Subgroups — C₂×#10588690 #11868236

Order 1152.

720 conjugacy classes of subgroups.

The subgroup lattice has 273 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#63431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63392	C₂	1	2	yes	maximal
4	#10588690	C₂	1	8	yes	maximal, direct product factor
5	C₂×#63366	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#5007	—	3	1	no	maximal
8	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₉	—	2	1	no	—
10	C₂²×Dic₁₈	—	2	1	no	—
11	C₄×D₄×C₉	—	2	2	no	—
12	C₄×D₄×C₉	C₂²	1	4	yes	—
13	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
15	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₉	—	2	1	no	—
17	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
18	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
19	#63431	—	2	2	no	—
20	#63431	C₂²	1	4	yes	complemented
21	#63392	C₂²	1	8	yes	—
22	#63366	C₂²	1	8	yes	complemented
23	C₄⋊(C₉⋊C₈)	C₂²	1	4	yes	complemented
24	C₂×C₃×C₄×D₄	D₃	1	1	yes	characteristic
25	C₂×#869	—	3	1	no	—
26	C₂×#832	—	3	2	no	—
27	#5007	—	3	8	no	—
28	C₂×#807	—	3	2	no	—
29	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
30	C₂⁴×C₉	—	2	1	no	—
31	C₂×D₄×C₉	—	2	1	no	—
32	C₂×D₄×C₉	—	2	2	no	—
33	C₂×D₄×C₉	Q₈	1	4	yes	complemented
34	C₂×D₄×C₉	C₂³	1	2	yes	—
35	C₂²×C₄×C₉	—	2	1	no	—
36	C₂²×C₄×C₉	—	2	4	no	—
37	C₂²×C₄×C₉	—	4	4	no	—
38	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂²×Dic₉	—	2	2	no	—
42	C₂×Dic₁₈	—	2	4	no	—
43	C₂×Dic₁₈	—	4	4	no	—
44	C₉×C₄⋊C₄	—	2	1	no	—
45	C₉×C₄⋊C₄	C₂³	1	2	yes	—
46	C₉×C₂²⋊C₄	—	2	4	no	—
47	C₄²×C₉	D₄	1	2	yes	—
48	C₄²×C₉	C₂³	1	2	yes	—
49	C₄⋊Dic₉	—	2	2	no	—
50	C₄⋊Dic₉	—	2	4	no	—
51	C₄⋊Dic₉	C₂³	1	4	yes	—
52	C₂×C₉⋊C₈	—	2	4	no	—
53	C₂×C₉⋊C₈	C₂³	1	4	yes	—
54	C₂×#474	—	9	1	no	Sylow
55	C₂²×C₃×D₄	C₂×D₃	1	1	yes	characteristic
56	C₂³×C₃×C₄	—	2	1	no	—
57	C₂²×Dic₆	—	6	1	no	—
58	C₃×C₄×D₄	—	2	2	no	—
59	C₃×C₄×D₄	C₂×D₃	1	4	yes	—
60	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
61	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
62	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
63	C₂×C₄⋊Dic₃	—	3	2	no	—
64	C₂×C₄⋊Dic₃	—	6	1	no	—
65	C₂²×C₃⋊C₈	—	3	2	no	—
66	#869	—	3	4	no	—
67	#869	—	6	2	no	—
68	#832	—	3	8	no	—
69	#807	—	3	8	no	—
70	C₄⋊(C₃⋊C₈)	—	3	4	no	—
71	C₂³×C₉	—	2	4	no	—
72	C₂³×C₉	—	2	2	no	—
73	C₂³×C₉	—	4	2	no	—
74	C₂³×C₉	—	4	2	no	—
75	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
76	D₄×C₉	—	2	4	no	—
77	D₄×C₉	—	2	2	no	—
78	D₄×C₉	C₂×Q₈	1	4	yes	complemented
79	C₂×C₄×C₉	—	2	2	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	4	2	no	—
82	C₂×C₄×C₉	—	4	8	no	—
83	C₂×C₄×C₉	D₈	1	4	yes	—
84	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
88	C₂×C₄×C₉	#41	1	2	yes	—
89	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
90	C₂×Dic₉	—	2	4	no	—
91	C₂×Dic₉	—	4	4	no	—
92	Dic₁₈	—	4	8	no	—
93	C₉⋊C₈	—	2	4	no	—
94	C₂×C₄⋊Q₈	—	9	1	no	—
95	C₂×C₄×D₄	D₉	1	1	yes	characteristic
96	#474	—	9	8	no	—
97	C₂×#107	—	9	2	no	—
98	C₂×C₄⋊C₈	—	9	1	no	—
99	C₂×#102	—	9	2	no	—
100	C₂⁴×C₃	—	2	1	no	—
101	C₂×C₃×D₄	—	2	2	no	—
102	C₂×C₃×D₄	—	2	1	no	—
103	C₂×C₃×D₄	Dic₆	1	4	yes	—
104	C₂×C₃×D₄	C₂²×D₃	1	2	yes	—
105	C₂²×C₃×C₄	—	2	1	no	—
106	C₂²×C₃×C₄	—	2	4	no	—
107	C₂²×C₃×C₄	—	4	4	no	—
108	C₂²×C₃×C₄	#67	1	1	yes	characteristic
109	C₂²×C₃×C₄	#67	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
111	C₂²×Dic₃	—	6	2	no	—
112	C₂×Dic₆	—	6	4	no	—
113	C₂×Dic₆	—	12	4	no	—
114	C₃×C₄⋊C₄	—	2	1	no	—
115	C₃×C₄⋊C₄	C₂²×D₃	1	2	yes	—
116	C₃×C₂²⋊C₄	—	2	4	no	—
117	C₃×C₄²	#67	1	2	yes	—
118	C₃×C₄²	C₂²×D₃	1	2	yes	—
119	C₄⋊Dic₃	—	3	4	no	—
120	C₄⋊Dic₃	—	6	4	no	—
121	C₄⋊Dic₃	—	6	2	no	—
122	C₂×C₃⋊C₈	—	3	4	no	—
123	C₂×C₃⋊C₈	—	6	4	no	—
124	C₂²×C₉	—	2	4	no	—
125	C₂²×C₉	—	4	2	no	—
126	C₂²×C₉	—	4	2	no	—
127	C₂²×C₉	—	4	8	no	—
128	C₂²×C₉	C₂²⋊Q₈	1	2	yes	—
129	C₂²×C₉	C₂×D₈	1	2	yes	—
130	C₂²×C₉	#137	1	2	yes	—
131	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
132	C₄×C₉	—	2	2	no	—
133	C₄×C₉	—	4	2	no	—
134	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
135	C₄×C₉	C₂×D₈	1	4	yes	—
136	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
137	C₄×C₉	C₂×#41	1	1	yes	characteristic
138	Dic₉	—	4	4	no	—
139	C₂²×Q₈	—	18	1	no	—
140	C₂²×D₄	C₂×D₉	1	1	yes	characteristic
141	C₂³×C₄	—	2	1	no	—
142	C₂²×C₈	—	9	2	no	—
143	C₄⋊Q₈	—	9	4	no	—
144	C₄⋊Q₈	—	18	2	no	—
145	C₄×D₄	—	2	2	no	—
146	C₄×D₄	C₂×D₉	1	4	yes	—
147	C₂×C₄⋊C₄	—	9	2	no	—
148	C₂×C₄⋊C₄	—	18	1	no	—
149	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	—	2	1	no	—
151	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
152	#107	—	9	8	no	—
153	C₄⋊C₈	—	9	4	no	—
154	#102	—	9	8	no	—
155	C₂³×C₃	—	2	4	no	—
156	C₂³×C₃	—	2	2	no	—
157	C₂³×C₃	—	4	2	no	—
158	C₂³×C₃	—	4	2	no	—
159	C₂³×C₃	C₂×#67	1	1	yes	characteristic
160	C₃×D₄	—	2	4	no	—
161	C₃×D₄	—	2	2	no	—
162	C₃×D₄	C₂×Dic₆	1	4	yes	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	4	2	no	—
166	C₂×C₃×C₄	—	4	8	no	—
167	C₂×C₃×C₄	#213	1	4	yes	—
168	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
169	C₂×C₃×C₄	#235	1	2	yes	—
170	C₂×C₃×C₄	C₂×#67	1	2	yes	—
171	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×#67	1	2	yes	—
173	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
174	C₂×Dic₃	—	6	4	no	—
175	C₂×Dic₃	—	12	4	no	—
176	Dic₆	—	12	8	no	—
177	C₃⋊C₈	—	6	4	no	—
178	C₂×C₉	—	4	4	no	—
179	C₂×C₉	#474	1	4	yes	complemented
180	C₂×C₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
181	C₂×C₉	C₂²×D₈	1	1	yes	characteristic
182	C₂×C₉	C₂×#137	1	1	yes	characteristic
183	C₂⁴	—	2	1	no	—
184	C₂×Q₈	—	18	4	no	—
185	C₂×Q₈	—	36	4	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	Dic₁₈	1	4	yes	complemented
189	C₂×D₄	C₂²×D₉	1	2	yes	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	—	18	2	no	—
194	C₂²×C₄	#611	1	1	yes	characteristic
195	C₂²×C₄	#611	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
197	C₂×C₈	—	9	4	no	—
198	C₂×C₈	—	18	4	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	9	4	no	—
201	C₄⋊C₄	—	18	2	no	—
202	C₄⋊C₄	—	18	4	no	—
203	C₄⋊C₄	C₂²×D₉	1	2	yes	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₄²	#611	1	2	yes	—
206	C₄²	C₂²×D₉	1	2	yes	—
207	C₂²×C₃	—	2	4	no	—
208	C₂²×C₃	—	4	8	no	—
209	C₂²×C₃	—	4	2	no	—
210	C₂²×C₃	—	4	2	no	—
211	C₂²×C₃	#924	1	2	yes	—
212	C₂²×C₃	C₂×#213	1	2	yes	—
213	C₂²×C₃	#951	1	2	yes	—
214	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	—	4	2	no	—
217	C₃×C₄	#924	1	2	yes	—
218	C₃×C₄	C₂×#213	1	4	yes	—
219	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
220	C₃×C₄	C₂×#235	1	1	yes	characteristic
221	Dic₃	—	12	4	no	—
222	C₉	C₂×#474	1	1	yes	characteristic, complemented, Sylow
223	C₂³	—	2	2	no	—
224	C₂³	—	2	4	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	C₂×#611	1	1	yes	characteristic
228	Q₈	—	36	8	no	—
229	D₄	—	2	2	no	—
230	D₄	—	2	4	no	—
231	D₄	C₂×Dic₁₈	1	4	yes	complemented
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	—	18	4	no	—
237	C₂×C₄	—	36	4	no	—
238	C₂×C₄	#3673	1	4	yes	—
239	C₂×C₄	C₂×Dic₁₈	1	2	yes	—
240	C₂×C₄	#3697	1	2	yes	—
241	C₂×C₄	C₂×#611	1	2	yes	—
242	C₂×C₄	C₂×#611	1	2	yes	—
243	C₂×C₄	C₂×#611	1	1	yes	characteristic
244	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
245	C₈	—	18	4	no	—
246	C₂×C₃	—	4	4	no	—
247	C₂×C₃	#5007	1	4	yes	—
248	C₂×C₃	C₂×#924	1	1	yes	characteristic
249	C₂×C₃	C₂²×#213	1	1	yes	characteristic
250	C₂×C₃	C₂×#951	1	1	yes	characteristic
251	C₂²	—	2	4	no	—
252	C₂²	—	4	8	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	#63486	1	2	yes	—
256	C₂²	C₂×#3673	1	2	yes	—
257	C₂²	#63511	1	2	yes	—
258	C₂²	C₂²×#611	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	36	4	no	—
262	C₄	#63486	1	2	yes	—
263	C₄	C₂×#3673	1	4	yes	—
264	C₄	C₂²×Dic₁₈	1	1	yes	characteristic
265	C₄	C₂×#3697	1	1	yes	characteristic
266	C₃	C₂×#5007	1	1	yes	characteristic
267	C₂	—	4	4	no	—
268	C₂	#10588690	1	4	yes	direct product factor
269	C₂	C₂×#63486	1	1	yes	characteristic
270	C₂	C₂²×#3673	1	1	yes	characteristic
271	C₂	C₂×#63511	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#611
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₄×D₄×C₉	C₂
socle	C₂³×C₃	C₂×#67
prime radical	C₂²×C₃×D₄	C₂×D₃
hypercenter	C₂×C₄×D₄	D₉
hypocenter	C₉	C₂×#474
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#63431	C₂	semidirect
4. #10588690	C₂	direct (N × H)
5. C₂×#63366	C₂	semidirect
6. C₂×C₄⋊(C₉⋊C₈)	C₂	semidirect
20. #63431	C₂²	semidirect
22. #63366	C₂²	semidirect
23. C₄⋊(C₉⋊C₈)	C₂²	semidirect
33. C₂×D₄×C₉	Q₈	semidirect
78. D₄×C₉	C₂×Q₈	semidirect
179. C₂×C₉	#474	semidirect
188. C₂×D₄	Dic₁₈	semidirect
222. C₉	C₂×#474	semidirect
231. D₄	C₂×Dic₁₈	semidirect
268. C₂	#10588690	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×C₄×D₄×C₉	2
C₂×#63431	2
C₂×#63392	2
#10588690	2
C₂×#63366	2
C₂×C₄⋊(C₉⋊C₈)	2
C₂×#5007	3

Subgroups — C2×#10588690 #11868236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588690 #11868236