Subgroups — #83330

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	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	#4999	C₂	1	2	yes	maximal
6	#4997	C₂	1	2	yes	maximal
7	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
8	C₄×#208	C₂	1	1	yes	maximal, characteristic
9	#2962	—	3	1	no	maximal, Sylow
10	C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
12	C₃×#125	C₄	1	2	yes	—
13	C₃×C₂²⋊Q₈	C₄	1	2	yes	—
14	C₃×C₂²⋊Q₈	C₄	1	2	yes	—
15	C₃×C₄×Q₈	C₂²	1	2	yes	—
16	C₃×C₄×Q₈	C₂²	1	2	yes	—
17	C₃×#117	C₂²	1	2	yes	—
18	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
19	C₂×#208	C₂²	1	1	yes	characteristic
20	C₂×#208	C₂²	1	1	yes	characteristic
21	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
22	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
24	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
25	C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
26	C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
27	#803	C₂²	1	2	yes	—
28	#803	C₂²	1	4	yes	—
29	C₄×C₃⋊C₈	C₂²	1	2	yes	—
30	C₄×C₃⋊C₈	C₂²	1	4	yes	—
31	#533	D₃	1	1	yes	characteristic
32	#446	—	3	2	no	—
33	#446	—	3	4	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	—	3	1	no	—
39	C₂×C₃×Q₈	C₂×C₄	1	2	yes	—
40	C₂×C₃×Q₈	C₂×C₄	1	2	yes	—
41	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
44	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
47	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
48	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
51	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
52	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
53	C₃×C₄²	C₂³	1	1	yes	characteristic
54	C₃×C₄²	C₂³	1	4	yes	—
55	C₃×C₄²	C₂³	1	1	yes	characteristic
56	#208	—	2	2	no	—
57	#208	Q₈	1	4	yes	complemented
58	C₂×C₃⋊C₈	—	2	4	no	—
59	C₂×C₃⋊C₈	C₂³	1	2	yes	—
60	C₂×C₃⋊C₈	C₂³	1	2	yes	—
61	C₂×C₃⋊C₈	C₂³	1	2	yes	—
62	C₂×C₃⋊C₈	C₂³	1	2	yes	—
63	C₂×M₁₆	—	3	1	no	—
64	C₂×M₁₆	—	3	1	no	—
65	C₂²×C₈	—	3	2	no	—
66	C₄⋊Q₈	Dic₃	1	1	yes	characteristic
67	C₄⋊Q₈	Dic₃	1	1	yes	characteristic
68	#125	Dic₃	1	2	yes	—
69	C₂²⋊Q₈	Dic₃	1	2	yes	—
70	C₂²⋊Q₈	Dic₃	1	2	yes	—
71	C₄×Q₈	C₂×D₃	1	2	yes	—
72	C₄×Q₈	C₂×D₃	1	2	yes	—
73	#117	C₂×D₃	1	2	yes	—
74	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
75	C₄⋊C₈	—	3	1	no	—
76	C₄⋊C₈	—	3	1	no	—
77	C₄⋊C₈	—	3	2	no	—
78	C₄⋊C₈	—	3	4	no	—
79	C₄⋊C₈	—	3	4	no	—
80	#97	—	3	2	no	—
81	#97	—	3	4	no	—
82	C₄×C₈	—	3	2	no	—
83	C₄×C₈	—	3	4	no	—
84	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
85	C₃×Q₈	—	2	4	no	—
86	C₃×Q₈	—	4	2	no	—
87	C₂×C₃×C₄	—	2	2	no	—
88	C₂×C₃×C₄	—	4	1	no	—
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	C₂²×C₄	1	4	yes	—
93	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
94	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
97	C₂×C₃×C₄	#41	1	2	yes	—
98	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
99	C₃⋊C₈	—	2	2	no	—
100	C₃⋊C₈	—	2	2	no	—
101	C₃⋊C₈	—	2	2	no	—
102	C₃⋊C₈	C₂×Q₈	1	4	yes	—
103	C₂×Q₈	C₂×Dic₃	1	2	yes	—
104	C₂×Q₈	C₂×Dic₃	1	2	yes	—
105	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
106	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
107	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
108	M₁₆	—	3	4	no	—
109	M₁₆	—	6	2	no	—
110	C₂×C₈	—	3	2	no	—
111	C₂×C₈	—	3	2	no	—
112	C₂×C₈	—	3	2	no	—
113	C₂×C₈	—	3	2	no	—
114	C₂×C₈	—	6	4	no	—
115	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
116	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
117	C₄⋊C₄	C₂×Dic₃	1	4	yes	—
118	C₄⋊C₄	C₂×Dic₃	1	4	yes	—
119	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
120	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
121	C₂²⋊C₄	C₂×Dic₃	1	4	yes	—
122	C₄²	C₂×Dic₃	1	1	yes	characteristic
123	C₄²	C₂×Dic₃	1	1	yes	characteristic
124	C₄²	C₂²×D₃	1	1	yes	characteristic
125	C₄²	C₂²×D₃	1	4	yes	—
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	4	no	—
133	C₃×C₄	—	2	1	no	—
134	C₃×C₄	—	2	1	no	—
135	C₃×C₄	—	2	4	no	—
136	C₃×C₄	C₄×Q₈	1	2	yes	—
137	C₃×C₄	#131	1	2	yes	—
138	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
139	C₃×C₄	C₂×#41	1	1	yes	characteristic
140	C₂³	C₂²×Dic₃	1	1	yes	characteristic
141	Q₈	—	2	4	no	—
142	Q₈	—	4	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	D₃×Q₈	1	2	yes	—
146	C₂×C₄	#239	1	2	yes	—
147	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
148	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
149	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
150	C₂×C₄	C₂²×Dic₃	1	4	yes	—
151	C₂×C₄	C₂²×Dic₃	1	4	yes	—
152	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
153	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
154	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
155	C₈	—	3	4	no	—
156	C₈	—	6	2	no	—
157	C₈	—	6	2	no	—
158	C₈	—	6	2	no	—
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	Q₈×Dic₃	1	2	yes	—
166	C₂²	C₂³×Dic₃	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	4	no	—
172	C₄	Q₈×Dic₃	1	2	yes	—
173	C₄	#948	1	2	yes	—
174	C₄	C₂×D₃×Q₈	1	1	yes	characteristic
175	C₄	C₂×#239	1	1	yes	characteristic
176	C₃	#2962	1	1	yes	characteristic, complemented, Sylow
177	C₂	—	2	2	no	—
178	C₂	C₂×Q₈×Dic₃	1	1	yes	characteristic
179	C₂	C₂×#948	1	1	yes	characteristic
180	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	#2962
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
57. #208	Q₈	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
C₃×#533	2
#5024	2
Q₈×C₃⋊C₈	2
#4999	2
#4997	2
C₂×C₄⋊(C₃⋊C₈)	2
C₄×#208	2
#2962	3