Group #13143064 — subgroups

Subgroups — #10588773⋊C₃ #13143064

Order 1728.

388 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.

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

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11721394	C₂	1	1	yes	maximal, characteristic
3	#63369⋊C₃	C₂	1	4	yes	maximal, complemented
4	C₄×3₋¹⁺²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#5088	—	3	1	no	maximal
6	#10588773	C₃	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₈×3₋¹⁺²	—	2	1	no	—
8	C₂×C₄⋊AGL(1,ℤ/9)	—	2	1	no	—
9	C₂×C₄⋊AGL(1,ℤ/9)	—	2	1	no	—
10	C₂×C₄⋊AGL(1,ℤ/9)	C₂²	1	2	yes	—
11	C₄⋊C₄×3₋¹⁺²	—	2	1	no	—
12	C₄⋊C₄×3₋¹⁺²	C₂²	1	2	yes	—
13	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
14	C₂×3₋¹⁺²⋊C₈	C₂²	1	2	yes	—
15	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic
16	C₃×#874	—	3	1	no	—
17	C₃×#809	—	3	4	no	—
18	C₃×C₄×C₃⋊C₈	—	3	1	no	—
19	C₉×C₄⋊Q₈	—	2	1	no	—
20	C₉×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
21	#63436	C₂×C₃	1	1	yes	characteristic
22	#63369	C₂×C₃	1	4	yes	complemented
23	C₄×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²×AGL(1,ℤ/9)	—	2	2	no	—
25	Q₈×3₋¹⁺²	—	4	2	no	—
26	C₂×C₄×3₋¹⁺²	—	2	2	no	—
27	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
28	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
29	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
30	C₄⋊AGL(1,ℤ/9)	—	2	4	no	—
31	C₄⋊AGL(1,ℤ/9)	—	4	2	no	—
32	C₄⋊AGL(1,ℤ/9)	—	4	2	no	—
33	3₋¹⁺²⋊C₈	—	2	2	no	—
34	C₃×#486	—	9	1	no	—
35	#5088	—	3	1	no	—
36	C₂×C₃²×Q₈	—	2	1	no	—
37	C₂×C₃×D₁₂	—	3	2	no	—
38	C₂×C₃×D₁₂	—	6	1	no	—
39	C₂×C₃×D₁₂	—	6	1	no	—
40	C₃²×C₄⋊C₄	—	2	1	no	—
41	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
42	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
43	C₂×C₃×C₃⋊C₈	—	3	2	no	—
44	C₂×Q₈×C₉	—	2	2	no	—
45	C₂×Q₈×C₉	—	2	1	no	—
46	C₂×D₃₆	—	2	1	no	—
47	C₂×D₃₆	—	2	1	no	—
48	C₂×D₃₆	C₂²×C₃	1	2	yes	—
49	C₉×C₄⋊C₄	—	2	2	no	—
50	C₉×C₄⋊C₄	—	2	2	no	—
51	C₉×C₄⋊C₄	—	2	1	no	—
52	C₉×C₄⋊C₄	C₂²×C₃	1	2	yes	—
53	C₄²×C₉	—	2	1	no	—
54	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
56	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
57	C₂×AGL(1,ℤ/9)	—	4	2	no	—
58	C₂×AGL(1,ℤ/9)	—	4	2	no	—
59	C₂×AGL(1,ℤ/9)	—	4	2	no	—
60	C₄×3₋¹⁺²	—	4	2	no	—
61	C₄×3₋¹⁺²	D₈	1	2	yes	—
62	C₄×3₋¹⁺²	SD₁₆	1	2	yes	—
63	C₄×3₋¹⁺²	#41	1	2	yes	—
64	C₃×C₄⋊Q₈	—	3	1	no	—
65	C₃×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
66	C₃×#127	—	9	1	no	—
67	#874	—	3	1	no	—
68	C₃×#102	—	9	4	no	—
69	C₃×C₄×C₈	—	9	1	no	—
70	#809	—	3	4	no	—
71	C₄×C₃⋊C₈	—	3	1	no	—
72	C₂²×C₃×D₃	—	6	2	no	—
73	C₃²×Q₈	—	4	2	no	—
74	C₂×C₃²×C₄	—	2	2	no	—
75	C₂×C₃²×C₄	#67	1	1	yes	characteristic
76	C₂×C₃²×C₄	#67	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
78	C₃×D₁₂	—	6	4	no	—
79	C₃×D₁₂	—	12	2	no	—
80	C₃×D₁₂	—	12	2	no	—
81	C₂²×D₉	—	2	2	no	—
82	C₃×C₃⋊C₈	—	6	2	no	—
83	Q₈×C₉	—	4	4	no	—
84	Q₈×C₉	—	4	2	no	—
85	C₂×C₄×C₉	—	2	4	no	—
86	C₂×C₄×C₉	—	2	2	no	—
87	C₂×C₄×C₉	—	2	1	no	—
88	C₂×C₄×C₉	—	2	1	no	—
89	C₂×C₄×C₉	—	2	1	no	—
90	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
91	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
92	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
93	D₃₆	—	2	4	no	—
94	D₃₆	—	4	2	no	—
95	D₃₆	—	4	2	no	—
96	C₉⋊C₈	—	2	2	no	—
97	#486	—	9	1	no	Sylow
98	C₂×3₋¹⁺²	#124	1	1	yes	characteristic
99	C₂×3₋¹⁺²	C₂×D₈	1	1	yes	characteristic
100	C₂×3₋¹⁺²	C₂×SD₁₆	1	1	yes	characteristic
101	AGL(1,ℤ/9)	—	8	2	no	—
102	C₂×C₃×Q₈	—	2	1	no	—
103	C₂×C₃×Q₈	—	6	1	no	—
104	C₂×C₃×D₄	—	9	2	no	—
105	C₂×C₃×D₄	—	18	1	no	—
106	C₂×C₃×D₄	—	18	1	no	—
107	C₂×D₁₂	—	3	2	no	—
108	C₂×D₁₂	—	6	1	no	—
109	C₂×D₁₂	—	6	1	no	—
110	C₂×C₃×C₈	—	9	2	no	—
111	C₃×C₄⋊C₄	—	2	1	no	—
112	C₃×C₄⋊C₄	—	3	2	no	—
113	C₃×C₄⋊C₄	—	6	1	no	—
114	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
115	C₃×C₄²	—	3	1	no	—
116	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
117	C₂×C₃⋊C₈	—	3	2	no	—
118	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
119	C₂×C₃×D₃	—	12	2	no	—
120	C₂×C₃×D₃	—	12	2	no	—
121	C₂×C₃×D₃	—	12	2	no	—
122	C₃²×C₄	—	4	2	no	—
123	C₃²×C₄	#213	1	2	yes	—
124	C₃²×C₄	#215	1	2	yes	—
125	C₃²×C₄	#239	1	2	yes	—
126	C₂²×C₉	—	2	1	no	—
127	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
128	C₂×D₉	—	4	2	no	—
129	C₂×D₉	—	4	2	no	—
130	C₂×D₉	—	4	2	no	—
131	C₄×C₉	—	2	2	no	—
132	C₄×C₉	—	2	2	no	—
133	C₄×C₉	—	2	2	no	—
134	C₄×C₉	—	4	4	no	—
135	C₄×C₉	—	4	2	no	—
136	C₄×C₉	C₃×D₈	1	2	yes	—
137	C₄×C₉	C₃×SD₁₆	1	2	yes	—
138	C₄×C₉	C₃×#41	1	2	yes	—
139	C₄⋊Q₈	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
140	#127	—	9	1	no	—
141	#102	—	9	4	no	—
142	C₄×C₈	—	9	1	no	—
143	3₋¹⁺²	#486	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₃	—	18	2	no	—
145	C₂²×D₃	—	6	2	no	—
146	C₃×Q₈	—	4	2	no	—
147	C₃×Q₈	—	12	2	no	—
148	C₃×D₄	—	18	4	no	—
149	C₃×D₄	—	36	2	no	—
150	C₃×D₄	—	36	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	3	1	no	—
153	C₂×C₃×C₄	—	3	1	no	—
154	C₂×C₃×C₄	—	3	1	no	—
155	C₂×C₃×C₄	—	6	2	no	—
156	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
157	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
159	D₁₂	—	6	4	no	—
160	D₁₂	—	12	2	no	—
161	D₁₂	—	12	2	no	—
162	C₃×C₈	—	18	2	no	—
163	C₃⋊C₈	—	6	2	no	—
164	C₂×C₃²	C₂×#213	1	1	yes	characteristic
165	C₂×C₃²	C₂×#215	1	1	yes	characteristic
166	C₂×C₃²	#947	1	1	yes	characteristic
167	C₃×D₃	—	24	2	no	—
168	C₂×C₉	—	2	1	no	—
169	C₂×C₉	—	2	1	no	—
170	C₂×C₉	—	2	1	no	—
171	C₂×C₉	C₃×#124	1	1	yes	characteristic
172	C₂×C₉	C₂×C₃×D₈	1	1	yes	characteristic
173	C₂×C₉	C₂×C₃×SD₁₆	1	1	yes	characteristic
174	D₉	—	8	2	no	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×D₄	—	9	2	no	—
177	C₂×D₄	—	18	1	no	—
178	C₂×D₄	—	18	1	no	—
179	C₂×C₈	—	9	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	C₂×AGL(1,ℤ/9)	1	2	yes	—
182	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
183	C₂²×C₃	—	3	1	no	—
184	C₂²×C₃	—	36	2	no	—
185	C₂²×C₃	—	36	2	no	—
186	C₂²×C₃	—	36	2	no	—
187	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
188	C₂×D₃	—	12	2	no	—
189	C₂×D₃	—	12	2	no	—
190	C₂×D₃	—	12	2	no	—
191	C₃×C₄	—	3	2	no	—
192	C₃×C₄	—	3	2	no	—
193	C₃×C₄	—	3	2	no	—
194	C₃×C₄	—	4	2	no	—
195	C₃×C₄	—	12	2	no	—
196	C₃×C₄	C₃×#213	1	2	yes	—
197	C₃×C₄	C₃×#215	1	2	yes	—
198	C₃×C₄	C₃×#239	1	2	yes	—
199	C₃²	#5088	1	1	yes	characteristic
200	C₉	—	2	1	no	—
201	C₉	C₃×#486	1	1	yes	characteristic, complemented
202	C₂³	—	18	2	no	—
203	Q₈	—	4	2	no	—
204	D₄	—	18	4	no	—
205	D₄	—	36	2	no	—
206	D₄	—	36	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	#611⋊C₃	1	1	yes	characteristic
209	C₂×C₄	#611⋊C₃	1	1	yes	characteristic
210	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
211	C₈	—	18	2	no	—
212	C₂×C₃	—	3	1	no	—
213	C₂×C₃	—	3	1	no	—
214	C₂×C₃	—	3	1	no	—
215	C₂×C₃	—	72	2	no	—
216	C₂×C₃	C₂×C₃×#213	1	1	yes	characteristic
217	C₂×C₃	C₂×C₃×#215	1	1	yes	characteristic
218	C₂×C₃	C₃×#947	1	1	yes	characteristic
219	D₃	—	24	2	no	—
220	C₂²	—	36	2	no	—
221	C₂²	—	36	2	no	—
222	C₂²	—	36	2	no	—
223	C₂²	C₂×#611⋊C₃	1	1	yes	characteristic
224	C₄	—	4	2	no	—
225	C₄	#3673⋊C₃	1	2	yes	—
226	C₄	#3675⋊C₃	1	2	yes	—
227	C₄	#88724	1	2	yes	—
228	C₃	—	3	1	no	—
229	C₃	C₃×#5088	1	1	yes	characteristic
230	C₂	—	72	2	no	—
231	C₂	C₂×#3673⋊C₃	1	1	yes	characteristic
232	C₂	C₂×#3675⋊C₃	1	1	yes	characteristic
233	C₂	#63509⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

#11721394

C₂

yes

maximal, characteristic

#63369⋊C₃

C₂

yes

maximal, complemented

C₄×3₋¹⁺²⋊C₈

C₂

yes

maximal, characteristic, complemented

C₃×#5088

—

maximal

#10588773

C₃

yes

maximal, characteristic, complemented

C₂×Q₈×3₋¹⁺²

—

C₂×C₄⋊AGL(1,ℤ/9)

—

C₂×C₄⋊AGL(1,ℤ/9)

—

C₂×C₄⋊AGL(1,ℤ/9)

C₂²

yes

—

C₄⋊C₄×3₋¹⁺²

—

C₄⋊C₄×3₋¹⁺²

C₂²

yes

—

C₄²×3₋¹⁺²

C₂²

yes

characteristic

C₂×3₋¹⁺²⋊C₈

C₂²

yes

—

C₃²×C₄⋊Q₈

D₃

yes

characteristic

C₃×#874

—

C₃×#809

—

C₃×C₄×C₃⋊C₈

—

C₉×C₄⋊Q₈

—

C₉×C₄⋊Q₈

C₂×C₃

yes

characteristic, complemented

#63436

C₂×C₃

yes

characteristic

#63369

C₂×C₃

yes

complemented

C₄×C₉⋊C₈

C₂×C₃

yes

characteristic, complemented

C₂²×AGL(1,ℤ/9)

—

Q₈×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

D₄

yes

characteristic

C₂×C₄×3₋¹⁺²

D₄

yes

characteristic

C₂×C₄×3₋¹⁺²

C₂³

yes

—

—

—

3₋¹⁺²⋊C₈

—

C₃×#486

—

#5088

—

C₂×C₃²×Q₈

—

C₂×C₃×D₁₂

—

C₂×C₃×D₁₂

—

C₂×C₃×D₁₂

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₂×C₃×C₃⋊C₈

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂×D₃₆

—

C₂×D₃₆

—

C₂×D₃₆

C₂²×C₃

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

C₂²×C₃

yes

—

C₄²×C₉

—

C₄²×C₉

C₂²×C₃

yes

characteristic

C₂×C₉⋊C₈

C₂²×C₃

yes

—

C₂²×3₋¹⁺²

C₂×D₄

yes

—

—

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

D₈

yes

—

C₄×3₋¹⁺²

SD₁₆

yes

—

C₄×3₋¹⁺²

#41

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

C₃×D₃

yes

—

—

—

C₃×C₄×C₈

—

#809

—

C₄×C₃⋊C₈

—

C₂²×C₃×D₃

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

yes

C₂×C₃²×C₄

yes

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₃×D₁₂

—

C₃×D₁₂

—

C₃×D₁₂

—

C₂²×D₉

—

C₃×C₃⋊C₈

—

Q₈×C₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×D₄

yes

characteristic

C₂×C₄×C₉

C₃×D₄

yes

characteristic

C₂×C₄×C₉

C₂³×C₃

yes

characteristic

D₃₆

—

D₃₆

—

D₃₆

—

C₉⋊C₈

—

—

C₂×3₋¹⁺²

yes

C₂×3₋¹⁺²

C₂×D₈

yes

characteristic

100

C₂×3₋¹⁺²

C₂×SD₁₆

yes

characteristic

101

AGL(1,ℤ/9)

—

102

C₂×C₃×Q₈

—

103

C₂×C₃×Q₈

—

104

C₂×C₃×D₄

—

105

C₂×C₃×D₄

—

106

C₂×C₃×D₄

—

107

C₂×D₁₂

—

108

C₂×D₁₂

—

109

C₂×D₁₂

—

110

C₂×C₃×C₈

—

111

C₃×C₄⋊C₄

—

112

C₃×C₄⋊C₄

—

113

C₃×C₄⋊C₄

—

114

C₃×C₄⋊C₄

C₂×C₃×D₃

yes

—

115

C₃×C₄²

—

116

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

117

C₂×C₃⋊C₈

—

118

C₂²×C₃²

C₂×#67

yes

characteristic

119

C₂×C₃×D₃

—

120

C₂×C₃×D₃

—

121

C₂×C₃×D₃

—

122

C₃²×C₄

—

123

C₃²×C₄

#213

yes

—

124

C₃²×C₄

#215

yes

—

125

C₃²×C₄

#239

yes

—

126

C₂²×C₉

—

127

C₂²×C₉

C₂×C₃×D₄

yes

characteristic

128

C₂×D₉

—

129

C₂×D₉

—

130

C₂×D₉

—

131

C₄×C₉

—

132

C₄×C₉

—

133

C₄×C₉

—

134

C₄×C₉

—

135

C₄×C₉

—

136

C₄×C₉

C₃×D₈

yes

—

137

C₄×C₉

C₃×SD₁₆

yes

—

138

C₄×C₉

C₃×#41

yes

—

139

C₄⋊Q₈

AGL(1,ℤ/9)

yes

characteristic, complemented

140

#127

—

141

#102

—

142

C₄×C₈

—

143

3₋¹⁺²

#486

yes

characteristic, complemented, Sylow

144

C₂³×C₃

—

145

C₂²×D₃

—

146

C₃×Q₈

—

147

C₃×Q₈

—

148

C₃×D₄

—

149

C₃×D₄

—

150

C₃×D₄

—

151

C₂×C₃×C₄

—

152

C₂×C₃×C₄

—

153

C₂×C₃×C₄

—

154

C₂×C₃×C₄

—

155

C₂×C₃×C₄

—

156

C₂×C₃×C₄

C₃×#67

yes

characteristic

157

C₂×C₃×C₄

C₃×#67

yes

characteristic

158

C₂×C₃×C₄

C₂²×C₃×D₃

yes

characteristic

159

D₁₂

—

160

D₁₂

—

161

D₁₂

—

162

C₃×C₈

—

163

C₃⋊C₈

—

164

C₂×C₃²

C₂×#213

yes

characteristic

165

C₂×C₃²

C₂×#215

yes

characteristic

166

C₂×C₃²

#947

yes

characteristic

167

C₃×D₃

—

168

C₂×C₉

—

169

C₂×C₉

—

170

C₂×C₉

—

171

C₂×C₉

C₃×#124

yes

characteristic

172

C₂×C₉

C₂×C₃×D₈

yes

characteristic

173

C₂×C₉

C₂×C₃×SD₁₆

yes

characteristic

174

D₉

—

175

C₂×Q₈

—

176

C₂×D₄

—

177

C₂×D₄

—

178

C₂×D₄

—

179

C₂×C₈

—

180

C₄⋊C₄

—

181

C₄⋊C₄

C₂×AGL(1,ℤ/9)

yes

—

182

C₄²

C₂×AGL(1,ℤ/9)

yes

characteristic

183

C₂²×C₃

—

184

C₂²×C₃

—

185

C₂²×C₃

—

186

C₂²×C₃

—

187

C₂²×C₃

C₂×C₃×#67

yes

characteristic

188

C₂×D₃

—

189

C₂×D₃

—

190

C₂×D₃

—

191

C₃×C₄

—

192

C₃×C₄

—

193

C₃×C₄

—

194

C₃×C₄

—

195

C₃×C₄

—

196

C₃×C₄

C₃×#213

yes

—

197

C₃×C₄

C₃×#215

yes

—

198

C₃×C₄

C₃×#239

yes

—

199

C₃²

#5088

yes

characteristic

200

C₉

—

201

C₉

C₃×#486

yes

characteristic, complemented

202

C₂³

—

203

Q₈

—

204

D₄

—

205

D₄

—

206

D₄

—

207

C₂×C₄

—

208

C₂×C₄

#611⋊C₃

yes

characteristic

209

C₂×C₄

#611⋊C₃

yes

characteristic

210

C₂×C₄

C₂²×AGL(1,ℤ/9)

yes

characteristic

211

C₈

—

212

C₂×C₃

—

213

C₂×C₃

—

214

C₂×C₃

—

215

C₂×C₃

—

216

C₂×C₃

C₂×C₃×#213

yes

characteristic

217

C₂×C₃

C₂×C₃×#215

yes

characteristic

218

C₂×C₃

C₃×#947

yes

characteristic

219

D₃

—

220

C₂²

—

221

C₂²

—

222

C₂²

—

223

C₂²

C₂×#611⋊C₃

yes

characteristic

224

C₄

—

225

C₄

#3673⋊C₃

yes

—

226

C₄

#3675⋊C₃

yes

—

227

C₄

#88724

yes

—

228

C₃

—

229

C₃

C₃×#5088

yes

characteristic

230

C₂

—

231

C₂

C₂×#3673⋊C₃

yes

characteristic

232

C₂

C₂×#3675⋊C₃

yes

characteristic

233

C₂

#63509⋊C₃

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#611⋊C₃
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	3₋¹⁺²×C₄⋊Q₈	C₂
socle	C₂²×C₃	C₂×C₃×#67
prime radical	#11721394	C₂
hypercenter	C₄⋊Q₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#486
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×#611⋊C₃

commutator, cosocle

C₂×C₄×C₉

C₂³×C₃

Frattini

C₂×C₃×C₄

C₂²×C₃×D₃

Fitting

3₋¹⁺²×C₄⋊Q₈

C₂

socle

C₂²×C₃

C₂×C₃×#67

C₂

C₄⋊Q₈

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. 3₋¹⁺²×C₄⋊Q₈	C₂	semidirect
3. #63369⋊C₃	C₂	semidirect
4. C₄×3₋¹⁺²⋊C₈	C₂	semidirect
6. #10588773	C₃	semidirect
20. C₉×C₄⋊Q₈	C₂×C₃	semidirect
22. #63369	C₂×C₃	semidirect
23. C₄×C₉⋊C₈	C₂×C₃	semidirect
139. C₄⋊Q₈	AGL(1,ℤ/9)	semidirect
143. 3₋¹⁺²	#486	semidirect
201. C₉	C₃×#486	semidirect

H ≅ G/N

type

1. 3₋¹⁺²×C₄⋊Q₈

C₂

semidirect

3. #63369⋊C₃

C₂

semidirect

4. C₄×3₋¹⁺²⋊C₈

C₂

semidirect

6. #10588773

C₃

semidirect

20. C₉×C₄⋊Q₈

C₂×C₃

semidirect

22. #63369

C₂×C₃

semidirect

23. C₄×C₉⋊C₈

C₂×C₃

semidirect

139. C₄⋊Q₈

AGL(1,ℤ/9)

semidirect

143. 3₋¹⁺²

#486

semidirect

201. C₉

C₃×#486

semidirect

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

subgroup	index
3₋¹⁺²×C₄⋊Q₈	2
#11721394	2
#63369⋊C₃	2
C₄×3₋¹⁺²⋊C₈	2
C₃×#5088	3
#10588773	3

subgroup

index

3₋¹⁺²×C₄⋊Q₈

C₄×3₋¹⁺²⋊C₈

Subgroups — #10588773⋊C₃ #13143064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588773⋊C3 #13143064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588773⋊C₃ #13143064