Subgroups — #11774331

Order 1152.

372 conjugacy classes of subgroups.

The subgroup lattice has 237 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	#10588605	C₂	1	1	yes	maximal, characteristic
3	#10588605	C₂	1	1	yes	maximal, characteristic
4	#10588212	C₂	1	4	yes	maximal, complemented
5	#71621	—	3	1	no	maximal
6	C₂²×Q₈×C₉	C₄	1	1	yes	characteristic
7	C₄×Q₈×C₉	—	2	1	no	—
8	C₄×Q₈×C₉	—	2	1	no	—
9	C₄×Q₈×C₉	C₄	1	2	yes	—
10	C₄×Q₈×C₉	C₂²	1	2	yes	—
11	C₂×C₉×C₄⋊C₄	—	2	1	no	—
12	C₂×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₂×C₄²×C₉	—	2	1	no	—
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	#63389	—	2	1	no	—
16	#63389	—	2	1	no	—
17	C₄⋊(C₉⋊C₈)	—	2	1	no	—
18	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
19	C₄×C₉⋊C₈	—	2	1	no	—
20	C₄×C₉⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃×C₄×Q₈	D₃	1	1	yes	characteristic
22	#4925	—	3	1	no	—
23	#4925	—	3	1	no	—
24	#4535	—	3	4	no	—
25	C₂×Q₈×C₉	—	2	1	no	—
26	C₂×Q₈×C₉	—	2	2	no	—
27	C₂×Q₈×C₉	C₈	1	4	yes	complemented
28	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₉	—	2	1	no	—
30	C₂²×C₄×C₉	—	2	1	no	—
31	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
34	C₉×C₄⋊C₄	—	2	2	no	—
35	C₉×C₄⋊C₄	—	2	1	no	—
36	C₉×C₄⋊C₄	—	2	2	no	—
37	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₄²×C₉	—	2	2	no	—
39	C₄²×C₉	—	2	2	no	—
40	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₉	D₄	1	1	yes	characteristic
42	C₄²×C₉	D₄	1	1	yes	characteristic
43	C₄²×C₉	C₂³	1	1	yes	characteristic
44	C₂×C₉⋊C₈	—	2	2	no	—
45	C₂×C₉⋊C₈	—	2	2	no	—
46	#1479	—	9	1	no	Sylow
47	C₂²×C₃×Q₈	Dic₃	1	1	yes	characteristic
48	C₃×C₄×Q₈	—	2	1	no	—
49	C₃×C₄×Q₈	—	2	1	no	—
50	C₃×C₄×Q₈	Dic₃	1	2	yes	—
51	C₃×C₄×Q₈	C₂×D₃	1	2	yes	—
52	C₂×C₃×C₄⋊C₄	—	2	1	no	—
53	C₂×C₃×C₄⋊C₄	Dic₃	1	1	yes	characteristic
54	C₂×C₃×C₄²	—	2	1	no	—
55	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
56	#830	—	6	1	no	—
57	#830	—	6	1	no	—
58	C₄⋊(C₃⋊C₈)	—	3	2	no	—
59	C₄⋊(C₃⋊C₈)	—	6	1	no	—
60	C₄×C₃⋊C₈	—	3	2	no	—
61	C₄×C₃⋊C₈	—	6	1	no	—
62	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
63	Q₈×C₉	—	2	4	no	—
64	Q₈×C₉	—	2	2	no	—
65	Q₈×C₉	C₂×C₈	1	4	yes	—
66	C₂×C₄×C₉	—	2	4	no	—
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	2	no	—
70	C₂×C₄×C₉	—	4	1	no	—
71	C₂×C₄×C₉	—	4	2	no	—
72	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
76	C₂×C₄×C₉	M₁₆	1	2	yes	—
77	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
80	C₉⋊C₈	—	4	2	no	—
81	C₉⋊C₈	—	4	2	no	—
82	C₂×C₄×Q₈	D₉	1	1	yes	characteristic
83	#431	—	9	1	no	—
84	#431	—	9	1	no	—
85	Q₈⋊C₈	—	9	4	no	—
86	C₂×C₃×Q₈	—	2	2	no	—
87	C₂×C₃×Q₈	—	2	1	no	—
88	C₂×C₃×Q₈	C₃⋊C₈	1	4	yes	—
89	C₂×C₃×Q₈	C₂×Dic₃	1	2	yes	—
90	C₂²×C₃×C₄	—	2	1	no	—
91	C₂²×C₃×C₄	—	2	1	no	—
92	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
93	C₂²×C₃×C₄	#67	1	1	yes	characteristic
94	C₂²×C₃×C₄	#67	1	1	yes	characteristic
95	C₃×C₄⋊C₄	—	2	2	no	—
96	C₃×C₄⋊C₄	—	2	1	no	—
97	C₃×C₄⋊C₄	—	2	2	no	—
98	C₃×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
99	C₃×C₄²	—	2	2	no	—
100	C₃×C₄²	—	2	2	no	—
101	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
102	C₃×C₄²	#67	1	1	yes	characteristic
103	C₃×C₄²	#67	1	1	yes	characteristic
104	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₃⋊C₈	—	6	2	no	—
106	C₂×C₃⋊C₈	—	6	2	no	—
107	C₂²×C₉	—	2	1	no	—
108	C₂²×C₉	—	2	1	no	—
109	C₂²×C₉	C₂²⋊C₈	1	2	yes	—
110	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₄×C₉	—	2	4	no	—
112	C₄×C₉	—	2	1	no	—
113	C₄×C₉	—	2	1	no	—
114	C₄×C₉	—	2	1	no	—
115	C₄×C₉	—	4	2	no	—
116	C₄×C₉	C₂²⋊C₈	1	2	yes	—
117	C₄×C₉	C₂²×C₈	1	1	yes	characteristic
118	C₄×C₉	C₂×M₁₆	1	1	yes	characteristic
119	C₄×C₉	#137	1	2	yes	—
120	C₂²×Q₈	Dic₉	1	1	yes	characteristic
121	C₄×Q₈	—	2	1	no	—
122	C₄×Q₈	—	2	1	no	—
123	C₄×Q₈	Dic₉	1	2	yes	—
124	C₄×Q₈	C₂×D₉	1	2	yes	—
125	C₂×C₄⋊C₄	—	2	1	no	—
126	C₂×C₄⋊C₄	Dic₉	1	1	yes	characteristic
127	C₂×C₄²	—	2	1	no	—
128	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
129	C₄⋊C₈	—	9	2	no	—
130	C₄⋊C₈	—	18	1	no	—
131	C₂²⋊C₈	—	18	1	no	—
132	C₂²⋊C₈	—	18	1	no	—
133	C₄×C₈	—	9	2	no	—
134	C₄×C₈	—	18	1	no	—
135	C₂³×C₃	#217	1	1	yes	characteristic
136	C₃×Q₈	—	2	4	no	—
137	C₃×Q₈	—	2	2	no	—
138	C₃×Q₈	C₂×C₃⋊C₈	1	4	yes	—
139	C₂×C₃×C₄	—	2	4	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	4	1	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
146	C₂×C₃×C₄	#208	1	2	yes	—
147	C₂×C₃×C₄	#217	1	1	yes	characteristic
148	C₂×C₃×C₄	#217	1	1	yes	characteristic
149	C₂×C₃×C₄	#217	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
153	C₃⋊C₈	—	12	2	no	—
154	C₃⋊C₈	—	12	2	no	—
155	C₂×C₉	—	2	2	no	—
156	C₂×C₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
157	C₂×C₉	#419	1	1	yes	characteristic
158	C₂×C₉	#421	1	1	yes	characteristic
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	1	no	—
161	C₂×Q₈	C₉⋊C₈	1	4	yes	complemented
162	C₂×Q₈	C₂×Dic₉	1	2	yes	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
166	C₂²×C₄	#611	1	1	yes	characteristic
167	C₂²×C₄	#611	1	1	yes	characteristic
168	C₂×C₈	—	18	2	no	—
169	C₂×C₈	—	18	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	C₂×Dic₉	1	2	yes	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	C₂×Dic₉	1	1	yes	characteristic
177	C₄²	#611	1	1	yes	characteristic
178	C₄²	#611	1	1	yes	characteristic
179	C₄²	C₂²×D₉	1	1	yes	characteristic
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	#830	1	2	yes	—
183	C₂²×C₃	C₂×#217	1	1	yes	characteristic
184	C₃×C₄	—	2	4	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	4	2	no	—
189	C₃×C₄	#830	1	2	yes	—
190	C₃×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
191	C₃×C₄	C₂×#208	1	1	yes	characteristic
192	C₃×C₄	#942	1	2	yes	—
193	C₉	#1479	1	1	yes	characteristic, complemented, Sylow
194	C₂³	#3676	1	1	yes	characteristic
195	Q₈	—	2	4	no	—
196	Q₈	—	2	2	no	—
197	Q₈	C₂×C₉⋊C₈	1	4	yes	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	C₂×C₉⋊C₈	1	2	yes	—
205	C₂×C₄	#3667	1	2	yes	—
206	C₂×C₄	#3676	1	1	yes	characteristic
207	C₂×C₄	#3676	1	1	yes	characteristic
208	C₂×C₄	#3676	1	1	yes	characteristic
209	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
210	C₂×C₄	C₂×#611	1	1	yes	characteristic
211	C₂×C₄	C₂×#611	1	1	yes	characteristic
212	C₈	—	36	2	no	—
213	C₈	—	36	2	no	—
214	C₂×C₃	—	2	2	no	—
215	C₂×C₃	C₂×#830	1	1	yes	characteristic
216	C₂×C₃	#5225	1	1	yes	characteristic
217	C₂×C₃	#5236	1	1	yes	characteristic
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	#63389	1	2	yes	—
221	C₂²	C₂×#3676	1	1	yes	characteristic
222	C₄	—	2	4	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	4	2	no	—
227	C₄	#63389	1	2	yes	—
228	C₄	C₂²×C₉⋊C₈	1	1	yes	characteristic
229	C₄	C₂×#3667	1	1	yes	characteristic
230	C₄	#63505	1	2	yes	—
231	C₃	#71621	1	1	yes	characteristic
232	C₂	—	2	2	no	—
233	C₂	C₂×#63389	1	1	yes	characteristic
234	C₂	#10588906	1	1	yes	characteristic
235	C₂	#10588914	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#611
commutator	C₄×C₉	C₂²×C₈
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₂×C₄×Q₈×C₉	C₂
socle	C₂²×C₃	C₂×#217
prime radical	C₂³×C₃	#217
cosocle	C₄²×C₉	C₂³
hypercenter	C₂×C₄×Q₈	D₉
hypocenter	C₉	#1479
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
4. #10588212	C₂	semidirect
27. C₂×Q₈×C₉	C₈	semidirect
161. C₂×Q₈	C₉⋊C₈	semidirect
193. C₉	#1479	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₉	2
#10588605	2
#10588212	2
#71621	3