Subgroups — #11796938

Order 1152.

330 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	C₂×#63372	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63371	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588242	C₂	1	2	yes	maximal, complemented
5	#10588240	C₂	1	2	yes	maximal, complemented
6	#72081	—	3	1	no	maximal
7	C₂×C₉×#41	—	2	1	no	—
8	C₉×#135	—	2	1	no	—
9	C₉×#135	—	2	1	no	—
10	C₉×#135	C₄	1	2	yes	—
11	C₉×#135	C₄	1	2	yes	—
12	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₉×SD₁₆	—	2	1	no	—
14	C₂×C₉×D₈	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
16	C₂×#3667	—	2	1	no	—
17	#63372	—	2	1	no	—
18	#63372	C₂²	1	2	yes	—
19	#63371	—	2	1	no	—
20	#63371	C₂²	1	2	yes	—
21	C₂×C₉⋊C₁₆	—	2	1	no	—
22	C₂×C₃×#135	D₃	1	1	yes	characteristic
23	C₂×#819	—	3	1	no	—
24	C₂×#812	—	3	1	no	—
25	#4565	—	3	2	no	—
26	#4563	—	3	2	no	—
27	C₉×#41	—	2	2	no	—
28	C₉×#41	—	2	2	no	—
29	C₉×#41	—	4	1	no	—
30	C₉×#41	—	4	1	no	—
31	C₂×Q₈×C₉	—	2	1	no	—
32	C₂×D₄×C₉	—	2	1	no	—
33	C₂×D₄×C₉	—	4	1	no	—
34	C₂²×C₄×C₉	—	4	1	no	—
35	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
36	C₉×Q₁₆	—	2	1	no	—
37	C₉×Q₁₆	C₂×C₄	1	2	yes	—
38	C₉×SD₁₆	—	2	2	no	—
39	C₉×SD₁₆	—	2	2	no	—
40	C₉×D₈	—	2	1	no	—
41	C₉×D₈	C₂×C₄	1	2	yes	—
42	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×C₉	D₄	1	1	yes	characteristic
45	C₂×C₈×C₉	D₄	1	1	yes	characteristic
46	C₂×C₈×C₉	D₄	1	1	yes	characteristic
47	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
48	#3667	—	2	2	no	—
49	#3667	—	4	1	no	—
50	C₂×C₉⋊C₈	—	4	1	no	—
51	C₉⋊C₁₆	—	2	2	no	—
52	#2148	—	9	1	no	Sylow
53	C₂×C₃×#41	—	2	1	no	—
54	C₃×#135	—	2	1	no	—
55	C₃×#135	—	2	1	no	—
56	C₃×#135	Dic₃	1	2	yes	—
57	C₃×#135	Dic₃	1	2	yes	—
58	C₂×C₃×Q₁₆	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃×SD₁₆	—	2	1	no	—
60	C₂×C₃×D₈	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
62	C₂×#208	—	6	1	no	—
63	#819	—	3	2	no	—
64	#819	—	6	1	no	—
65	#812	—	3	2	no	—
66	#812	—	6	1	no	—
67	C₂×C₃⋊C₁₆	—	6	1	no	—
68	C₂³×C₉	—	4	1	no	—
69	C₂³×C₉	SD₁₆	1	1	yes	characteristic
70	Q₈×C₉	—	2	2	no	—
71	Q₈×C₉	—	4	1	no	—
72	D₄×C₉	—	2	2	no	—
73	D₄×C₉	—	4	2	no	—
74	D₄×C₉	—	4	2	no	—
75	D₄×C₉	—	4	1	no	—
76	C₂×C₄×C₉	—	4	1	no	—
77	C₂×C₄×C₉	—	4	2	no	—
78	C₂×C₄×C₉	—	4	2	no	—
79	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₉	D₈	1	1	yes	characteristic
82	C₂×C₄×C₉	D₈	1	1	yes	characteristic
83	C₂×C₄×C₉	SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂×D₄	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	C₂²×C₄	1	1	yes	characteristic
88	C₈×C₉	C₂×D₄	1	1	yes	characteristic
89	C₉⋊C₈	—	4	2	no	—
90	C₂×#135	D₉	1	1	yes	characteristic
91	C₂×M₃₂	—	9	1	no	—
92	C₂×#108	—	9	1	no	—
93	#362	—	9	2	no	—
94	#361	—	9	2	no	—
95	C₃×#41	—	2	2	no	—
96	C₃×#41	—	2	2	no	—
97	C₃×#41	—	4	1	no	—
98	C₃×#41	—	4	1	no	—
99	C₂×C₃×Q₈	—	2	1	no	—
100	C₂×C₃×D₄	—	2	1	no	—
101	C₂×C₃×D₄	—	4	1	no	—
102	C₂²×C₃×C₄	—	4	1	no	—
103	C₂²×C₃×C₄	#67	1	1	yes	characteristic
104	C₃×Q₁₆	—	2	1	no	—
105	C₃×Q₁₆	C₂×Dic₃	1	2	yes	—
106	C₃×SD₁₆	—	2	2	no	—
107	C₃×SD₁₆	—	2	2	no	—
108	C₃×D₈	—	2	1	no	—
109	C₃×D₈	C₂×Dic₃	1	2	yes	—
110	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
111	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
112	C₂×C₃×C₈	#67	1	1	yes	characteristic
113	C₂×C₃×C₈	#67	1	1	yes	characteristic
114	C₂×C₃×C₈	#67	1	1	yes	characteristic
115	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
116	#208	—	6	2	no	—
117	#208	—	12	1	no	—
118	C₂×C₃⋊C₈	—	12	1	no	—
119	C₃⋊C₁₆	—	6	2	no	—
120	C₂²×C₉	—	4	1	no	—
121	C₂²×C₉	—	4	2	no	—
122	C₂²×C₉	—	8	2	no	—
123	C₂²×C₉	#102	1	1	yes	characteristic
124	C₂²×C₉	#102	1	1	yes	characteristic
125	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
126	C₄×C₉	—	4	2	no	—
127	C₄×C₉	#102	1	1	yes	characteristic
128	C₄×C₉	#102	1	1	yes	characteristic
129	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₉	C₂×D₈	1	1	yes	characteristic
131	C₂×#41	—	2	1	no	—
132	#135	—	2	1	no	—
133	#135	—	2	1	no	—
134	#135	Dic₉	1	2	yes	—
135	#135	Dic₉	1	2	yes	—
136	C₂×Q₁₆	C₂×D₉	1	1	yes	characteristic
137	C₂×SD₁₆	—	2	1	no	—
138	C₂×D₈	C₂×D₉	1	1	yes	characteristic
139	C₂×M₁₆	—	18	1	no	—
140	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
141	M₃₂	—	9	2	no	—
142	M₃₂	—	18	1	no	—
143	C₂×C₁₆	—	18	1	no	—
144	#108	—	9	2	no	—
145	#108	—	18	1	no	—
146	C₂³×C₃	—	4	1	no	—
147	C₂³×C₃	#214	1	1	yes	characteristic
148	C₃×Q₈	—	2	2	no	—
149	C₃×Q₈	—	4	1	no	—
150	C₃×D₄	—	2	2	no	—
151	C₃×D₄	—	4	2	no	—
152	C₃×D₄	—	4	2	no	—
153	C₃×D₄	—	4	1	no	—
154	C₂×C₃×C₄	—	4	1	no	—
155	C₂×C₃×C₄	—	4	2	no	—
156	C₂×C₃×C₄	—	4	2	no	—
157	C₂×C₃×C₄	#213	1	1	yes	characteristic
158	C₂×C₃×C₄	#213	1	1	yes	characteristic
159	C₂×C₃×C₄	#214	1	1	yes	characteristic
160	C₂×C₃×C₄	#217	1	1	yes	characteristic
161	C₂×C₃×C₄	#217	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
163	C₃×C₈	#217	1	1	yes	characteristic
164	C₃×C₈	#217	1	1	yes	characteristic
165	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
166	C₃×C₈	C₂×#67	1	1	yes	characteristic
167	C₃⋊C₈	—	12	2	no	—
168	C₂×C₉	—	2	1	no	—
169	C₂×C₉	—	2	1	no	—
170	C₂×C₉	—	2	1	no	—
171	C₂×C₉	—	8	2	no	—
172	C₂×C₉	C₂×#102	1	1	yes	characteristic
173	#41	—	2	2	no	—
174	#41	—	2	2	no	—
175	#41	—	4	1	no	—
176	#41	—	4	1	no	—
177	C₂×Q₈	—	2	1	no	—
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	4	1	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	#611	1	1	yes	characteristic
182	Q₁₆	—	2	1	no	—
183	Q₁₆	C₂×Dic₉	1	2	yes	—
184	SD₁₆	—	2	2	no	—
185	SD₁₆	—	2	2	no	—
186	D₈	—	2	1	no	—
187	D₈	C₂×Dic₉	1	2	yes	—
188	M₁₆	—	18	2	no	—
189	M₁₆	—	36	1	no	—
190	C₂×C₈	—	36	1	no	—
191	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
192	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
193	C₂×C₈	#611	1	1	yes	characteristic
194	C₂×C₈	#611	1	1	yes	characteristic
195	C₂×C₈	#611	1	1	yes	characteristic
196	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
197	C₁₆	—	18	2	no	—
198	C₂²×C₃	—	4	1	no	—
199	C₂²×C₃	—	4	2	no	—
200	C₂²×C₃	—	8	2	no	—
201	C₂²×C₃	#832	1	1	yes	characteristic
202	C₂²×C₃	#832	1	1	yes	characteristic
203	C₂²×C₃	C₂×#214	1	1	yes	characteristic
204	C₃×C₄	—	4	2	no	—
205	C₃×C₄	#832	1	1	yes	characteristic
206	C₃×C₄	#832	1	1	yes	characteristic
207	C₃×C₄	C₂×#213	1	1	yes	characteristic
208	C₃×C₄	C₂×#217	1	1	yes	characteristic
209	C₉	#2148	1	1	yes	characteristic, complemented, Sylow
210	C₂³	—	4	1	no	—
211	C₂³	#3672	1	1	yes	characteristic
212	Q₈	—	2	2	no	—
213	Q₈	—	4	1	no	—
214	D₄	—	2	2	no	—
215	D₄	—	4	2	no	—
216	D₄	—	4	2	no	—
217	D₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	#3672	1	1	yes	characteristic
222	C₂×C₄	#3673	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	—	36	2	no	—
228	C₈	#3676	1	1	yes	characteristic
229	C₈	#3676	1	1	yes	characteristic
230	C₈	C₂²×Dic₉	1	1	yes	characteristic
231	C₈	C₂×#611	1	1	yes	characteristic
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	8	2	no	—
236	C₂×C₃	C₂×#832	1	1	yes	characteristic
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	8	2	no	—
240	C₂²	#63392	1	1	yes	characteristic
241	C₂²	#63392	1	1	yes	characteristic
242	C₂²	C₂×#3672	1	1	yes	characteristic
243	C₄	—	4	2	no	—
244	C₄	#63392	1	1	yes	characteristic
245	C₄	#63392	1	1	yes	characteristic
246	C₄	C₂×#3673	1	1	yes	characteristic
247	C₄	C₂×#3676	1	1	yes	characteristic
248	C₃	#72081	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	#2148
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. C₂×#63372	C₂	semidirect
3. C₂×#63371	C₂	semidirect
4. #10588242	C₂	semidirect
5. #10588240	C₂	semidirect
209. C₉	#2148	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
C₂×#63372	2
C₂×#63371	2
#10588242	2
#10588240	2
#72081	3