Subgroups — C₂×#10592416 #11861044

Order 1152.

1392 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	C₂×#64113	C₂	1	1	yes	maximal, characteristic
2	C₂×#64110	C₂	1	2	yes	maximal, complemented
3	C₂×C₃²×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#63646	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#63632	C₂	1	2	yes	maximal, complemented
6	#10592416	C₂	1	8	yes	maximal, direct product factor
7	C₂×#4717	—	3	4	no	maximal
8	C₂×#3828	C₂²	1	2	yes	—
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
12	C₂×#3748	C₂²	1	2	yes	—
13	C₂×#3746	C₄	1	1	yes	characteristic, complemented
14	#64113	C₄	1	4	yes	complemented
15	#64113	C₂²	1	4	yes	—
16	C₂×#3745	C₄	1	1	yes	characteristic, complemented
17	C₂×#3744	C₄	1	2	yes	complemented
18	#64110	—	2	4	no	—
19	#64110	C₂²	1	8	yes	complemented
20	C₂×#3743	—	2	2	no	—
21	C₂×C₈×C₃⋊D₃	—	2	2	no	—
22	C₃²×#97	C₂²	1	4	yes	complemented
23	#63646	C₂²	1	4	yes	complemented
24	#63632	C₂²	1	8	yes	complemented
25	C₂×#904	—	3	4	no	—
26	C₂×#901	—	3	8	no	—
27	C₂×C₃×#97	D₃	1	4	yes	complemented
28	C₂×#819	—	3	4	no	—
29	C₂×#805	—	3	8	no	—
30	#4717	—	3	32	no	—
31	C₂²×C₃²×C₄	—	2	1	no	—
32	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₂×#638	—	2	2	no	—
34	#3828	—	2	2	no	—
35	#3828	—	2	2	no	—
36	#3828	C₂×C₄	1	4	yes	—
37	#3828	C₂³	1	4	yes	—
38	C₂×C₄⋊(C₃⋊D₃)	C₂×C₄	1	2	yes	—
39	C₂×C₄×C₃⋊D₃	—	2	2	no	—
40	C₂×#634	C₂×C₄	1	2	yes	—
41	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
44	C₂×C₃²×C₈	D₄	1	2	yes	—
45	C₂×C₃²×C₈	D₄	1	2	yes	—
46	C₂×C₃²×C₈	C₂³	1	2	yes	—
47	C₂×C₃²×C₈	C₂³	1	2	yes	—
48	C₃²×C₄²	—	2	1	no	—
49	C₃²×C₄²	C₂³	1	2	yes	—
50	#3748	—	2	2	no	—
51	#3748	C₂³	1	4	yes	—
52	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
53	#3746	C₂×C₄	1	4	yes	complemented
54	#3745	C₂×C₄	1	4	yes	complemented
55	#3744	C₂×C₄	1	8	yes	complemented
56	#3743	—	2	8	no	—
57	C₈×C₃⋊D₃	—	2	8	no	—
58	C₂×#444	—	9	1	no	Sylow
59	C₂×#235	—	3	8	no	—
60	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
61	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
62	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
63	C₂×#208	—	3	8	no	—
64	C₂×Dic₁₂	—	3	4	no	—
65	#904	—	3	16	no	—
66	#904	—	3	16	no	—
67	C₂×D₂₄	—	3	4	no	—
68	C₂×#204	—	3	8	no	—
69	#901	—	3	32	no	—
70	#901	—	6	16	no	—
71	C₂×ΓL(1,𝔽₂₅)	—	6	8	no	—
72	C₂×D₃×C₈	—	6	8	no	—
73	C₃×#97	C₂×D₃	1	16	yes	complemented
74	#819	—	3	16	no	—
75	#805	—	3	32	no	—
76	C₂³×C₃²	#41	1	1	yes	characteristic
77	C₂²×C₃⋊D₃	—	2	2	no	—
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	—	4	2	no	—
80	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	#41	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
86	#638	—	2	4	no	—
87	#638	—	2	4	no	—
88	C₂×C₃⋊Dic₃	—	2	2	no	—
89	C₄⋊(C₃⋊D₃)	—	2	2	no	—
90	C₄⋊(C₃⋊D₃)	C₂²×C₄	1	4	yes	—
91	C₄×C₃⋊D₃	—	2	4	no	—
92	C₄×C₃⋊D₃	—	2	4	no	—
93	#634	—	2	2	no	—
94	#634	C₂²×C₄	1	4	yes	—
95	C₃²×C₈	C₂²×C₄	1	2	yes	—
96	C₃²×C₈	C₂²×C₄	1	2	yes	—
97	C₃²×C₈	C₂×D₄	1	4	yes	—
98	C₃⋊(C₃⋊C₈)	—	2	4	no	—
99	C₂×#135	—	9	1	no	—
100	C₂×#131	—	9	2	no	—
101	#444	—	9	8	no	—
102	C₂×#108	—	9	1	no	—
103	C₂×#104	—	9	2	no	—
104	C₂×#97	C₃⋊D₃	1	1	yes	characteristic, complemented
105	C₂²×C₃×C₄	—	2	4	no	—
106	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
107	C₂×#67	—	6	8	no	—
108	#235	—	3	16	no	—
109	#235	—	3	16	no	—
110	#235	—	6	8	no	—
111	#235	—	6	8	no	—
112	C₂×D₁₂	—	3	8	no	—
113	C₂×C₄×D₃	—	6	8	no	—
114	C₂×Dic₆	—	3	8	no	—
115	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
116	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
117	C₂×C₃×C₈	C₄×D₃	1	8	yes	—
118	C₂×C₃×C₈	D₁₂	1	8	yes	—
119	C₂×C₃×C₈	D₁₂	1	8	yes	—
120	C₂×C₃×C₈	C₂²×D₃	1	8	yes	—
121	C₂×C₃×C₈	C₂²×D₃	1	8	yes	—
122	C₃×C₄²	—	2	4	no	—
123	C₃×C₄²	C₂²×D₃	1	8	yes	—
124	#208	—	3	16	no	—
125	#208	—	6	8	no	—
126	C₂×C₃⋊C₈	—	6	8	no	—
127	Dic₁₂	—	3	16	no	—
128	D₂₄	—	3	16	no	—
129	#204	—	3	32	no	—
130	ΓL(1,𝔽₂₅)	—	6	32	no	—
131	D₃×C₈	—	6	32	no	—
132	C₂²×C₃²	—	2	2	no	—
133	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
134	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
135	C₂²×C₃²	C₂×#41	1	1	yes	characteristic
136	C₂×C₃⋊D₃	—	2	4	no	—
137	C₂×C₃⋊D₃	—	4	4	no	—
138	C₃²×C₄	—	4	2	no	—
139	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
140	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
141	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
142	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
143	C₃⋊Dic₃	—	2	4	no	—
144	C₂×#41	—	9	2	no	—
145	#135	—	9	4	no	—
146	#135	—	9	4	no	—
147	C₂×Q₁₆	—	9	1	no	—
148	C₂×SD₁₆	—	9	2	no	—
149	C₂×D₈	—	9	1	no	—
150	#131	—	9	8	no	—
151	#131	—	18	4	no	—
152	C₂×M₁₆	—	9	2	no	—
153	C₂×M₁₆	—	18	2	no	—
154	C₂²×C₈	—	18	2	no	—
155	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
156	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
157	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
158	#108	—	9	4	no	—
159	#104	—	9	8	no	—
160	#97	C₂×C₃⋊D₃	1	4	yes	complemented
161	C₂³×C₃	#235	1	4	yes	—
162	C₂²×D₃	—	6	8	no	—
163	C₂×C₃×C₄	—	2	8	no	—
164	C₂×C₃×C₄	—	4	8	no	—
165	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
166	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
167	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
168	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
169	C₂×C₃×C₄	#235	1	4	yes	—
170	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
171	#67	—	6	16	no	—
172	#67	—	6	16	no	—
173	C₂×Dic₃	—	6	8	no	—
174	D₁₂	—	3	16	no	—
175	D₁₂	—	6	8	no	—
176	C₄×D₃	—	6	16	no	—
177	C₄×D₃	—	6	16	no	—
178	Dic₆	—	3	16	no	—
179	Dic₆	—	6	8	no	—
180	C₃×C₈	C₂×C₄×D₃	1	8	yes	—
181	C₃×C₈	C₂×C₄×D₃	1	8	yes	—
182	C₃×C₈	C₂×D₁₂	1	16	yes	—
183	C₃⋊C₈	—	6	16	no	—
184	C₂×C₃²	—	2	1	no	—
185	C₂×C₃²	—	2	1	no	—
186	C₂×C₃²	#444	1	2	yes	complemented
187	C₂×C₃²	C₂×C₄×D₄	1	1	yes	characteristic
188	C₃⋊D₃	—	4	4	no	—
189	#41	—	9	4	no	—
190	#41	—	9	4	no	—
191	#41	—	18	2	no	—
192	#41	—	18	2	no	—
193	C₂×Q₈	—	9	2	no	—
194	C₂×D₄	—	9	2	no	—
195	C₂×D₄	—	18	2	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	18	2	no	—
198	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
199	Q₁₆	—	9	4	no	—
200	SD₁₆	—	9	8	no	—
201	D₈	—	9	4	no	—
202	M₁₆	—	9	4	no	—
203	M₁₆	—	18	2	no	—
204	M₁₆	—	18	8	no	—
205	C₂×C₈	—	18	2	no	—
206	C₂×C₈	—	18	8	no	—
207	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
208	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
209	C₂×C₈	C₄×C₃⋊D₃	1	2	yes	—
210	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
211	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
212	C₂×C₈	C₂²×C₃⋊D₃	1	2	yes	—
213	C₂×C₈	C₂²×C₃⋊D₃	1	2	yes	—
214	C₄²	—	2	1	no	—
215	C₄²	C₂²×C₃⋊D₃	1	2	yes	—
216	C₂²×C₃	—	2	8	no	—
217	C₂²×C₃	C₄×D₁₂	1	4	yes	—
218	C₂²×C₃	C₄×D₁₂	1	4	yes	—
219	C₂²×C₃	C₂×#235	1	4	yes	—
220	C₂×D₃	—	6	16	no	—
221	C₂×D₃	—	12	16	no	—
222	C₃×C₄	—	4	8	no	—
223	C₃×C₄	C₄×D₁₂	1	4	yes	—
224	C₃×C₄	C₄×D₁₂	1	4	yes	—
225	C₃×C₄	C₂²×C₄×D₃	1	4	yes	—
226	C₃×C₄	C₂²×D₁₂	1	4	yes	—
227	Dic₃	—	6	16	no	—
228	C₃²	C₂×#444	1	1	yes	characteristic, complemented, Sylow
229	C₂³	—	18	2	no	—
230	C₂³	#3828	1	1	yes	characteristic
231	Q₈	—	9	4	no	—
232	Q₈	—	18	2	no	—
233	D₄	—	9	4	no	—
234	D₄	—	18	4	no	—
235	D₄	—	18	4	no	—
236	D₄	—	18	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	18	2	no	—
240	C₂×C₄	—	18	4	no	—
241	C₂×C₄	—	18	4	no	—
242	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
243	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
246	C₂×C₄	#3828	1	1	yes	characteristic
247	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
248	C₈	—	18	4	no	—
249	C₈	C₂×C₄×C₃⋊D₃	1	2	yes	—
250	C₈	C₂×C₄×C₃⋊D₃	1	2	yes	—
251	C₈	C₂×C₄⋊(C₃⋊D₃)	1	4	yes	—
252	C₂×C₃	—	2	4	no	—
253	C₂×C₃	—	2	4	no	—
254	C₂×C₃	#4717	1	8	yes	complemented
255	C₂×C₃	C₂×C₄×D₁₂	1	4	yes	—
256	D₃	—	12	16	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	18	4	no	—
259	C₂²	—	36	4	no	—
260	C₂²	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
261	C₂²	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
262	C₂²	C₂×#3828	1	1	yes	characteristic
263	C₄	—	4	2	no	—
264	C₄	—	18	4	no	—
265	C₄	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
266	C₄	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
267	C₄	C₂²×C₄×C₃⋊D₃	1	1	yes	characteristic
268	C₄	C₂²×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
269	C₃	C₂×#4717	1	4	yes	complemented
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	36	4	no	—
273	C₂	#10592416	1	2	yes	direct product factor
274	C₂	C₂×C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×C₃⋊D₃
Fitting	C₂×C₃²×#97	C₂
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×#64113	C₂
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	C₂×#97	C₃⋊D₃
hypocenter	C₃²	C₂×#444
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. C₂×#64110	C₂	semidirect
3. C₂×C₃²×#97	C₂	semidirect
4. C₂×#63646	C₂	semidirect
5. C₂×#63632	C₂	semidirect
6. #10592416	C₂	direct (N × H)
13. C₂×#3746	C₄	semidirect
14. #64113	C₄	semidirect
16. C₂×#3745	C₄	semidirect
17. C₂×#3744	C₄	semidirect
19. #64110	C₂²	semidirect
22. C₃²×#97	C₂²	semidirect
23. #63646	C₂²	semidirect
24. #63632	C₂²	semidirect
27. C₂×C₃×#97	D₃	semidirect
53. #3746	C₂×C₄	semidirect
54. #3745	C₂×C₄	semidirect
55. #3744	C₂×C₄	semidirect
73. C₃×#97	C₂×D₃	semidirect
104. C₂×#97	C₃⋊D₃	semidirect
160. #97	C₂×C₃⋊D₃	semidirect
186. C₂×C₃²	#444	semidirect
228. C₃²	C₂×#444	semidirect
254. C₂×C₃	#4717	semidirect
269. C₃	C₂×#4717	semidirect
273. C₂	#10592416	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×#64113	2
C₂×#64110	2
C₂×C₃²×#97	2
C₂×#63646	2
C₂×#63632	2
#10592416	2
C₂×#4717	3

Subgroups — C2×#10592416 #11861044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10592416 #11861044