Group #13142928 — subgroups

Subgroups — #10588737⋊C₃ #13142928

Order 1728.

410 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 100 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×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×#3674⋊C₃	C₂	1	2	yes	maximal, complemented
3	C₂×#3672⋊C₃	C₂	1	2	yes	maximal
4	#63431⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#63362⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5060	—	3	1	no	maximal
7	#10588737	C₃	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₂×D₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	C₂×3₋¹⁺²⋊Q₈	C₂²	1	2	yes	—
11	C₂²⋊C₄×3₋¹⁺²	—	2	2	no	—
12	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
13	#3674⋊C₃	—	2	4	no	—
14	#3672⋊C₃	—	2	4	no	—
15	C₄⋊(3₋¹⁺²⋊C₄)	—	2	2	no	—
16	C₂×3₋¹⁺²⋊C₈	C₂²	1	2	yes	—
17	C₃²×#124	D₃	1	1	yes	characteristic
18	C₂×C₃×#216	—	3	2	no	—
19	C₂×C₃×#214	—	3	2	no	—
20	C₃×#869	—	3	1	no	—
21	C₃×#803	—	3	1	no	—
22	C₉×#124	—	2	1	no	—
23	C₉×#124	C₂×C₃	1	1	yes	characteristic
24	C₂×#3674	C₂×C₃	1	2	yes	complemented
25	C₂×#3672	C₂×C₃	1	2	yes	—
26	#63431	C₂×C₃	1	1	yes	characteristic, complemented
27	#63362	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂³×3₋¹⁺²	—	2	1	no	—
29	Q₈×3₋¹⁺²	—	2	2	no	—
30	D₄×3₋¹⁺²	—	2	2	no	—
31	C₂×C₄×3₋¹⁺²	—	2	1	no	—
32	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
33	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
34	C₂×3₋¹⁺²⋊C₄	—	2	2	no	—
35	3₋¹⁺²⋊Q₈	—	2	2	no	—
36	3₋¹⁺²⋊Q₈	—	2	2	no	—
37	3₋¹⁺²⋊C₈	D₄	1	4	yes	—
38	C₃×#497	—	9	1	no	—
39	#5060	—	3	1	no	—
40	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
41	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃×Dic₆	—	3	2	no	—
43	C₃²×C₂²⋊C₄	—	2	2	no	—
44	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
45	C₃×#216	—	6	4	no	—
46	C₃×#214	—	6	4	no	—
47	C₃×C₄⋊Dic₃	—	6	2	no	—
48	C₂×C₃×C₃⋊C₈	—	3	2	no	—
49	C₂×Q₈×C₉	—	2	1	no	—
50	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
51	C₂×D₄×C₉	—	2	1	no	—
52	C₂×D₄×C₉	C₂²×C₃	1	1	yes	characteristic
53	C₂×Dic₁₈	C₂²×C₃	1	2	yes	—
54	C₉×C₂²⋊C₄	—	2	4	no	—
55	C₉×C₂²⋊C₄	—	2	2	no	—
56	C₄²×C₉	—	2	1	no	—
57	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
58	#3674	—	2	4	no	—
59	#3672	—	2	4	no	—
60	C₄⋊Dic₉	—	2	2	no	—
61	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
62	C₂²×3₋¹⁺²	—	4	1	no	—
63	C₂²×3₋¹⁺²	—	4	2	no	—
64	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
65	C₄×3₋¹⁺²	—	2	2	no	—
66	C₄×3₋¹⁺²	—	4	1	no	—
67	C₄×3₋¹⁺²	C₂×D₄	1	2	yes	—
68	3₋¹⁺²⋊C₄	—	4	2	no	—
69	C₂×C₃×Q₁₆	—	9	2	no	—
70	C₂×C₃×SD₁₆	—	9	2	no	—
71	C₃×C₄⋊Q₈	—	9	1	no	—
72	C₃×#124	—	3	1	no	—
73	C₃×#124	C₃×D₃	1	1	yes	characteristic
74	C₂×#216	—	3	2	no	—
75	C₂×#214	—	3	2	no	—
76	#869	—	3	1	no	—
77	C₃×#97	—	9	1	no	—
78	#803	—	3	1	no	—
79	C₂³×C₃²	—	2	1	no	—
80	C₃²×Q₈	—	2	2	no	—
81	C₃²×D₄	—	2	2	no	—
82	C₂×C₃²×C₄	—	2	1	no	—
83	C₂×C₃²×C₄	#67	1	2	yes	—
84	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂×C₃×Dic₃	—	6	2	no	—
86	C₃×Dic₆	—	6	2	no	—
87	C₃×Dic₆	—	6	2	no	—
88	C₂³×C₉	—	2	2	no	—
89	C₂³×C₉	—	2	1	no	—
90	C₃×C₃⋊C₈	—	3	4	no	—
91	Q₈×C₉	—	2	2	no	—
92	Q₈×C₉	—	4	2	no	—
93	D₄×C₉	—	2	2	no	—
94	D₄×C₉	—	4	2	no	—
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	2	1	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
100	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
101	C₂×Dic₉	—	2	2	no	—
102	Dic₁₈	—	2	2	no	—
103	Dic₁₈	—	2	2	no	—
104	C₉⋊C₈	C₃×D₄	1	4	yes	—
105	#497	—	9	1	no	Sylow
106	C₂×3₋¹⁺²	—	8	1	no	—
107	C₂×3₋¹⁺²	#127	1	1	yes	characteristic
108	C₂×3₋¹⁺²	#137	1	2	yes	—
109	C₂×C₃×Q₈	—	3	1	no	—
110	C₂×C₃×Q₈	—	9	2	no	—
111	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
112	C₂×C₃×D₄	—	3	1	no	—
113	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
114	C₂×Dic₆	—	3	2	no	—
115	C₃×Q₁₆	—	18	4	no	—
116	C₃×SD₁₆	—	18	4	no	—
117	C₂×C₃×C₈	—	9	2	no	—
118	C₃×C₄⋊C₄	—	18	2	no	—
119	C₃×C₂²⋊C₄	—	2	2	no	—
120	C₃×C₂²⋊C₄	—	6	2	no	—
121	C₃×C₄²	—	3	1	no	—
122	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
123	#216	—	6	4	no	—
124	#214	—	6	4	no	—
125	C₄⋊Dic₃	—	6	2	no	—
126	C₂×C₃⋊C₈	—	3	2	no	—
127	C₂²×C₃²	—	4	1	no	—
128	C₂²×C₃²	—	4	2	no	—
129	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
130	C₃²×C₄	—	2	2	no	—
131	C₃²×C₄	—	4	1	no	—
132	C₃²×C₄	D₃×D₄	1	2	yes	—
133	C₃×Dic₃	—	12	2	no	—
134	C₂²×C₉	—	2	1	no	—
135	C₂²×C₉	—	4	2	no	—
136	C₂²×C₉	—	4	4	no	—
137	C₂²×C₉	—	4	1	no	—
138	C₂²×C₉	—	4	2	no	—
139	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
140	C₄×C₉	—	2	2	no	—
141	C₄×C₉	—	2	2	no	—
142	C₄×C₉	—	4	2	no	—
143	C₄×C₉	—	4	2	no	—
144	C₄×C₉	—	4	1	no	—
145	C₄×C₉	C₂×C₃×D₄	1	2	yes	—
146	Dic₉	—	4	2	no	—
147	C₂×Q₁₆	—	9	2	no	—
148	C₂×SD₁₆	—	9	2	no	—
149	C₄⋊Q₈	—	9	1	no	—
150	#124	AGL(1,ℤ/9)	1	1	yes	characteristic
151	#97	—	9	1	no	—
152	3₋¹⁺²	#497	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₃	—	2	1	no	—
154	C₂³×C₃	—	6	1	no	—
155	C₃×Q₈	—	2	2	no	—
156	C₃×Q₈	—	6	2	no	—
157	C₃×Q₈	—	18	2	no	—
158	C₃×Q₈	—	18	2	no	—
159	C₃×D₄	—	2	2	no	—
160	C₃×D₄	—	6	2	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	3	2	no	—
163	C₂×C₃×C₄	—	3	1	no	—
164	C₂×C₃×C₄	—	6	1	no	—
165	C₂×C₃×C₄	—	18	2	no	—
166	C₂×C₃×C₄	C₃×#67	1	2	yes	—
167	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
168	C₂×Dic₃	—	6	2	no	—
169	Dic₆	—	6	2	no	—
170	Dic₆	—	6	2	no	—
171	C₃×C₈	—	9	4	no	—
172	C₃⋊C₈	—	3	4	no	—
173	C₂×C₃²	—	8	1	no	—
174	C₂×C₃²	#940	1	1	yes	characteristic
175	C₂×C₃²	#951	1	2	yes	—
176	C₂×C₉	—	2	1	no	—
177	C₂×C₉	—	2	2	no	—
178	C₂×C₉	—	8	2	no	—
179	C₂×C₉	—	8	1	no	—
180	C₂×C₉	C₃×#127	1	1	yes	characteristic
181	C₂×C₉	C₃×#137	1	2	yes	—
182	C₂×Q₈	—	9	2	no	—
183	C₂×Q₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
184	C₂×D₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
185	Q₁₆	—	18	4	no	—
186	SD₁₆	—	18	4	no	—
187	C₂×C₈	—	9	2	no	—
188	C₄⋊C₄	—	18	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
191	C₂²×C₃	—	3	1	no	—
192	C₂²×C₃	—	4	1	no	—
193	C₂²×C₃	—	4	2	no	—
194	C₂²×C₃	—	12	1	no	—
195	C₂²×C₃	—	12	2	no	—
196	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	3	2	no	—
199	C₃×C₄	—	4	1	no	—
200	C₃×C₄	—	6	2	no	—
201	C₃×C₄	—	12	1	no	—
202	C₃×C₄	—	36	2	no	—
203	C₃×C₄	C₃×D₃×D₄	1	2	yes	—
204	Dic₃	—	12	2	no	—
205	C₃²	#5060	1	1	yes	characteristic
206	C₉	—	2	1	no	—
207	C₉	C₃×#497	1	1	yes	characteristic, complemented
208	C₂³	—	2	1	no	—
209	Q₈	—	2	2	no	—
210	Q₈	—	18	2	no	—
211	Q₈	—	18	2	no	—
212	D₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	18	2	no	—
215	C₂×C₄	#611⋊C₃	1	2	yes	—
216	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
217	C₈	—	9	4	no	—
218	C₂×C₃	—	3	1	no	—
219	C₂×C₃	—	3	2	no	—
220	C₂×C₃	—	8	1	no	—
221	C₂×C₃	—	24	1	no	—
222	C₂×C₃	C₃×#940	1	1	yes	characteristic
223	C₂×C₃	C₃×#951	1	2	yes	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₂×#611⋊C₃	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	4	1	no	—
229	C₄	—	36	2	no	—
230	C₄	D₄×AGL(1,ℤ/9)	1	2	yes	—
231	C₃	—	3	1	no	—
232	C₃	C₃×#5060	1	1	yes	characteristic
233	C₂	—	8	1	no	—
234	C₂	#63502⋊C₃	1	1	yes	characteristic
235	C₂	#63511⋊C₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×#124

C₂

yes

maximal, characteristic

C₂×#3674⋊C₃

C₂

yes

maximal, complemented

C₂×#3672⋊C₃

C₂

yes

maximal

#63431⋊C₃

C₂

yes

maximal, characteristic, complemented

#63362⋊C₃

C₂

yes

maximal, characteristic, complemented

C₃×#5060

—

maximal

#10588737

C₃

yes

maximal, characteristic, complemented

C₂×Q₈×3₋¹⁺²

C₂²

yes

characteristic

C₂×D₄×3₋¹⁺²

C₂²

yes

characteristic

C₂×3₋¹⁺²⋊Q₈

C₂²

yes

—

C₂²⋊C₄×3₋¹⁺²

—

C₄²×3₋¹⁺²

C₂²

yes

characteristic

#3674⋊C₃

—

#3672⋊C₃

—

C₄⋊(3₋¹⁺²⋊C₄)

—

C₂×3₋¹⁺²⋊C₈

C₂²

yes

—

C₃²×#124

D₃

yes

characteristic

C₂×C₃×#216

—

C₂×C₃×#214

—

—

—

—

C₂×C₃

yes

characteristic

C₂×#3674

C₂×C₃

yes

complemented

C₂×#3672

C₂×C₃

yes

—

#63431

C₂×C₃

yes

characteristic, complemented

#63362

C₂×C₃

yes

characteristic, complemented

C₂³×3₋¹⁺²

—

Q₈×3₋¹⁺²

—

D₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

D₄

yes

—

C₂×C₄×3₋¹⁺²

C₂³

yes

characteristic

C₂×3₋¹⁺²⋊C₄

—

3₋¹⁺²⋊Q₈

—

3₋¹⁺²⋊Q₈

—

3₋¹⁺²⋊C₈

D₄

yes

—

C₃×#497

—

#5060

—

C₂×C₃²×Q₈

C₂×D₃

yes

characteristic

C₂×C₃²×D₄

C₂×D₃

yes

characteristic

C₂×C₃×Dic₆

—

C₃²×C₂²⋊C₄

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₃×#216

—

C₃×#214

—

C₃×C₄⋊Dic₃

—

C₂×C₃×C₃⋊C₈

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

C₂²×C₃

yes

characteristic

C₂×D₄×C₉

—

C₂×D₄×C₉

C₂²×C₃

yes

characteristic

C₂×Dic₁₈

C₂²×C₃

yes

—

C₉×C₂²⋊C₄

—

C₉×C₂²⋊C₄

—

C₄²×C₉

—

C₄²×C₉

C₂²×C₃

yes

characteristic

#3674

—

#3672

—

C₄⋊Dic₉

—

C₂×C₉⋊C₈

C₂²×C₃

yes

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

C₂×D₄

yes

characteristic

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₂×D₄

yes

—

3₋¹⁺²⋊C₄

—

C₂×C₃×Q₁₆

—

C₂×C₃×SD₁₆

—

C₃×C₄⋊Q₈

—

C₃×#124

—

C₃×#124

C₃×D₃

yes

—

—

—

C₃×#97

—

#803

—

C₂³×C₃²

—

C₃²×Q₈

—

C₃²×D₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×Dic₆

—

C₂³×C₉

—

C₂³×C₉

—

C₃×C₃⋊C₈

—

Q₈×C₉

—

Q₈×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×D₄

yes

—

100

C₂×C₄×C₉

C₂³×C₃

yes

characteristic

101

C₂×Dic₉

—

102

Dic₁₈

—

103

Dic₁₈

—

104

C₉⋊C₈

C₃×D₄

yes

—

105

#497

—

Sylow

106

C₂×3₋¹⁺²

—

107

C₂×3₋¹⁺²

#127

yes

characteristic

108

C₂×3₋¹⁺²

#137

yes

—

109

C₂×C₃×Q₈

—

110

C₂×C₃×Q₈

—

111

C₂×C₃×Q₈

C₂×C₃×D₃

yes

characteristic

112

C₂×C₃×D₄

—

113

C₂×C₃×D₄

C₂×C₃×D₃

yes

characteristic

114

C₂×Dic₆

—

115

C₃×Q₁₆

—

116

C₃×SD₁₆

—

117

C₂×C₃×C₈

—

118

C₃×C₄⋊C₄

—

119

C₃×C₂²⋊C₄

—

120

C₃×C₂²⋊C₄

—

121

C₃×C₄²

—

122

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

123

#216

—

124

#214

—

125

C₄⋊Dic₃

—

126

C₂×C₃⋊C₈

—

127

C₂²×C₃²

—

128

C₂²×C₃²

—

129

C₂²×C₃²

C₂×#67

yes

characteristic

130

C₃²×C₄

—

131

C₃²×C₄

—

132

C₃²×C₄

D₃×D₄

yes

—

133

C₃×Dic₃

—

134

C₂²×C₉

—

135

C₂²×C₉

—

136

C₂²×C₉

—

137

C₂²×C₉

—

138

C₂²×C₉

—

139

C₂²×C₉

C₂×C₃×D₄

yes

characteristic

140

C₄×C₉

—

141

C₄×C₉

—

142

C₄×C₉

—

143

C₄×C₉

—

144

C₄×C₉

—

145

C₄×C₉

C₂×C₃×D₄

yes

—

146

Dic₉

—

147

C₂×Q₁₆

—

148

C₂×SD₁₆

—

149

C₄⋊Q₈

—

150

yes

151

—

152

3₋¹⁺²

#497

yes

characteristic, complemented, Sylow

153

C₂³×C₃

—

154

C₂³×C₃

—

155

C₃×Q₈

—

156

C₃×Q₈

—

157

C₃×Q₈

—

158

C₃×Q₈

—

159

C₃×D₄

—

160

C₃×D₄

—

161

C₂×C₃×C₄

—

162

C₂×C₃×C₄

—

163

C₂×C₃×C₄

—

164

C₂×C₃×C₄

—

165

C₂×C₃×C₄

—

166

C₂×C₃×C₄

C₃×#67

yes

—

167

C₂×C₃×C₄

C₂²×C₃×D₃

yes

characteristic

168

C₂×Dic₃

—

169

Dic₆

—

170

Dic₆

—

171

C₃×C₈

—

172

C₃⋊C₈

—

173

C₂×C₃²

—

174

C₂×C₃²

#940

yes

characteristic

175

C₂×C₃²

#951

yes

—

176

C₂×C₉

—

177

C₂×C₉

—

178

C₂×C₉

—

179

C₂×C₉

—

180

C₂×C₉

C₃×#127

yes

characteristic

181

C₂×C₉

C₃×#137

yes

—

182

C₂×Q₈

—

183

C₂×Q₈

C₂×AGL(1,ℤ/9)

yes

characteristic

184

C₂×D₄

C₂×AGL(1,ℤ/9)

yes

characteristic

185

Q₁₆

—

186

SD₁₆

—

187

C₂×C₈

—

188

C₄⋊C₄

—

189

C₂²⋊C₄

—

190

C₄²

C₂×AGL(1,ℤ/9)

yes

characteristic

191

C₂²×C₃

—

192

C₂²×C₃

—

193

C₂²×C₃

—

194

C₂²×C₃

—

195

C₂²×C₃

—

196

C₂²×C₃

C₂×C₃×#67

yes

characteristic

197

C₃×C₄

—

198

C₃×C₄

—

199

C₃×C₄

—

200

C₃×C₄

—

201

C₃×C₄

—

202

C₃×C₄

—

203

C₃×C₄

C₃×D₃×D₄

yes

—

204

Dic₃

—

205

C₃²

#5060

yes

characteristic

206

C₉

—

207

C₉

C₃×#497

yes

characteristic, complemented

208

C₂³

—

209

Q₈

—

210

Q₈

—

211

Q₈

—

212

D₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

#611⋊C₃

yes

—

216

C₂×C₄

C₂²×AGL(1,ℤ/9)

yes

characteristic

217

C₈

—

218

C₂×C₃

—

219

C₂×C₃

—

220

C₂×C₃

—

221

C₂×C₃

—

222

C₂×C₃

C₃×#940

yes

characteristic

223

C₂×C₃

C₃×#951

yes

—

224

C₂²

—

225

C₂²

—

226

C₂²

C₂×#611⋊C₃

yes

characteristic

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

D₄×AGL(1,ℤ/9)

yes

—

231

C₃

—

232

C₃

C₃×#5060

yes

characteristic

233

C₂

—

234

C₂

#63502⋊C₃

yes

characteristic

235

C₂

#63511⋊C₃

yes

—

	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×#124	C₂
socle	C₂²×C₃	C₂×C₃×#67
prime radical	C₂×C₃²×D₄	C₂×D₃
hypercenter	#124	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#497
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

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×#124

C₂

socle

C₂²×C₃

C₂×C₃×#67

prime radical

C₂×C₃²×D₄

C₂×D₃

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #124

N	H ≅ G/N	type
2. C₂×#3674⋊C₃	C₂	semidirect
4. #63431⋊C₃	C₂	semidirect
5. #63362⋊C₃	C₂	semidirect
7. #10588737	C₃	semidirect
24. C₂×#3674	C₂×C₃	semidirect
26. #63431	C₂×C₃	semidirect
27. #63362	C₂×C₃	semidirect
152. 3₋¹⁺²	#497	semidirect
207. C₉	C₃×#497	semidirect

H ≅ G/N

type

2. C₂×#3674⋊C₃

C₂

semidirect

4. #63431⋊C₃

C₂

semidirect

5. #63362⋊C₃

C₂

semidirect

7. #10588737

C₃

semidirect

24. C₂×#3674

C₂×C₃

semidirect

26. #63431

C₂×C₃

semidirect

27. #63362

C₂×C₃

semidirect

152. 3₋¹⁺²

#497

semidirect

207. C₉

C₃×#497

semidirect

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

subgroup	index
3₋¹⁺²×#124	2
C₂×#3674⋊C₃	2
C₂×#3672⋊C₃	2
#63431⋊C₃	2
#63362⋊C₃	2
C₃×#5060	3
#10588737	3

subgroup

index

3₋¹⁺²×#124

C₂×#3674⋊C₃

C₂×#3672⋊C₃

Subgroups — #10588737⋊C₃ #13142928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588737⋊C3 #13142928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588737⋊C₃ #13142928