Subgroups — #83331

Order 384.

322 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×#533	C₂	1	1	yes	maximal, characteristic
2	#5024	C₂	1	2	yes	maximal, complemented
3	#5024	C₂	1	4	yes	maximal, complemented
4	Q₈×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	#5000	C₂	1	4	yes	maximal
6	#4926	C₂	1	1	yes	maximal, characteristic
7	#4925	C₂	1	1	yes	maximal, characteristic
8	#2985	—	3	1	no	maximal, Sylow
9	C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₃×#125	C₄	1	2	yes	—
12	C₃×C₂²⋊Q₈	C₄	1	2	yes	—
13	C₃×C₂²⋊Q₈	C₄	1	2	yes	—
14	C₃×C₄×Q₈	C₂²	1	2	yes	—
15	C₃×C₄×Q₈	C₂²	1	2	yes	—
16	C₃×#117	C₂²	1	2	yes	—
17	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
18	#830	C₂²	1	2	yes	—
19	#830	C₂²	1	2	yes	—
20	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
21	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
22	C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
23	C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
24	#803	C₂²	1	2	yes	—
25	#803	C₂²	1	4	yes	—
26	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
27	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
28	C₄×C₃⋊C₈	C₂²	1	4	yes	—
29	#533	D₃	1	1	yes	characteristic
30	#446	—	3	2	no	—
31	#446	—	3	4	no	—
32	C₈×Q₈	—	3	2	no	—
33	#433	—	3	4	no	—
34	#432	—	3	1	no	—
35	#431	—	3	1	no	—
36	C₂×C₃×Q₈	C₂×C₄	1	2	yes	—
37	C₂×C₃×Q₈	C₂×C₄	1	2	yes	—
38	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
41	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
44	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
45	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₃×C₂²⋊C₄	C₂×C₄	1	4	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	4	yes	—
52	C₃×C₄²	C₂³	1	1	yes	characteristic
53	C₂×C₃⋊C₈	C₂³	1	2	yes	—
54	C₂×C₃⋊C₈	C₂³	1	2	yes	—
55	C₂×C₃⋊C₈	C₂³	1	4	yes	—
56	C₄⋊Q₈	Dic₃	1	1	yes	characteristic
57	C₄⋊Q₈	Dic₃	1	1	yes	characteristic
58	#125	Dic₃	1	2	yes	—
59	C₂²⋊Q₈	Dic₃	1	2	yes	—
60	C₂²⋊Q₈	Dic₃	1	2	yes	—
61	C₄×Q₈	C₂×D₃	1	2	yes	—
62	C₄×Q₈	C₂×D₃	1	2	yes	—
63	#117	C₂×D₃	1	2	yes	—
64	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
65	C₄⋊C₈	—	3	2	no	—
66	C₄⋊C₈	—	3	2	no	—
67	C₄⋊C₈	—	3	4	no	—
68	C₄⋊C₈	—	3	4	no	—
69	C₂²⋊C₈	—	3	2	no	—
70	C₂²⋊C₈	—	3	2	no	—
71	#97	—	3	2	no	—
72	#97	—	3	4	no	—
73	C₄×C₈	—	3	1	no	—
74	C₄×C₈	—	3	1	no	—
75	C₄×C₈	—	3	4	no	—
76	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
77	C₃×Q₈	—	2	4	no	—
78	C₃×Q₈	—	4	2	no	—
79	C₂×C₃×C₄	—	2	2	no	—
80	C₂×C₃×C₄	—	2	2	no	—
81	C₂×C₃×C₄	—	4	1	no	—
82	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
87	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
88	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
90	C₃⋊C₈	—	2	2	no	—
91	C₃⋊C₈	—	2	2	no	—
92	C₃⋊C₈	—	2	4	no	—
93	C₂×Q₈	C₂×Dic₃	1	2	yes	—
94	C₂×Q₈	C₂×Dic₃	1	2	yes	—
95	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
96	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
98	C₂×C₈	—	3	2	no	—
99	C₂×C₈	—	3	2	no	—
100	C₂×C₈	—	3	4	no	—
101	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
102	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
103	C₄⋊C₄	C₂×Dic₃	1	4	yes	—
104	C₄⋊C₄	C₂×Dic₃	1	4	yes	—
105	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
106	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
107	C₂²⋊C₄	C₂×Dic₃	1	4	yes	—
108	C₄²	C₂×Dic₃	1	1	yes	characteristic
109	C₄²	C₂×Dic₃	1	1	yes	characteristic
110	C₄²	C₂²×D₃	1	1	yes	characteristic
111	C₄²	C₂²×D₃	1	4	yes	—
112	C₄²	C₂²×D₃	1	1	yes	characteristic
113	C₂²×C₃	—	2	1	no	—
114	C₂²×C₃	—	2	1	no	—
115	C₂²×C₃	—	2	1	no	—
116	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
117	C₃×C₄	—	2	1	no	—
118	C₃×C₄	—	2	1	no	—
119	C₃×C₄	—	2	4	no	—
120	C₃×C₄	—	2	1	no	—
121	C₃×C₄	—	2	1	no	—
122	C₃×C₄	—	2	4	no	—
123	C₃×C₄	#131	1	2	yes	—
124	C₃×C₄	2₋¹⁺⁴	1	2	yes	—
125	C₂³	C₂²×Dic₃	1	1	yes	characteristic
126	Q₈	—	2	4	no	—
127	Q₈	—	4	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
132	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
133	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
134	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
135	C₂×C₄	C₂²×Dic₃	1	4	yes	—
136	C₂×C₄	C₂²×Dic₃	1	4	yes	—
137	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
139	C₈	—	6	2	no	—
140	C₈	—	6	2	no	—
141	C₈	—	6	4	no	—
142	C₂×C₃	—	4	1	no	—
143	C₂×C₃	#519	1	1	yes	characteristic
144	C₂×C₃	C₂×#131	1	1	yes	characteristic
145	C₂×C₃	#568	1	1	yes	characteristic
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	C₂³×Dic₃	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	4	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	4	no	—
156	C₄	#948	1	2	yes	—
157	C₄	#1007	1	2	yes	—
158	C₃	#2985	1	1	yes	characteristic, complemented, Sylow
159	C₂	—	4	1	no	—
160	C₂	#5812	1	1	yes	characteristic
161	C₂	C₂×#948	1	1	yes	characteristic
162	C₂	#5819	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#533	C₂
prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#533	D₃
hypocenter	C₃	#2985
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
2. #5024	C₂	semidirect
3. #5024	C₂	semidirect
4. Q₈×C₃⋊C₈	C₂	semidirect
158. C₃	#2985	semidirect

Maximal subgroups

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

subgroup	index
C₃×#533	2
#5024	2
Q₈×C₃⋊C₈	2
#5000	2
#4926	2
#4925	2
#2985	3