Finite Group Databasebeta

Subgroups — #11721222

Order 864.

200 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×D₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#88635	C₂	1	1	yes	maximal, characteristic
3	#88548	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#216	C₂	1	1	yes	maximal, characteristic, complemented
5	#88427	C₂	1	1	yes	maximal, characteristic, complemented
6	#88416	C₂	1	1	yes	maximal, characteristic, complemented
7	#88413	C₂	1	1	yes	maximal, characteristic, complemented
8	#63948	—	3	1	no	maximal
9	#63480	—	3	1	no	maximal
10	C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₄⋊(C₃⋊D₉)	C₂²	1	1	yes	characteristic
12	C₃×C₄×D₉	—	2	1	no	—
13	C₃×C₄×D₉	C₂²	1	1	yes	characteristic
14	C₉×Dic₆	C₂²	1	1	yes	characteristic
15	C₃×Dic₁₈	—	2	1	no	—
16	C₃×Dic₁₈	C₂²	1	1	yes	characteristic
17	#6192	—	2	1	no	—
18	(C₃⋊D₉)⋊C₄	—	2	1	no	—
19	C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
20	C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×D₃×Q₈	—	3	1	no	—
22	#3799	—	3	1	no	—
23	#3755	—	3	1	no	—
24	C₃×#216	D₃	1	1	yes	characteristic
25	#3718	—	3	1	no	—
26	#3717	—	3	1	no	—
27	#3711	—	3	1	no	—
28	#3701	—	3	1	no	—
29	Q₈×D₉	D₃	1	1	yes	characteristic, complemented
30	C₉×Q₁₆	—	3	1	no	—
31	#3675	—	3	1	no	—
32	#3675	—	6	1	no	—
33	#3674	—	3	1	no	—
34	#3664	—	3	1	no	—
35	#3663	—	3	1	no	—
36	C₂×C₃⋊D₉	—	2	1	no	—
37	C₂×C₃×D₉	D₄	1	1	yes	characteristic
38	C₃×C₄×C₉	—	2	1	no	—
39	C₃×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₉×Dic₃	—	2	1	no	—
41	C₃×Dic₉	—	2	1	no	—
42	C₃×Dic₉	D₄	1	1	yes	characteristic
43	#942	—	9	1	no	—
44	#918	—	9	1	no	—
45	C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
46	C₄⋊(C₃⋊D₃)	—	3	1	no	—
47	C₃×C₄×D₃	—	3	1	no	—
48	C₃×C₄×D₃	—	6	1	no	—
49	C₃×Dic₆	—	3	1	no	—
50	C₃×Dic₆	—	6	1	no	—
51	C₃×Dic₆	C₂×D₃	1	1	yes	characteristic
52	#626	—	6	1	no	—
53	#624	—	6	1	no	—
54	C₃⋊(C₃⋊C₈)	—	3	1	no	—
55	C₃×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
56	Q₈×C₉	—	2	1	no	—
57	Q₈×C₉	—	3	1	no	—
58	Q₈×C₉	C₂×D₃	1	1	yes	characteristic
59	D₃₆	—	3	1	no	—
60	D₃₆	—	6	1	no	—
61	D₃₆	—	6	1	no	—
62	C₄×D₉	—	2	1	no	—
63	C₄×D₉	—	6	1	no	—
64	C₄×D₉	C₂×D₃	1	1	yes	characteristic
65	Dic₁₈	—	2	1	no	—
66	Dic₁₈	C₂×D₃	1	1	yes	characteristic
67	C₈×C₉	—	3	1	no	—
68	C₉⋊C₈	—	3	1	no	—
69	C₉⋊C₈	—	6	1	no	—
70	C₂×C₃×C₉	C₂×D₄	1	1	yes	characteristic
71	C₃⋊D₉	—	4	1	no	—
72	C₃×D₉	—	2	1	no	—
73	C₂×C₃×Q₈	—	9	1	no	—
74	#239	—	9	1	no	—
75	D₃×Q₈	—	3	1	no	—
76	#235	—	9	1	no	—
77	C₃×Q₁₆	—	3	1	no	—
78	#216	—	9	1	no	—
79	#216	—	9	1	no	—
80	#216	D₉	1	1	yes	characteristic, complemented
81	#215	—	9	1	no	—
82	#215	—	9	1	no	—
83	#215	—	9	1	no	—
84	#215	—	18	1	no	—
85	#208	—	9	1	no	—
86	#204	—	9	1	no	—
87	ΓL(1,𝔽₂₅)	—	9	1	no	—
88	C₂×C₃⋊D₃	—	6	1	no	—
89	C₂×C₃×D₃	—	3	1	no	—
90	C₃²×C₄	—	2	1	no	—
91	C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
92	C₃×Dic₃	—	2	1	no	—
93	C₃×Dic₃	—	3	1	no	—
94	C₃×Dic₃	—	6	1	no	—
95	C₂×D₉	—	6	1	no	—
96	C₂×D₉	—	12	1	no	—
97	C₂×D₉	#67	1	1	yes	characteristic
98	C₄×C₉	—	2	1	no	—
99	C₄×C₉	—	2	1	no	—
100	C₄×C₉	—	4	1	no	—
101	C₄×C₉	—	6	1	no	—
102	C₄×C₉	C₂²×D₃	1	1	yes	characteristic
103	Dic₉	—	2	1	no	—
104	Dic₉	#67	1	1	yes	characteristic
105	#137	—	27	1	no	Sylow
106	C₃×C₉	#137	1	1	yes	characteristic, complemented, Sylow
107	C₃×Q₈	—	2	1	no	—
108	C₃×Q₈	—	3	1	no	—
109	C₃×Q₈	—	9	1	no	—
110	C₃×Q₈	—	18	1	no	—
111	C₃×Q₈	C₂×D₉	1	1	yes	characteristic
112	C₃×Q₈	D₃²	1	1	yes	characteristic
113	C₂×C₃×C₄	—	9	1	no	—
114	C₂×C₃×C₄	—	18	1	no	—
115	#67	—	18	1	no	—
116	D₁₂	—	9	1	no	—
117	D₁₂	—	9	1	no	—
118	D₁₂	—	18	1	no	—
119	D₁₂	—	18	1	no	—
120	C₄×D₃	—	3	1	no	—
121	C₄×D₃	—	6	1	no	—
122	C₄×D₃	—	18	1	no	—
123	C₄×D₃	—	18	1	no	—
124	Dic₆	—	3	1	no	—
125	Dic₆	—	6	1	no	—
126	Dic₆	C₂×D₉	1	1	yes	characteristic
127	C₃×C₈	—	3	1	no	—
128	C₃⋊C₈	—	9	1	no	—
129	C₃⋊C₈	—	9	1	no	—
130	C₃⋊C₈	—	18	1	no	—
131	C₃⋊C₈	C₂×D₉	1	1	yes	characteristic
132	C₂×C₃²	D₃×D₄	1	1	yes	characteristic
133	C₃⋊D₃	—	12	1	no	—
134	C₃×D₃	—	6	1	no	—
135	C₂×C₉	—	2	1	no	—
136	C₂×C₉	C₂×#67	1	1	yes	characteristic
137	D₉	—	2	1	no	—
138	D₉	—	12	1	no	—
139	D₉	—	24	1	no	—
140	#41	—	27	1	no	—
141	C₂×Q₈	—	9	1	no	—
142	Q₁₆	—	3	1	no	—
143	Q₁₆	—	27	1	no	—
144	SD₁₆	—	27	1	no	—
145	SD₁₆	—	27	1	no	—
146	M₁₆	—	27	1	no	—
147	C₂²×C₃	—	9	1	no	—
148	C₂×D₃	—	3	1	no	—
149	C₂×D₃	—	18	1	no	—
150	C₂×D₃	—	18	1	no	—
151	C₂×D₃	—	36	1	no	—
152	C₃×C₄	—	2	1	no	—
153	C₃×C₄	—	2	1	no	—
154	C₃×C₄	—	2	1	no	—
155	C₃×C₄	—	4	1	no	—
156	C₃×C₄	—	6	1	no	—
157	C₃×C₄	—	9	1	no	—
158	C₃×C₄	—	18	1	no	—
159	C₃×C₄	C₂²×D₉	1	1	yes	characteristic
160	C₃×C₄	C₂×D₃²	1	1	yes	characteristic
161	Dic₃	—	2	1	no	—
162	Dic₃	—	3	1	no	—
163	Dic₃	—	6	1	no	—
164	C₃²	#918	1	1	yes	characteristic
165	C₉	—	2	1	no	—
166	C₉	#942	1	1	yes	characteristic, complemented
167	Q₈	—	3	1	no	—
168	Q₈	—	9	1	no	—
169	Q₈	—	18	1	no	—
170	Q₈	D₃×D₉	1	1	yes	characteristic
171	D₄	—	27	1	no	—
172	D₄	—	54	1	no	—
173	C₂×C₄	—	9	1	no	—
174	C₂×C₄	—	18	1	no	—
175	C₂×C₄	—	54	1	no	—
176	C₈	—	3	1	no	—
177	C₈	—	27	1	no	—
178	C₂×C₃	—	2	1	no	—
179	C₂×C₃	—	18	1	no	—
180	C₂×C₃	D₄×D₉	1	1	yes	characteristic
181	C₂×C₃	D₃×#67	1	1	yes	characteristic
182	D₃	—	6	1	no	—
183	D₃	—	36	1	no	—
184	D₃	—	36	1	no	—
185	D₃	—	72	1	no	—
186	C₂²	—	9	1	no	—
187	C₂²	—	54	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	6	1	no	—
190	C₄	—	9	1	no	—
191	C₄	—	18	1	no	—
192	C₄	C₂×D₃×D₉	1	1	yes	characteristic
193	C₃	—	2	1	no	—
194	C₃	#63480	1	1	yes	characteristic, complemented
195	C₃	#63948	1	1	yes	characteristic
196	C₂	—	18	1	no	—
197	C₂	—	108	1	no	—
198	C₂	D₉×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₉×#67
commutator, cosocle	C₃×C₄×C₉	C₂³
Frattini	C₃×C₄	C₂×D₃²
Fitting	C₃×Q₈×C₉	C₂²
socle	C₂×C₃²	D₃×D₄
prime radical	#88635	C₂
hypercenter	Q₈	D₃×D₉
hypocenter	C₃×C₉	#137
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×Q₈×D₉	C₂	semidirect
3. #88548	C₂	semidirect
4. C₉×#216	C₂	semidirect
5. #88427	C₂	semidirect
6. #88416	C₂	semidirect
7. #88413	C₂	semidirect
29. Q₈×D₉	D₃	semidirect
80. #216	D₉	semidirect
106. C₃×C₉	#137	semidirect
166. C₉	#942	semidirect
194. C₃	#63480	semidirect

Maximal subgroups

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

subgroup	index
C₃×Q₈×D₉	2
#88635	2
#88548	2
C₉×#216	2
#88427	2
#88416	2
#88413	2
#63948	3
#63480	3