Subgroups — C₄⋊C₄×C₃⋊(C₃⋊C₈) #11784251

Order 1152.

888 conjugacy classes of subgroups.

The subgroup lattice has 207 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal
4	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#10589390	C₂	1	2	yes	maximal
6	C₄⋊C₄×C₃⋊C₈	—	3	4	no	maximal
7	C₂×C₃²×C₄⋊C₄	C₄	1	2	yes	—
8	C₂×C₃²×C₄²	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
11	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₃²×#95	C₄	1	2	yes	—
14	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	4	yes	complemented
15	C₄×C₃⋊(C₃⋊C₈)	—	2	4	no	—
16	C₄×C₃⋊(C₃⋊C₈)	C₄	1	4	yes	complemented
17	C₃×C₄×C₄⋊C₄	D₃	1	4	yes	—
18	C₂×C₄⋊(C₃⋊C₈)	—	3	4	no	—
19	C₂×C₄×C₃⋊C₈	—	3	4	no	—
20	C₂×C₄×C₃⋊C₈	—	3	8	no	—
21	#4524	—	3	8	no	—
22	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₂×C₄	1	4	yes	—
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
26	C₃²×C₄⋊C₄	C₈	1	8	yes	complemented
27	C₃²×C₄²	—	2	2	no	—
28	C₃²×C₄²	—	2	2	no	—
29	C₃²×C₄²	C₂×C₄	1	2	yes	—
30	C₃²×C₄²	C₂×C₄	1	2	yes	—
31	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
32	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
33	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
34	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
35	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
36	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
37	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
38	C₂×C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
39	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	—
40	C₈×C₄⋊C₄	—	9	1	no	Sylow
41	C₂×C₃×C₄⋊C₄	Dic₃	1	8	yes	—
42	C₂×C₃×C₄²	C₂×D₃	1	8	yes	—
43	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
44	C₂²×C₃⋊C₈	—	3	4	no	—
45	C₂²×C₃⋊C₈	—	3	8	no	—
46	C₂²×C₃⋊C₈	—	3	4	no	—
47	C₃×#95	Dic₃	1	8	yes	—
48	C₄⋊(C₃⋊C₈)	—	3	16	no	—
49	C₄×C₃⋊C₈	—	3	16	no	—
50	C₄×C₃⋊C₈	—	6	16	no	—
51	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	—	2	4	no	—
53	C₂×C₃²×C₄	—	2	2	no	—
54	C₂×C₃²×C₄	—	2	4	no	—
55	C₂×C₃²×C₄	C₄²	1	2	yes	—
56	C₂×C₃²×C₄	C₄²	1	2	yes	—
57	C₂×C₃²×C₄	C₂×C₈	1	8	yes	—
58	C₂×C₃²×C₄	C₂×C₈	1	4	yes	—
59	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₃²×C₄	#41	1	1	yes	characteristic
64	C₂×C₃²×C₄	#41	1	1	yes	characteristic
65	C₃⋊(C₃⋊C₈)	—	2	2	no	—
66	C₃⋊(C₃⋊C₈)	—	2	4	no	—
67	C₃⋊(C₃⋊C₈)	C₄⋊C₄	1	4	yes	direct product factor
68	C₂×C₄⋊C₈	—	9	1	no	—
69	C₂×C₄×C₈	—	9	1	no	—
70	C₂×C₄×C₈	—	9	2	no	—
71	C₄×C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic
72	#336	—	9	2	no	—
73	C₂²×C₃×C₄	C₂×Dic₃	1	16	yes	—
74	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
75	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
76	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
77	C₃×C₄⋊C₄	C₃⋊C₈	1	32	yes	complemented
78	C₃×C₄²	—	2	8	no	—
79	C₃×C₄²	—	2	8	no	—
80	C₃×C₄²	C₄×D₃	1	8	yes	—
81	C₃×C₄²	C₄×D₃	1	8	yes	—
82	C₂×C₃⋊C₈	—	3	8	no	—
83	C₂×C₃⋊C₈	—	3	8	no	—
84	C₂×C₃⋊C₈	—	3	8	no	—
85	C₂×C₃⋊C₈	—	3	8	no	—
86	C₂×C₃⋊C₈	—	3	16	no	—
87	C₂×C₃⋊C₈	—	6	8	no	—
88	C₂×C₃⋊C₈	—	6	4	no	—
89	C₂×C₃⋊C₈	—	6	4	no	—
90	C₂×C₃⋊C₈	—	6	8	no	—
91	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
92	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
93	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
94	C₂²×C₃²	C₂²×C₈	1	2	yes	—
95	C₂²×C₃²	#131	1	2	yes	—
96	C₃²×C₄	—	2	2	no	—
97	C₃²×C₄	—	2	8	no	—
98	C₃²×C₄	C₄×C₈	1	4	yes	complemented
99	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₃²×C₄	#117	1	1	yes	characteristic
101	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
102	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
103	C₂²×C₈	—	9	1	no	—
104	C₂²×C₈	—	9	1	no	—
105	C₂²×C₈	—	9	2	no	—
106	C₂×C₄⋊C₄	C₃⋊Dic₃	1	2	yes	—
107	C₂×C₄²	C₂×C₃⋊D₃	1	2	yes	—
108	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
109	C₄⋊C₈	—	9	4	no	—
110	C₄×C₈	—	9	4	no	—
111	C₄×C₈	—	18	4	no	—
112	#95	C₃⋊Dic₃	1	2	yes	—
113	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
114	C₂×C₃×C₄	—	2	16	no	—
115	C₂×C₃×C₄	—	2	8	no	—
116	C₂×C₃×C₄	—	2	16	no	—
117	C₂×C₃×C₄	C₂×C₃⋊C₈	1	32	yes	—
118	C₂×C₃×C₄	C₂×C₃⋊C₈	1	16	yes	—
119	C₂×C₃×C₄	C₄×Dic₃	1	8	yes	—
120	C₂×C₃×C₄	C₄×Dic₃	1	8	yes	—
121	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
122	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
123	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
124	C₂×C₃×C₄	#237	1	4	yes	—
125	C₂×C₃×C₄	D₃×Q₈	1	4	yes	—
126	C₂×C₃×C₄	#239	1	4	yes	—
127	C₃⋊C₈	—	3	16	no	—
128	C₃⋊C₈	—	6	8	no	—
129	C₃⋊C₈	—	6	16	no	—
130	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
132	C₂×C₃²	#405	1	1	yes	characteristic
133	C₂×C₃²	C₈×D₄	1	2	yes	—
134	C₂×C₃²	C₈×Q₈	1	2	yes	—
135	C₂²×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
136	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
138	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
139	C₂×C₈	—	9	2	no	—
140	C₂×C₈	—	9	2	no	—
141	C₂×C₈	—	9	2	no	—
142	C₂×C₈	—	9	2	no	—
143	C₂×C₈	—	9	4	no	—
144	C₂×C₈	—	18	2	no	—
145	C₂×C₈	—	18	1	no	—
146	C₂×C₈	—	18	1	no	—
147	C₂×C₈	—	18	2	no	—
148	C₄⋊C₄	C₃⋊(C₃⋊C₈)	1	8	yes	direct product factor
149	C₄²	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	C₄×C₃⋊D₃	1	2	yes	—
152	C₄²	C₄×C₃⋊D₃	1	2	yes	—
153	C₂²×C₃	C₂²×C₃⋊C₈	1	8	yes	—
154	C₂²×C₃	C₂×C₄×Dic₃	1	4	yes	—
155	C₂²×C₃	D₄×Dic₃	1	4	yes	—
156	C₂²×C₃	Q₈×Dic₃	1	4	yes	—
157	C₂²×C₃	#948	1	8	yes	—
158	C₃×C₄	—	2	8	no	—
159	C₃×C₄	—	2	32	no	—
160	C₃×C₄	C₄×C₃⋊C₈	1	16	yes	complemented
161	C₃×C₄	#887	1	4	yes	—
162	C₃×C₄	D₃×C₄⋊C₄	1	4	yes	—
163	C₃×C₄	#892	1	4	yes	—
164	C₃×C₄	#893	1	4	yes	—
165	C₃²	C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
166	C₂³	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	8	yes	—
171	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	—
172	C₂×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
173	C₂×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
174	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
176	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
177	C₂×C₄	#3830	1	1	yes	characteristic
178	C₂×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
179	C₂×C₄	#3832	1	1	yes	characteristic
180	C₈	—	9	4	no	—
181	C₈	—	18	2	no	—
182	C₈	—	18	4	no	—
183	C₂×C₃	C₂×C₄×C₃⋊C₈	1	4	yes	—
184	C₂×C₃	Dic₃×C₄⋊C₄	1	4	yes	—
185	C₂×C₃	#4997	1	4	yes	—
186	C₂×C₃	D₄×C₃⋊C₈	1	8	yes	—
187	C₂×C₃	Q₈×C₃⋊C₈	1	8	yes	—
188	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	2	yes	—
189	C₂²	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
190	C₂²	D₄×C₃⋊Dic₃	1	1	yes	characteristic
191	C₂²	Q₈×C₃⋊Dic₃	1	1	yes	characteristic
192	C₂²	#64157	1	2	yes	—
193	C₄	—	2	2	no	—
194	C₄	—	2	8	no	—
195	C₄	C₄×C₃⋊(C₃⋊C₈)	1	4	yes	complemented
196	C₄	#64096	1	1	yes	characteristic
197	C₄	C₄⋊C₄×C₃⋊D₃	1	1	yes	characteristic
198	C₄	#64101	1	1	yes	characteristic
199	C₄	#64102	1	1	yes	characteristic
200	C₃	C₄⋊C₄×C₃⋊C₈	1	4	yes	complemented
201	C₂	C₂×C₄×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
202	C₂	C₄⋊C₄×C₃⋊Dic₃	1	1	yes	characteristic
203	C₂	#10592696	1	1	yes	characteristic
204	C₂	D₄×C₃⋊(C₃⋊C₈)	1	2	yes	—
205	C₂	Q₈×C₃⋊(C₃⋊C₈)	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×C₃⋊D₃
commutator	C₂×C₃²	C₂×C₄×C₈
Fitting	C₃²×C₄×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₄×C₄⋊C₄	C₃⋊D₃
hypocenter	C₃²	C₈×C₄⋊C₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. C₄⋊(C₃⋊(C₃⋊C₈))	C₄	semidirect
16. C₄×C₃⋊(C₃⋊C₈)	C₄	semidirect
26. C₃²×C₄⋊C₄	C₈	semidirect
67. C₃⋊(C₃⋊C₈)	C₄⋊C₄	direct (N × H)
77. C₃×C₄⋊C₄	C₃⋊C₈	semidirect
98. C₃²×C₄	C₄×C₈	semidirect
148. C₄⋊C₄	C₃⋊(C₃⋊C₈)	direct (N × H)
160. C₃×C₄	C₄×C₃⋊C₈	semidirect
165. C₃²	C₈×C₄⋊C₄	semidirect
195. C₄	C₄×C₃⋊(C₃⋊C₈)	semidirect
200. C₃	C₄⋊C₄×C₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×C₄⋊C₄	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₂×C₄×C₃⋊(C₃⋊C₈)	2
#10589390	2
C₄⋊C₄×C₃⋊C₈	3

Subgroups — C4⋊C4×C3⋊(C3⋊C8) #11784251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×C₃⋊(C₃⋊C₈) #11784251