Subgroups — #78469

Order 384.

380 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5479	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#5116	C₂	1	1	yes	maximal, characteristic, complemented
4	#5115	C₂	1	1	yes	maximal, characteristic
5	#5114	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#4697	C₂	1	2	yes	maximal, complemented
8	#4696	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#204	C₂	1	2	yes	maximal, complemented
11	#4683	C₂	1	1	yes	maximal, characteristic, complemented
12	#3168	—	3	1	no	maximal, Sylow
13	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
16	#930	C₂²	1	2	yes	—
17	#929	—	2	2	no	—
18	C₄×#67	—	2	2	no	—
19	#924	C₂²	1	2	yes	—
20	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
21	C₂×D₂₄	C₂²	1	1	yes	characteristic
22	C₂×#204	C₂²	1	2	yes	—
23	#876	—	2	2	no	—
24	#875	C₂²	1	2	yes	—
25	C₄×D₁₂	C₂²	1	2	yes	—
26	#872	—	2	2	no	—
27	#870	C₂²	1	2	yes	—
28	C₄×Dic₆	C₂²	1	2	yes	—
29	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
31	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
32	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
33	#821	C₂²	1	2	yes	—
34	#821	C₂²	1	2	yes	—
35	#818	C₂²	1	1	yes	characteristic
36	#818	C₂²	1	1	yes	characteristic
37	#817	C₂²	1	2	yes	—
38	#816	C₂²	1	2	yes	—
39	#816	C₂²	1	2	yes	—
40	#529	—	3	2	no	—
41	#499	—	3	1	no	—
42	#495	—	3	1	no	—
43	#488	—	3	2	no	—
44	#467	—	3	1	no	—
45	#466	—	3	1	no	—
46	#465	—	3	2	no	—
47	C₄×Q₁₆	—	3	1	no	—
48	C₄×SD₁₆	—	3	2	no	—
49	C₄×D₈	—	3	1	no	—
50	C₂×C₄×C₈	D₃	1	1	yes	characteristic, complemented
51	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
52	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₂×#67	—	2	2	no	—
55	C₂×D₁₂	C₂³	1	2	yes	—
56	C₂×C₄×D₃	—	2	2	no	—
57	C₂×Dic₆	C₂³	1	2	yes	—
58	C₂×C₃×C₈	—	2	2	no	—
59	C₂×C₃×C₈	—	2	2	no	—
60	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
61	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
64	C₃×C₄²	D₄	1	1	yes	characteristic
65	C₃×C₄²	D₄	1	1	yes	characteristic
66	C₃×C₄²	C₂³	1	1	yes	characteristic
67	C₃×C₄²	C₂³	1	1	yes	characteristic
68	#217	—	2	2	no	—
69	(C₂×D₃)⋊C₄	—	2	2	no	—
70	(C₂×D₃)⋊C₄	—	2	2	no	—
71	(C₂×D₃)⋊C₄	—	2	2	no	—
72	(C₂×D₃)⋊C₄	—	2	2	no	—
73	C₄⋊Dic₃	C₂³	1	2	yes	—
74	C₄⋊Dic₃	C₂³	1	2	yes	—
75	Dic₃⋊C₄	—	2	2	no	—
76	Dic₃⋊C₄	—	2	2	no	—
77	Dic₃⋊C₄	—	2	2	no	—
78	Dic₃⋊C₄	—	2	2	no	—
79	C₄×Dic₃	—	2	2	no	—
80	Dic₁₂	—	2	2	no	—
81	D₂₄	—	2	2	no	—
82	#204	—	2	4	no	—
83	C₂×Q₁₆	—	3	1	no	—
84	C₂×SD₁₆	—	3	2	no	—
85	C₂×D₈	—	3	1	no	—
86	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
88	#126	—	6	2	no	—
89	#125	—	3	2	no	—
90	#124	—	3	2	no	—
91	#123	—	6	2	no	—
92	C₂²⋊Q₈	—	3	2	no	—
93	#121	—	3	2	no	—
94	C₄×Q₈	—	3	2	no	—
95	C₄×D₄	—	3	2	no	—
96	C₄×D₄	—	6	2	no	—
97	#117	—	6	2	no	—
98	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
99	#107	—	3	1	no	—
100	#107	—	3	1	no	—
101	#106	—	3	2	no	—
102	#103	—	3	2	no	—
103	#103	—	3	2	no	—
104	#102	—	3	2	no	—
105	#102	—	3	2	no	—
106	C₄×C₈	C₂×D₃	1	1	yes	characteristic
107	C₄×C₈	C₂×D₃	1	1	yes	characteristic
108	C₄×C₈	C₂×D₃	1	1	yes	characteristic
109	C₄×C₈	C₂×D₃	1	1	yes	characteristic
110	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×D₃	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
122	#67	—	4	4	no	—
123	C₂×Dic₃	—	2	2	no	—
124	C₂×Dic₃	—	2	2	no	—
125	C₂×Dic₃	—	2	2	no	—
126	D₁₂	—	2	2	no	—
127	D₁₂	—	2	2	no	—
128	C₄×D₃	—	4	4	no	—
129	Dic₆	—	2	2	no	—
130	Dic₆	—	2	2	no	—
131	C₃×C₈	—	2	1	no	—
132	C₃×C₈	—	2	1	no	—
133	C₃×C₈	#41	1	2	yes	—
134	C₃×C₈	#41	1	2	yes	—
135	C₂×Q₈	—	3	2	no	—
136	C₂×D₄	—	3	2	no	—
137	C₂×D₄	—	6	2	no	—
138	C₂²×C₄	—	6	2	no	—
139	C₂²×C₄	D₁₂	1	1	yes	characteristic
140	C₂²×C₄	D₁₂	1	1	yes	characteristic
141	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
142	Q₁₆	—	6	2	no	—
143	SD₁₆	—	6	4	no	—
144	D₈	—	6	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
148	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
149	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
150	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
151	C₄⋊C₄	—	3	2	no	—
152	C₄⋊C₄	—	3	2	no	—
153	C₄⋊C₄	—	6	2	no	—
154	C₄⋊C₄	—	6	2	no	—
155	C₄⋊C₄	—	6	2	no	—
156	C₄⋊C₄	—	6	2	no	—
157	C₂²⋊C₄	—	6	2	no	—
158	C₂²⋊C₄	—	6	2	no	—
159	C₂²⋊C₄	—	6	2	no	—
160	C₂²⋊C₄	—	6	2	no	—
161	C₂²⋊C₄	—	6	2	no	—
162	C₄²	—	6	2	no	—
163	C₄²	D₁₂	1	1	yes	characteristic
164	C₄²	D₁₂	1	1	yes	characteristic
165	C₄²	C₂²×D₃	1	1	yes	characteristic
166	C₄²	C₂²×D₃	1	1	yes	characteristic
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	#135	1	2	yes	—
170	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
171	C₂×D₃	—	4	2	no	—
172	C₂×D₃	—	4	2	no	—
173	C₂×D₃	—	4	2	no	—
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	#135	1	2	yes	—
179	C₃×C₄	C₂×#41	1	1	yes	characteristic
180	C₃×C₄	C₂×#41	1	1	yes	characteristic
181	Dic₃	—	4	2	no	—
182	Dic₃	—	4	2	no	—
183	Dic₃	—	4	2	no	—
184	C₂³	—	6	2	no	—
185	C₂³	C₂×D₁₂	1	1	yes	characteristic
186	Q₈	—	6	2	no	—
187	Q₈	—	6	2	no	—
188	D₄	—	6	2	no	—
189	D₄	—	6	2	no	—
190	D₄	—	12	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	6	2	no	—
196	C₂×C₄	—	6	2	no	—
197	C₂×C₄	—	6	2	no	—
198	C₂×C₄	—	12	4	no	—
199	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
200	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
201	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
202	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
203	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
204	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	#235	1	2	yes	—
208	C₈	#235	1	2	yes	—
209	C₂×C₃	—	2	2	no	—
210	C₂×C₃	#532	1	1	yes	characteristic
211	C₂×C₃	C₂×#135	1	1	yes	characteristic
212	C₂×C₃	C₂×#135	1	1	yes	characteristic
213	D₃	—	8	2	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	12	2	no	—
217	C₂²	—	12	2	no	—
218	C₂²	—	12	2	no	—
219	C₂²	#904	1	2	yes	—
220	C₂²	C₂²×D₁₂	1	1	yes	characteristic
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	12	2	no	—
226	C₄	—	12	2	no	—
227	C₄	—	12	2	no	—
228	C₄	#904	1	2	yes	—
229	C₄	C₂×#235	1	1	yes	characteristic
230	C₄	C₂×#235	1	1	yes	characteristic
231	C₃	#3168	1	1	yes	characteristic, complemented, Sylow
232	C₂	—	2	2	no	—
233	C₂	—	24	2	no	—
234	C₂	#5477	1	1	yes	characteristic
235	C₂	C₂×#904	1	1	yes	characteristic
236	C₂	C₂×#904	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#5115	C₂
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	#3168
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #5479	C₂	semidirect
2. C₂×C₃×C₄×C₈	C₂	semidirect
3. #5116	C₂	semidirect
5. #5114	C₂	semidirect
6. C₄×Dic₁₂	C₂	semidirect
7. #4697	C₂	semidirect
8. #4696	C₂	semidirect
9. C₄×D₂₄	C₂	semidirect
10. C₄×#204	C₂	semidirect
11. #4683	C₂	semidirect
50. C₂×C₄×C₈	D₃	semidirect
231. C₃	#3168	semidirect

Maximal subgroups

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

subgroup	index
#5479	2
C₂×C₃×C₄×C₈	2
#5116	2
#5115	2
#5114	2
C₄×Dic₁₂	2
#4697	2
#4696	2
C₄×D₂₄	2
C₄×#204	2
#4683	2
#3168	3