Subgroups — #11797135

Order 1152.

333 conjugacy classes of subgroups.

The subgroup lattice has 261 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63639	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
3	C₃×#5106	C₂	1	1	yes	maximal, characteristic
4	#10589138	C₂	1	4	yes	maximal, complemented
5	#70685	—	3	1	no	maximal
6	#70650	—	3	1	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₃×#926	—	2	1	no	—
9	C₂×C₃×#208	—	2	1	no	—
10	#63639	C₄	1	2	yes	complemented
11	#63639	C₂²	1	2	yes	—
12	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	2	yes	complemented
13	C₃×#819	—	2	1	no	—
14	C₃×#819	C₂²	1	2	yes	—
15	C₃×#818	—	2	1	no	—
16	C₃×#817	C₂²	1	2	yes	—
17	C₂×C₃×#108	—	3	1	no	—
18	C₃×#427	—	3	1	no	—
19	C₂×#819	D₃	1	1	yes	characteristic
20	#5106	D₃	1	1	yes	characteristic
21	C₂×#812	—	3	1	no	—
22	C₂×#812	—	3	1	no	—
23	C₂×#812	—	6	1	no	—
24	#4494	—	3	4	no	—
25	#4487	—	3	4	no	—
26	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
31	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
33	C₃×#217	—	4	1	no	—
34	C₃×C₄⋊Dic₃	—	2	2	no	—
35	C₃×C₄×Dic₃	—	4	1	no	—
36	C₃×#208	—	2	2	no	—
37	C₃×#208	—	4	1	no	—
38	C₂×C₃×C₃⋊C₈	—	4	1	no	—
39	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	4	yes	—
40	#2155	—	9	1	no	Sylow
41	C₂×C₃×M₁₆	—	6	1	no	—
42	C₂²×C₃×C₈	—	2	1	no	—
43	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
45	C₃×#117	—	6	1	no	—
46	#926	—	2	1	no	—
47	C₂×#208	—	2	1	no	—
48	C₃×#108	—	3	2	no	—
49	C₃×#108	—	6	1	no	—
50	C₃×#107	—	6	1	no	—
51	C₃×#106	—	3	2	no	—
52	#819	—	2	1	no	—
53	#819	C₂×D₃	1	2	yes	—
54	#818	—	2	1	no	—
55	#817	C₂×D₃	1	2	yes	—
56	#812	—	3	2	no	—
57	#812	—	3	2	no	—
58	#812	—	3	2	no	—
59	#812	—	3	2	no	—
60	#812	—	6	2	no	—
61	#812	—	6	2	no	—
62	C₂×C₃⋊C₁₆	—	3	2	no	—
63	C₂×C₃⋊C₁₆	—	3	2	no	—
64	C₂×C₃⋊C₁₆	—	6	2	no	—
65	C₂³×C₃²	SD₁₆	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
69	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
70	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×Dic₃	—	4	2	no	—
73	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
74	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
75	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
77	C₃×C₃⋊C₈	—	4	2	no	—
78	C₂×M₃₂	—	9	1	no	—
79	C₂×#108	—	3	1	no	—
80	#427	—	3	1	no	—
81	#365	—	9	4	no	—
82	C₂²×C₃×C₄	—	2	1	no	—
83	C₂²×C₃×C₄	#67	1	1	yes	characteristic
84	C₂²×C₃×C₄	#67	1	1	yes	characteristic
85	C₃×M₁₆	—	6	2	no	—
86	C₃×M₁₆	—	12	1	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	2	1	no	—
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	—	12	1	no	—
94	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
95	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
96	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
97	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
98	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
99	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
100	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
101	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
102	C₂×C₃×C₈	#67	1	1	yes	characteristic
103	C₂×C₃×C₈	#67	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
106	C₃×C₄⋊C₄	—	6	2	no	—
107	C₃×C₂²⋊C₄	—	12	1	no	—
108	C₃×C₄²	—	12	1	no	—
109	#217	—	4	1	no	—
110	C₄⋊Dic₃	—	2	2	no	—
111	C₄×Dic₃	—	4	1	no	—
112	#208	—	2	2	no	—
113	#208	—	4	1	no	—
114	C₂×C₃⋊C₈	—	4	1	no	—
115	C₃⋊C₁₆	—	3	4	no	—
116	C₃⋊C₁₆	—	3	4	no	—
117	C₃⋊C₁₆	—	6	4	no	—
118	C₂²×C₃²	—	4	1	no	—
119	C₂²×C₃²	#106	1	1	yes	characteristic
120	C₂²×C₃²	#106	1	1	yes	characteristic
121	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
122	C₃²×C₄	#106	1	1	yes	characteristic
123	C₃²×C₄	#106	1	1	yes	characteristic
124	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
126	C₃×Dic₃	—	8	2	no	—
127	C₂×M₁₆	—	6	1	no	—
128	C₂²×C₈	D₃²	1	1	yes	characteristic
129	#117	—	6	1	no	—
130	M₃₂	—	9	2	no	—
131	M₃₂	—	9	2	no	—
132	C₂×C₁₆	—	9	2	no	—
133	#108	—	3	2	no	—
134	#108	—	6	1	no	—
135	#107	—	6	1	no	—
136	#106	—	3	2	no	—
137	C₂³×C₃	—	2	1	no	—
138	C₂³×C₃	#214	1	1	yes	characteristic
139	C₂³×C₃	#215	1	1	yes	characteristic
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	12	2	no	—
147	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
148	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
149	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
150	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
151	C₂×C₃×C₄	#214	1	1	yes	characteristic
152	C₂×C₃×C₄	#214	1	1	yes	characteristic
153	C₂×C₃×C₄	#214	1	1	yes	characteristic
154	C₂×C₃×C₄	#215	1	1	yes	characteristic
155	C₂×C₃×C₄	#215	1	1	yes	characteristic
156	C₂×C₃×C₄	#215	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
159	C₂×Dic₃	—	4	2	no	—
160	C₃×C₈	—	2	1	no	—
161	C₃×C₈	—	2	1	no	—
162	C₃×C₈	—	2	1	no	—
163	C₃×C₈	—	2	1	no	—
164	C₃×C₈	—	12	2	no	—
165	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
166	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
167	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
168	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
169	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
170	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
171	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
172	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
173	C₃⋊C₈	—	4	2	no	—
174	C₂×C₃²	—	2	1	no	—
175	C₂×C₃²	—	2	1	no	—
176	C₂×C₃²	—	2	1	no	—
177	C₂×C₃²	C₂×#106	1	1	yes	characteristic
178	C₂²×C₄	#625	1	1	yes	characteristic
179	M₁₆	—	6	2	no	—
180	M₁₆	—	12	1	no	—
181	C₂×C₈	—	12	1	no	—
182	C₂×C₈	#624	1	1	yes	characteristic
183	C₂×C₈	#624	1	1	yes	characteristic
184	C₂×C₈	#625	1	1	yes	characteristic
185	C₂×C₈	#627	1	1	yes	characteristic
186	C₂×C₈	#627	1	1	yes	characteristic
187	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
188	C₄⋊C₄	—	6	2	no	—
189	C₂²⋊C₄	—	12	1	no	—
190	C₄²	—	12	1	no	—
191	C₁₆	—	9	4	no	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	2	1	no	—
195	C₂²×C₃	—	4	1	no	—
196	C₂²×C₃	—	4	1	no	—
197	C₂²×C₃	—	4	2	no	—
198	C₂²×C₃	#808	1	1	yes	characteristic
199	C₂²×C₃	#808	1	1	yes	characteristic
200	C₂²×C₃	#808	1	1	yes	characteristic
201	C₂²×C₃	#808	1	1	yes	characteristic
202	C₂²×C₃	C₂×#214	1	1	yes	characteristic
203	C₂²×C₃	C₂×#215	1	1	yes	characteristic
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	24	2	no	—
209	C₃×C₄	#808	1	1	yes	characteristic
210	C₃×C₄	#808	1	1	yes	characteristic
211	C₃×C₄	#808	1	1	yes	characteristic
212	C₃×C₄	#808	1	1	yes	characteristic
213	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
214	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
215	C₃×C₄	C₂×#214	1	1	yes	characteristic
216	C₃×C₄	C₂×#215	1	1	yes	characteristic
217	Dic₃	—	8	2	no	—
218	C₃²	#2155	1	1	yes	characteristic, complemented, Sylow
219	C₂³	#3715	1	1	yes	characteristic
220	C₂×C₄	—	12	2	no	—
221	C₂×C₄	#3715	1	1	yes	characteristic
222	C₂×C₄	#3715	1	1	yes	characteristic
223	C₂×C₄	#3715	1	1	yes	characteristic
224	C₂×C₄	#3724	1	1	yes	characteristic
225	C₂×C₄	#3724	1	1	yes	characteristic
226	C₂×C₄	C₂×#625	1	1	yes	characteristic
227	C₈	—	12	2	no	—
228	C₈	#3724	1	1	yes	characteristic
229	C₈	#3724	1	1	yes	characteristic
230	C₈	C₂×#624	1	1	yes	characteristic
231	C₈	C₂×#627	1	1	yes	characteristic
232	C₂×C₃	—	2	2	no	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	2	1	no	—
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	4	1	no	—
241	C₂×C₃	—	4	1	no	—
242	C₂×C₃	C₂×#808	1	1	yes	characteristic
243	C₂×C₃	C₂×#808	1	1	yes	characteristic
244	C₂²	—	4	1	no	—
245	C₂²	#63574	1	1	yes	characteristic
246	C₂²	#63574	1	1	yes	characteristic
247	C₂²	C₂×#3715	1	1	yes	characteristic
248	C₄	—	24	2	no	—
249	C₄	#63574	1	1	yes	characteristic
250	C₄	#63574	1	1	yes	characteristic
251	C₄	C₂×#3715	1	1	yes	characteristic
252	C₄	C₂×#3724	1	1	yes	characteristic
253	C₃	—	2	1	no	—
254	C₃	#70650	1	1	yes	characteristic, complemented
255	C₃	#70685	1	1	yes	characteristic, complemented
256	C₂	—	2	1	no	—
257	C₂	—	2	1	no	—
258	C₂	—	2	1	no	—
259	C₂	C₂×#63574	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#63574
commutator	C₃²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂×C₃²	C₂×#106
prime radical	C₂³×C₃²	SD₁₆
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#2155
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. #10589138	C₂	semidirect
10. #63639	C₄	semidirect
12. C₂×C₃⋊(C₃⋊C₁₆)	C₄	semidirect
218. C₃²	#2155	semidirect
254. C₃	#70650	semidirect
255. C₃	#70685	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63639	2
C₂×C₃×#819	2
C₃×#5106	2
#10589138	2
#70685	3
#70650	3