Subgroups — #82521

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#41	C₂	1	1	yes	maximal, characteristic
2	#5024	C₂	1	2	yes	maximal, complemented
3	#5012	C₂	1	2	yes	maximal, complemented
4	#5011	C₂	1	4	yes	maximal, complemented
5	#4926	C₂	1	2	yes	maximal, complemented
6	#4925	C₂	1	1	yes	maximal, characteristic, complemented
7	#4924	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
9	#2989	—	3	1	no	maximal, Sylow
10	C₂×C₃×#41	C₄	1	1	yes	characteristic
11	C₃×C₄×Q₈	C₄	1	1	yes	characteristic
12	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×D₄	C₄	1	2	yes	—
14	C₃×C₄×D₄	C₄	1	1	yes	characteristic
15	C₃×C₄×D₄	C₂²	1	2	yes	—
16	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
17	C₃×#117	C₄	1	2	yes	—
18	C₃×#117	C₄	1	1	yes	characteristic
19	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄²	C₂²	1	2	yes	—
21	C₂×#208	C₂²	1	4	yes	—
22	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
23	#830	C₂²	1	2	yes	—
24	#830	C₂²	1	4	yes	—
25	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
26	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
27	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
28	C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
29	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
30	#803	C₂²	1	4	yes	—
31	C₄×C₃⋊C₈	C₂²	1	2	yes	—
32	C₄×#41	D₃	1	1	yes	characteristic
33	#446	—	3	2	no	—
34	#436	—	3	2	no	—
35	#435	—	3	4	no	—
36	#432	—	3	2	no	—
37	#431	—	3	1	no	—
38	#423	—	3	2	no	—
39	C₂×C₄⋊C₈	—	3	1	no	—
40	C₃×#41	—	2	4	no	—
41	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
43	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
45	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
46	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
49	C₂²×C₃×C₄	C₂³	1	2	yes	—
50	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
51	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
52	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
54	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
55	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
56	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
57	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
58	C₃×C₄²	C₂×C₄	1	2	yes	—
59	C₃×C₄²	C₂×C₄	1	2	yes	—
60	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
61	C₃×C₄²	C₂³	1	1	yes	characteristic
62	C₃×C₄²	C₂³	1	2	yes	—
63	C₃×C₄²	C₂³	1	1	yes	characteristic
64	#208	—	4	4	no	—
65	C₂×C₃⋊C₈	—	4	2	no	—
66	C₂×C₃⋊C₈	C₂³	1	2	yes	—
67	C₂×C₃⋊C₈	C₂³	1	2	yes	—
68	C₂×C₃⋊C₈	C₂³	1	4	yes	—
69	C₂×#41	Dic₃	1	1	yes	characteristic
70	C₂×M₁₆	—	3	4	no	—
71	C₂²×C₈	—	3	2	no	—
72	C₄×Q₈	Dic₃	1	1	yes	characteristic
73	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
74	C₄×D₄	Dic₃	1	2	yes	—
75	C₄×D₄	Dic₃	1	1	yes	characteristic
76	C₄×D₄	C₂×D₃	1	2	yes	—
77	C₄×D₄	C₂×D₃	1	1	yes	characteristic
78	#117	Dic₃	1	2	yes	—
79	#117	Dic₃	1	1	yes	characteristic
80	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
81	C₂×C₄²	C₂×D₃	1	2	yes	—
82	C₄⋊C₈	—	3	2	no	—
83	C₄⋊C₈	—	3	2	no	—
84	C₄⋊C₈	—	3	1	no	—
85	C₄⋊C₈	—	3	4	no	—
86	C₄⋊C₈	—	3	1	no	—
87	C₂²⋊C₈	—	3	2	no	—
88	C₂²⋊C₈	—	3	4	no	—
89	#97	—	3	4	no	—
90	C₄×C₈	—	3	2	no	—
91	C₂³×C₃	C₂²×C₄	1	2	yes	—
92	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
93	C₃×Q₈	—	2	2	no	—
94	C₃×D₄	—	2	4	no	—
95	C₃×D₄	—	2	2	no	—
96	C₂×C₃×C₄	—	2	4	no	—
97	C₂×C₃×C₄	—	2	1	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	1	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	4	1	no	—
102	C₂×C₃×C₄	—	4	2	no	—
103	C₂×C₃×C₄	M₁₆	1	4	yes	complemented
104	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
107	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
109	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
110	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
113	C₃⋊C₈	—	2	4	no	—
114	C₃⋊C₈	—	2	2	no	—
115	C₃⋊C₈	—	2	2	no	—
116	#41	—	2	4	no	—
117	C₂×Q₈	C₂×Dic₃	1	1	yes	characteristic
118	C₂×D₄	C₂×Dic₃	1	2	yes	—
119	C₂×D₄	C₂×Dic₃	1	1	yes	characteristic
120	C₂²×C₄	C₂×Dic₃	1	2	yes	—
121	C₂²×C₄	C₂×Dic₃	1	2	yes	—
122	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
123	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₃	1	2	yes	—
126	M₁₆	—	12	4	no	—
127	C₂×C₈	—	3	2	no	—
128	C₂×C₈	—	3	2	no	—
129	C₂×C₈	—	3	4	no	—
130	C₂×C₈	—	12	2	no	—
131	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
132	C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
133	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
134	C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
135	C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
136	C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
137	C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
138	C₄²	C₂×Dic₃	1	1	yes	characteristic
139	C₄²	C₂×Dic₃	1	2	yes	—
140	C₄²	C₂×Dic₃	1	2	yes	—
141	C₄²	C₂×Dic₃	1	1	yes	characteristic
142	C₄²	C₂²×D₃	1	1	yes	characteristic
143	C₄²	C₂²×D₃	1	2	yes	—
144	C₄²	C₂²×D₃	1	1	yes	characteristic
145	C₂²×C₃	—	2	1	no	—
146	C₂²×C₃	—	2	1	no	—
147	C₂²×C₃	—	2	2	no	—
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	—	2	2	no	—
150	C₂²×C₃	C₂×M₁₆	1	2	yes	—
151	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
152	C₃×C₄	—	2	1	no	—
153	C₃×C₄	—	2	2	no	—
154	C₃×C₄	—	2	2	no	—
155	C₃×C₄	—	2	1	no	—
156	C₃×C₄	—	2	1	no	—
157	C₃×C₄	—	2	2	no	—
158	C₃×C₄	C₂×M₁₆	1	2	yes	—
159	C₃×C₄	C₂×M₁₆	1	2	yes	—
160	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
161	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
162	C₂³	C₂²×Dic₃	1	2	yes	—
163	C₂³	C₂²×Dic₃	1	1	yes	characteristic
164	Q₈	—	2	2	no	—
165	D₄	—	2	4	no	—
166	D₄	—	2	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	#208	1	4	yes	complemented
175	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
176	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₃	1	2	yes	—
178	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
179	C₂×C₄	C₂²×Dic₃	1	2	yes	—
180	C₂×C₄	C₂²×Dic₃	1	2	yes	—
181	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
183	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
184	C₈	—	6	4	no	—
185	C₈	—	6	2	no	—
186	C₈	—	6	2	no	—
187	C₂×C₃	—	2	2	no	—
188	C₂×C₃	—	4	2	no	—
189	C₂×C₃	#520	1	1	yes	characteristic
190	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
191	C₂×C₃	#568	1	1	yes	characteristic
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	C₂×#208	1	2	yes	—
198	C₂²	C₂³×Dic₃	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	C₂×#208	1	2	yes	—
206	C₄	C₂×#208	1	2	yes	—
207	C₄	#1009	1	1	yes	characteristic
208	C₄	#1010	1	1	yes	characteristic
209	C₃	#2989	1	1	yes	characteristic, complemented, Sylow
210	C₂	—	2	2	no	—
211	C₂	—	4	2	no	—
212	C₂	C₂²×#208	1	1	yes	characteristic
213	C₂	#5819	1	1	yes	characteristic
214	C₂	#5827	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×C₄×#41	C₂
prime radical	C₂×C₃×#41	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₄×#41	D₃
hypocenter	C₃	#2989
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. #5024	C₂	semidirect
3. #5012	C₂	semidirect
4. #5011	C₂	semidirect
5. #4926	C₂	semidirect
6. #4925	C₂	semidirect
7. #4924	C₂	semidirect
8. C₂×C₄⋊(C₃⋊C₈)	C₂	semidirect
103. C₂×C₃×C₄	M₁₆	semidirect
174. C₂×C₄	#208	semidirect
209. C₃	#2989	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#41	2
#5024	2
#5012	2
#5011	2
#4926	2
#4925	2
#4924	2
C₂×C₄⋊(C₃⋊C₈)	2
#2989	3