Subgroups — C₄⋊C₄×Dic₆ #73636

Order 384.

568 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₆	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	#4975	C₂	1	2	yes	maximal
5	Dic₃×C₄⋊C₄	C₂	1	2	yes	maximal
6	#4935	C₂	1	1	yes	maximal, characteristic
7	#4928	C₂	1	2	yes	maximal
8	#4647	C₂	1	4	yes	maximal
9	Q₈×C₄⋊C₄	—	3	1	no	maximal, Sylow
10	C₂²×Dic₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄²	C₂²	1	2	yes	—
14	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
20	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
21	C₂×Dic₃⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄×Dic₃	C₂²	1	4	yes	—
23	C₂×C₄×Dic₃	C₂²	1	2	yes	—
24	C₄×Dic₆	—	2	8	no	—
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	1	no	—
29	C₂×C₄×Q₈	—	3	2	no	—
30	#389	—	3	2	no	—
31	#389	—	3	4	no	—
32	#384	—	3	2	no	—
33	#384	—	3	1	no	—
34	C₄×C₄⋊C₄	—	3	2	no	—
35	C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
36	C₂²×C₃×C₄	C₂³	1	2	yes	—
37	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
38	C₂²×C₃×C₄	C₂³	1	2	yes	—
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
41	C₂²×Dic₃	C₂³	1	4	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	D₄	1	4	yes	—
46	C₂×Dic₆	Q₈	1	4	yes	—
47	C₃×C₄⋊C₄	—	2	2	no	—
48	C₃×C₄⋊C₄	Q₈	1	4	yes	complemented
49	C₃×C₄²	—	2	4	no	—
50	C₃×C₄²	C₂×C₄	1	4	yes	—
51	C₄⋊Dic₃	—	2	4	no	—
52	C₄⋊Dic₃	—	2	4	no	—
53	C₄⋊Dic₃	—	4	2	no	—
54	C₄⋊Dic₃	C₂×C₄	1	4	yes	—
55	Dic₃⋊C₄	—	2	8	no	—
56	Dic₃⋊C₄	—	4	4	no	—
57	Dic₃⋊C₄	C₂×C₄	1	8	yes	—
58	C₄×Dic₃	—	2	8	no	—
59	C₄×Dic₃	C₂×C₄	1	8	yes	—
60	C₂²×Q₈	—	3	1	no	—
61	C₄×Q₈	—	3	8	no	—
62	C₄×Q₈	—	6	8	no	—
63	C₂×C₄⋊C₄	—	3	2	no	—
64	C₂×C₄⋊C₄	—	3	1	no	—
65	C₂×C₄⋊C₄	—	3	4	no	—
66	C₂×C₄⋊C₄	—	3	2	no	—
67	C₂×C₄⋊C₄	—	3	2	no	—
68	C₂×C₄⋊C₄	—	3	4	no	—
69	C₂×C₄⋊C₄	—	3	2	no	—
70	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
71	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
72	C₂×C₄²	—	3	2	no	—
73	C₂×C₄²	—	3	4	no	—
74	C₂×C₄²	C₂×D₃	1	2	yes	—
75	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
76	#95	—	3	4	no	—
77	#95	C₂×D₃	1	2	yes	—
78	C₂³×C₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₃×C₄	—	2	2	no	—
80	C₂×C₃×C₄	—	2	1	no	—
81	C₂×C₃×C₄	—	2	1	no	—
82	C₂×C₃×C₄	—	2	2	no	—
83	C₂×C₃×C₄	—	2	2	no	—
84	C₂×C₃×C₄	—	2	2	no	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
87	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
88	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
89	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
90	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
91	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
92	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
93	C₂×C₃×C₄	#41	1	2	yes	—
94	C₂×Dic₃	—	2	2	no	—
95	C₂×Dic₃	—	2	2	no	—
96	C₂×Dic₃	—	2	4	no	—
97	C₂×Dic₃	—	2	4	no	—
98	C₂×Dic₃	—	2	4	no	—
99	C₂×Dic₃	C₂²×C₄	1	4	yes	—
100	C₂×Dic₃	C₂²×C₄	1	4	yes	—
101	C₂×Dic₃	C₂×D₄	1	4	yes	—
102	C₂×Dic₃	C₂×Q₈	1	4	yes	—
103	Dic₆	C₄⋊C₄	1	16	yes	direct product factor
104	C₂×Q₈	—	3	4	no	—
105	C₂×Q₈	—	3	4	no	—
106	C₂×Q₈	—	3	4	no	—
107	C₂²×C₄	—	3	2	no	—
108	C₂²×C₄	—	3	2	no	—
109	C₂²×C₄	—	3	4	no	—
110	C₂²×C₄	C₂²×D₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	3	8	no	—
117	C₄⋊C₄	—	3	4	no	—
118	C₄⋊C₄	—	6	8	no	—
119	C₄⋊C₄	—	6	4	no	—
120	C₄⋊C₄	—	6	4	no	—
121	C₄⋊C₄	—	12	4	no	—
122	C₄⋊C₄	—	12	2	no	—
123	C₄⋊C₄	Dic₆	1	4	yes	direct product factor
124	C₄²	—	2	4	no	—
125	C₄²	—	3	8	no	—
126	C₄²	—	6	8	no	—
127	C₄²	C₄×D₃	1	4	yes	—
128	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
129	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
130	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
131	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
132	C₂²×C₃	C₂×#41	1	1	yes	characteristic
133	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
135	C₃×C₄	—	2	2	no	—
136	C₃×C₄	—	2	4	no	—
137	C₃×C₄	—	4	2	no	—
138	C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
139	C₃×C₄	C₄×Q₈	1	4	yes	complemented
140	Dic₃	—	2	4	no	—
141	Dic₃	—	4	4	no	—
142	Dic₃	C₂×C₄⋊C₄	1	8	yes	—
143	C₂³	C₂³×D₃	1	1	yes	characteristic
144	Q₈	—	3	16	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	3	4	no	—
153	C₂×C₄	—	3	4	no	—
154	C₂×C₄	—	3	4	no	—
155	C₂×C₄	—	3	4	no	—
156	C₂×C₄	—	6	2	no	—
157	C₂×C₄	—	6	2	no	—
158	C₂×C₄	—	6	4	no	—
159	C₂×C₄	—	6	4	no	—
160	C₂×C₄	—	6	4	no	—
161	C₂×C₄	C₂×Dic₆	1	4	yes	—
162	C₂×C₄	C₂×Dic₆	1	2	yes	—
163	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
164	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
165	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
166	C₂×C₄	#235	1	2	yes	—
167	C₂×C₄	D₃×D₄	1	2	yes	—
168	C₂×C₄	D₃×Q₈	1	2	yes	—
169	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
171	C₂×C₃	#520	1	1	yes	characteristic
172	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
173	C₂×C₃	#550	1	1	yes	characteristic
174	C₂×C₃	#557	1	1	yes	characteristic
175	C₂×C₃	Q₈²	1	1	yes	characteristic
176	C₂²	C₂²×Dic₆	1	1	yes	characteristic
177	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
178	C₂²	C₂×#235	1	1	yes	characteristic
179	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
180	C₂²	C₂×D₃×Q₈	1	1	yes	characteristic
181	C₂²	#1009	1	1	yes	characteristic
182	C₂²	#1010	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	4	no	—
185	C₄	—	3	8	no	—
186	C₄	—	4	2	no	—
187	C₄	—	6	4	no	—
188	C₄	—	12	4	no	—
189	C₄	C₄×Dic₆	1	4	yes	complemented
190	C₄	D₃×C₄⋊C₄	1	4	yes	—
191	C₃	Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
192	C₂	C₂×C₄×Dic₆	1	1	yes	characteristic
193	C₂	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic
194	C₂	#5523	1	1	yes	characteristic
195	C₂	D₄×Dic₆	1	1	yes	characteristic
196	C₂	#5553	1	1	yes	characteristic
197	C₂	Q₈×Dic₆	1	1	yes	characteristic
198	C₂	#5567	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
25. C₄×Dic₆	C₄	semidirect
48. C₃×C₄⋊C₄	Q₈	semidirect
103. Dic₆	C₄⋊C₄	direct (N × H)
123. C₄⋊C₄	Dic₆	direct (N × H)
139. C₃×C₄	C₄×Q₈	semidirect
189. C₄	C₄×Dic₆	semidirect
191. C₃	Q₈×C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₆	2
C₃×C₄×C₄⋊C₄	2
#4975	2
Dic₃×C₄⋊C₄	2
#4935	2
#4928	2
#4647	2
Q₈×C₄⋊C₄	3

Subgroups — C4⋊C4×Dic6 #73636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×Dic₆ #73636