Group #13153334 — subgroups

Subgroups — #10589824⋊C₃ #13153334

Order 1728.

670 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 254 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
2	#63506⋊C₃	C₂	1	4	yes	maximal
3	#63450⋊C₃	C₂	1	4	yes	maximal
4	#63449⋊C₃	C₂	1	4	yes	maximal
5	C₄×3₋¹⁺²⋊Q₈	C₂	1	2	yes	maximal
6	C₃×#5722	—	3	1	no	maximal
7	#10589824	C₃	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×3₋¹⁺²	C₂²	1	2	yes	—
9	C₂×3₋¹⁺²⋊Q₈	C₂²	1	4	yes	—
10	C₂×3₋¹⁺²⋊Q₈	C₂²	1	2	yes	—
11	C₄⋊C₄×3₋¹⁺²	C₂²	1	4	yes	—
12	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
13	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	2	yes	—
14	(3₋¹⁺²⋊C₄)⋊C₄	C₂²	1	8	yes	—
15	(3₋¹⁺²⋊C₄)⋊C₄	C₂²	1	4	yes	—
16	C₄×3₋¹⁺²⋊C₄	C₂²	1	4	yes	—
17	C₄×3₋¹⁺²⋊C₄	C₂²	1	4	yes	—
18	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic
19	C₃×#944	—	3	4	no	—
20	C₃×#888	—	3	4	no	—
21	C₃×#887	—	3	4	no	—
22	C₃×C₄×Dic₆	—	3	2	no	—
23	C₉×C₄⋊Q₈	—	2	1	no	—
24	C₉×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
25	#63506	C₂×C₃	1	4	yes	—
26	#63450	C₂×C₃	1	4	yes	—
27	#63449	C₂×C₃	1	4	yes	—
28	C₄×Dic₁₈	C₂×C₃	1	2	yes	—
29	Q₈×3₋¹⁺²	—	4	2	no	—
30	C₂×C₄×3₋¹⁺²	C₂³	1	4	yes	—
31	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
32	C₂×C₄×3₋¹⁺²	C₂³	1	2	yes	—
33	C₂×3₋¹⁺²⋊C₄	C₂³	1	4	yes	—
34	C₂×3₋¹⁺²⋊C₄	C₂³	1	4	yes	—
35	3₋¹⁺²⋊Q₈	—	4	4	no	—
36	3₋¹⁺²⋊Q₈	Q₈	1	8	yes	complemented
37	C₃×Q₈²	—	9	1	no	—
38	#5722	—	3	1	no	—
39	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
40	C₂×C₃×Dic₆	—	3	2	no	—
41	C₂×C₃×Dic₆	—	3	4	no	—
42	C₃²×C₄⋊C₄	C₂×D₃	1	4	yes	—
43	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
44	C₃×C₄⋊Dic₃	—	3	2	no	—
45	C₃×Dic₃⋊C₄	—	3	8	no	—
46	C₃×Dic₃⋊C₄	—	3	4	no	—
47	C₃×C₄×Dic₃	—	3	4	no	—
48	C₃×C₄×Dic₃	—	3	4	no	—
49	C₂×Q₈×C₉	—	2	2	no	—
50	C₂×Q₈×C₉	C₂²×C₃	1	2	yes	—
51	C₂×Dic₁₈	C₂²×C₃	1	4	yes	—
52	C₂×Dic₁₈	C₂²×C₃	1	2	yes	—
53	C₉×C₄⋊C₄	—	2	4	no	—
54	C₉×C₄⋊C₄	C₂²×C₃	1	4	yes	—
55	C₄²×C₉	—	2	1	no	—
56	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
57	C₄⋊Dic₉	C₂²×C₃	1	2	yes	—
58	Dic₉⋊C₄	C₂²×C₃	1	4	yes	—
59	Dic₉⋊C₄	C₂²×C₃	1	8	yes	—
60	C₄×Dic₉	C₂²×C₃	1	4	yes	—
61	C₄×Dic₉	C₂²×C₃	1	4	yes	—
62	C₂²×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
63	C₄×3₋¹⁺²	—	2	1	no	—
64	C₄×3₋¹⁺²	—	2	4	no	—
65	C₄×3₋¹⁺²	C₂×Q₈	1	4	yes	—
66	3₋¹⁺²⋊C₄	—	2	4	no	—
67	3₋¹⁺²⋊C₄	C₂×Q₈	1	8	yes	—
68	C₃×C₄⋊Q₈	—	3	1	no	—
69	C₃×C₄⋊Q₈	—	9	4	no	—
70	C₃×C₄⋊Q₈	—	9	4	no	—
71	C₃×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
72	C₃×C₄×Q₈	—	9	2	no	—
73	C₃×C₄×Q₈	—	9	4	no	—
74	#944	—	3	4	no	—
75	#888	—	3	4	no	—
76	#887	—	3	4	no	—
77	C₄×Dic₆	—	3	2	no	—
78	C₃²×Q₈	—	4	2	no	—
79	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
80	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
82	C₂×C₃×Dic₃	—	3	4	no	—
83	C₂×C₃×Dic₃	—	3	4	no	—
84	C₃×Dic₆	—	3	8	no	—
85	C₃×Dic₆	—	12	4	no	—
86	Q₈×C₉	—	4	2	no	—
87	Q₈×C₉	—	4	4	no	—
88	C₂×C₄×C₉	—	2	2	no	—
89	C₂×C₄×C₉	—	2	1	no	—
90	C₂×C₄×C₉	—	2	4	no	—
91	C₂×C₄×C₉	C₂³×C₃	1	4	yes	—
92	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
93	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
94	C₂×Dic₉	C₂³×C₃	1	4	yes	—
95	C₂×Dic₉	C₂³×C₃	1	4	yes	—
96	Dic₁₈	—	4	4	no	—
97	Dic₁₈	C₃×Q₈	1	8	yes	complemented
98	Q₈²	—	9	1	no	Sylow
99	C₂×3₋¹⁺²	C₂²×Q₈	1	2	yes	—
100	C₂×3₋¹⁺²	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂×C₃×Q₈	—	3	2	no	—
102	C₂×C₃×Q₈	—	9	2	no	—
103	C₂×C₃×Q₈	—	9	4	no	—
104	C₂×C₃×Q₈	C₂×C₃×D₃	1	2	yes	—
105	C₂×Dic₆	—	3	2	no	—
106	C₂×Dic₆	—	3	4	no	—
107	C₃×C₄⋊C₄	—	3	4	no	—
108	C₃×C₄⋊C₄	—	9	4	no	—
109	C₃×C₄⋊C₄	—	9	8	no	—
110	C₃×C₄⋊C₄	—	9	2	no	—
111	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
112	C₃×C₄²	—	3	1	no	—
113	C₃×C₄²	—	9	4	no	—
114	C₃×C₄²	—	9	4	no	—
115	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
116	C₄⋊Dic₃	—	3	2	no	—
117	Dic₃⋊C₄	—	3	4	no	—
118	Dic₃⋊C₄	—	3	8	no	—
119	C₄×Dic₃	—	3	4	no	—
120	C₄×Dic₃	—	3	4	no	—
121	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
122	C₃²×C₄	—	2	4	no	—
123	C₃²×C₄	—	2	1	no	—
124	C₃²×C₄	D₃×Q₈	1	4	yes	—
125	C₃×Dic₃	—	3	8	no	—
126	C₃×Dic₃	—	6	4	no	—
127	C₂²×C₉	—	2	1	no	—
128	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
129	C₄×C₉	—	2	2	no	—
130	C₄×C₉	—	2	4	no	—
131	C₄×C₉	—	2	1	no	—
132	C₄×C₉	—	2	4	no	—
133	C₄×C₉	—	4	4	no	—
134	C₄×C₉	C₂×C₃×Q₈	1	4	yes	—
135	Dic₉	—	2	4	no	—
136	Dic₉	C₂×C₃×Q₈	1	8	yes	—
137	C₄⋊Q₈	—	9	4	no	—
138	C₄⋊Q₈	—	9	4	no	—
139	C₄⋊Q₈	AGL(1,ℤ/9)	1	1	yes	characteristic
140	C₄×Q₈	—	9	4	no	—
141	C₄×Q₈	—	9	2	no	—
142	3₋¹⁺²	Q₈²	1	1	yes	characteristic, complemented, Sylow
143	C₃×Q₈	—	4	2	no	—
144	C₃×Q₈	—	9	8	no	—
145	C₃×Q₈	—	12	2	no	—
146	C₃×Q₈	—	36	4	no	—
147	C₂×C₃×C₄	—	3	2	no	—
148	C₂×C₃×C₄	—	3	1	no	—
149	C₂×C₃×C₄	—	3	4	no	—
150	C₂×C₃×C₄	—	9	4	no	—
151	C₂×C₃×C₄	—	9	4	no	—
152	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
153	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
155	C₂×Dic₃	—	3	4	no	—
156	C₂×Dic₃	—	3	4	no	—
157	Dic₆	—	3	8	no	—
158	Dic₆	—	12	4	no	—
159	C₂×C₃²	#1004	1	1	yes	characteristic
160	C₂×C₃²	C₂×D₃×Q₈	1	2	yes	—
161	C₂×C₉	—	2	1	no	—
162	C₂×C₉	—	2	2	no	—
163	C₂×C₉	C₂²×C₃×Q₈	1	2	yes	—
164	C₂×C₉	C₃×H(5,𝔽₂)	1	1	yes	characteristic
165	C₂×Q₈	—	9	2	no	—
166	C₂×Q₈	—	9	4	no	—
167	C₂×Q₈	C₂×AGL(1,ℤ/9)	1	2	yes	—
168	C₄⋊C₄	—	9	8	no	—
169	C₄⋊C₄	—	9	4	no	—
170	C₄⋊C₄	—	9	2	no	—
171	C₄⋊C₄	C₂×AGL(1,ℤ/9)	1	4	yes	—
172	C₄²	—	9	4	no	—
173	C₄²	—	9	4	no	—
174	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
175	C₂²×C₃	—	3	1	no	—
176	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
177	C₃×C₄	—	2	4	no	—
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	3	4	no	—
180	C₃×C₄	—	6	1	no	—
181	C₃×C₄	—	6	4	no	—
182	C₃×C₄	—	9	8	no	—
183	C₃×C₄	—	18	4	no	—
184	C₃×C₄	C₃×D₃×Q₈	1	4	yes	—
185	Dic₃	—	3	8	no	—
186	Dic₃	—	6	4	no	—
187	C₃²	#5722	1	1	yes	characteristic
188	C₉	—	2	1	no	—
189	C₉	C₃×Q₈²	1	1	yes	characteristic, complemented
190	Q₈	—	4	2	no	—
191	Q₈	—	9	8	no	—
192	Q₈	—	36	4	no	—
193	C₂×C₄	—	9	4	no	—
194	C₂×C₄	—	9	4	no	—
195	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	4	yes	—
196	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
197	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	2	yes	—
198	C₂×C₃	—	3	1	no	—
199	C₂×C₃	—	3	2	no	—
200	C₂×C₃	C₃×#1004	1	1	yes	characteristic
201	C₂×C₃	C₂×C₃×D₃×Q₈	1	2	yes	—
202	C₂²	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	4	no	—
205	C₄	—	9	8	no	—
206	C₄	—	18	4	no	—
207	C₄	Q₈×AGL(1,ℤ/9)	1	4	yes	—
208	C₃	—	3	1	no	—
209	C₃	C₃×#5722	1	1	yes	characteristic
210	C₂	#63710⋊C₃	1	1	yes	characteristic
211	C₂	C₂×Q₈×AGL(1,ℤ/9)	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×C₄⋊Q₈

C₂

yes

maximal, characteristic

#63506⋊C₃

C₂

yes

maximal

#63450⋊C₃

C₂

yes

maximal

#63449⋊C₃

C₂

yes

maximal

C₄×3₋¹⁺²⋊Q₈

C₂

yes

—

C₃

yes

maximal, characteristic, complemented

C₂×Q₈×3₋¹⁺²

C₂²

yes

—

C₂×3₋¹⁺²⋊Q₈

C₂²

yes

—

C₂×3₋¹⁺²⋊Q₈

C₂²

yes

—

C₄⋊C₄×3₋¹⁺²

C₂²

yes

—

C₄²×3₋¹⁺²

C₂²

yes

characteristic

C₄⋊(3₋¹⁺²⋊C₄)

C₂²

yes

—

(3₋¹⁺²⋊C₄)⋊C₄

C₂²

yes

—

(3₋¹⁺²⋊C₄)⋊C₄

C₂²

yes

—

C₄×3₋¹⁺²⋊C₄

C₂²

yes

—

C₄×3₋¹⁺²⋊C₄

C₂²

yes

—

C₃²×C₄⋊Q₈

D₃

yes

—

—

—

C₃×C₄×Dic₆

—

C₉×C₄⋊Q₈

—

C₉×C₄⋊Q₈

C₂×C₃

yes

characteristic

#63506

C₂×C₃

yes

—

#63450

C₂×C₃

yes

—

#63449

C₂×C₃

yes

—

C₄×Dic₁₈

C₂×C₃

yes

—

Q₈×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₂³

yes

—

C₂×C₄×3₋¹⁺²

C₂³

yes

characteristic

C₂×C₄×3₋¹⁺²

C₂³

yes

—

C₂×3₋¹⁺²⋊C₄

C₂³

yes

—

C₂×3₋¹⁺²⋊C₄

C₂³

yes

—

3₋¹⁺²⋊Q₈

—

3₋¹⁺²⋊Q₈

Q₈

yes

complemented

C₃×Q₈²

—

#5722

—

C₂×C₃²×Q₈

C₂×D₃

yes

—

C₂×C₃×Dic₆

—

C₂×C₃×Dic₆

—

C₃²×C₄⋊C₄

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₃×C₄⋊Dic₃

—

C₃×Dic₃⋊C₄

—

C₃×Dic₃⋊C₄

—

C₃×C₄×Dic₃

—

C₃×C₄×Dic₃

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

C₂²×C₃

yes

—

C₂×Dic₁₈

C₂²×C₃

yes

—

C₂×Dic₁₈

C₂²×C₃

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

C₂²×C₃

yes

—

C₄²×C₉

—

C₄²×C₉

C₂²×C₃

yes

characteristic

C₄⋊Dic₉

C₂²×C₃

yes

—

Dic₉⋊C₄

C₂²×C₃

yes

—

Dic₉⋊C₄

C₂²×C₃

yes

—

C₄×Dic₉

C₂²×C₃

yes

—

C₄×Dic₉

C₂²×C₃

yes

—

C₂²×3₋¹⁺²

C₂⁴

yes

characteristic

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₂×Q₈

yes

—

3₋¹⁺²⋊C₄

—

3₋¹⁺²⋊C₄

C₂×Q₈

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

C₃×D₃

yes

characteristic

C₃×C₄×Q₈

—

C₃×C₄×Q₈

—

#944

—

#888

—

#887

—

C₄×Dic₆

—

C₃²×Q₈

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×Dic₆

—

Q₈×C₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂³×C₃

yes

—

C₂×C₄×C₉

C₂³×C₃

yes

characteristic

C₂×C₄×C₉

C₂³×C₃

yes

—

C₂×Dic₉

C₂³×C₃

yes

—

C₂×Dic₉

C₂³×C₃

yes

—

Dic₁₈

—

Dic₁₈

C₃×Q₈

yes

complemented

Q₈²

—

Sylow

C₂×3₋¹⁺²

C₂²×Q₈

yes

—

100

C₂×3₋¹⁺²

H(5,𝔽₂)

yes

characteristic

101

C₂×C₃×Q₈

—

102

C₂×C₃×Q₈

—

103

C₂×C₃×Q₈

—

104

C₂×C₃×Q₈

C₂×C₃×D₃

yes

—

105

C₂×Dic₆

—

106

C₂×Dic₆

—

107

C₃×C₄⋊C₄

—

108

C₃×C₄⋊C₄

—

109

C₃×C₄⋊C₄

—

110

C₃×C₄⋊C₄

—

111

C₃×C₄⋊C₄

C₂×C₃×D₃

yes

—

112

C₃×C₄²

—

113

C₃×C₄²

—

114

C₃×C₄²

—

115

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

116

C₄⋊Dic₃

—

117

Dic₃⋊C₄

—

118

Dic₃⋊C₄

—

119

C₄×Dic₃

—

120

C₄×Dic₃

—

121

C₂²×C₃²

C₂³×D₃

yes

characteristic

122

C₃²×C₄

—

123

C₃²×C₄

—

124

C₃²×C₄

D₃×Q₈

yes

—

125

C₃×Dic₃

—

126

C₃×Dic₃

—

127

C₂²×C₉

—

128

C₂²×C₉

C₂⁴×C₃

yes

characteristic

129

C₄×C₉

—

130

C₄×C₉

—

131

C₄×C₉

—

132

C₄×C₉

—

133

C₄×C₉

—

134

C₄×C₉

C₂×C₃×Q₈

yes

—

135

Dic₉

—

136

Dic₉

C₂×C₃×Q₈

yes

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

C₄⋊Q₈

AGL(1,ℤ/9)

yes

characteristic

140

C₄×Q₈

—

141

C₄×Q₈

—

142

3₋¹⁺²

Q₈²

yes

characteristic, complemented, Sylow

143

C₃×Q₈

—

144

C₃×Q₈

—

145

C₃×Q₈

—

146

C₃×Q₈

—

147

C₂×C₃×C₄

—

148

C₂×C₃×C₄

—

149

C₂×C₃×C₄

—

150

C₂×C₃×C₄

—

151

C₂×C₃×C₄

—

152

C₂×C₃×C₄

C₂²×C₃×D₃

yes

—

153

C₂×C₃×C₄

C₂²×C₃×D₃

yes

characteristic

154

C₂×C₃×C₄

C₂²×C₃×D₃

yes

—

155

C₂×Dic₃

—

156

C₂×Dic₃

—

157

Dic₆

—

158

Dic₆

—

159

C₂×C₃²

#1004

yes

characteristic

160

C₂×C₃²

C₂×D₃×Q₈

yes

—

161

C₂×C₉

—

162

C₂×C₉

—

163

C₂×C₉

C₂²×C₃×Q₈

yes

—

164

C₂×C₉

C₃×H(5,𝔽₂)

yes

characteristic

165

C₂×Q₈

—

166

C₂×Q₈

—

167

C₂×Q₈

C₂×AGL(1,ℤ/9)

yes

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

C₂×AGL(1,ℤ/9)

yes

—

172

C₄²

—

173

C₄²

—

174

C₄²

C₂×AGL(1,ℤ/9)

yes

characteristic

175

C₂²×C₃

—

176

C₂²×C₃

C₂³×C₃×D₃

yes

characteristic

177

C₃×C₄

—

178

C₃×C₄

—

179

C₃×C₄

—

180

C₃×C₄

—

181

C₃×C₄

—

182

C₃×C₄

—

183

C₃×C₄

—

184

C₃×C₄

C₃×D₃×Q₈

yes

—

185

Dic₃

—

186

Dic₃

—

187

C₃²

#5722

yes

characteristic

188

C₉

—

189

C₉

C₃×Q₈²

yes

characteristic, complemented

190

Q₈

—

191

Q₈

—

192

Q₈

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

C₂²×AGL(1,ℤ/9)

yes

—

196

C₂×C₄

C₂²×AGL(1,ℤ/9)

yes

characteristic

197

C₂×C₄

C₂²×AGL(1,ℤ/9)

yes

—

198

C₂×C₃

—

199

C₂×C₃

—

200

C₂×C₃

C₃×#1004

yes

characteristic

201

C₂×C₃

C₂×C₃×D₃×Q₈

yes

—

202

C₂²

C₂³×AGL(1,ℤ/9)

yes

characteristic

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

Q₈×AGL(1,ℤ/9)

yes

—

208

C₃

—

209

C₃

C₃×#5722

yes

characteristic

210

C₂

#63710⋊C₃

yes

characteristic

211

C₂

C₂×Q₈×AGL(1,ℤ/9)

yes

—

	subgroup	G/H
center	C₂²	C₂³×AGL(1,ℤ/9)
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini, socle	C₂²×C₃	C₂³×C₃×D₃
Fitting	3₋¹⁺²×C₄⋊Q₈	C₂
prime radical	C₂²×C₃²	C₂³×D₃
hypercenter	C₄⋊Q₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×Q₈²
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂³×AGL(1,ℤ/9)

commutator, cosocle

C₂²×C₉

C₂⁴×C₃

Frattini, socle

C₂²×C₃

C₂³×C₃×D₃

Fitting

3₋¹⁺²×C₄⋊Q₈

C₂

prime radical

C₂²×C₃²

C₂³×D₃

hypercenter

C₄⋊Q₈

AGL(1,ℤ/9)

hypocenter

C₉

C₃×Q₈²

LCS

G ▷ C₂²×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
7. #10589824	C₃	semidirect
36. 3₋¹⁺²⋊Q₈	Q₈	semidirect
97. Dic₁₈	C₃×Q₈	semidirect
142. 3₋¹⁺²	Q₈²	semidirect
189. C₉	C₃×Q₈²	semidirect

H ≅ G/N

type

7. #10589824

C₃

semidirect

36. 3₋¹⁺²⋊Q₈

Q₈

semidirect

97. Dic₁₈

C₃×Q₈

semidirect

142. 3₋¹⁺²

Q₈²

semidirect

189. C₉

C₃×Q₈²

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
#63506⋊C₃	2
#63450⋊C₃	2
#63449⋊C₃	2
C₄×3₋¹⁺²⋊Q₈	2
C₃×#5722	3
#10589824	3

subgroup

index

3₋¹⁺²×C₄⋊Q₈

C₄×3₋¹⁺²⋊Q₈

Subgroups — #10589824⋊C₃ #13153334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10589824⋊C3 #13153334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10589824⋊C₃ #13153334