Subgroups — C₂×#10596154 #11910205

Order 1152.

4224 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#64156	C₂	1	2	yes	maximal, complemented
3	C₂×#64146	C₂	1	2	yes	maximal, complemented
4	#10596154	C₂	1	16	yes	maximal, direct product factor
5	C₂×#64094	C₂	1	4	yes	maximal, complemented
6	C₂×#64091	C₂	1	4	yes	maximal, complemented
7	C₂×#64084	C₂	1	2	yes	maximal, complemented
8	C₂×#5683	—	3	4	no	maximal
9	C₂⁴×C₃⋊D₃	C₂²	1	2	yes	complemented
10	C₂²×C₃²×Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₂²×C₃²×D₄	C₂²	1	1	yes	characteristic
12	C₂²×#638	C₂²	1	4	yes	—
13	C₂²×C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
14	C₂²×C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
15	C₂²×#634	C₂²	1	2	yes	complemented
16	C₃²×#124	C₂²	1	8	yes	complemented
17	C₂×C₃²×C₂²⋊C₄	C₂²	1	4	yes	complemented
18	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic, complemented
19	C₂×#3757	C₂²	1	2	yes	complemented
20	#64156	C₂²	1	16	yes	complemented
21	#64146	C₂²	1	16	yes	complemented
22	C₂×#3752	C₂²	1	4	yes	complemented
23	C₂×#3752	C₂²	1	4	yes	complemented
24	C₂×#3752	C₂²	1	4	yes	complemented
25	C₂×C₄×C₃⋊Dic₃	C₂²	1	2	yes	complemented
26	#64094	C₂²	1	32	yes	complemented
27	#64091	C₂²	1	32	yes	complemented
28	#64084	C₂²	1	16	yes	complemented
29	C₂×C₃×#124	D₃	1	4	yes	complemented
30	C₂×#947	—	3	8	no	—
31	C₂×#937	—	3	8	no	—
32	#5683	—	3	64	no	—
33	C₂×#885	—	3	16	no	—
34	C₂×#882	—	3	16	no	—
35	C₂×#875	—	3	8	no	—
36	C₂⁴×C₃²	C₂³	1	2	yes	—
37	C₂³×C₃⋊D₃	—	4	4	no	—
38	C₂³×C₃⋊D₃	—	4	8	no	—
39	C₂³×C₃⋊D₃	C₂³	1	4	yes	—
40	C₂³×C₃⋊D₃	C₂³	1	8	yes	complemented
41	C₂×C₃²×Q₈	—	4	2	no	—
42	C₂×C₃²×Q₈	C₂³	1	4	yes	complemented
43	C₂×C₃²×D₄	—	4	2	no	—
44	C₂×C₃²×D₄	C₂³	1	4	yes	—
45	C₂²×C₃²×C₄	C₂³	1	2	yes	—
46	C₂²×C₃²×C₄	C₂³	1	2	yes	—
47	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
48	C₂×#638	—	4	8	no	—
49	C₂×#638	C₂³	1	16	yes	—
50	C₂²×C₃⋊Dic₃	C₂³	1	4	yes	—
51	C₂×C₄⋊(C₃⋊D₃)	—	4	4	no	—
52	C₂×C₄⋊(C₃⋊D₃)	—	4	4	no	—
53	C₂×C₄⋊(C₃⋊D₃)	C₂³	1	8	yes	—
54	C₂×C₄⋊(C₃⋊D₃)	C₂³	1	8	yes	—
55	C₂×#634	—	4	4	no	—
56	C₂×#634	C₂³	1	8	yes	complemented
57	C₃²×C₂²⋊C₄	C₂³	1	16	yes	complemented
58	C₃²×C₄²	C₂³	1	4	yes	complemented
59	#3757	C₂³	1	8	yes	complemented
60	#3752	C₂³	1	16	yes	complemented
61	#3752	C₂³	1	16	yes	complemented
62	#3752	C₂³	1	16	yes	complemented
63	C₄×C₃⋊Dic₃	C₂³	1	8	yes	complemented
64	C₂×H(3,𝔽₄)	—	9	1	no	Sylow
65	C₂⁴×D₃	—	3	8	no	—
66	C₂²×C₃×Q₈	C₂×D₃	1	4	yes	complemented
67	C₂²×C₃×D₄	C₂×D₃	1	4	yes	—
68	C₂²×#67	—	3	16	no	—
69	C₂²×D₁₂	—	3	8	no	—
70	C₂²×D₁₂	—	3	8	no	—
71	C₂²×Dic₆	—	3	8	no	—
72	C₃×#124	C₂×D₃	1	32	yes	complemented
73	C₂×C₃×C₂²⋊C₄	C₂×D₃	1	16	yes	complemented
74	C₂×C₃×C₄²	C₂×D₃	1	4	yes	complemented
75	C₂×#217	—	3	8	no	—
76	#947	—	3	64	no	—
77	#937	—	3	64	no	—
78	C₂×(C₂×D₃)⋊C₄	—	3	16	no	—
79	C₂×(C₂×D₃)⋊C₄	—	3	16	no	—
80	C₂×(C₂×D₃)⋊C₄	—	3	16	no	—
81	C₂×C₄×Dic₃	—	3	8	no	—
82	#885	—	3	128	no	—
83	#882	—	3	128	no	—
84	#875	—	3	64	no	—
85	C₂³×C₃²	—	2	8	no	—
86	C₂³×C₃²	—	2	4	no	—
87	C₂³×C₃²	C₂⁴	1	4	yes	—
88	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
89	C₂²×C₃⋊D₃	—	2	8	no	—
90	C₂²×C₃⋊D₃	—	2	8	no	—
91	C₂²×C₃⋊D₃	—	2	8	no	—
92	C₂²×C₃⋊D₃	—	4	8	no	—
93	C₂²×C₃⋊D₃	—	4	16	no	—
94	C₂²×C₃⋊D₃	—	4	32	no	—
95	C₂²×C₃⋊D₃	C₂⁴	1	8	yes	—
96	C₃²×Q₈	—	4	4	no	—
97	C₃²×D₄	—	4	4	no	—
98	C₂×C₃²×C₄	—	2	4	no	—
99	C₂×C₃²×C₄	—	2	4	no	—
100	C₂×C₃²×C₄	—	2	2	no	—
101	C₂×C₃²×C₄	C₂⁴	1	4	yes	—
102	C₂×C₃²×C₄	C₂⁴	1	2	yes	—
103	C₂×C₃²×C₄	C₂⁴	1	4	yes	—
104	#638	—	4	16	no	—
105	C₂×C₃⋊Dic₃	—	2	8	no	—
106	C₂×C₃⋊Dic₃	C₂⁴	1	8	yes	—
107	C₄⋊(C₃⋊D₃)	—	4	8	no	—
108	C₄⋊(C₃⋊D₃)	—	4	8	no	—
109	#634	—	4	8	no	—
110	H(3,𝔽₄)	—	9	16	no	—
111	C₂×#124	—	9	2	no	—
112	C₂×#124	—	9	4	no	—
113	C₂×#124	—	9	2	no	—
114	C₂×#124	C₃⋊D₃	1	1	yes	characteristic, complemented
115	C₂×#120	—	9	2	no	—
116	C₂×#120	—	9	4	no	—
117	C₂⁴×C₃	C₂²×D₃	1	8	yes	—
118	C₂³×D₃	—	3	16	no	—
119	C₂³×D₃	—	3	32	no	—
120	C₂³×D₃	—	12	16	no	—
121	C₂³×D₃	—	12	32	no	—
122	C₂×C₃×Q₈	—	4	8	no	—
123	C₂×C₃×Q₈	C₂²×D₃	1	16	yes	complemented
124	C₂×C₃×D₄	—	4	8	no	—
125	C₂×C₃×D₄	C₂²×D₃	1	16	yes	—
126	C₂²×C₃×C₄	C₂²×D₃	1	8	yes	—
127	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
128	C₂²×C₃×C₄	C₂²×D₃	1	8	yes	—
129	C₂×#67	—	3	64	no	—
130	C₂×#67	—	12	32	no	—
131	C₂²×Dic₃	—	3	16	no	—
132	C₂×D₁₂	—	3	32	no	—
133	C₂×D₁₂	—	3	32	no	—
134	C₂×D₁₂	—	12	16	no	—
135	C₂×D₁₂	—	12	16	no	—
136	C₂×Dic₆	—	3	32	no	—
137	C₂×Dic₆	—	12	16	no	—
138	C₃×C₂²⋊C₄	C₂²×D₃	1	64	yes	complemented
139	C₃×C₄²	C₂²×D₃	1	16	yes	complemented
140	#217	—	3	32	no	—
141	(C₂×D₃)⋊C₄	—	3	64	no	—
142	(C₂×D₃)⋊C₄	—	3	64	no	—
143	(C₂×D₃)⋊C₄	—	3	64	no	—
144	C₄×Dic₃	—	3	32	no	—
145	C₂²×C₃²	—	2	8	no	—
146	C₂²×C₃²	—	2	4	no	—
147	C₂²×C₃²	—	4	8	no	—
148	C₂²×C₃²	H(5,𝔽₂)	1	2	yes	—
149	C₂²×C₃²	H(5,𝔽₂)	1	4	yes	—
150	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
151	C₂×C₃⋊D₃	—	2	8	no	—
152	C₂×C₃⋊D₃	—	2	8	no	—
153	C₂×C₃⋊D₃	—	2	8	no	—
154	C₂×C₃⋊D₃	—	4	32	no	—
155	C₂×C₃⋊D₃	—	4	16	no	—
156	C₃²×C₄	—	2	4	no	—
157	C₃²×C₄	—	2	4	no	—
158	C₃²×C₄	—	2	2	no	—
159	C₃⋊Dic₃	—	2	8	no	—
160	C₂⁵	—	9	2	no	—
161	C₂²×Q₈	—	9	2	no	—
162	C₂²×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
163	C₂²×D₄	—	9	4	no	—
164	C₂²×D₄	—	9	2	no	—
165	C₂²×D₄	—	9	2	no	—
166	C₂²×D₄	C₂×C₃⋊D₃	1	1	yes	characteristic
167	#124	—	9	16	no	—
168	#124	—	9	32	no	—
169	#124	—	9	16	no	—
170	#124	C₂×C₃⋊D₃	1	8	yes	complemented
171	#120	—	9	16	no	—
172	#120	—	9	32	no	—
173	C₂×C₂²⋊C₄	—	9	2	no	—
174	C₂×C₂²⋊C₄	—	9	4	no	—
175	C₂×C₂²⋊C₄	—	9	4	no	—
176	C₂×C₂²⋊C₄	—	9	4	no	—
177	C₂×C₂²⋊C₄	C₂×C₃⋊D₃	1	4	yes	complemented
178	C₂×C₄²	—	9	2	no	—
179	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
180	C₂³×C₃	—	2	16	no	—
181	C₂³×C₃	—	2	32	no	—
182	C₂³×C₃	C₂³×D₃	1	16	yes	—
183	C₂³×C₃	C₂³×D₃	1	4	yes	—
184	C₂²×D₃	—	3	32	no	—
185	C₂²×D₃	—	6	32	no	—
186	C₂²×D₃	—	6	32	no	—
187	C₂²×D₃	—	6	32	no	—
188	C₂²×D₃	—	12	32	no	—
189	C₂²×D₃	—	12	64	no	—
190	C₂²×D₃	—	12	128	no	—
191	C₃×Q₈	—	4	16	no	—
192	C₃×D₄	—	4	16	no	—
193	C₂×C₃×C₄	—	2	16	no	—
194	C₂×C₃×C₄	—	2	8	no	—
195	C₂×C₃×C₄	—	2	16	no	—
196	C₂×C₃×C₄	C₂³×D₃	1	16	yes	—
197	C₂×C₃×C₄	C₂³×D₃	1	8	yes	—
198	C₂×C₃×C₄	C₂³×D₃	1	16	yes	—
199	#67	—	12	64	no	—
200	C₂×Dic₃	—	3	32	no	—
201	C₂×Dic₃	—	6	32	no	—
202	D₁₂	—	12	32	no	—
203	D₁₂	—	12	32	no	—
204	Dic₆	—	12	32	no	—
205	C₂×C₃²	—	4	4	no	—
206	C₂×C₃²	H(3,𝔽₄)	1	4	yes	complemented
207	C₂×C₃²	C₂×H(5,𝔽₂)	1	1	yes	characteristic
208	C₂×C₃²	C₂×H(5,𝔽₂)	1	2	yes	—
209	C₃⋊D₃	—	4	8	no	—
210	C₂⁴	—	9	4	no	—
211	C₂⁴	—	9	8	no	—
212	C₂⁴	—	36	4	no	—
213	C₂⁴	—	36	8	no	—
214	C₂⁴	C₂²×C₃⋊D₃	1	2	yes	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	—	9	8	no	—
217	C₂×Q₈	—	36	4	no	—
218	C₂×Q₈	C₂²×C₃⋊D₃	1	4	yes	complemented
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	9	16	no	—
221	C₂×D₄	—	9	8	no	—
222	C₂×D₄	—	9	8	no	—
223	C₂×D₄	—	36	4	no	—
224	C₂×D₄	—	36	4	no	—
225	C₂×D₄	—	36	8	no	—
226	C₂×D₄	C₂²×C₃⋊D₃	1	4	yes	—
227	C₂²×C₄	—	9	4	no	—
228	C₂²×C₄	C₂²×C₃⋊D₃	1	2	yes	—
229	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₃⋊D₃	1	2	yes	—
231	C₂²⋊C₄	—	9	8	no	—
232	C₂²⋊C₄	—	9	16	no	—
233	C₂²⋊C₄	—	9	16	no	—
234	C₂²⋊C₄	—	9	16	no	—
235	C₂²⋊C₄	C₂²×C₃⋊D₃	1	16	yes	complemented
236	C₄²	—	9	8	no	—
237	C₄²	C₂²×C₃⋊D₃	1	4	yes	complemented
238	C₂²×C₃	—	2	16	no	—
239	C₂²×C₃	—	2	32	no	—
240	C₂²×C₃	—	4	32	no	—
241	C₂²×C₃	#1004	1	8	yes	—
242	C₂²×C₃	#1009	1	16	yes	—
243	C₂²×C₃	C₂⁴×D₃	1	4	yes	—
244	C₂×D₃	—	6	32	no	—
245	C₂×D₃	—	6	32	no	—
246	C₂×D₃	—	6	32	no	—
247	C₂×D₃	—	12	128	no	—
248	C₂×D₃	—	12	64	no	—
249	C₃×C₄	—	2	16	no	—
250	C₃×C₄	—	2	8	no	—
251	C₃×C₄	—	2	16	no	—
252	Dic₃	—	6	32	no	—
253	C₃²	C₂×H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	2	4	no	—
255	C₂³	—	2	8	no	—
256	C₂³	—	9	8	no	—
257	C₂³	—	18	8	no	—
258	C₂³	—	18	8	no	—
259	C₂³	—	18	8	no	—
260	C₂³	—	36	8	no	—
261	C₂³	—	36	16	no	—
262	C₂³	—	36	32	no	—
263	C₂³	C₂³×C₃⋊D₃	1	4	yes	—
264	C₂³	C₂³×C₃⋊D₃	1	1	yes	characteristic
265	Q₈	—	4	4	no	—
266	Q₈	—	36	8	no	—
267	D₄	—	4	4	no	—
268	D₄	—	36	8	no	—
269	D₄	—	36	8	no	—
270	D₄	—	36	16	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	9	8	no	—
275	C₂×C₄	—	18	8	no	—
276	C₂×C₄	C₂³×C₃⋊D₃	1	4	yes	—
277	C₂×C₄	C₂³×C₃⋊D₃	1	2	yes	—
278	C₂×C₄	C₂³×C₃⋊D₃	1	4	yes	—
279	C₂×C₃	—	4	16	no	—
280	C₂×C₃	#5683	1	16	yes	complemented
281	C₂×C₃	C₂×#1004	1	4	yes	—
282	C₂×C₃	C₂×#1009	1	8	yes	—
283	D₃	—	12	32	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	8	no	—
286	C₂²	—	4	8	no	—
287	C₂²	—	18	8	no	—
288	C₂²	—	18	8	no	—
289	C₂²	—	18	8	no	—
290	C₂²	—	36	32	no	—
291	C₂²	—	36	16	no	—
292	C₂²	#64361	1	2	yes	—
293	C₂²	#64366	1	4	yes	—
294	C₂²	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	4	no	—
297	C₄	—	2	4	no	—
298	C₄	—	18	8	no	—
299	C₃	C₂×#5683	1	4	yes	complemented
300	C₂	—	4	4	no	—
301	C₂	—	36	8	no	—
302	C₂	#10596154	1	4	yes	direct product factor
303	C₂	C₂×#64361	1	1	yes	characteristic
304	C₂	C₂×#64366	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂⁵
Frattini	C₂²	C₂⁴×C₃⋊D₃
Fitting	C₂×C₃²×#124	C₂
socle	C₂³×C₃²	C₂⁴
hypercenter	C₂×#124	C₃⋊D₃
hypocenter	C₃²	C₂×H(3,𝔽₄)
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×#124	C₂	semidirect
2. C₂×#64156	C₂	semidirect
3. C₂×#64146	C₂	semidirect
4. #10596154	C₂	direct (N × H)
5. C₂×#64094	C₂	semidirect
6. C₂×#64091	C₂	semidirect
7. C₂×#64084	C₂	semidirect
9. C₂⁴×C₃⋊D₃	C₂²	semidirect
10. C₂²×C₃²×Q₈	C₂²	semidirect
15. C₂²×#634	C₂²	semidirect
16. C₃²×#124	C₂²	semidirect
17. C₂×C₃²×C₂²⋊C₄	C₂²	semidirect
18. C₂×C₃²×C₄²	C₂²	semidirect
19. C₂×#3757	C₂²	semidirect
20. #64156	C₂²	semidirect
21. #64146	C₂²	semidirect
22. C₂×#3752	C₂²	semidirect
23. C₂×#3752	C₂²	semidirect
24. C₂×#3752	C₂²	semidirect
25. C₂×C₄×C₃⋊Dic₃	C₂²	semidirect
26. #64094	C₂²	semidirect
27. #64091	C₂²	semidirect
28. #64084	C₂²	semidirect
29. C₂×C₃×#124	D₃	semidirect
40. C₂³×C₃⋊D₃	C₂³	semidirect
42. C₂×C₃²×Q₈	C₂³	semidirect
56. C₂×#634	C₂³	semidirect
57. C₃²×C₂²⋊C₄	C₂³	semidirect
58. C₃²×C₄²	C₂³	semidirect
59. #3757	C₂³	semidirect
60. #3752	C₂³	semidirect
61. #3752	C₂³	semidirect
62. #3752	C₂³	semidirect
63. C₄×C₃⋊Dic₃	C₂³	semidirect
66. C₂²×C₃×Q₈	C₂×D₃	semidirect
72. C₃×#124	C₂×D₃	semidirect
73. C₂×C₃×C₂²⋊C₄	C₂×D₃	semidirect
74. C₂×C₃×C₄²	C₂×D₃	semidirect
114. C₂×#124	C₃⋊D₃	semidirect
123. C₂×C₃×Q₈	C₂²×D₃	semidirect
138. C₃×C₂²⋊C₄	C₂²×D₃	semidirect
139. C₃×C₄²	C₂²×D₃	semidirect
162. C₂²×Q₈	C₂×C₃⋊D₃	semidirect
170. #124	C₂×C₃⋊D₃	semidirect
177. C₂×C₂²⋊C₄	C₂×C₃⋊D₃	semidirect
179. C₂×C₄²	C₂×C₃⋊D₃	semidirect
206. C₂×C₃²	H(3,𝔽₄)	semidirect
218. C₂×Q₈	C₂²×C₃⋊D₃	semidirect
235. C₂²⋊C₄	C₂²×C₃⋊D₃	semidirect
237. C₄²	C₂²×C₃⋊D₃	semidirect
253. C₃²	C₂×H(3,𝔽₄)	semidirect
280. C₂×C₃	#5683	semidirect
299. C₃	C₂×#5683	semidirect
302. C₂	#10596154	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×C₃²×#124	2
C₂×#64156	2
C₂×#64146	2
#10596154	2
C₂×#64094	2
C₂×#64091	2
C₂×#64084	2
C₂×#5683	3

Subgroups — C2×#10596154 #11910205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10596154 #11910205