C₂³×D₃	1	1	yes	characteristic
155	Q₈	—	2	16	no	—
156	Q₈	—	3	16	no	—
157	Q₈	—	6	8	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	3	8	no	—
161	C₂×C₄	—	3	4	no	—
162	C₂×C₄	—	3	4	no	—
163	C₂×C₄	—	3	4	no	—
164	C₂×C₄	—	3	4	no	—
165	C₂×C₄	—	6	4	no	—
166	C₂×C₄	—	6	4	no	—
167	C₂×C₄	—	6	4	no	—
168	C₂×C₄	#237	1	4	yes	—
169	C₂×C₄	D₃×Q₈	1	4	yes	—
170	C₂×C₄	C₂³×D₃	1	8	yes	—
171	C₂×C₄	C₂³×D₃	1	2	yes	—
172	C₂×C₄	C₂³×D₃	1	2	yes	—
173	C₂×C₄	C₂³×D₃	1	2	yes	—
174	C₂×C₃	#557	1	4	yes	complemented
175	C₂×C₃	C₂³×Q₈	1	1	yes	characteristic
176	C₂×C₃	C₂²×#41	1	1	yes	characteristic
177	C₂×C₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
178	C₂²	C₂×#237	1	2	yes	—
179	C₂²	C₂×D₃×Q₈	1	2	yes	—
180	C₂²	#1007	1	2	yes	—
181	C₂²	C₂⁴×D₃	1	1	yes	characteristic
182	C₄	—	2	8	no	—
183	C₄	—	2	2	no	—
184	C₄	—	3	8	no	—
185	C₄	—	6	4	no	—
186	C₄	—	6	4	no	—
187	C₄	—	6	4	no	—
188	C₄	C₂×#237	1	4	yes	—
189	C₄	C₂×D₃×Q₈	1	4	yes	—
190	C₃	C₂×#557	1	1	yes	characteristic, complemented, Sylow
191	C₂	#5721	1	4	yes	direct product factor
192	C₂	C₂²×#237	1	1	yes	characteristic
193	C₂	C₂²×D₃×Q₈	1	1	yes	characteristic
194	C₂	C₂×#1007	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂×C₃×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₃
hypocenter	C₃	C₂×#557
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
4. #5721	C₂	direct (N × H)
55. C₂×Dic₆	Q₈	semidirect
114. Dic₆	C₂×Q₈	semidirect
174. C₂×C₃	#557	semidirect
190. C₃	C₂×#557	semidirect
191. C₂	#5721	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
C₂×Q₈×Dic₃	2
C₂×#944	2
#5721	2
C₂×#890	2
C₂×#889	2
C₂×#887	2
C₂×C₄×Dic₆	2
C₂×#557	3

Subgroups — C2×#5721 #86202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#5721 #86202