Subgroups — C₂×#10588763 #11865620

Order 1152.

720 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3674	C₂	1	2	yes	maximal
3	C₂×C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#63395	C₂	1	2	yes	maximal
5	#10588763	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	C₂×#5090	—	3	1	no	maximal
8	C₂²×Q₈×C₉	C₂²	1	2	yes	—
9	C₂²×Dic₁₈	C₂²	1	1	yes	characteristic
10	C₉×C₄⋊Q₈	—	2	2	no	—
11	C₉×C₄⋊Q₈	C₂²	1	4	yes	—
12	C₂×C₉×C₄⋊C₄	—	2	1	no	—
13	C₂×C₉×C₄⋊C₄	—	2	1	no	—
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₂×#3674	—	2	8	no	—
16	C₂×#3674	C₂²	1	8	yes	—
17	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
18	C₂×Dic₉⋊C₄	—	2	1	no	—
19	C₂×C₄×Dic₉	—	2	1	no	—
20	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
21	C₄×Dic₁₈	—	2	2	no	—
22	C₄×Dic₁₈	C₂²	1	4	yes	—
23	#63395	C₂²	1	8	yes	—
24	C₄⋊(C₉⋊C₈)	C₂²	1	4	yes	—
25	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
26	C₂²×#216	—	3	2	no	—
27	C₂×C₄×Dic₆	—	3	1	no	—
28	C₂×#835	—	3	2	no	—
29	#5090	—	3	8	no	—
30	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
31	C₂×Q₈×C₉	—	2	2	no	—
32	C₂×Q₈×C₉	—	2	4	no	—
33	C₂×Q₈×C₉	—	2	4	no	—
34	C₂×Q₈×C₉	C₂³	1	4	yes	—
35	C₂²×C₄×C₉	—	2	2	no	—
36	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₂²×Dic₉	—	2	1	no	—
40	C₂²×Dic₉	—	2	1	no	—
41	C₂×Dic₁₈	—	2	1	no	—
42	C₂×Dic₁₈	—	2	2	no	—
43	C₂×Dic₁₈	D₄	1	4	yes	—
44	C₂×Dic₁₈	C₂³	1	2	yes	—
45	C₉×C₄⋊C₄	—	2	4	no	—
46	C₉×C₄⋊C₄	—	2	4	no	—
47	C₄²×C₉	D₄	1	2	yes	—
48	C₄²×C₉	C₂³	1	2	yes	—
49	#3674	—	2	16	no	—
50	C₄⋊Dic₉	—	2	1	no	—
51	C₄⋊Dic₉	C₂³	1	2	yes	—
52	Dic₉⋊C₄	—	2	4	no	—
53	C₄×Dic₉	—	2	4	no	—
54	C₂×C₉⋊C₈	—	2	4	no	—
55	C₂×C₉⋊C₈	C₂³	1	4	yes	—
56	C₂×#462	—	9	1	no	Sylow
57	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	—
58	C₂²×Dic₆	—	3	1	no	—
59	C₃×C₄⋊Q₈	—	2	2	no	—
60	C₃×C₄⋊Q₈	C₂×D₃	1	4	yes	—
61	C₂×C₃×C₄⋊C₄	—	2	1	no	—
62	C₂×C₃×C₄⋊C₄	—	2	1	no	—
63	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
64	C₂×#216	—	3	8	no	—
65	C₂×#216	—	6	8	no	—
66	C₂×C₄⋊Dic₃	—	3	1	no	—
67	C₂×Dic₃⋊C₄	—	6	1	no	—
68	C₂×C₄×Dic₃	—	6	1	no	—
69	C₂²×C₃⋊C₈	—	3	2	no	—
70	C₄×Dic₆	—	3	4	no	—
71	C₄×Dic₆	—	6	2	no	—
72	#835	—	3	8	no	—
73	C₄⋊(C₃⋊C₈)	—	3	4	no	—
74	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
75	Q₈×C₉	—	2	4	no	—
76	Q₈×C₉	—	2	4	no	—
77	Q₈×C₉	—	4	4	no	—
78	C₂×C₄×C₉	—	2	2	no	—
79	C₂×C₄×C₉	—	2	4	no	—
80	C₂×C₄×C₉	—	4	4	no	—
81	C₂×C₄×C₉	Q₁₆	1	4	yes	complemented
82	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₉	#41	1	2	yes	—
87	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
88	C₂×Dic₉	—	2	4	no	—
89	C₂×Dic₉	—	2	2	no	—
90	C₂×Dic₉	—	2	2	no	—
91	C₂×Dic₉	—	4	2	no	—
92	Dic₁₈	—	2	4	no	—
93	Dic₁₈	—	2	2	no	—
94	Dic₁₈	C₂×D₄	1	4	yes	—
95	C₉⋊C₈	—	2	4	no	—
96	C₂²×Q₁₆	—	9	2	no	—
97	C₂×C₄⋊Q₈	D₉	1	1	yes	characteristic
98	C₂×C₄×Q₈	—	9	1	no	—
99	#462	—	9	8	no	—
100	C₂×C₄⋊C₈	—	9	1	no	—
101	C₂×#103	—	9	2	no	—
102	C₂×C₃×Q₈	—	2	2	no	—
103	C₂×C₃×Q₈	—	2	4	no	—
104	C₂×C₃×Q₈	—	2	4	no	—
105	C₂×C₃×Q₈	C₂²×D₃	1	4	yes	—
106	C₂²×C₃×C₄	—	2	2	no	—
107	C₂²×C₃×C₄	#67	1	1	yes	characteristic
108	C₂²×C₃×C₄	#67	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₂²×Dic₃	—	6	1	no	—
111	C₂²×Dic₃	—	6	1	no	—
112	C₂×Dic₆	—	3	4	no	—
113	C₂×Dic₆	—	3	2	no	—
114	C₂×Dic₆	—	6	2	no	—
115	C₂×Dic₆	—	6	1	no	—
116	C₃×C₄⋊C₄	—	2	4	no	—
117	C₃×C₄⋊C₄	—	2	4	no	—
118	C₃×C₄²	#67	1	2	yes	—
119	C₃×C₄²	C₂²×D₃	1	2	yes	—
120	#216	—	6	16	no	—
121	C₄⋊Dic₃	—	3	2	no	—
122	C₄⋊Dic₃	—	6	1	no	—
123	Dic₃⋊C₄	—	6	4	no	—
124	C₄×Dic₃	—	6	4	no	—
125	C₂×C₃⋊C₈	—	3	4	no	—
126	C₂×C₃⋊C₈	—	6	4	no	—
127	C₂²×C₉	#121	1	2	yes	—
128	C₂²×C₉	C₂×Q₁₆	1	2	yes	—
129	C₂²×C₉	#137	1	2	yes	—
130	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₉	—	2	2	no	—
132	C₄×C₉	—	4	4	no	—
133	C₄×C₉	#121	1	2	yes	—
134	C₄×C₉	C₂×Q₁₆	1	4	yes	complemented
135	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₉	C₂×#41	1	1	yes	characteristic
137	Dic₉	—	2	4	no	—
138	Dic₉	—	4	2	no	—
139	C₂²×Q₈	—	9	1	no	—
140	C₂²×Q₈	C₂×D₉	1	2	yes	—
141	C₂×Q₁₆	—	9	8	no	—
142	C₂×Q₁₆	—	18	8	no	—
143	C₂²×C₈	—	9	2	no	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	C₂×D₉	1	4	yes	—
146	C₄×Q₈	—	9	4	no	—
147	C₄×Q₈	—	18	2	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	9	1	no	—
151	C₂×C₄⋊C₄	—	18	1	no	—
152	C₂×C₄²	—	18	1	no	—
153	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
154	C₄⋊C₈	—	9	4	no	—
155	#103	—	9	8	no	—
156	C₂³×C₃	C₂×#67	1	1	yes	characteristic
157	C₃×Q₈	—	2	4	no	—
158	C₃×Q₈	—	2	4	no	—
159	C₃×Q₈	—	4	4	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	—	4	4	no	—
163	C₂×C₃×C₄	#216	1	4	yes	—
164	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
165	C₂×C₃×C₄	#237	1	2	yes	—
166	C₂×C₃×C₄	C₂×#67	1	2	yes	—
167	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×#67	1	2	yes	—
169	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
170	C₂×Dic₃	—	6	4	no	—
171	C₂×Dic₃	—	6	2	no	—
172	C₂×Dic₃	—	6	2	no	—
173	C₂×Dic₃	—	12	2	no	—
174	Dic₆	—	3	4	no	—
175	Dic₆	—	6	4	no	—
176	Dic₆	—	6	2	no	—
177	C₃⋊C₈	—	6	4	no	—
178	C₂×C₉	#462	1	4	yes	complemented
179	C₂×C₉	C₂×#121	1	1	yes	characteristic
180	C₂×C₉	C₂²×Q₁₆	1	1	yes	characteristic
181	C₂×C₉	C₂×#137	1	1	yes	characteristic
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	9	4	no	—
186	C₂×Q₈	—	9	2	no	—
187	C₂×Q₈	—	18	2	no	—
188	C₂×Q₈	—	18	1	no	—
189	C₂×Q₈	C₂²×D₉	1	4	yes	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	18	1	no	—
192	C₂²×C₄	—	18	1	no	—
193	C₂²×C₄	#611	1	1	yes	characteristic
194	C₂²×C₄	#611	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
196	Q₁₆	—	18	16	no	—
197	C₂×C₈	—	9	4	no	—
198	C₂×C₈	—	18	4	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	9	2	no	—
202	C₄⋊C₄	—	18	1	no	—
203	C₄⋊C₄	—	18	4	no	—
204	C₄²	—	18	4	no	—
205	C₄²	#611	1	2	yes	—
206	C₄²	C₂²×D₉	1	2	yes	—
207	C₂²×C₃	#938	1	2	yes	—
208	C₂²×C₃	#942	1	2	yes	—
209	C₂²×C₃	C₂×#216	1	2	yes	—
210	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	4	4	no	—
213	C₃×C₄	#938	1	2	yes	—
214	C₃×C₄	C₂×#216	1	4	yes	—
215	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
216	C₃×C₄	C₂×#237	1	1	yes	characteristic
217	Dic₃	—	6	4	no	—
218	Dic₃	—	12	2	no	—
219	C₉	C₂×#462	1	1	yes	characteristic, complemented, Sylow
220	C₂³	C₂×#611	1	1	yes	characteristic
221	Q₈	—	2	4	no	—
222	Q₈	—	2	4	no	—
223	Q₈	—	4	4	no	—
224	Q₈	—	9	4	no	—
225	Q₈	—	18	4	no	—
226	Q₈	—	18	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	18	4	no	—
231	C₂×C₄	—	18	2	no	—
232	C₂×C₄	—	18	2	no	—
233	C₂×C₄	—	36	2	no	—
234	C₂×C₄	#3674	1	4	yes	complemented
235	C₂×C₄	D₄×D₉	1	2	yes	—
236	C₂×C₄	#3699	1	2	yes	—
237	C₂×C₄	C₂×#611	1	2	yes	—
238	C₂×C₄	C₂×#611	1	1	yes	characteristic
239	C₂×C₄	C₂×#611	1	2	yes	—
240	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
241	C₈	—	18	4	no	—
242	C₂×C₃	#5090	1	4	yes	—
243	C₂×C₃	C₂×#938	1	1	yes	characteristic
244	C₂×C₃	C₂×#942	1	1	yes	characteristic
245	C₂×C₃	C₂²×#216	1	1	yes	characteristic
246	C₂²	#63500	1	2	yes	—
247	C₂²	C₂×#3674	1	2	yes	—
248	C₂²	#63505	1	2	yes	—
249	C₂²	C₂²×#611	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	4	4	no	—
252	C₄	—	18	4	no	—
253	C₄	—	36	2	no	—
254	C₄	#63500	1	2	yes	—
255	C₄	C₂×#3674	1	4	yes	complemented
256	C₄	C₂×D₄×D₉	1	1	yes	characteristic
257	C₄	C₂×#3699	1	1	yes	characteristic
258	C₃	C₂×#5090	1	1	yes	characteristic
259	C₂	#10588763	1	4	yes	direct product factor
260	C₂	C₂×#63500	1	1	yes	characteristic
261	C₂	C₂²×#3674	1	1	yes	characteristic
262	C₂	C₂×#63505	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×C₉×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃	C₂×#67
hypercenter	C₂×C₄⋊Q₈	D₉
hypocenter	C₉	C₂×#462
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
5. #10588763	C₂	direct (N × H)
81. C₂×C₄×C₉	Q₁₆	semidirect
134. C₄×C₉	C₂×Q₁₆	semidirect
178. C₂×C₉	#462	semidirect
219. C₉	C₂×#462	semidirect
234. C₂×C₄	#3674	semidirect
255. C₄	C₂×#3674	semidirect
259. C₂	#10588763	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×C₉×C₄⋊Q₈	2
C₂²×#3674	2
C₂×C₄×Dic₁₈	2
C₂×#63395	2
#10588763	2
C₂×C₄⋊(C₉⋊C₈)	2
C₂×#5090	3

Subgroups — C2×#10588763 #11865620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588763 #11865620