Subgroups — #78552

Order 384.

392 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5522	C₂	1	1	yes	maximal, characteristic
2	C₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
3	#5129	C₂	1	1	yes	maximal, characteristic
4	#5118	C₂	1	1	yes	maximal, characteristic, complemented
5	#5110	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#4997	C₂	1	1	yes	maximal, characteristic, complemented
8	#4709	C₂	1	2	yes	maximal, complemented
9	#4705	C₂	1	2	yes	maximal, complemented
10	#4690	C₂	1	2	yes	maximal, complemented
11	#4685	C₂	1	2	yes	maximal, complemented
12	#2925	—	3	1	no	maximal, Sylow
13	C₂×#235	C₂²	1	1	yes	characteristic
14	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
16	C₃×#117	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
18	C₂×#208	C₂²	1	1	yes	characteristic
19	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	#893	C₄	1	1	yes	characteristic
21	#893	C₄	1	1	yes	characteristic
22	#887	C₄	1	1	yes	characteristic
23	#887	C₄	1	1	yes	characteristic
24	#67⋊C₄	C₄	1	2	yes	—
25	#67⋊C₄	C₄	1	2	yes	—
26	#872	C₂²	1	2	yes	—
27	C₄²×D₃	C₂²	1	2	yes	—
28	C₃×#97	C₂²	1	2	yes	—
29	C₃×C₄×C₈	C₂²	1	2	yes	—
30	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
32	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
34	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
35	Dic₃⋊C₈	C₂²	1	2	yes	—
36	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₃⋊C₈	C₂²	1	2	yes	—
38	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
39	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
40	#803	C₂²	1	2	yes	—
41	C₄×C₃⋊C₈	C₂²	1	2	yes	—
42	C₄×#41	—	3	1	no	—
43	#433	—	3	2	no	—
44	#433	—	3	2	no	—
45	#432	—	3	2	no	—
46	#431	—	3	2	no	—
47	#423	—	3	1	no	—
48	C₂×C₄⋊C₈	—	3	1	no	—
49	#408	—	3	1	no	—
50	#408	—	3	1	no	—
51	#405	—	3	1	no	—
52	#405	D₃	1	1	yes	characteristic, complemented
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₂×#67	C₂×C₄	1	2	yes	—
55	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
57	#235	—	2	2	no	—
58	#235	—	2	2	no	—
59	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
61	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
62	C₂×C₄×D₃	C₂³	1	2	yes	—
63	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
64	C₃×M₁₆	—	2	2	no	—
65	C₂×C₃×C₈	—	2	2	no	—
66	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂³	1	2	yes	—
68	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
69	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
70	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
71	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
72	C₃×C₄²	C₂³	1	2	yes	—
73	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
74	(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
75	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
76	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
77	C₄×Dic₃	C₂×C₄	1	2	yes	—
78	C₄×Dic₃	C₂×C₄	1	2	yes	—
79	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₃	C₂³	1	1	yes	characteristic
82	C₄×Dic₃	C₂³	1	1	yes	characteristic
83	#208	—	2	2	no	—
84	C₂×C₃⋊C₈	—	2	2	no	—
85	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
86	C₂×C₃⋊C₈	C₂³	1	2	yes	—
87	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
88	C₂×#41	—	3	1	no	—
89	C₂×M₁₆	—	3	1	no	—
90	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
91	C₂²×C₈	—	3	1	no	—
92	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
93	C₄×Q₈	—	3	1	no	—
94	C₄×Q₈	—	3	1	no	—
95	C₄×D₄	—	3	1	no	—
96	C₄×D₄	—	3	1	no	—
97	C₄×D₄	—	3	2	no	—
98	C₄×D₄	—	3	2	no	—
99	#117	—	3	2	no	—
100	#117	C₂×D₃	1	1	yes	characteristic
101	C₂×C₄²	—	3	1	no	—
102	C₂×C₄²	—	3	2	no	—
103	C₄⋊C₈	—	3	1	no	—
104	C₄⋊C₈	—	3	1	no	—
105	C₄⋊C₈	—	3	1	no	—
106	C₄⋊C₈	—	3	1	no	—
107	C₄⋊C₈	—	3	2	no	—
108	C₄⋊C₈	—	3	2	no	—
109	C₂²⋊C₈	—	3	2	no	—
110	C₂²⋊C₈	—	3	2	no	—
111	C₂²⋊C₈	—	3	2	no	—
112	C₂²⋊C₈	—	3	2	no	—
113	#97	—	3	2	no	—
114	#97	C₂×D₃	1	2	yes	—
115	C₄×C₈	—	3	2	no	—
116	C₄×C₈	C₂×D₃	1	2	yes	—
117	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²×D₃	C₂²×C₄	1	2	yes	—
119	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
120	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
121	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₃×C₄	#41	1	2	yes	—
123	C₂×C₃×C₄	#41	1	2	yes	—
124	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
125	#67	—	2	4	no	—
126	C₂×Dic₃	—	2	2	no	—
127	C₂×Dic₃	—	4	1	no	—
128	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₃	C₂²×C₄	1	2	yes	—
130	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
133	D₁₂	—	2	2	no	—
134	C₄×D₃	—	2	2	no	—
135	C₄×D₃	—	2	2	no	—
136	C₄×D₃	—	2	4	no	—
137	C₄×D₃	—	4	2	no	—
138	Dic₆	—	2	2	no	—
139	C₃×C₈	—	2	1	no	—
140	C₃×C₈	—	2	1	no	—
141	C₃×C₈	—	2	2	no	—
142	C₃⋊C₈	—	2	1	no	—
143	C₃⋊C₈	—	2	1	no	—
144	C₃⋊C₈	—	2	2	no	—
145	#41	—	6	2	no	—
146	#41	—	6	2	no	—
147	C₂×Q₈	—	3	1	no	—
148	C₂×D₄	—	3	2	no	—
149	C₂×D₄	—	3	1	no	—
150	C₂²×C₄	—	3	1	no	—
151	C₂²×C₄	—	3	1	no	—
152	C₂²×C₄	—	3	2	no	—
153	C₂²×C₄	—	3	2	no	—
154	C₂²×C₄	—	3	2	no	—
155	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
156	M₁₆	—	2	2	no	—
157	M₁₆	—	6	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	3	1	no	—
160	C₂×C₈	—	3	2	no	—
161	C₂×C₈	—	3	1	no	—
162	C₂×C₈	—	6	2	no	—
163	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
164	C₂×C₈	C₂²×D₃	1	2	yes	—
165	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
166	C₄⋊C₄	—	3	2	no	—
167	C₄⋊C₄	—	3	2	no	—
168	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
169	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
170	C₂²⋊C₄	—	3	2	no	—
171	C₂²⋊C₄	—	3	2	no	—
172	C₂²⋊C₄	C₄×D₃	1	2	yes	—
173	C₄²	—	3	1	no	—
174	C₄²	—	3	2	no	—
175	C₄²	—	3	1	no	—
176	C₄²	—	3	2	no	—
177	C₄²	—	3	1	no	—
178	C₄²	—	3	1	no	—
179	C₄²	C₂²×D₃	1	2	yes	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	#117	1	2	yes	—
183	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
184	C₂×D₃	—	2	2	no	—
185	C₂×D₃	—	2	2	no	—
186	C₂×D₃	—	2	2	no	—
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	#117	1	2	yes	—
190	C₃×C₄	C₂×#41	1	1	yes	characteristic
191	C₃×C₄	C₂×#41	1	1	yes	characteristic
192	Dic₃	—	2	2	no	—
193	Dic₃	—	2	1	no	—
194	Dic₃	—	2	1	no	—
195	Dic₃	—	2	1	no	—
196	Dic₃	#131	1	2	yes	—
197	C₂³	—	3	2	no	—
198	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
199	Q₈	—	6	2	no	—
200	D₄	—	6	4	no	—
201	D₄	—	6	2	no	—
202	C₂×C₄	—	3	1	no	—
203	C₂×C₄	—	3	1	no	—
204	C₂×C₄	—	3	1	no	—
205	C₂×C₄	—	3	1	no	—
206	C₂×C₄	—	3	2	no	—
207	C₂×C₄	—	6	4	no	—
208	C₂×C₄	—	6	2	no	—
209	C₂×C₄	—	6	2	no	—
210	C₂×C₄	—	6	2	no	—
211	C₂×C₄	—	12	2	no	—
212	C₂×C₄	—	12	1	no	—
213	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
214	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
215	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
216	C₂×C₄	#235	1	2	yes	—
217	C₂×C₄	#235	1	2	yes	—
218	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	2	no	—
222	C₈	—	6	1	no	—
223	C₈	—	6	1	no	—
224	C₈	—	6	2	no	—
225	C₂×C₃	—	2	2	no	—
226	C₂×C₃	C₂×#117	1	1	yes	characteristic
227	C₂×C₃	C₂×#131	1	1	yes	characteristic
228	C₂×C₃	#568	1	1	yes	characteristic
229	D₃	—	4	2	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	6	2	no	—
233	C₂²	—	6	2	no	—
234	C₂²	—	6	2	no	—
235	C₂²	#872	1	2	yes	—
236	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	3	2	no	—
240	C₄	—	6	2	no	—
241	C₄	—	6	1	no	—
242	C₄	—	6	1	no	—
243	C₄	—	6	1	no	—
244	C₄	#872	1	2	yes	—
245	C₄	C₂×#235	1	1	yes	characteristic
246	C₄	C₂×#235	1	1	yes	characteristic
247	C₃	#2925	1	1	yes	characteristic, complemented, Sylow
248	C₂	—	2	2	no	—
249	C₂	—	12	2	no	—
250	C₂	C₂×#872	1	1	yes	characteristic
251	C₂	D₃×#131	1	1	yes	characteristic
252	C₂	#5750	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#405	C₂
prime radical	C₂×#235	C₂²
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#405	D₃
hypocenter	C₃	#2925
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. C₃×#405	C₂	semidirect
4. #5118	C₂	semidirect
6. C₂×Dic₃⋊C₈	C₂	semidirect
7. #4997	C₂	semidirect
8. #4709	C₂	semidirect
9. #4705	C₂	semidirect
10. #4690	C₂	semidirect
11. #4685	C₂	semidirect
52. #405	D₃	semidirect
247. C₃	#2925	semidirect

Maximal subgroups

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

subgroup	index
#5522	2
C₃×#405	2
#5129	2
#5118	2
#5110	2
C₂×Dic₃⋊C₈	2
#4997	2
#4709	2
#4705	2
#4690	2
#4685	2
#2925	3