Subgroups — #73716

Order 384.

452 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#870	C₂	1	2	yes	maximal
2	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#4975	C₂	1	2	yes	maximal
4	#4972	C₂	1	2	yes	maximal
5	#4935	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
7	#4928	C₂	1	2	yes	maximal
8	#4657	C₂	1	4	yes	maximal
9	#2552	—	3	1	no	maximal, Sylow
10	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄²	C₂²	1	2	yes	—
13	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
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	4	yes	—
22	C₂×C₄×Dic₃	C₂²	1	2	yes	—
23	#870	—	2	8	no	—
24	C₃×#95	C₂²	1	2	yes	—
25	C₃⋊#95	C₂²	1	2	yes	—
26	C₃⋊#95	C₂²	1	4	yes	—
27	C₂×#125	—	3	2	no	—
28	#398	—	3	4	no	—
29	#389	—	3	2	no	—
30	#389	—	3	2	no	—
31	#384	—	3	2	no	—
32	#384	—	3	1	no	—
33	C₄×C₄⋊C₄	—	3	1	no	—
34	C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
35	C₂²×C₃×C₄	C₂³	1	2	yes	—
36	C₂²×C₃×C₄	C₂³	1	2	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	4	yes	—
41	C₂²×Dic₃	C₂³	1	2	yes	—
42	C₂²×Dic₃	C₂³	1	2	yes	—
43	C₃×C₄⋊C₄	—	2	2	no	—
44	C₃×C₄⋊C₄	—	2	2	no	—
45	C₃×C₄²	—	2	4	no	—
46	C₃×C₄²	Q₈	1	4	yes	—
47	C₄⋊Dic₃	—	2	4	no	—
48	C₄⋊Dic₃	—	2	4	no	—
49	C₄⋊Dic₃	—	4	2	no	—
50	C₄⋊Dic₃	D₄	1	4	yes	—
51	C₄⋊Dic₃	Q₈	1	4	yes	—
52	Dic₃⋊C₄	—	2	8	no	—
53	Dic₃⋊C₄	—	2	8	no	—
54	Dic₃⋊C₄	—	4	4	no	—
55	C₄×Dic₃	—	2	4	no	—
56	#125	—	6	8	no	—
57	C₂×C₄⋊C₄	—	3	1	no	—
58	C₂×C₄⋊C₄	—	3	1	no	—
59	C₂×C₄⋊C₄	—	3	2	no	—
60	C₂×C₄⋊C₄	—	3	4	no	—
61	C₂×C₄⋊C₄	—	3	4	no	—
62	C₂×C₄⋊C₄	—	3	2	no	—
63	C₂×C₄⋊C₄	—	3	4	no	—
64	C₂×C₄⋊C₄	—	3	2	no	—
65	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
66	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
67	C₂×C₄²	—	3	2	no	—
68	C₂×C₄²	C₂×D₃	1	2	yes	—
69	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
70	#95	—	3	2	no	—
71	#95	—	3	4	no	—
72	#95	C₂×D₃	1	2	yes	—
73	C₂³×C₃	C₂⁴	1	1	yes	characteristic
74	C₂×C₃×C₄	—	2	1	no	—
75	C₂×C₃×C₄	—	2	1	no	—
76	C₂×C₃×C₄	—	2	2	no	—
77	C₂×C₃×C₄	—	2	2	no	—
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	—	2	2	no	—
80	C₂×C₃×C₄	—	2	2	no	—
81	C₂×C₃×C₄	—	2	2	no	—
82	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
84	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
85	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
86	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
87	C₂×C₃×C₄	#41	1	2	yes	—
88	C₂×C₃×C₄	#41	1	2	yes	—
89	C₂×Dic₃	—	2	2	no	—
90	C₂×Dic₃	—	2	2	no	—
91	C₂×Dic₃	—	2	2	no	—
92	C₂×Dic₃	—	2	2	no	—
93	C₂×Dic₃	—	2	4	no	—
94	C₂×Dic₃	—	2	4	no	—
95	C₂×Dic₃	—	2	4	no	—
96	C₂×Dic₃	C₂×D₄	1	4	yes	—
97	C₂×Dic₃	C₂×Q₈	1	4	yes	—
98	C₂²×C₄	—	3	2	no	—
99	C₂²×C₄	—	3	2	no	—
100	C₂²×C₄	—	3	4	no	—
101	C₂²×C₄	C₂²×D₃	1	2	yes	—
102	C₂²×C₄	C₂²×D₃	1	2	yes	—
103	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	3	4	no	—
109	C₄⋊C₄	—	3	4	no	—
110	C₄⋊C₄	—	6	8	no	—
111	C₄⋊C₄	—	6	8	no	—
112	C₄⋊C₄	—	6	4	no	—
113	C₄⋊C₄	—	6	4	no	—
114	C₄⋊C₄	—	12	4	no	—
115	C₄⋊C₄	—	12	2	no	—
116	C₄²	—	2	4	no	—
117	C₄²	—	6	4	no	—
118	C₄²	Dic₆	1	4	yes	—
119	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
120	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂²×C₃	C₂×#41	1	1	yes	characteristic
123	C₂²×C₃	C₂×#41	1	1	yes	characteristic
124	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
126	C₃×C₄	—	2	2	no	—
127	C₃×C₄	—	4	2	no	—
128	C₃×C₄	—	4	2	no	—
129	C₃×C₄	#125	1	4	yes	complemented
130	C₃×C₄	C₄⋊Q₈	1	4	yes	—
131	Dic₃	—	2	4	no	—
132	Dic₃	—	2	4	no	—
133	Dic₃	—	4	4	no	—
134	C₂³	C₂³×D₃	1	1	yes	characteristic
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	3	4	no	—
144	C₂×C₄	—	3	4	no	—
145	C₂×C₄	—	6	2	no	—
146	C₂×C₄	—	6	2	no	—
147	C₂×C₄	—	6	2	no	—
148	C₂×C₄	—	6	2	no	—
149	C₂×C₄	—	6	4	no	—
150	C₂×C₄	—	6	4	no	—
151	C₂×C₄	—	6	4	no	—
152	C₂×C₄	C₂×Dic₆	1	2	yes	—
153	C₂×C₄	C₂×Dic₆	1	2	yes	—
154	C₂×C₄	C₂×Dic₆	1	2	yes	—
155	C₂×C₄	#235	1	2	yes	—
156	C₂×C₄	#235	1	2	yes	—
157	C₂×C₄	D₃×D₄	1	2	yes	—
158	C₂×C₄	D₃×Q₈	1	2	yes	—
159	C₂×C₃	C₂×#125	1	1	yes	characteristic
160	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
161	C₂×C₃	#544	1	1	yes	characteristic
162	C₂×C₃	#547	1	1	yes	characteristic
163	C₂×C₃	#550	1	1	yes	characteristic
164	C₂×C₃	#554	1	1	yes	characteristic
165	C₂×C₃	#557	1	1	yes	characteristic
166	C₂²	C₂²×Dic₆	1	1	yes	characteristic
167	C₂²	C₂×#235	1	1	yes	characteristic
168	C₂²	C₂×#235	1	1	yes	characteristic
169	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
170	C₂²	C₂×D₃×Q₈	1	1	yes	characteristic
171	C₂²	#1009	1	1	yes	characteristic
172	C₂²	#1010	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	4	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	6	4	no	—
177	C₄	—	6	4	no	—
178	C₄	—	12	4	no	—
179	C₄	#870	1	4	yes	complemented
180	C₄	#888	1	4	yes	—
181	C₃	#2552	1	1	yes	characteristic, complemented, Sylow
182	C₂	C₂×#870	1	1	yes	characteristic
183	C₂	C₂×#888	1	1	yes	characteristic
184	C₂	#5519	1	1	yes	characteristic
185	C₂	#5551	1	1	yes	characteristic
186	C₂	#5553	1	1	yes	characteristic
187	C₂	#5567	1	1	yes	characteristic
188	C₂	#5579	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×C₄×C₄⋊C₄	C₂
hypercenter	C₄×C₄⋊C₄	D₃
hypocenter	C₃	#2552
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
129. C₃×C₄	#125	semidirect
179. C₄	#870	semidirect
181. C₃	#2552	semidirect

Maximal subgroups

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

subgroup	index
C₂×#870	2
C₃×C₄×C₄⋊C₄	2
#4975	2
#4972	2
#4935	2
C₄×C₄⋊Dic₃	2
#4928	2
#4657	2
#2552	3