Subgroups — #69484

Order 384.

260 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal
4	C₂×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#4524	C₂	1	2	yes	maximal
6	#1939	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊C₄	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄⋊C₄	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄²	C₂²	1	2	yes	—
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×#95	C₄	1	2	yes	—
15	C₄⋊(C₃⋊C₈)	—	2	2	no	—
16	C₄⋊(C₃⋊C₈)	—	2	4	no	—
17	C₄×C₃⋊C₈	—	2	2	no	—
18	C₂×C₄⋊C₈	—	3	2	no	—
19	C₂×C₄⋊C₈	—	3	1	no	—
20	C₂×C₄×C₈	—	3	1	no	—
21	C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
22	#336	—	3	2	no	—
23	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₃×C₄⋊C₄	—	2	2	no	—
29	C₃×C₄⋊C₄	C₈	1	4	yes	complemented
30	C₃×C₄²	—	2	2	no	—
31	C₃×C₄²	—	2	2	no	—
32	C₃×C₄²	D₄	1	2	yes	—
33	C₃×C₄²	Q₈	1	2	yes	—
34	C₂×C₃⋊C₈	—	2	1	no	—
35	C₂×C₃⋊C₈	—	2	1	no	—
36	C₂×C₃⋊C₈	—	2	1	no	—
37	C₂×C₃⋊C₈	—	2	1	no	—
38	C₂×C₃⋊C₈	—	2	2	no	—
39	C₂×C₃⋊C₈	—	2	2	no	—
40	C₂×C₃⋊C₈	—	2	2	no	—
41	C₂×C₃⋊C₈	D₄	1	2	yes	—
42	C₂×C₃⋊C₈	Q₈	1	2	yes	—
43	C₂²×C₈	—	3	1	no	—
44	C₂²×C₈	—	3	1	no	—
45	C₂²×C₈	—	3	2	no	—
46	C₂×C₄⋊C₄	Dic₃	1	1	yes	characteristic
47	C₂×C₄⋊C₄	Dic₃	1	1	yes	characteristic
48	C₂×C₄²	C₂×D₃	1	2	yes	—
49	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
50	C₄⋊C₈	—	6	4	no	—
51	C₄⋊C₈	—	6	2	no	—
52	C₄×C₈	—	6	2	no	—
53	#95	Dic₃	1	2	yes	—
54	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃×C₄	—	2	2	no	—
56	C₂×C₃×C₄	—	2	1	no	—
57	C₂×C₃×C₄	—	2	1	no	—
58	C₂×C₃×C₄	—	2	2	no	—
59	C₂×C₃×C₄	—	2	2	no	—
60	C₂×C₃×C₄	—	2	2	no	—
61	C₂×C₃×C₄	—	2	2	no	—
62	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
63	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
64	C₂×C₃×C₄	C₂×C₈	1	4	yes	—
65	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
66	C₂×C₃×C₄	M₁₆	1	2	yes	—
67	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₃×C₄	#41	1	1	yes	characteristic
72	C₂×C₃×C₄	#41	1	1	yes	characteristic
73	C₃⋊C₈	—	2	2	no	—
74	C₃⋊C₈	—	2	2	no	—
75	C₃⋊C₈	—	4	2	no	—
76	C₂²×C₄	C₂×Dic₃	1	2	yes	—
77	C₂²×C₄	C₂×Dic₃	1	2	yes	—
78	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
79	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
80	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
81	C₂×C₈	—	3	2	no	—
82	C₂×C₈	—	3	2	no	—
83	C₂×C₈	—	6	1	no	—
84	C₂×C₈	—	6	1	no	—
85	C₂×C₈	—	6	1	no	—
86	C₂×C₈	—	6	1	no	—
87	C₂×C₈	—	6	2	no	—
88	C₂×C₈	—	6	2	no	—
89	C₂×C₈	—	6	2	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	C₃⋊C₈	1	4	yes	complemented
92	C₄²	—	2	2	no	—
93	C₄²	—	2	2	no	—
94	C₄²	Dic₆	1	2	yes	—
95	C₄²	D₁₂	1	2	yes	—
96	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
98	C₂²×C₃	C₄×Q₈	1	1	yes	characteristic
99	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
100	C₂²×C₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×C₃	#131	1	1	yes	characteristic
102	C₂²×C₃	#131	1	1	yes	characteristic
103	C₃×C₄	—	2	2	no	—
104	C₃×C₄	—	2	4	no	—
105	C₃×C₄	—	4	2	no	—
106	C₃×C₄	C₄⋊C₈	1	4	yes	complemented
107	C₃×C₄	#121	1	1	yes	characteristic
108	C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
109	C₃×C₄	#125	1	1	yes	characteristic
110	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
111	C₂³	C₂²×Dic₃	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	C₂×C₃⋊C₈	1	4	yes	—
120	C₂×C₄	C₂×C₃⋊C₈	1	2	yes	—
121	C₂×C₄	#208	1	2	yes	—
122	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
123	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
124	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
125	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
126	C₂×C₄	D₃×D₄	1	1	yes	characteristic
127	C₂×C₄	#237	1	1	yes	characteristic
128	C₂×C₄	D₃×Q₈	1	1	yes	characteristic
129	C₂×C₄	#239	1	1	yes	characteristic
130	C₈	—	6	2	no	—
131	C₈	—	6	2	no	—
132	C₈	—	12	2	no	—
133	C₂×C₃	#389	1	1	yes	characteristic
134	C₂×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
135	C₂×C₃	#424	1	1	yes	characteristic
136	C₂×C₃	C₈×D₄	1	1	yes	characteristic
137	C₂×C₃	#436	1	1	yes	characteristic
138	C₂×C₃	C₈×Q₈	1	1	yes	characteristic
139	C₂×C₃	#446	1	1	yes	characteristic
140	C₂²	C₂²×C₃⋊C₈	1	1	yes	characteristic
141	C₂²	C₂×#208	1	1	yes	characteristic
142	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
143	C₂²	D₄×Dic₃	1	1	yes	characteristic
144	C₂²	Q₈×Dic₃	1	1	yes	characteristic
145	C₂²	#948	1	1	yes	characteristic
146	C₂²	#948	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	4	no	—
149	C₄	—	4	2	no	—
150	C₄	C₄⋊(C₃⋊C₈)	1	4	yes	complemented
151	C₄	#888	1	1	yes	characteristic
152	C₄	#890	1	1	yes	characteristic
153	C₄	#895	1	1	yes	characteristic
154	C₄	#897	1	1	yes	characteristic
155	C₃	#1939	1	1	yes	characteristic, complemented, Sylow
156	C₂	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
157	C₂	#4984	1	1	yes	characteristic
158	C₂	#4999	1	1	yes	characteristic
159	C₂	D₄×C₃⋊C₈	1	1	yes	characteristic
160	C₂	#5012	1	1	yes	characteristic
161	C₂	Q₈×C₃⋊C₈	1	1	yes	characteristic
162	C₂	#5024	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₈
Fitting	C₃×C₄×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₄×C₄⋊C₄	D₃
hypocenter	C₃	#1939
LCS	G ▷ C₂²×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
29. C₃×C₄⋊C₄	C₈	semidirect
91. C₄⋊C₄	C₃⋊C₈	semidirect
106. C₃×C₄	C₄⋊C₈	semidirect
150. C₄	C₄⋊(C₃⋊C₈)	semidirect
155. C₃	#1939	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₄⋊C₄	2
C₂×C₄⋊(C₃⋊C₈)	2
C₂×C₄×C₃⋊C₈	2
#4524	2
#1939	3