Subgroups — #82302

Order 384.

456 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
2	#5207	C₂	1	2	yes	maximal, complemented
3	C₂×#830	C₂	1	2	yes	maximal, complemented
4	#5011	C₂	1	8	yes	maximal, complemented
5	#4926	C₂	1	2	yes	maximal, complemented
6	#2973	—	3	1	no	maximal, Sylow
7	C₂²×C₃×D₄	C₄	1	1	yes	characteristic
8	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₃×C₄×D₄	C₄	1	4	yes	—
11	C₃×C₄×D₄	C₂²	1	4	yes	—
12	C₂×C₃×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₂×C₃×C₂²⋊C₄	C₄	1	1	yes	characteristic
14	C₂×C₃×C₂²⋊C₄	C₄	1	1	yes	characteristic
15	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
16	C₂×#208	C₂²	1	4	yes	complemented
17	C₂²×C₃⋊C₈	C₂²	1	4	yes	complemented
18	#830	C₂²	1	4	yes	—
19	#830	C₂²	1	4	yes	complemented
20	#830	C₂²	1	4	yes	complemented
21	C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	complemented
22	#803	C₂²	1	4	yes	complemented
23	C₂×C₄×D₄	D₃	1	1	yes	characteristic
24	#435	—	3	8	no	—
25	#432	—	3	2	no	—
26	#407	—	3	2	no	—
27	C₂×C₂²⋊C₈	—	3	2	no	—
28	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
29	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×D₄	—	2	4	no	—
31	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
32	C₂²×C₃×C₄	—	4	2	no	—
33	C₂²×C₃×C₄	—	4	2	no	—
34	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
39	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
40	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
41	C₂²×C₃×C₄	C₂³	1	2	yes	—
42	C₂²×C₃×C₄	C₂³	1	2	yes	—
43	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
44	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
45	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
46	C₃×C₄²	C₂×C₄	1	2	yes	—
47	C₃×C₄²	C₂³	1	2	yes	—
48	#208	—	4	4	no	—
49	C₂×C₃⋊C₈	—	4	4	no	—
50	C₂×C₃⋊C₈	C₂³	1	8	yes	—
51	C₂²×D₄	Dic₃	1	1	yes	characteristic
52	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
53	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
54	C₂×M₁₆	—	3	4	no	—
55	C₂²×C₈	—	3	4	no	—
56	C₄×D₄	Dic₃	1	4	yes	—
57	C₄×D₄	C₂×D₃	1	4	yes	—
58	C₂×C₄⋊C₄	Dic₃	1	1	yes	characteristic
59	C₂×C₂²⋊C₄	Dic₃	1	1	yes	characteristic
60	C₂×C₂²⋊C₄	Dic₃	1	1	yes	characteristic
61	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
62	C₄⋊C₈	—	3	4	no	—
63	C₂²⋊C₈	—	3	4	no	—
64	C₂²⋊C₈	—	3	4	no	—
65	C₂²⋊C₈	—	3	4	no	—
66	#97	—	3	4	no	—
67	C₂³×C₃	—	2	2	no	—
68	C₂³×C₃	—	4	1	no	—
69	C₂³×C₃	—	4	1	no	—
70	C₂³×C₃	—	4	1	no	—
71	C₂³×C₃	—	4	1	no	—
72	C₂³×C₃	M₁₆	1	4	yes	complemented
73	C₂³×C₃	C₂²×C₄	1	2	yes	—
74	C₂³×C₃	C₂²×C₄	1	2	yes	—
75	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
76	C₃×D₄	—	2	8	no	—
77	C₂×C₃×C₄	—	2	2	no	—
78	C₂×C₃×C₄	—	2	4	no	—
79	C₂×C₃×C₄	—	2	4	no	—
80	C₂×C₃×C₄	—	2	2	no	—
81	C₂×C₃×C₄	—	4	2	no	—
82	C₂×C₃×C₄	—	4	2	no	—
83	C₂×C₃×C₄	—	4	1	no	—
84	C₂×C₃×C₄	—	4	1	no	—
85	C₂×C₃×C₄	—	4	1	no	—
86	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
88	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
89	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
90	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
91	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
92	C₃⋊C₈	—	2	8	no	—
93	C₂⁴	C₂×Dic₃	1	1	yes	characteristic
94	C₂⁴	C₂×Dic₃	1	1	yes	characteristic
95	C₂×D₄	—	2	4	no	—
96	C₂×D₄	C₂×Dic₃	1	4	yes	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	—	4	2	no	—
99	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
100	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
101	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
102	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
103	C₂²×C₄	C₂×Dic₃	1	2	yes	—
104	C₂²×C₄	C₂×Dic₃	1	2	yes	—
105	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₃	1	2	yes	—
107	C₂²×C₄	C₂²×D₃	1	2	yes	—
108	M₁₆	—	12	4	no	—
109	C₂×C₈	—	3	8	no	—
110	C₂×C₈	—	12	4	no	—
111	C₄⋊C₄	C₂×Dic₃	1	4	yes	—
112	C₂²⋊C₄	C₂×Dic₃	1	4	yes	—
113	C₂²⋊C₄	C₂×Dic₃	1	4	yes	—
114	C₄²	C₂×Dic₃	1	2	yes	—
115	C₄²	C₂²×D₃	1	2	yes	—
116	C₂²×C₃	—	2	1	no	—
117	C₂²×C₃	—	2	2	no	—
118	C₂²×C₃	—	2	1	no	—
119	C₂²×C₃	—	2	2	no	—
120	C₂²×C₃	—	2	2	no	—
121	C₂²×C₃	—	2	2	no	—
122	C₂²×C₃	—	2	2	no	—
123	C₂²×C₃	—	4	4	no	—
124	C₂²×C₃	—	4	4	no	—
125	C₂²×C₃	C₂×M₁₆	1	2	yes	—
126	C₂²×C₃	C₂×M₁₆	1	4	yes	—
127	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
128	C₃×C₄	—	2	1	no	—
129	C₃×C₄	—	2	2	no	—
130	C₃×C₄	—	2	2	no	—
131	C₃×C₄	—	2	2	no	—
132	C₃×C₄	—	2	2	no	—
133	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
134	C₂³	—	2	2	no	—
135	C₂³	—	4	1	no	—
136	C₂³	—	4	1	no	—
137	C₂³	—	4	1	no	—
138	C₂³	—	4	1	no	—
139	C₂³	#208	1	4	yes	complemented
140	C₂³	C₂²×Dic₃	1	2	yes	—
141	C₂³	C₂²×Dic₃	1	2	yes	—
142	C₂³	C₂²×Dic₃	1	1	yes	characteristic
143	D₄	—	2	8	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
154	C₂×C₄	C₂²×Dic₃	1	2	yes	—
155	C₂×C₄	C₂²×Dic₃	1	2	yes	—
156	C₂×C₄	C₂²×Dic₃	1	2	yes	—
157	C₂×C₄	C₂²×Dic₃	1	2	yes	—
158	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
159	C₈	—	6	8	no	—
160	C₂×C₃	—	2	2	no	—
161	C₂×C₃	—	2	4	no	—
162	C₂×C₃	—	4	2	no	—
163	C₂×C₃	#518	1	1	yes	characteristic
164	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
165	C₂×C₃	#568	1	1	yes	characteristic
166	C₂²	—	2	1	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	4	4	no	—
174	C₂²	—	4	4	no	—
175	C₂²	C₂×#208	1	2	yes	—
176	C₂²	C₂×#208	1	4	yes	—
177	C₂²	C₂³×Dic₃	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	#1004	1	2	yes	—
184	C₃	#2973	1	1	yes	characteristic, complemented, Sylow
185	C₂	—	2	2	no	—
186	C₂	—	2	4	no	—
187	C₂	—	4	2	no	—
188	C₂	C₂²×#208	1	1	yes	characteristic
189	C₂	#5798	1	1	yes	characteristic
190	C₂	#5819	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×C₄×D₄	C₂
prime radical	C₂²×C₃×D₄	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×C₄×D₄	D₃
hypocenter	C₃	#2973
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #5207	C₂	semidirect
3. C₂×#830	C₂	semidirect
4. #5011	C₂	semidirect
5. #4926	C₂	semidirect
16. C₂×#208	C₂²	semidirect
17. C₂²×C₃⋊C₈	C₂²	semidirect
19. #830	C₂²	semidirect
20. #830	C₂²	semidirect
21. C₄⋊(C₃⋊C₈)	C₂²	semidirect
22. #803	C₂²	semidirect
72. C₂³×C₃	M₁₆	semidirect
139. C₂³	#208	semidirect
184. C₃	#2973	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×D₄	2
#5207	2
C₂×#830	2
#5011	2
#4926	2
#2973	3