Subgroups — C₂×#10589119 #11796625

Order 1152.

593 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#3746	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#818	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#10589119	C₂	1	4	yes	maximal, direct product factor
5	C₂×#4503	—	3	1	no	maximal
6	C₂×#4492	—	3	1	no	maximal
7	C₂²×#634	—	2	1	no	—
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#3746	—	2	1	no	—
10	C₂×#3746	—	2	2	no	—
11	C₂×#3746	C₄	1	4	yes	complemented
12	C₂×#3746	C₂²	1	2	yes	—
13	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
14	C₃×#818	—	2	1	no	—
15	C₃×#818	C₂²	1	2	yes	—
16	C₂×C₃×C₃⋊C₁₆	—	2	2	no	—
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
18	C₂²×Dic₁₂	—	3	1	no	—
19	C₂²×Dic₁₂	—	3	1	no	—
20	C₂²×Dic₁₂	—	6	1	no	—
21	C₂²×C₃×C₁₆	—	3	1	no	—
22	C₂×C₃×#107	—	3	1	no	—
23	C₂×#818	D₃	1	1	yes	characteristic
24	C₂²×C₃⋊C₁₆	D₃	1	1	yes	characteristic
25	#4503	—	3	4	no	—
26	#4492	—	3	4	no	—
27	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
28	C₂²×C₃⋊Dic₃	—	4	1	no	—
29	C₂×#634	—	2	4	no	—
30	C₂×#634	—	2	2	no	—
31	C₂×#634	—	4	1	no	—
32	C₂×#634	—	4	2	no	—
33	C₂²×C₃×Dic₃	—	4	1	no	—
34	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
36	C₂×C₃²×C₈	D₄	1	2	yes	—
37	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
38	#3746	—	2	4	no	—
39	#3746	—	2	2	no	—
40	#3746	C₂×C₄	1	4	yes	complemented
41	C₃×C₄⋊Dic₃	—	2	2	no	—
42	C₃×C₄⋊Dic₃	—	4	1	no	—
43	C₃×C₃⋊C₁₆	—	2	2	no	—
44	C₂×#358	—	9	1	no	Sylow
45	C₂²×Dic₆	—	6	2	no	—
46	C₂²×Dic₆	—	6	1	no	—
47	C₂²×Dic₆	—	6	1	no	—
48	C₂²×C₃×C₈	—	2	1	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃×C₄⋊C₄	—	6	1	no	—
52	C₂×C₄⋊Dic₃	—	2	1	no	—
53	C₂×Dic₁₂	—	3	4	no	—
54	C₂×Dic₁₂	—	3	4	no	—
55	C₂×Dic₁₂	—	3	2	no	—
56	C₂×Dic₁₂	—	3	2	no	—
57	C₂×Dic₁₂	—	6	4	no	—
58	C₂×Dic₁₂	—	6	2	no	—
59	C₂×Dic₁₂	—	6	2	no	—
60	C₂×Dic₁₂	—	6	4	no	—
61	C₂×Dic₁₂	—	6	1	no	—
62	C₂×Dic₁₂	—	6	2	no	—
63	C₂×Dic₁₂	—	6	1	no	—
64	C₂×Dic₁₂	—	6	2	no	—
65	C₂×C₃×C₁₆	—	3	2	no	—
66	C₂×C₃×C₁₆	—	6	2	no	—
67	C₃×#107	—	3	2	no	—
68	C₃×#107	—	6	1	no	—
69	#818	—	2	1	no	—
70	#818	C₂×D₃	1	2	yes	—
71	C₂×C₃⋊C₁₆	—	2	2	no	—
72	C₂×C₃⋊C₁₆	C₂×D₃	1	2	yes	—
73	C₂³×C₃²	D₈	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
75	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
76	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
77	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃⋊Dic₃	—	4	4	no	—
79	C₂×C₃⋊Dic₃	—	4	2	no	—
80	#634	—	2	4	no	—
81	#634	—	4	4	no	—
82	#634	—	4	2	no	—
83	C₂×C₃×Dic₃	—	4	2	no	—
84	C₂×C₃×Dic₃	—	8	2	no	—
85	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
86	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂²×Q₁₆	—	9	1	no	—
89	C₂²×C₁₆	—	3	1	no	—
90	C₂×#107	—	3	1	no	—
91	#358	—	9	4	no	—
92	C₂²×C₃×C₄	—	2	1	no	—
93	C₂²×C₃×C₄	—	12	1	no	—
94	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
95	C₂²×C₃×C₄	#67	1	1	yes	characteristic
96	C₂²×Dic₃	—	4	1	no	—
97	C₂²×Dic₃	—	12	2	no	—
98	C₂²×Dic₃	—	12	1	no	—
99	C₂²×Dic₃	—	12	1	no	—
100	C₂×Dic₆	—	6	8	no	—
101	C₂×Dic₆	—	6	4	no	—
102	C₂×Dic₆	—	6	4	no	—
103	C₂×Dic₆	—	6	4	no	—
104	C₂×Dic₆	—	6	2	no	—
105	C₂×Dic₆	—	6	2	no	—
106	C₂×Dic₆	—	12	2	no	—
107	C₂×Dic₆	—	12	4	no	—
108	C₂×Dic₆	—	12	2	no	—
109	C₂×Dic₆	—	12	1	no	—
110	C₂×Dic₆	—	12	1	no	—
111	C₂×Dic₆	—	12	2	no	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₂×C₃×C₈	—	2	1	no	—
114	C₂×C₃×C₈	—	2	2	no	—
115	C₂×C₃×C₈	—	2	1	no	—
116	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
117	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
118	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
119	C₂×C₃×C₈	D₁₂	1	2	yes	—
120	C₂×C₃×C₈	#67	1	1	yes	characteristic
121	C₂×C₃×C₈	#67	1	2	yes	—
122	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
123	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
124	C₃×C₄⋊C₄	—	6	2	no	—
125	C₃×C₄⋊C₄	—	12	1	no	—
126	C₄⋊Dic₃	—	2	2	no	—
127	C₄⋊Dic₃	—	4	1	no	—
128	Dic₁₂	—	3	4	no	—
129	Dic₁₂	—	3	4	no	—
130	Dic₁₂	—	6	4	no	—
131	Dic₁₂	—	6	8	no	—
132	Dic₁₂	—	6	4	no	—
133	Dic₁₂	—	6	4	no	—
134	Dic₁₂	—	6	4	no	—
135	Dic₁₂	—	6	2	no	—
136	Dic₁₂	—	6	2	no	—
137	C₃×C₁₆	—	6	2	no	—
138	C₃⋊C₁₆	—	2	2	no	—
139	C₂²×C₃²	#102	1	2	yes	—
140	C₂²×C₃²	SD₃₂	1	2	yes	—
141	C₂²×C₃²	Q₃₂	1	2	yes	—
142	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
143	C₃²×C₄	#102	1	2	yes	—
144	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
146	C₃⋊Dic₃	—	4	4	no	—
147	C₃×Dic₃	—	8	2	no	—
148	C₂²×Q₈	—	18	1	no	—
149	C₂×Q₁₆	—	9	4	no	—
150	C₂×Q₁₆	—	9	2	no	—
151	C₂×Q₁₆	—	18	1	no	—
152	C₂×Q₁₆	—	18	2	no	—
153	C₂²×C₈	D₃²	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	6	1	no	—
155	C₂×C₁₆	—	3	2	no	—
156	C₂×C₁₆	—	6	2	no	—
157	#107	—	3	2	no	—
158	#107	—	6	1	no	—
159	C₂³×C₃	—	2	1	no	—
160	C₂³×C₃	D₂₄	1	1	yes	characteristic
161	C₂³×C₃	#213	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	12	2	no	—
167	C₂×C₃×C₄	—	24	2	no	—
168	C₂×C₃×C₄	#204	1	2	yes	—
169	C₂×C₃×C₄	D₂₄	1	1	yes	characteristic
170	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
171	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
172	C₂×C₃×C₄	#213	1	1	yes	characteristic
173	C₂×C₃×C₄	#215	1	2	yes	—
174	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
176	C₂×Dic₃	—	4	2	no	—
177	C₂×Dic₃	—	8	2	no	—
178	C₂×Dic₃	—	12	8	no	—
179	C₂×Dic₃	—	12	4	no	—
180	C₂×Dic₃	—	12	4	no	—
181	C₂×Dic₃	—	12	4	no	—
182	C₂×Dic₃	—	12	2	no	—
183	C₂×Dic₃	—	12	2	no	—
184	Dic₆	—	6	8	no	—
185	Dic₆	—	6	4	no	—
186	Dic₆	—	6	4	no	—
187	Dic₆	—	12	8	no	—
188	Dic₆	—	12	4	no	—
189	Dic₆	—	12	4	no	—
190	Dic₆	—	12	4	no	—
191	Dic₆	—	12	2	no	—
192	Dic₆	—	12	2	no	—
193	C₃×C₈	—	2	2	no	—
194	C₃×C₈	—	2	1	no	—
195	C₃×C₈	—	2	1	no	—
196	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
197	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
198	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
199	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
200	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
201	C₃×C₈	C₂×#67	1	1	yes	characteristic
202	C₂×C₃²	#358	1	4	yes	complemented
203	C₂×C₃²	C₂×#102	1	1	yes	characteristic
204	C₂×C₃²	C₂×SD₃₂	1	1	yes	characteristic
205	C₂×C₃²	C₂×Q₃₂	1	1	yes	characteristic
206	C₂×Q₈	—	18	4	no	—
207	C₂×Q₈	—	18	2	no	—
208	C₂×Q₈	—	36	1	no	—
209	C₂×Q₈	—	36	2	no	—
210	C₂²×C₄	—	12	1	no	—
211	C₂²×C₄	—	36	1	no	—
212	C₂²×C₄	#626	1	1	yes	characteristic
213	Q₁₆	—	9	4	no	—
214	Q₁₆	—	18	4	no	—
215	Q₁₆	—	18	2	no	—
216	C₂×C₈	#624	1	2	yes	—
217	C₂×C₈	#626	1	1	yes	characteristic
218	C₂×C₈	#626	1	2	yes	—
219	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
220	C₄⋊C₄	—	6	2	no	—
221	C₄⋊C₄	—	12	1	no	—
222	C₁₆	—	6	2	no	—
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	#800	1	2	yes	—
228	C₂²×C₃	Dic₂₄	1	2	yes	—
229	C₂²×C₃	#809	1	2	yes	—
230	C₂²×C₃	#821	1	2	yes	—
231	C₂²×C₃	#827	1	2	yes	—
232	C₂²×C₃	#829	1	2	yes	—
233	C₂²×C₃	C₂×D₂₄	1	1	yes	characteristic
234	C₂²×C₃	C₂×#213	1	1	yes	characteristic
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	24	2	no	—
239	C₃×C₄	#809	1	2	yes	—
240	C₃×C₄	#821	1	2	yes	—
241	C₃×C₄	C₂×#204	1	1	yes	characteristic
242	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
243	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
244	C₃×C₄	C₂×#215	1	1	yes	characteristic
245	Dic₃	—	8	2	no	—
246	Dic₃	—	12	8	no	—
247	Dic₃	—	12	4	no	—
248	Dic₃	—	12	4	no	—
249	C₃²	C₂×#358	1	1	yes	characteristic, complemented, Sylow
250	C₂³	#3714	1	1	yes	characteristic
251	Q₈	—	18	4	no	—
252	Q₈	—	36	4	no	—
253	Q₈	—	36	2	no	—
254	C₂×C₄	—	12	2	no	—
255	C₂×C₄	—	24	2	no	—
256	C₂×C₄	—	36	4	no	—
257	C₂×C₄	—	36	2	no	—
258	C₂×C₄	#3714	1	1	yes	characteristic
259	C₂×C₄	#3717	1	2	yes	—
260	C₂×C₄	#3722	1	2	yes	—
261	C₂×C₄	C₂×#626	1	1	yes	characteristic
262	C₈	#3722	1	2	yes	—
263	C₈	C₂×#624	1	1	yes	characteristic
264	C₈	C₂×#626	1	1	yes	characteristic
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	2	4	no	—
269	C₂×C₃	#4492	1	4	yes	complemented
270	C₂×C₃	#4503	1	4	yes	complemented
271	C₂×C₃	C₂×#800	1	1	yes	characteristic
272	C₂×C₃	C₂×Dic₂₄	1	1	yes	characteristic
273	C₂×C₃	C₂×#809	1	1	yes	characteristic
274	C₂×C₃	C₂×#821	1	1	yes	characteristic
275	C₂×C₃	C₂×#827	1	1	yes	characteristic
276	C₂×C₃	C₂×#829	1	1	yes	characteristic
277	C₂²	#63548	1	2	yes	—
278	C₂²	#63551	1	2	yes	—
279	C₂²	#63564	1	2	yes	—
280	C₂²	C₂×#3714	1	1	yes	characteristic
281	C₄	—	24	2	no	—
282	C₄	—	36	4	no	—
283	C₄	#63564	1	2	yes	—
284	C₄	C₂×#3717	1	1	yes	characteristic
285	C₄	C₂×#3722	1	1	yes	characteristic
286	C₃	—	2	1	no	—
287	C₃	C₂×#4492	1	1	yes	characteristic, complemented
288	C₃	C₂×#4503	1	1	yes	characteristic, complemented
289	C₂	#10589119	1	4	yes	direct product factor
290	C₂	C₂×#63548	1	1	yes	characteristic
291	C₂	C₂×#63551	1	1	yes	characteristic
292	C₂	C₂×#63564	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#3714
commutator	C₃²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle, prime radical	C₂³×C₃²	D₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	C₂×#358
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. #10589119	C₂	direct (N × H)
11. C₂×#3746	C₄	semidirect
40. #3746	C₂×C₄	semidirect
202. C₂×C₃²	#358	semidirect
249. C₃²	C₂×#358	semidirect
269. C₂×C₃	#4492	semidirect
270. C₂×C₃	#4503	semidirect
287. C₃	C₂×#4492	semidirect
288. C₃	C₂×#4503	semidirect
289. C₂	#10589119	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#3746	2
C₂×C₃×#818	2
C₂²×C₃×C₃⋊C₁₆	2
#10589119	2
C₂×#4503	3
C₂×#4492	3

Subgroups — C2×#10589119 #11796625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10589119 #11796625