Subgroups — #11796847

Order 1152.

336 conjugacy classes of subgroups.

The subgroup lattice has 255 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#135	C₂	1	1	yes	maximal, characteristic
2	#10588787	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63371	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588243	C₂	1	4	yes	maximal, complemented
5	#72076	—	3	1	no	maximal
6	C₂×C₉×#41	—	2	1	no	—
7	C₉×#135	—	2	1	no	—
8	C₉×#135	—	2	1	no	—
9	C₉×#135	C₄	1	2	yes	complemented
10	C₉×#135	C₂²	1	2	yes	—
11	C₂×C₉×Q₁₆	C₄	1	1	yes	characteristic, complemented
12	C₂×C₉×SD₁₆	—	2	1	no	—
13	C₂×C₉×D₈	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
15	#63488	—	2	1	no	—
16	#63378	—	2	1	no	—
17	#63377	C₂²	1	2	yes	complemented
18	#63371	—	2	1	no	—
19	#63371	C₂²	1	2	yes	complemented
20	C₂×C₉⋊C₁₆	—	2	1	no	—
21	C₂×C₃×#135	D₃	1	1	yes	characteristic
22	#5106	—	3	1	no	—
23	C₂×#812	—	3	1	no	—
24	#4566	—	3	4	no	—
25	C₉×#41	—	2	2	no	—
26	C₉×#41	—	2	2	no	—
27	C₉×#41	—	4	1	no	—
28	C₉×#41	—	4	1	no	—
29	C₂×Q₈×C₉	—	2	1	no	—
30	C₂×D₄×C₉	—	2	1	no	—
31	C₂×D₄×C₉	—	4	1	no	—
32	C₂²×C₄×C₉	—	4	1	no	—
33	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
34	C₉×Q₁₆	—	2	1	no	—
35	C₉×Q₁₆	C₂×C₄	1	2	yes	—
36	C₉×SD₁₆	—	2	2	no	—
37	C₉×SD₁₆	—	2	2	no	—
38	C₉×D₈	—	2	1	no	—
39	C₉×D₈	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×C₉	D₄	1	1	yes	characteristic
43	C₂×C₈×C₉	D₄	1	1	yes	characteristic
44	C₂×C₈×C₉	D₄	1	1	yes	characteristic
45	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
46	#3676	—	4	1	no	—
47	C₄⋊Dic₉	—	2	2	no	—
48	C₄×Dic₉	—	4	1	no	—
49	C₉⋊C₁₆	—	2	2	no	—
50	#2145	—	9	1	no	Sylow
51	C₂×C₃×#41	—	2	1	no	—
52	C₃×#135	—	2	1	no	—
53	C₃×#135	—	2	1	no	—
54	C₃×#135	Dic₃	1	2	yes	—
55	C₃×#135	C₂×D₃	1	2	yes	—
56	C₂×C₃×Q₁₆	Dic₃	1	1	yes	characteristic
57	C₂×C₃×SD₁₆	—	2	1	no	—
58	C₂×C₃×D₈	Dic₃	1	1	yes	characteristic
59	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
60	#926	—	6	1	no	—
61	#818	—	6	1	no	—
62	#817	—	3	2	no	—
63	#812	—	3	2	no	—
64	#812	—	6	1	no	—
65	C₂×C₃⋊C₁₆	—	6	1	no	—
66	C₂³×C₉	—	4	1	no	—
67	C₂³×C₉	D₈	1	1	yes	characteristic
68	Q₈×C₉	—	2	2	no	—
69	Q₈×C₉	—	4	1	no	—
70	D₄×C₉	—	2	2	no	—
71	D₄×C₉	—	4	2	no	—
72	D₄×C₉	—	4	2	no	—
73	D₄×C₉	—	4	1	no	—
74	C₂×C₄×C₉	—	4	1	no	—
75	C₂×C₄×C₉	—	4	2	no	—
76	C₂×C₄×C₉	—	4	2	no	—
77	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₉	D₈	1	1	yes	characteristic
80	C₂×C₄×C₉	SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₉	SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₉	—	4	2	no	—
84	C₈×C₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₈×C₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₈×C₉	C₂²×C₄	1	1	yes	characteristic
87	C₈×C₉	C₂×D₄	1	1	yes	characteristic
88	C₂×#135	D₉	1	1	yes	characteristic
89	C₂×M₃₂	—	9	1	no	—
90	#427	—	9	1	no	—
91	#360	—	9	4	no	—
92	C₃×#41	—	2	2	no	—
93	C₃×#41	—	2	2	no	—
94	C₃×#41	—	4	1	no	—
95	C₃×#41	—	4	1	no	—
96	C₂×C₃×Q₈	—	2	1	no	—
97	C₂×C₃×D₄	—	2	1	no	—
98	C₂×C₃×D₄	—	4	1	no	—
99	C₂²×C₃×C₄	—	4	1	no	—
100	C₂²×C₃×C₄	#67	1	1	yes	characteristic
101	C₃×Q₁₆	—	2	1	no	—
102	C₃×Q₁₆	C₂×Dic₃	1	2	yes	—
103	C₃×SD₁₆	—	2	2	no	—
104	C₃×SD₁₆	—	2	2	no	—
105	C₃×D₈	—	2	1	no	—
106	C₃×D₈	C₂×Dic₃	1	2	yes	—
107	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
109	C₂×C₃×C₈	#67	1	1	yes	characteristic
110	C₂×C₃×C₈	#67	1	1	yes	characteristic
111	C₂×C₃×C₈	#67	1	1	yes	characteristic
112	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
113	#217	—	12	1	no	—
114	C₄⋊Dic₃	—	6	2	no	—
115	C₄×Dic₃	—	12	1	no	—
116	C₃⋊C₁₆	—	6	2	no	—
117	C₂²×C₉	—	4	1	no	—
118	C₂²×C₉	—	4	2	no	—
119	C₂²×C₉	—	8	2	no	—
120	C₂²×C₉	#102	1	1	yes	characteristic
121	C₂²×C₉	#102	1	1	yes	characteristic
122	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
123	C₄×C₉	—	4	2	no	—
124	C₄×C₉	#102	1	1	yes	characteristic
125	C₄×C₉	#102	1	1	yes	characteristic
126	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₄×C₉	C₂×SD₁₆	1	1	yes	characteristic
128	Dic₉	—	8	2	no	—
129	C₂×#41	—	2	1	no	—
130	#135	—	2	1	no	—
131	#135	—	2	1	no	—
132	#135	Dic₉	1	2	yes	complemented
133	#135	C₂×D₉	1	2	yes	—
134	C₂×Q₁₆	Dic₉	1	1	yes	characteristic, complemented
135	C₂×SD₁₆	—	2	1	no	—
136	C₂×D₈	Dic₉	1	1	yes	characteristic, complemented
137	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
138	#117	—	18	1	no	—
139	M₃₂	—	9	2	no	—
140	M₃₂	—	18	1	no	—
141	C₂×C₁₆	—	18	1	no	—
142	#107	—	18	1	no	—
143	#106	—	9	2	no	—
144	C₂³×C₃	—	4	1	no	—
145	C₂³×C₃	#213	1	1	yes	characteristic
146	C₃×Q₈	—	2	2	no	—
147	C₃×Q₈	—	4	1	no	—
148	C₃×D₄	—	2	2	no	—
149	C₃×D₄	—	4	2	no	—
150	C₃×D₄	—	4	2	no	—
151	C₃×D₄	—	4	1	no	—
152	C₂×C₃×C₄	—	4	1	no	—
153	C₂×C₃×C₄	—	4	2	no	—
154	C₂×C₃×C₄	—	4	2	no	—
155	C₂×C₃×C₄	#213	1	1	yes	characteristic
156	C₂×C₃×C₄	#214	1	1	yes	characteristic
157	C₂×C₃×C₄	#214	1	1	yes	characteristic
158	C₂×C₃×C₄	#217	1	1	yes	characteristic
159	C₂×C₃×C₄	#217	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
161	C₂×Dic₃	—	12	2	no	—
162	C₃×C₈	#217	1	1	yes	characteristic
163	C₃×C₈	#217	1	1	yes	characteristic
164	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
165	C₃×C₈	C₂×#67	1	1	yes	characteristic
166	C₂×C₉	—	2	1	no	—
167	C₂×C₉	—	2	1	no	—
168	C₂×C₉	—	2	1	no	—
169	C₂×C₉	—	8	2	no	—
170	C₂×C₉	C₂×#102	1	1	yes	characteristic
171	#41	—	2	2	no	—
172	#41	—	2	2	no	—
173	#41	—	4	1	no	—
174	#41	—	4	1	no	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	—	4	1	no	—
178	C₂²×C₄	—	4	1	no	—
179	C₂²×C₄	#611	1	1	yes	characteristic
180	Q₁₆	—	2	1	no	—
181	Q₁₆	C₂×Dic₉	1	2	yes	—
182	SD₁₆	—	2	2	no	—
183	SD₁₆	—	2	2	no	—
184	D₈	—	2	1	no	—
185	D₈	C₂×Dic₉	1	2	yes	—
186	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
187	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
188	C₂×C₈	#611	1	1	yes	characteristic
189	C₂×C₈	#611	1	1	yes	characteristic
190	C₂×C₈	#611	1	1	yes	characteristic
191	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
192	C₄⋊C₄	—	18	2	no	—
193	C₂²⋊C₄	—	36	1	no	—
194	C₄²	—	36	1	no	—
195	C₁₆	—	18	2	no	—
196	C₂²×C₃	—	4	1	no	—
197	C₂²×C₃	—	4	2	no	—
198	C₂²×C₃	—	8	2	no	—
199	C₂²×C₃	#832	1	1	yes	characteristic
200	C₂²×C₃	#832	1	1	yes	characteristic
201	C₂²×C₃	C₂×#213	1	1	yes	characteristic
202	C₃×C₄	—	4	2	no	—
203	C₃×C₄	#832	1	1	yes	characteristic
204	C₃×C₄	#832	1	1	yes	characteristic
205	C₃×C₄	C₂×#214	1	1	yes	characteristic
206	C₃×C₄	C₂×#217	1	1	yes	characteristic
207	Dic₃	—	24	2	no	—
208	C₉	#2145	1	1	yes	characteristic, complemented, Sylow
209	C₂³	—	4	1	no	—
210	C₂³	#3673	1	1	yes	characteristic
211	Q₈	—	2	2	no	—
212	Q₈	—	4	1	no	—
213	D₄	—	2	2	no	—
214	D₄	—	4	2	no	—
215	D₄	—	4	2	no	—
216	D₄	—	4	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	36	2	no	—
221	C₂×C₄	#3672	1	1	yes	characteristic
222	C₂×C₄	#3672	1	1	yes	characteristic
223	C₂×C₄	#3673	1	1	yes	characteristic
224	C₂×C₄	#3676	1	1	yes	characteristic
225	C₂×C₄	#3676	1	1	yes	characteristic
226	C₂×C₄	C₂×#611	1	1	yes	characteristic
227	C₈	#3676	1	1	yes	characteristic
228	C₈	#3676	1	1	yes	characteristic
229	C₈	C₂²×Dic₉	1	1	yes	characteristic
230	C₈	C₂×#611	1	1	yes	characteristic
231	C₂×C₃	—	2	1	no	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	8	2	no	—
235	C₂×C₃	C₂×#832	1	1	yes	characteristic
236	C₂²	—	4	1	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	8	2	no	—
239	C₂²	#63392	1	1	yes	characteristic
240	C₂²	#63392	1	1	yes	characteristic
241	C₂²	C₂×#3673	1	1	yes	characteristic
242	C₄	—	4	2	no	—
243	C₄	—	72	2	no	—
244	C₄	#63392	1	1	yes	characteristic
245	C₄	#63392	1	1	yes	characteristic
246	C₄	C₂×#3672	1	1	yes	characteristic
247	C₄	C₂×#3676	1	1	yes	characteristic
248	C₃	#72076	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	8	2	no	—
253	C₂	C₂×#63392	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#63392
commutator	C₈×C₉	C₂²×C₄
Frattini	C₂×C₃×C₈	C₂²×D₃
Fitting	C₂×C₉×#135	C₂
socle	C₂×C₃	C₂×#832
prime radical	C₂×C₃×#135	D₃
cosocle	C₂×C₈×C₉	C₂³
hypercenter	C₂×#135	D₉
hypocenter	C₉	#2145
LCS	G ▷ C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
2. #10588787	C₂	semidirect
3. C₂×#63371	C₂	semidirect
4. #10588243	C₂	semidirect
9. C₉×#135	C₄	semidirect
11. C₂×C₉×Q₁₆	C₄	semidirect
13. C₂×C₉×D₈	C₄	semidirect
17. #63377	C₂²	semidirect
19. #63371	C₂²	semidirect
132. #135	Dic₉	semidirect
134. C₂×Q₁₆	Dic₉	semidirect
136. C₂×D₈	Dic₉	semidirect
208. C₉	#2145	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#135	2
#10588787	2
C₂×#63371	2
#10588243	2
#72076	3