Subgroups — C₂×#10590624 #11899397

Order 1152.

869 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64110	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#812	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#63544	C₂	1	2	yes	maximal, complemented
5	#10590624	C₂	1	8	yes	maximal, direct product factor
6	C₂×#63542	C₂	1	2	yes	maximal, complemented
7	C₂×#4907	—	3	1	no	maximal
8	C₂×#4901	—	3	1	no	maximal
9	C₂×#3828	C₄	1	1	yes	characteristic
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3748	C₄	1	1	yes	characteristic
12	#64110	C₄	1	4	yes	—
13	#64110	C₂²	1	4	yes	—
14	C₂×#3743	C₄	1	2	yes	—
15	C₂×C₈×C₃⋊D₃	C₂²	1	2	yes	—
16	C₃×#812	C₂²	1	4	yes	complemented
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
19	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₃⋊C₁₆	C₂²	1	4	yes	complemented
21	#63544	C₂²	1	8	yes	complemented
22	#63542	C₂²	1	8	yes	complemented
23	C₂×#901	—	3	1	no	—
24	C₂×#901	—	3	1	no	—
25	C₂×#901	—	6	1	no	—
26	C₂×C₃×M₃₂	—	3	1	no	—
27	C₂²×C₃×C₁₆	—	3	1	no	—
28	C₂×#812	D₃	1	1	yes	characteristic, complemented
29	C₂²×C₃⋊C₁₆	D₃	1	1	yes	characteristic, complemented
30	#4907	—	3	8	no	—
31	#4901	—	3	8	no	—
32	C₂×#798	—	3	2	no	—
33	C₂×#798	—	3	2	no	—
34	C₂×D₃×C₁₆	—	3	2	no	—
35	C₂×D₃×C₁₆	—	3	2	no	—
36	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×#638	C₈	1	2	yes	—
38	#3828	C₈	1	4	yes	—
39	#3828	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊(C₃⋊D₃)	C₈	1	1	yes	characteristic
41	C₂×C₄×C₃⋊D₃	C₂×C₄	1	2	yes	—
42	C₂×#634	C₈	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
45	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₃²×C₈	C₂³	1	2	yes	—
47	#3748	C₂×C₄	1	4	yes	—
48	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
49	#3743	C₂×C₄	1	8	yes	—
50	C₈×C₃⋊D₃	C₂×C₄	1	4	yes	—
51	C₈×C₃⋊D₃	C₂³	1	4	yes	—
52	C₃×C₃⋊C₁₆	C₂³	1	2	yes	—
53	C₃×C₃⋊C₁₆	C₂³	1	4	yes	—
54	C₃×C₃⋊C₁₆	C₂³	1	2	yes	—
55	C₂×#504	—	9	1	no	Sylow
56	C₂×#235	—	3	1	no	—
57	C₂×#235	—	3	1	no	—
58	C₂×#235	—	6	1	no	—
59	C₂²×C₃×C₈	—	2	1	no	—
60	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
62	C₂×#208	—	3	1	no	—
63	C₂×#208	—	3	1	no	—
64	C₂×#208	—	6	1	no	—
65	#901	—	3	4	no	—
66	#901	—	3	4	no	—
67	#901	—	3	4	no	—
68	#901	—	3	4	no	—
69	#901	—	6	4	no	—
70	#901	—	6	4	no	—
71	C₂×ΓL(1,𝔽₂₅)	—	3	2	no	—
72	C₂×ΓL(1,𝔽₂₅)	—	3	2	no	—
73	C₂×ΓL(1,𝔽₂₅)	—	6	2	no	—
74	C₂×D₃×C₈	—	3	2	no	—
75	C₂×D₃×C₈	—	3	2	no	—
76	C₂×D₃×C₈	—	6	2	no	—
77	C₃×M₃₂	—	3	4	no	—
78	C₂×C₃×C₁₆	—	3	1	no	—
79	C₂×C₃×C₁₆	—	3	1	no	—
80	C₂×C₃×C₁₆	—	3	4	no	—
81	C₂×C₃×C₁₆	—	3	2	no	—
82	#812	C₂×D₃	1	4	yes	complemented
83	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
84	C₂×C₃⋊C₁₆	C₂×D₃	1	2	yes	—
85	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
86	C₂×C₃⋊C₁₆	C₂×D₃	1	4	yes	complemented
87	#798	—	3	8	no	—
88	#798	—	3	8	no	—
89	D₃×C₁₆	—	3	8	no	—
90	D₃×C₁₆	—	3	8	no	—
91	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
92	C₂²×C₃⋊D₃	C₂×C₈	1	2	yes	—
93	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
95	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
97	#638	C₂×C₈	1	8	yes	—
98	C₂×C₃⋊Dic₃	C₂×C₈	1	2	yes	—
99	C₄⋊(C₃⋊D₃)	C₂×C₈	1	4	yes	—
100	C₄×C₃⋊D₃	C₂×C₈	1	4	yes	—
101	C₄×C₃⋊D₃	C₂²×C₄	1	4	yes	—
102	#634	C₂×C₈	1	4	yes	—
103	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₃²×C₈	C₂²×C₄	1	2	yes	—
105	C₃²×C₈	C₂⁴	1	1	yes	characteristic
106	C₃⋊(C₃⋊C₈)	C₂²×C₄	1	4	yes	—
107	C₂×#131	—	9	1	no	—
108	#504	—	9	8	no	—
109	C₂×M₃₂	—	3	1	no	—
110	C₂×M₃₂	—	9	2	no	—
111	C₂²×C₁₆	—	3	1	no	—
112	C₂²×C₁₆	—	9	2	no	—
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
116	C₂×#67	—	3	2	no	—
117	C₂×#67	—	3	2	no	—
118	C₂×#67	—	6	2	no	—
119	#235	—	3	4	no	—
120	#235	—	3	4	no	—
121	#235	—	3	4	no	—
122	#235	—	3	4	no	—
123	#235	—	6	4	no	—
124	#235	—	6	4	no	—
125	C₂×D₁₂	—	3	1	no	—
126	C₂×D₁₂	—	3	1	no	—
127	C₂×D₁₂	—	6	1	no	—
128	C₂×C₄×D₃	—	3	2	no	—
129	C₂×C₄×D₃	—	3	2	no	—
130	C₂×C₄×D₃	—	6	2	no	—
131	C₂×Dic₆	—	3	1	no	—
132	C₂×Dic₆	—	3	1	no	—
133	C₂×Dic₆	—	6	1	no	—
134	C₂×C₃×C₈	—	2	1	no	—
135	C₂×C₃×C₈	—	2	2	no	—
136	C₂×C₃×C₈	—	2	2	no	—
137	C₂×C₃×C₈	—	2	1	no	—
138	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
139	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
140	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
141	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
142	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
143	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
144	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
145	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
146	#208	—	3	4	no	—
147	#208	—	3	4	no	—
148	#208	—	6	4	no	—
149	C₂×C₃⋊C₈	—	3	2	no	—
150	C₂×C₃⋊C₈	—	3	2	no	—
151	C₂×C₃⋊C₈	—	6	2	no	—
152	ΓL(1,𝔽₂₅)	—	3	8	no	—
153	ΓL(1,𝔽₂₅)	—	3	8	no	—
154	ΓL(1,𝔽₂₅)	—	6	8	no	—
155	D₃×C₈	—	3	4	no	—
156	D₃×C₈	—	3	4	no	—
157	D₃×C₈	—	3	4	no	—
158	D₃×C₈	—	3	4	no	—
159	D₃×C₈	—	6	4	no	—
160	D₃×C₈	—	6	4	no	—
161	C₃×C₁₆	—	3	2	no	—
162	C₃×C₁₆	—	3	4	no	—
163	C₃×C₁₆	—	3	2	no	—
164	C₃⋊C₁₆	C₂²×D₃	1	2	yes	—
165	C₃⋊C₁₆	C₂²×D₃	1	4	yes	—
166	C₃⋊C₁₆	C₂²×D₃	1	2	yes	—
167	C₂²×C₃²	—	2	2	no	—
168	C₂²×C₃²	C₂²×C₈	1	2	yes	—
169	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
170	C₂×C₃⋊D₃	—	2	4	no	—
171	C₂×C₃⋊D₃	C₂²×C₈	1	4	yes	—
172	C₃²×C₄	C₂²×C₈	1	2	yes	—
173	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
174	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
175	C₃⋊Dic₃	C₂²×C₈	1	4	yes	—
176	C₂×#41	—	9	1	no	—
177	#131	—	9	4	no	—
178	#131	—	9	4	no	—
179	C₂×M₁₆	—	9	1	no	—
180	C₂×M₁₆	—	9	2	no	—
181	C₂²×C₈	—	9	2	no	—
182	C₂²×C₈	D₃²	1	1	yes	characteristic
183	M₃₂	—	3	4	no	—
184	M₃₂	—	9	8	no	—
185	C₂×C₁₆	—	3	4	no	—
186	C₂×C₁₆	—	3	1	no	—
187	C₂×C₁₆	—	3	1	no	—
188	C₂×C₁₆	—	3	2	no	—
189	C₂×C₁₆	—	9	8	no	—
190	C₂³×C₃	—	2	1	no	—
191	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
192	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
193	C₂²×D₃	—	3	2	no	—
194	C₂²×D₃	—	3	2	no	—
195	C₂²×D₃	—	6	2	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
201	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
202	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
203	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
204	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
207	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
208	#67	—	3	8	no	—
209	#67	—	3	8	no	—
210	#67	—	6	8	no	—
211	C₂×Dic₃	—	3	2	no	—
212	C₂×Dic₃	—	3	2	no	—
213	C₂×Dic₃	—	6	2	no	—
214	D₁₂	—	3	4	no	—
215	D₁₂	—	3	4	no	—
216	D₁₂	—	6	4	no	—
217	C₄×D₃	—	3	4	no	—
218	C₄×D₃	—	3	4	no	—
219	C₄×D₃	—	3	4	no	—
220	C₄×D₃	—	3	4	no	—
221	C₄×D₃	—	6	4	no	—
222	C₄×D₃	—	6	4	no	—
223	Dic₆	—	3	4	no	—
224	Dic₆	—	3	4	no	—
225	Dic₆	—	6	4	no	—
226	C₃×C₈	—	2	1	no	—
227	C₃×C₈	—	2	1	no	—
228	C₃×C₈	—	2	2	no	—
229	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
230	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
231	C₃×C₈	C₂×C₄×D₃	1	2	yes	—
232	C₃×C₈	C₂×C₄×D₃	1	2	yes	—
233	C₃×C₈	C₂³×D₃	1	1	yes	characteristic
234	C₃×C₈	C₂³×D₃	1	1	yes	characteristic
235	C₃⋊C₈	—	3	4	no	—
236	C₃⋊C₈	—	3	4	no	—
237	C₃⋊C₈	—	6	4	no	—
238	C₂×C₃²	—	2	2	no	—
239	C₂×C₃²	#504	1	2	yes	complemented
240	C₂×C₃²	C₂³×C₈	1	1	yes	characteristic
241	C₃⋊D₃	—	2	4	no	—
242	#41	—	9	4	no	—
243	#41	—	9	4	no	—
244	C₂×Q₈	—	9	1	no	—
245	C₂×D₄	—	9	2	no	—
246	C₂×D₄	—	9	1	no	—
247	C₂²×C₄	—	9	2	no	—
248	C₂²×C₄	#624	1	1	yes	characteristic
249	M₁₆	—	9	4	no	—
250	M₁₆	—	9	8	no	—
251	C₂×C₈	—	9	2	no	—
252	C₂×C₈	—	9	4	no	—
253	C₂×C₈	—	9	4	no	—
254	C₂×C₈	#624	1	1	yes	characteristic
255	C₂×C₈	#624	1	2	yes	—
256	C₂×C₈	C₂×D₃²	1	2	yes	—
257	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
258	C₁₆	—	3	2	no	—
259	C₁₆	—	3	2	no	—
260	C₁₆	—	3	4	no	—
261	C₂²×C₃	—	2	2	no	—
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	2	no	—
265	C₂²×C₃	—	4	2	no	—
266	C₂²×C₃	C₂×D₃×C₈	1	2	yes	—
267	C₂²×C₃	C₂×D₃×C₈	1	2	yes	—
268	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
269	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
270	C₂×D₃	—	3	4	no	—
271	C₂×D₃	—	3	4	no	—
272	C₂×D₃	—	6	4	no	—
273	C₂×D₃	—	6	4	no	—
274	C₂×D₃	—	6	4	no	—
275	C₂×D₃	—	12	4	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	2	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	C₂×D₃×C₈	1	2	yes	—
280	C₃×C₄	C₂×D₃×C₈	1	2	yes	—
281	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
282	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
283	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
284	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
285	Dic₃	—	3	4	no	—
286	Dic₃	—	3	4	no	—
287	Dic₃	—	6	4	no	—
288	C₃²	C₂×#504	1	1	yes	characteristic, complemented, Sylow
289	C₂³	—	9	2	no	—
290	C₂³	#3710	1	1	yes	characteristic
291	Q₈	—	9	4	no	—
292	D₄	—	9	8	no	—
293	D₄	—	9	4	no	—
294	C₂×C₄	—	9	2	no	—
295	C₂×C₄	—	9	4	no	—
296	C₂×C₄	—	9	4	no	—
297	C₂×C₄	#3710	1	1	yes	characteristic
298	C₂×C₄	#3710	1	2	yes	—
299	C₂×C₄	C₂×#624	1	1	yes	characteristic
300	C₂×C₄	C₂×#624	1	2	yes	—
301	C₈	—	9	4	no	—
302	C₈	C₂×#624	1	1	yes	characteristic
303	C₈	C₂×#624	1	2	yes	—
304	C₈	C₂²×D₃²	1	1	yes	characteristic
305	C₂×C₃	—	2	2	no	—
306	C₂×C₃	—	2	2	no	—
307	C₂×C₃	—	2	2	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	4	2	no	—
310	C₂×C₃	#4901	1	2	yes	complemented
311	C₂×C₃	#4907	1	2	yes	complemented
312	C₂×C₃	C₂²×D₃×C₈	1	1	yes	characteristic
313	C₂×C₃	C₂²×D₃×C₈	1	1	yes	characteristic
314	D₃	—	6	4	no	—
315	D₃	—	6	4	no	—
316	D₃	—	12	4	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	9	4	no	—
319	C₂²	—	18	4	no	—
320	C₂²	C₂×#3710	1	2	yes	—
321	C₂²	C₂×#3710	1	1	yes	characteristic
322	C₄	—	9	4	no	—
323	C₄	C₂×#3710	1	1	yes	characteristic
324	C₄	C₂×#3710	1	2	yes	—
325	C₄	C₂²×#624	1	1	yes	characteristic
326	C₃	—	2	1	no	—
327	C₃	C₂×#4901	1	1	yes	characteristic, complemented
328	C₃	C₂×#4907	1	1	yes	characteristic, complemented
329	C₂	—	2	2	no	—
330	C₂	—	18	4	no	—
331	C₂	#10590624	1	2	yes	direct product factor
332	C₂	C₂²×#3710	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃²
commutator	C₂×C₃²	C₂³×C₈
Frattini	C₈	C₂²×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×#3828	C₄
cosocle	C₃²×C₈	C₂⁴
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	C₂×#504
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#812	C₂	semidirect
3. C₂²×C₃×C₃⋊C₁₆	C₂	semidirect
4. C₂×#63544	C₂	semidirect
5. #10590624	C₂	direct (N × H)
6. C₂×#63542	C₂	semidirect
16. C₃×#812	C₂²	semidirect
20. C₂×C₃×C₃⋊C₁₆	C₂²	semidirect
21. #63544	C₂²	semidirect
22. #63542	C₂²	semidirect
28. C₂×#812	D₃	semidirect
29. C₂²×C₃⋊C₁₆	D₃	semidirect
82. #812	C₂×D₃	semidirect
86. C₂×C₃⋊C₁₆	C₂×D₃	semidirect
239. C₂×C₃²	#504	semidirect
288. C₃²	C₂×#504	semidirect
310. C₂×C₃	#4901	semidirect
311. C₂×C₃	#4907	semidirect
327. C₃	C₂×#4901	semidirect
328. C₃	C₂×#4907	semidirect
331. C₂	#10590624	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#64110	2
C₂×C₃×#812	2
C₂²×C₃×C₃⋊C₁₆	2
C₂×#63544	2
#10590624	2
C₂×#63542	2
C₂×#4907	3
C₂×#4901	3

Subgroups — C2×#10590624 #11899397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10590624 #11899397