Subgroups — C₄×#887 #73615

Order 384.

684 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#887	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	Dic₃×C₄⋊C₄	C₂	1	2	yes	maximal
5	C₄×Dic₃⋊C₄	C₂	1	2	yes	maximal
6	C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
7	C₄²×Dic₃	C₂	1	2	yes	maximal
8	#4649	C₂	1	4	yes	maximal
9	C₄²×Q₈	—	3	1	no	maximal, Sylow
10	C₂²×Dic₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₄²	C₂²	1	2	yes	—
13	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₃⋊C₄	C₂²	1	2	yes	—
16	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
17	C₂×C₄×Dic₃	C₂²	1	2	yes	—
18	C₂×C₄×Dic₃	C₂²	1	4	yes	—
19	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₃	C₂²	1	4	yes	—
21	C₂×C₄×Dic₃	C₂²	1	2	yes	—
22	C₂×C₄×Dic₃	C₂²	1	2	yes	—
23	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
24	#887	C₄	1	16	yes	direct product factor
25	C₄×Dic₆	C₄	1	8	yes	complemented
26	C₃×#95	C₂²	1	2	yes	—
27	C₃⋊#95	C₂²	1	4	yes	—
28	C₂×C₄×Q₈	—	3	2	no	—
29	C₂×C₄×Q₈	—	3	1	no	—
30	C₄×C₄⋊C₄	—	3	2	no	—
31	C₄×C₄⋊C₄	—	3	2	no	—
32	C₄×C₄⋊C₄	—	3	4	no	—
33	C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
34	C₄³	—	3	2	no	—
35	C₄³	—	3	1	no	—
36	C₂²×C₃×C₄	C₂³	1	4	yes	—
37	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₂²×Dic₃	C₂³	1	2	yes	—
41	C₂²×Dic₃	C₂³	1	2	yes	—
42	C₂²×Dic₃	C₂³	1	2	yes	—
43	C₂²×Dic₃	C₂³	1	2	yes	—
44	C₂×Dic₆	C₂×C₄	1	4	yes	—
45	C₂×Dic₆	C₂×C₄	1	8	yes	—
46	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
47	C₃×C₄²	—	2	4	no	—
48	C₃×C₄²	C₂×C₄	1	4	yes	—
49	C₄⋊Dic₃	C₂×C₄	1	4	yes	—
50	Dic₃⋊C₄	C₂×C₄	1	8	yes	—
51	Dic₃⋊C₄	C₂×C₄	1	16	yes	—
52	C₄×Dic₃	—	2	2	no	—
53	C₄×Dic₃	—	2	4	no	—
54	C₄×Dic₃	—	2	8	no	—
55	C₄×Dic₃	C₂×C₄	1	8	yes	—
56	C₄×Dic₃	C₂×C₄	1	8	yes	—
57	C₄×Dic₃	C₂×C₄	1	16	yes	—
58	C₄×Dic₃	Q₈	1	4	yes	complemented
59	C₂²×Q₈	—	3	1	no	—
60	C₄×Q₈	—	3	16	no	—
61	C₄×Q₈	—	3	8	no	—
62	C₂×C₄⋊C₄	—	3	4	no	—
63	C₂×C₄⋊C₄	—	3	2	no	—
64	C₂×C₄⋊C₄	—	3	1	no	—
65	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
66	C₂×C₄²	—	3	4	no	—
67	C₂×C₄²	—	3	2	no	—
68	C₂×C₄²	—	3	2	no	—
69	C₂×C₄²	—	3	1	no	—
70	C₂×C₄²	—	3	4	no	—
71	C₂×C₄²	—	3	2	no	—
72	C₂×C₄²	—	3	1	no	—
73	C₂×C₄²	C₂×D₃	1	2	yes	—
74	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
75	#95	—	3	4	no	—
76	#95	C₂×D₃	1	2	yes	—
77	C₂³×C₃	C₂⁴	1	1	yes	characteristic
78	C₂×C₃×C₄	—	2	4	no	—
79	C₂×C₃×C₄	—	2	2	no	—
80	C₂×C₃×C₄	—	2	4	no	—
81	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
82	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
83	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
84	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
85	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
86	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
87	C₂×C₃×C₄	#41	1	2	yes	—
88	C₂×Dic₃	—	2	2	no	—
89	C₂×Dic₃	—	2	2	no	—
90	C₂×Dic₃	—	2	4	no	—
91	C₂×Dic₃	C₂²×C₄	1	4	yes	—
92	C₂×Dic₃	C₂²×C₄	1	4	yes	—
93	C₂×Dic₃	C₂²×C₄	1	4	yes	—
94	C₂×Dic₃	C₂²×C₄	1	8	yes	—
95	C₂×Dic₃	C₂²×C₄	1	4	yes	—
96	C₂×Dic₃	C₂×Q₈	1	4	yes	—
97	C₂×Dic₃	#41	1	4	yes	—
98	Dic₆	C₄²	1	16	yes	complemented
99	C₂×Q₈	—	3	8	no	—
100	C₂×Q₈	—	3	4	no	—
101	C₂²×C₄	—	3	2	no	—
102	C₂²×C₄	—	3	2	no	—
103	C₂²×C₄	—	3	2	no	—
104	C₂²×C₄	—	3	2	no	—
105	C₂²×C₄	C₂²×D₃	1	4	yes	—
106	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
108	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₄⋊C₄	—	3	16	no	—
110	C₄⋊C₄	—	3	8	no	—
111	C₄⋊C₄	—	3	4	no	—
112	C₄⋊C₄	C₄×D₃	1	8	yes	—
113	C₄²	—	2	4	no	—
114	C₄²	—	3	16	no	—
115	C₄²	—	3	8	no	—
116	C₄²	—	3	8	no	—
117	C₄²	—	3	4	no	—
118	C₄²	—	6	4	no	—
119	C₄²	—	6	8	no	—
120	C₄²	—	6	2	no	—
121	C₄²	C₄×D₃	1	4	yes	—
122	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
123	C₂²×C₃	C₂³×C₄	1	2	yes	—
124	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂²×C₃	C₂×#41	1	1	yes	characteristic
126	C₂²×C₃	C₂×#41	1	2	yes	—
127	C₃×C₄	—	2	2	no	—
128	C₃×C₄	—	2	8	no	—
129	C₃×C₄	C₂×C₄²	1	4	yes	—
130	C₃×C₄	C₄×Q₈	1	4	yes	complemented
131	Dic₃	—	2	4	no	—
132	Dic₃	—	2	4	no	—
133	Dic₃	C₂×C₄²	1	8	yes	—
134	Dic₃	C₄×Q₈	1	8	yes	complemented
135	C₂³	C₂³×D₃	1	1	yes	characteristic
136	Q₈	—	3	16	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	3	4	no	—
141	C₂×C₄	—	3	4	no	—
142	C₂×C₄	—	3	8	no	—
143	C₂×C₄	—	3	4	no	—
144	C₂×C₄	—	3	4	no	—
145	C₂×C₄	—	3	4	no	—
146	C₂×C₄	—	3	4	no	—
147	C₂×C₄	—	6	2	no	—
148	C₂×C₄	—	6	4	no	—
149	C₂×C₄	—	6	2	no	—
150	C₂×C₄	C₂×C₄×D₃	1	8	yes	—
151	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
152	C₂×C₄	C₂×C₄×D₃	1	4	yes	—
153	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
154	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
155	C₂×C₄	#237	1	2	yes	—
156	C₂×C₄	D₃×Q₈	1	2	yes	—
157	C₂×C₃	C₂²×C₄²	1	1	yes	characteristic
158	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
159	C₂×C₃	C₂×C₄×Q₈	1	2	yes	—
160	C₂×C₃	C₄×#41	1	1	yes	characteristic
161	C₂×C₃	C₄×#41	1	2	yes	—
162	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
163	C₂²	C₂²×C₄×D₃	1	2	yes	—
164	C₂²	C₂×#237	1	1	yes	characteristic
165	C₂²	C₂×D₃×Q₈	1	1	yes	characteristic
166	C₂²	D₃×#41	1	2	yes	—
167	C₄	—	2	2	no	—
168	C₄	—	2	8	no	—
169	C₄	—	3	8	no	—
170	C₄	—	3	8	no	—
171	C₄	—	6	4	no	—
172	C₄	—	6	4	no	—
173	C₄	C₄²×D₃	1	4	yes	—
174	C₄	#887	1	4	yes	direct product factor
175	C₃	C₄²×Q₈	1	1	yes	characteristic, complemented, Sylow
176	C₂	C₂×C₄²×D₃	1	1	yes	characteristic
177	C₂	C₂×#887	1	1	yes	characteristic
178	C₂	#5522	1	1	yes	characteristic
179	C₂	C₄×#237	1	2	yes	—
180	C₂	C₄×D₃×Q₈	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator	C₂×C₃	C₂²×C₄²
Frattini	C₂³	C₂³×D₃
Fitting	C₃×C₄×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₃	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₃
hypocenter	C₃	C₄²×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
24. #887	C₄	direct (N × H)
25. C₄×Dic₆	C₄	semidirect
58. C₄×Dic₃	Q₈	semidirect
98. Dic₆	C₄²	semidirect
130. C₃×C₄	C₄×Q₈	semidirect
134. Dic₃	C₄×Q₈	semidirect
174. C₄	#887	direct (N × H)
175. C₃	C₄²×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#887	2
C₂×C₄×Dic₆	2
C₃×C₄×C₄⋊C₄	2
Dic₃×C₄⋊C₄	2
C₄×Dic₃⋊C₄	2
C₄²×Dic₃	2
#4649	2
C₄²×Q₈	3

Subgroups — C4×#887 #73615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#887 #73615