Subgroups — #10596172

Order 576.

576 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#126	C₂	1	1	yes	maximal, characteristic
2	#64099	C₂	1	1	yes	maximal, characteristic, complemented
3	#64098	C₂	1	1	yes	maximal, characteristic, complemented
4	#64098	C₂	1	1	yes	maximal, characteristic, complemented
5	#64098	C₂	1	1	yes	maximal, characteristic, complemented
6	#64097	C₂	1	1	yes	maximal, characteristic, complemented
7	#64096	C₂	1	1	yes	maximal, characteristic, complemented
8	#64088	C₂	1	1	yes	maximal, characteristic
9	#64088	C₂	1	1	yes	maximal, characteristic
10	#64088	C₂	1	1	yes	maximal, characteristic
11	#64087	C₂	1	1	yes	maximal, characteristic
12	#64087	C₂	1	1	yes	maximal, characteristic
13	#64086	C₂	1	1	yes	maximal, characteristic
14	#64079	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#634	C₂	1	1	yes	maximal, characteristic, complemented
16	#5701	—	3	4	no	maximal
17	C₂²×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₂×#634	C₂²	1	1	yes	characteristic
19	C₂×#634	C₂²	1	1	yes	characteristic
20	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₃²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₃²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₃²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₃²×C₄²	C₂²	1	1	yes	characteristic
27	#3757	C₂²	1	1	yes	characteristic
28	#3757	C₂²	1	1	yes	characteristic
29	#3757	C₂²	1	1	yes	characteristic
30	C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
31	C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
32	C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
33	C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
34	C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
35	#3750	C₂²	1	1	yes	characteristic
36	#3750	C₂²	1	1	yes	characteristic
37	#3750	C₂²	1	1	yes	characteristic
38	#3750	C₂²	1	1	yes	characteristic
39	#3750	C₂²	1	1	yes	characteristic
40	#3750	C₂²	1	1	yes	characteristic
41	#3750	C₂²	1	1	yes	characteristic
42	#3750	C₂²	1	1	yes	characteristic
43	#3750	C₂²	1	1	yes	characteristic
44	#3750	C₂²	1	1	yes	characteristic
45	#3750	C₂²	1	1	yes	characteristic
46	#3750	C₂²	1	1	yes	characteristic
47	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
48	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
49	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
50	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
51	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
52	C₃×#126	D₃	1	4	yes	—
53	#890	—	3	4	no	—
54	#889	—	3	4	no	—
55	#889	—	3	4	no	—
56	#889	—	3	4	no	—
57	#888	—	3	4	no	—
58	#887	—	3	4	no	—
59	#879	—	3	4	no	—
60	#879	—	3	4	no	—
61	#879	—	3	4	no	—
62	#878	—	3	4	no	—
63	#878	—	3	4	no	—
64	#877	—	3	4	no	—
65	#870	—	3	4	no	—
66	C₄×Dic₆	—	3	4	no	—
67	C₂³×C₃²	C₂³	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
74	C₂×C₃⋊Dic₃	—	2	2	no	—
75	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
76	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
77	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
78	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
79	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
80	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
81	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
82	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
83	#634	—	2	2	no	—
84	#634	—	2	2	no	—
85	#541	—	9	1	no	Sylow
86	C₂²×Dic₃	—	3	4	no	—
87	C₂×Dic₆	—	3	4	no	—
88	C₂×Dic₆	—	3	4	no	—
89	C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
90	C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
91	C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
92	C₃×C₂²⋊C₄	C₂×D₃	1	4	yes	—
93	C₃×C₂²⋊C₄	C₂×D₃	1	4	yes	—
94	C₃×C₂²⋊C₄	C₂×D₃	1	4	yes	—
95	C₃×C₄²	C₂×D₃	1	4	yes	—
96	#217	—	3	4	no	—
97	#217	—	3	4	no	—
98	#217	—	3	4	no	—
99	C₄⋊Dic₃	—	3	4	no	—
100	C₄⋊Dic₃	—	3	4	no	—
101	C₄⋊Dic₃	—	3	4	no	—
102	C₄⋊Dic₃	—	3	4	no	—
103	C₄⋊Dic₃	—	3	4	no	—
104	Dic₃⋊C₄	—	3	4	no	—
105	Dic₃⋊C₄	—	3	4	no	—
106	Dic₃⋊C₄	—	3	4	no	—
107	Dic₃⋊C₄	—	3	4	no	—
108	Dic₃⋊C₄	—	3	4	no	—
109	Dic₃⋊C₄	—	3	4	no	—
110	Dic₃⋊C₄	—	3	4	no	—
111	Dic₃⋊C₄	—	3	4	no	—
112	Dic₃⋊C₄	—	3	4	no	—
113	Dic₃⋊C₄	—	3	4	no	—
114	Dic₃⋊C₄	—	3	4	no	—
115	Dic₃⋊C₄	—	3	4	no	—
116	C₄×Dic₃	—	3	4	no	—
117	C₄×Dic₃	—	3	4	no	—
118	C₄×Dic₃	—	3	4	no	—
119	C₄×Dic₃	—	3	4	no	—
120	C₄×Dic₃	—	3	4	no	—
121	C₂²×C₃²	—	2	1	no	—
122	C₂²×C₃²	—	2	1	no	—
123	C₂²×C₃²	—	2	1	no	—
124	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
125	C₃²×C₄	—	2	1	no	—
126	C₃²×C₄	—	2	1	no	—
127	C₃²×C₄	—	2	1	no	—
128	C₃²×C₄	—	2	1	no	—
129	C₃²×C₄	—	2	1	no	—
130	C₃²×C₄	—	2	1	no	—
131	C₃⋊Dic₃	—	2	1	no	—
132	C₃⋊Dic₃	—	2	1	no	—
133	C₃⋊Dic₃	—	2	1	no	—
134	C₃⋊Dic₃	—	2	1	no	—
135	C₃⋊Dic₃	—	2	1	no	—
136	C₃⋊Dic₃	—	2	1	no	—
137	C₃⋊Dic₃	—	2	1	no	—
138	C₃⋊Dic₃	#41	1	2	yes	—
139	C₄⋊Q₈	—	9	1	no	—
140	#126	—	9	1	no	—
141	#126	—	9	1	no	—
142	#126	—	9	1	no	—
143	#126	C₃⋊D₃	1	1	yes	characteristic
144	#125	—	9	1	no	—
145	#125	—	9	1	no	—
146	#125	—	9	1	no	—
147	#125	—	9	1	no	—
148	#125	—	9	1	no	—
149	C₂²⋊Q₈	—	9	1	no	—
150	C₂²⋊Q₈	—	9	1	no	—
151	C₄×Q₈	—	9	1	no	—
152	C₄×Q₈	—	9	1	no	—
153	#117	—	9	1	no	—
154	C₂³×C₃	C₂²×D₃	1	4	yes	—
155	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
156	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
157	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
158	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
159	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
160	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
161	C₂×Dic₃	—	3	4	no	—
162	C₂×Dic₃	—	3	4	no	—
163	C₂×Dic₃	—	3	4	no	—
164	C₂×Dic₃	—	3	4	no	—
165	C₂×Dic₃	—	3	4	no	—
166	C₂×Dic₃	—	3	4	no	—
167	C₂×Dic₃	—	3	4	no	—
168	C₂×Dic₃	—	3	4	no	—
169	C₂×Dic₃	—	6	8	no	—
170	Dic₆	—	6	8	no	—
171	Dic₆	—	6	8	no	—
172	C₂×C₃²	—	4	1	no	—
173	C₂×C₃²	C₂×#41	1	1	yes	characteristic
174	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
175	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
176	C₂×Q₈	—	9	1	no	—
177	C₂×Q₈	—	9	1	no	—
178	C₂²×C₄	—	9	1	no	—
179	C₄⋊C₄	—	9	1	no	—
180	C₄⋊C₄	—	9	1	no	—
181	C₄⋊C₄	—	9	1	no	—
182	C₄⋊C₄	—	9	1	no	—
183	C₄⋊C₄	—	9	1	no	—
184	C₄⋊C₄	—	9	1	no	—
185	C₄⋊C₄	—	9	1	no	—
186	C₄⋊C₄	—	9	1	no	—
187	C₄⋊C₄	—	9	1	no	—
188	C₄⋊C₄	—	9	1	no	—
189	C₄⋊C₄	—	9	1	no	—
190	C₄⋊C₄	—	9	1	no	—
191	C₄⋊C₄	—	9	1	no	—
192	C₄⋊C₄	—	9	1	no	—
193	C₄⋊C₄	—	9	1	no	—
194	C₄⋊C₄	—	9	1	no	—
195	C₄⋊C₄	—	9	1	no	—
196	C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
197	C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
198	C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
199	C₂²⋊C₄	—	9	1	no	—
200	C₂²⋊C₄	—	9	1	no	—
201	C₂²⋊C₄	—	9	1	no	—
202	C₂²⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
203	C₂²⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
204	C₂²⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
205	C₄²	—	9	1	no	—
206	C₄²	—	9	1	no	—
207	C₄²	—	9	1	no	—
208	C₄²	—	9	1	no	—
209	C₄²	—	9	1	no	—
210	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
211	C₂²×C₃	—	2	4	no	—
212	C₂²×C₃	—	2	4	no	—
213	C₂²×C₃	—	2	4	no	—
214	C₂²×C₃	C₂³×D₃	1	4	yes	—
215	C₃×C₄	—	2	4	no	—
216	C₃×C₄	—	2	4	no	—
217	C₃×C₄	—	2	4	no	—
218	C₃×C₄	—	2	4	no	—
219	C₃×C₄	—	2	4	no	—
220	C₃×C₄	—	2	4	no	—
221	Dic₃	—	3	8	no	—
222	Dic₃	—	6	4	no	—
223	Dic₃	—	6	4	no	—
224	Dic₃	—	6	4	no	—
225	Dic₃	—	6	4	no	—
226	Dic₃	—	6	4	no	—
227	Dic₃	—	6	4	no	—
228	Dic₃	—	6	4	no	—
229	C₃²	#541	1	1	yes	characteristic, complemented, Sylow
230	C₂³	C₂²×C₃⋊D₃	1	1	yes	characteristic
231	Q₈	—	18	2	no	—
232	Q₈	—	18	2	no	—
233	C₂×C₄	—	9	1	no	—
234	C₂×C₄	—	9	1	no	—
235	C₂×C₄	—	9	1	no	—
236	C₂×C₄	—	9	1	no	—
237	C₂×C₄	—	9	1	no	—
238	C₂×C₄	—	9	1	no	—
239	C₂×C₄	—	9	1	no	—
240	C₂×C₄	—	9	1	no	—
241	C₂×C₄	—	18	2	no	—
242	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
248	C₂×C₃	—	4	4	no	—
249	C₂×C₃	D₃×#41	1	4	yes	—
250	C₂×C₃	#1010	1	4	yes	—
251	C₂×C₃	#1010	1	4	yes	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	C₂³×C₃⋊D₃	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	9	2	no	—
263	C₄	—	18	1	no	—
264	C₄	—	18	1	no	—
265	C₄	—	18	1	no	—
266	C₄	—	18	1	no	—
267	C₄	—	18	1	no	—
268	C₄	—	18	1	no	—
269	C₄	—	18	1	no	—
270	C₃	#5701	1	4	yes	complemented
271	C₂	—	4	1	no	—
272	C₂	#41×C₃⋊D₃	1	1	yes	characteristic
273	C₂	#64367	1	1	yes	characteristic
274	C₂	#64367	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊D₃
commutator, socle, cosocle	C₂²×C₃²	C₂⁴
Fitting	C₃²×#126	C₂
prime radical	C₂³×C₃²	C₂³
hypercenter	#126	C₃⋊D₃
hypocenter	C₃²	#541
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ #126

Complements

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

N	H ≅ G/N	type
2. #64099	C₂	semidirect
3. #64098	C₂	semidirect
4. #64098	C₂	semidirect
5. #64098	C₂	semidirect
6. #64097	C₂	semidirect
7. #64096	C₂	semidirect
14. #64079	C₂	semidirect
15. C₄×#634	C₂	semidirect
229. C₃²	#541	semidirect
270. C₃	#5701	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#126	2
#64099	2
#64098	2
#64097	2
#64096	2
#64088	2
#64087	2
#64086	2
#64079	2
C₄×#634	2
#5701	3