Subgroups — #78508

Order 384.

356 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	#5470	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	#5122	C₂	1	2	yes	maximal
4	#5119	C₂	1	2	yes	maximal
5	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#4701	C₂	1	4	yes	maximal, complemented
7	#4679	C₂	1	2	yes	maximal, complemented
8	C₈×Dic₆	C₂	1	2	yes	maximal, complemented
9	#2962	—	3	1	no	maximal, Sylow
10	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
13	#926	C₂²	1	2	yes	—
14	#924	C₄	1	2	yes	—
15	#924	C₄	1	2	yes	—
16	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
17	#870	C₄	1	2	yes	—
18	#869	C₄	1	1	yes	characteristic
19	#869	C₄	1	1	yes	characteristic
20	C₄×Dic₆	C₂²	1	2	yes	—
21	C₄×Dic₆	C₂²	1	2	yes	—
22	C₃×#97	C₂²	1	2	yes	—
23	C₃×C₄×C₈	C₂²	1	2	yes	—
24	#815	C₂²	1	4	yes	—
25	Dic₃⋊C₈	C₂²	1	4	yes	—
26	Dic₃⋊C₈	C₂²	1	4	yes	—
27	C₈×Dic₃	C₂²	1	4	yes	—
28	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
29	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
30	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
31	#533	—	3	1	no	—
32	#446	—	3	4	no	—
33	#446	—	3	2	no	—
34	C₈×Q₈	—	3	2	no	—
35	#424	—	3	2	no	—
36	C₂×C₄⋊C₈	—	3	1	no	—
37	#405	—	3	2	no	—
38	C₄×M₁₆	D₃	1	1	yes	characteristic
39	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
42	C₂×Dic₆	C₂×C₄	1	2	yes	—
43	C₂×Dic₆	C₂×C₄	1	2	yes	—
44	C₃×M₁₆	—	2	2	no	—
45	C₃×M₁₆	Q₈	1	4	yes	complemented
46	C₂×C₃×C₈	C₂³	1	2	yes	—
47	C₂×C₃×C₈	C₂³	1	2	yes	—
48	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
49	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
50	C₃×C₄²	C₂³	1	1	yes	characteristic
51	C₃×C₄²	C₂³	1	1	yes	characteristic
52	#217	C₂×C₄	1	4	yes	—
53	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
54	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
55	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
57	Dic₃⋊C₄	C₂×C₄	1	4	yes	—
58	Dic₃⋊C₄	C₂×C₄	1	4	yes	—
59	C₄×Dic₃	C₂³	1	4	yes	—
60	C₂×C₃⋊C₈	—	2	4	no	—
61	C₂×C₃⋊C₈	C₂³	1	2	yes	—
62	C₂×C₃⋊C₈	C₂³	1	2	yes	—
63	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
64	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
65	C₂²×C₈	—	3	2	no	—
66	C₄⋊Q₈	—	3	1	no	—
67	C₄⋊Q₈	—	3	1	no	—
68	#125	—	3	2	no	—
69	C₂²⋊Q₈	—	3	2	no	—
70	C₂²⋊Q₈	—	3	2	no	—
71	C₄×Q₈	—	3	2	no	—
72	C₄×Q₈	—	3	2	no	—
73	#117	—	3	2	no	—
74	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
75	C₄⋊C₈	—	3	4	no	—
76	C₄⋊C₈	—	3	4	no	—
77	C₄⋊C₈	—	3	1	no	—
78	C₄⋊C₈	—	3	1	no	—
79	C₄⋊C₈	—	3	2	no	—
80	#97	—	3	4	no	—
81	#97	C₂×D₃	1	2	yes	—
82	C₄×C₈	—	3	4	no	—
83	C₄×C₈	C₂×D₃	1	2	yes	—
84	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	4	1	no	—
87	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
93	C₂×C₃×C₄	#41	1	2	yes	—
94	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
95	C₂×Dic₃	C₂²×C₄	1	4	yes	—
96	C₂×Dic₃	C₂²×C₄	1	4	yes	—
97	Dic₆	—	2	4	no	—
98	Dic₆	—	4	2	no	—
99	C₃×C₈	—	2	2	no	—
100	C₃×C₈	C₂×Q₈	1	4	yes	—
101	C₃⋊C₈	—	2	2	no	—
102	C₃⋊C₈	—	2	2	no	—
103	C₂×Q₈	—	3	2	no	—
104	C₂×Q₈	—	3	2	no	—
105	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
106	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
107	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	Dic₆	1	4	yes	complemented
110	C₂×C₈	—	3	2	no	—
111	C₂×C₈	—	3	2	no	—
112	C₂×C₈	—	6	4	no	—
113	C₂×C₈	C₂²×D₃	1	2	yes	—
114	C₂×C₈	C₂²×D₃	1	2	yes	—
115	C₄⋊C₄	—	3	2	no	—
116	C₄⋊C₄	—	3	2	no	—
117	C₄⋊C₄	—	3	4	no	—
118	C₄⋊C₄	—	3	4	no	—
119	C₄⋊C₄	—	3	2	no	—
120	C₄⋊C₄	—	3	2	no	—
121	C₂²⋊C₄	—	3	4	no	—
122	C₄²	—	3	4	no	—
123	C₄²	C₄×D₃	1	1	yes	characteristic
124	C₄²	C₄×D₃	1	1	yes	characteristic
125	C₄²	C₂²×D₃	1	1	yes	characteristic
126	C₄²	C₂²×D₃	1	1	yes	characteristic
127	C₂²×C₃	—	2	1	no	—
128	C₂²×C₃	—	2	1	no	—
129	C₂²×C₃	C₄×Q₈	1	2	yes	—
130	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
131	C₃×C₄	—	2	1	no	—
132	C₃×C₄	—	2	1	no	—
133	C₃×C₄	—	2	1	no	—
134	C₃×C₄	C₄×Q₈	1	2	yes	—
135	C₃×C₄	#131	1	2	yes	—
136	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
137	C₃×C₄	C₂×#41	1	1	yes	characteristic
138	Dic₃	—	2	4	no	—
139	Dic₃	—	2	4	no	—
140	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
141	Q₈	—	6	4	no	—
142	Q₈	—	12	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	3	4	no	—
145	C₂×C₄	—	3	4	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	C₂×Dic₆	1	2	yes	—
148	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
153	C₂×C₄	#235	1	2	yes	—
154	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
155	C₈	—	2	2	no	—
156	C₈	—	6	2	no	—
157	C₈	—	6	2	no	—
158	C₈	C₂×Dic₆	1	4	yes	—
159	C₂×C₃	—	2	2	no	—
160	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
161	C₂×C₃	C₂×#131	1	1	yes	characteristic
162	C₂×C₃	#568	1	1	yes	characteristic
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	C₄×Dic₆	1	2	yes	—
166	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	6	4	no	—
171	C₄	—	6	4	no	—
172	C₄	C₄×Dic₆	1	2	yes	—
173	C₄	#907	1	2	yes	—
174	C₄	C₂²×Dic₆	1	1	yes	characteristic
175	C₄	C₂×#235	1	1	yes	characteristic
176	C₃	#2962	1	1	yes	characteristic, complemented, Sylow
177	C₂	—	2	2	no	—
178	C₂	C₂×C₄×Dic₆	1	1	yes	characteristic
179	C₂	C₂×#907	1	1	yes	characteristic
180	C₂	#5745	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×C₄×M₁₆	C₂
prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₄×M₁₆	D₃
hypocenter	C₃	#2962
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
6. #4701	C₂	semidirect
7. #4679	C₂	semidirect
8. C₈×Dic₆	C₂	semidirect
45. C₃×M₁₆	Q₈	semidirect
109. M₁₆	Dic₆	semidirect
176. C₃	#2962	semidirect

Maximal subgroups

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

subgroup	index
#5470	2
C₃×C₄×M₁₆	2
#5122	2
#5119	2
C₂×C₄⋊(C₃⋊C₈)	2
#4701	2
#4679	2
C₈×Dic₆	2
#2962	3