Subgroups — #11796939

Order 1152.

660 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×C₃²×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×#63646	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63639	C₂	1	1	yes	maximal, characteristic, complemented
4	#10589431	C₂	1	2	yes	maximal, complemented
5	#10589429	C₂	1	2	yes	maximal, complemented
6	#72081	—	3	4	no	maximal
7	C₂×C₃²×#41	—	2	1	no	—
8	C₃²×#135	—	2	1	no	—
9	C₃²×#135	—	2	1	no	—
10	C₃²×#135	C₄	1	2	yes	—
11	C₃²×#135	C₄	1	2	yes	—
12	C₂×C₃²×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃²×SD₁₆	—	2	1	no	—
14	C₂×C₃²×D₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
16	C₂×#3748	—	2	1	no	—
17	#63646	—	2	1	no	—
18	#63646	C₂²	1	2	yes	—
19	#63639	—	2	1	no	—
20	#63639	C₂²	1	2	yes	—
21	C₂×C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
22	C₂×C₃×#135	D₃	1	4	yes	—
23	C₂×#819	—	3	4	no	—
24	C₂×#812	—	3	4	no	—
25	#4565	—	3	8	no	—
26	#4563	—	3	8	no	—
27	C₃²×#41	—	2	2	no	—
28	C₃²×#41	—	2	2	no	—
29	C₃²×#41	—	4	1	no	—
30	C₃²×#41	—	4	1	no	—
31	C₂×C₃²×Q₈	—	2	1	no	—
32	C₂×C₃²×D₄	—	2	1	no	—
33	C₂×C₃²×D₄	—	4	1	no	—
34	C₂²×C₃²×C₄	—	4	1	no	—
35	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
36	C₃²×Q₁₆	—	2	1	no	—
37	C₃²×Q₁₆	C₂×C₄	1	2	yes	—
38	C₃²×SD₁₆	—	2	2	no	—
39	C₃²×SD₁₆	—	2	2	no	—
40	C₃²×D₈	—	2	1	no	—
41	C₃²×D₈	C₂×C₄	1	2	yes	—
42	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
45	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
46	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
47	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
48	#3748	—	2	2	no	—
49	#3748	—	4	1	no	—
50	C₂×C₃⋊(C₃⋊C₈)	—	4	1	no	—
51	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
52	#2148	—	9	1	no	Sylow
53	C₂×C₃×#41	—	2	4	no	—
54	C₃×#135	—	2	4	no	—
55	C₃×#135	—	2	4	no	—
56	C₃×#135	Dic₃	1	8	yes	—
57	C₃×#135	Dic₃	1	8	yes	—
58	C₂×C₃×Q₁₆	C₂×D₃	1	4	yes	—
59	C₂×C₃×SD₁₆	—	2	4	no	—
60	C₂×C₃×D₈	C₂×D₃	1	4	yes	—
61	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
62	C₂×#208	—	6	4	no	—
63	#819	—	3	8	no	—
64	#819	—	6	4	no	—
65	#812	—	3	8	no	—
66	#812	—	6	4	no	—
67	C₂×C₃⋊C₁₆	—	6	4	no	—
68	C₂³×C₃²	—	4	1	no	—
69	C₂³×C₃²	SD₁₆	1	1	yes	characteristic
70	C₃²×Q₈	—	2	2	no	—
71	C₃²×Q₈	—	4	1	no	—
72	C₃²×D₄	—	2	2	no	—
73	C₃²×D₄	—	4	2	no	—
74	C₃²×D₄	—	4	2	no	—
75	C₃²×D₄	—	4	1	no	—
76	C₂×C₃²×C₄	—	4	1	no	—
77	C₂×C₃²×C₄	—	4	2	no	—
78	C₂×C₃²×C₄	—	4	2	no	—
79	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
82	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
83	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
89	C₃⋊(C₃⋊C₈)	—	4	2	no	—
90	C₂×#135	C₃⋊D₃	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	8	no	—
96	C₃×#41	—	2	8	no	—
97	C₃×#41	—	4	4	no	—
98	C₃×#41	—	4	4	no	—
99	C₂×C₃×Q₈	—	2	4	no	—
100	C₂×C₃×D₄	—	2	4	no	—
101	C₂×C₃×D₄	—	4	4	no	—
102	C₂²×C₃×C₄	—	4	4	no	—
103	C₂²×C₃×C₄	#67	1	4	yes	—
104	C₃×Q₁₆	—	2	4	no	—
105	C₃×Q₁₆	C₂×Dic₃	1	8	yes	—
106	C₃×SD₁₆	—	2	8	no	—
107	C₃×SD₁₆	—	2	8	no	—
108	C₃×D₈	—	2	4	no	—
109	C₃×D₈	C₂×Dic₃	1	8	yes	—
110	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
111	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
112	C₂×C₃×C₈	#67	1	4	yes	—
113	C₂×C₃×C₈	#67	1	4	yes	—
114	C₂×C₃×C₈	#67	1	4	yes	—
115	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
116	#208	—	6	8	no	—
117	#208	—	12	4	no	—
118	C₂×C₃⋊C₈	—	12	4	no	—
119	C₃⋊C₁₆	—	6	8	no	—
120	C₂²×C₃²	—	4	1	no	—
121	C₂²×C₃²	—	4	2	no	—
122	C₂²×C₃²	—	8	2	no	—
123	C₂²×C₃²	#102	1	1	yes	characteristic
124	C₂²×C₃²	#102	1	1	yes	characteristic
125	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
126	C₃²×C₄	—	4	2	no	—
127	C₃²×C₄	#102	1	1	yes	characteristic
128	C₃²×C₄	#102	1	1	yes	characteristic
129	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
131	C₂×#41	—	2	1	no	—
132	#135	—	2	1	no	—
133	#135	—	2	1	no	—
134	#135	C₃⋊Dic₃	1	2	yes	—
135	#135	C₃⋊Dic₃	1	2	yes	—
136	C₂×Q₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
137	C₂×SD₁₆	—	2	1	no	—
138	C₂×D₈	C₂×C₃⋊D₃	1	1	yes	characteristic
139	C₂×M₁₆	—	18	1	no	—
140	C₂²×C₈	C₂×C₃⋊D₃	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	4	no	—
147	C₂³×C₃	#214	1	4	yes	—
148	C₃×Q₈	—	2	8	no	—
149	C₃×Q₈	—	4	4	no	—
150	C₃×D₄	—	2	8	no	—
151	C₃×D₄	—	4	8	no	—
152	C₃×D₄	—	4	8	no	—
153	C₃×D₄	—	4	4	no	—
154	C₂×C₃×C₄	—	4	4	no	—
155	C₂×C₃×C₄	—	4	8	no	—
156	C₂×C₃×C₄	—	4	8	no	—
157	C₂×C₃×C₄	#213	1	4	yes	—
158	C₂×C₃×C₄	#213	1	4	yes	—
159	C₂×C₃×C₄	#214	1	4	yes	—
160	C₂×C₃×C₄	#217	1	4	yes	—
161	C₂×C₃×C₄	#217	1	4	yes	—
162	C₂×C₃×C₄	C₂×#67	1	4	yes	—
163	C₃×C₈	#217	1	4	yes	—
164	C₃×C₈	#217	1	4	yes	—
165	C₃×C₈	C₂²×Dic₃	1	4	yes	—
166	C₃×C₈	C₂×#67	1	4	yes	—
167	C₃⋊C₈	—	12	8	no	—
168	C₂×C₃²	—	2	1	no	—
169	C₂×C₃²	—	2	1	no	—
170	C₂×C₃²	—	2	1	no	—
171	C₂×C₃²	—	8	2	no	—
172	C₂×C₃²	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	#638	1	1	yes	characteristic
182	Q₁₆	—	2	1	no	—
183	Q₁₆	C₂×C₃⋊Dic₃	1	2	yes	—
184	SD₁₆	—	2	2	no	—
185	SD₁₆	—	2	2	no	—
186	D₈	—	2	1	no	—
187	D₈	C₂×C₃⋊Dic₃	1	2	yes	—
188	M₁₆	—	18	2	no	—
189	M₁₆	—	36	1	no	—
190	C₂×C₈	—	36	1	no	—
191	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
192	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
193	C₂×C₈	#638	1	1	yes	characteristic
194	C₂×C₈	#638	1	1	yes	characteristic
195	C₂×C₈	#638	1	1	yes	characteristic
196	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
197	C₁₆	—	18	2	no	—
198	C₂²×C₃	—	4	4	no	—
199	C₂²×C₃	—	4	8	no	—
200	C₂²×C₃	—	8	8	no	—
201	C₂²×C₃	#832	1	4	yes	—
202	C₂²×C₃	#832	1	4	yes	—
203	C₂²×C₃	C₂×#214	1	4	yes	—
204	C₃×C₄	—	4	8	no	—
205	C₃×C₄	#832	1	4	yes	—
206	C₃×C₄	#832	1	4	yes	—
207	C₃×C₄	C₂×#213	1	4	yes	—
208	C₃×C₄	C₂×#217	1	4	yes	—
209	C₃²	#2148	1	1	yes	characteristic, complemented, Sylow
210	C₂³	—	4	1	no	—
211	C₂³	#3754	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	#3753	1	1	yes	characteristic
222	C₂×C₄	#3753	1	1	yes	characteristic
223	C₂×C₄	#3754	1	1	yes	characteristic
224	C₂×C₄	#3757	1	1	yes	characteristic
225	C₂×C₄	#3757	1	1	yes	characteristic
226	C₂×C₄	C₂×#638	1	1	yes	characteristic
227	C₈	—	36	2	no	—
228	C₈	#3757	1	1	yes	characteristic
229	C₈	#3757	1	1	yes	characteristic
230	C₈	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
231	C₈	C₂×#638	1	1	yes	characteristic
232	C₂×C₃	—	2	4	no	—
233	C₂×C₃	—	2	4	no	—
234	C₂×C₃	—	2	4	no	—
235	C₂×C₃	—	8	8	no	—
236	C₂×C₃	C₂×#832	1	4	yes	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	8	2	no	—
240	C₂²	#63659	1	1	yes	characteristic
241	C₂²	#63659	1	1	yes	characteristic
242	C₂²	C₂×#3754	1	1	yes	characteristic
243	C₄	—	4	2	no	—
244	C₄	#63659	1	1	yes	characteristic
245	C₄	#63659	1	1	yes	characteristic
246	C₄	C₂×#3753	1	1	yes	characteristic
247	C₄	C₂×#3757	1	1	yes	characteristic
248	C₃	#72081	1	4	yes	complemented
249	C₂	—	2	1	no	—
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	8	2	no	—
253	C₂	C₂×#63659	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#63659
commutator	C₃²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×C₃⋊D₃
Fitting, prime radical	C₂×C₃²×#135	C₂
socle	C₂×C₃²	C₂×#102
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂×#135	C₃⋊D₃
hypocenter	C₃²	#2148
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
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×#63646	C₂	semidirect
3. C₂×#63639	C₂	semidirect
4. #10589431	C₂	semidirect
5. #10589429	C₂	semidirect
209. C₃²	#2148	semidirect
248. C₃	#72081	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#135	2
C₂×#63646	2
C₂×#63639	2
#10589431	2
#10589429	2
#72081	3