Subgroups — #11791869

Order 1152.

390 conjugacy classes of subgroups.

The subgroup lattice has 219 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#10588604	C₂	1	1	yes	maximal, characteristic
3	C₄×#3667	C₂	1	1	yes	maximal, characteristic
4	#10588207	C₂	1	4	yes	maximal
5	#71790	—	3	1	no	maximal
6	C₂²×Q₈×C₉	C₄	1	2	yes	—
7	C₉×C₄⋊Q₈	—	2	1	no	—
8	C₉×C₄⋊Q₈	—	2	1	no	—
9	C₉×C₄⋊Q₈	—	2	1	no	—
10	C₉×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₉×C₄⋊C₄	C₄	1	2	yes	—
12	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₂×#3667	C₂²	1	2	yes	—
15	C₂×#3667	C₂²	1	2	yes	—
16	C₄⋊(C₉⋊C₈)	—	2	1	no	—
17	C₄⋊(C₉⋊C₈)	—	2	1	no	—
18	#63362	—	2	1	no	—
19	C₄×C₉⋊C₈	—	2	1	no	—
20	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
21	#4924	—	3	1	no	—
22	C₄×#208	—	3	1	no	—
23	#4530	—	3	4	no	—
24	C₂×Q₈×C₉	—	2	2	no	—
25	C₂×Q₈×C₉	—	2	2	no	—
26	C₂×Q₈×C₉	—	4	4	no	—
27	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₉×C₄⋊C₄	—	2	2	no	—
32	C₉×C₄⋊C₄	—	2	2	no	—
33	C₉×C₄⋊C₄	—	2	2	no	—
34	C₉×C₄⋊C₄	—	2	2	no	—
35	C₄²×C₉	D₄	1	1	yes	characteristic
36	C₄²×C₉	D₄	1	1	yes	characteristic
37	C₄²×C₉	D₄	1	1	yes	characteristic
38	C₄²×C₉	D₄	1	1	yes	characteristic
39	#3667	—	2	2	no	—
40	#3667	—	2	4	no	—
41	#3667	Q₈	1	4	yes	complemented
42	C₂×C₉⋊C₈	—	2	2	no	—
43	C₂×C₉⋊C₈	—	2	2	no	—
44	#1997	—	9	1	no	Sylow
45	C₂²×C₃×Q₈	Dic₃	1	2	yes	—
46	C₃×C₄⋊Q₈	—	2	1	no	—
47	C₃×C₄⋊Q₈	—	2	1	no	—
48	C₃×C₄⋊Q₈	—	2	1	no	—
49	C₃×C₄⋊Q₈	—	2	1	no	—
50	C₂×C₃×C₄⋊C₄	Dic₃	1	2	yes	—
51	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
52	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
53	C₂×#208	—	3	2	no	—
54	C₂×#208	—	3	2	no	—
55	C₄⋊(C₃⋊C₈)	—	6	1	no	—
56	C₄⋊(C₃⋊C₈)	—	6	1	no	—
57	#803	—	6	1	no	—
58	C₄×C₃⋊C₈	—	6	1	no	—
59	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
60	Q₈×C₉	—	4	4	no	—
61	Q₈×C₉	—	4	4	no	—
62	C₂×C₄×C₉	—	2	2	no	—
63	C₂×C₄×C₉	—	2	2	no	—
64	C₂×C₄×C₉	—	2	4	no	—
65	C₂×C₄×C₉	—	4	4	no	—
66	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₉	#41	1	2	yes	—
74	C₉⋊C₈	—	2	4	no	—
75	C₉⋊C₈	—	4	2	no	—
76	C₂×C₄⋊Q₈	D₉	1	1	yes	characteristic
77	#423	—	9	1	no	—
78	C₄×M₁₆	—	9	1	no	—
79	#343	—	9	4	no	—
80	C₂×C₃×Q₈	—	2	2	no	—
81	C₂×C₃×Q₈	—	2	2	no	—
82	C₂×C₃×Q₈	—	4	4	no	—
83	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
86	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₃×C₄⋊C₄	—	2	2	no	—
88	C₃×C₄⋊C₄	—	2	2	no	—
89	C₃×C₄⋊C₄	—	2	2	no	—
90	C₃×C₄⋊C₄	—	2	2	no	—
91	C₃×C₄²	#67	1	1	yes	characteristic
92	C₃×C₄²	#67	1	1	yes	characteristic
93	C₃×C₄²	#67	1	1	yes	characteristic
94	C₃×C₄²	#67	1	1	yes	characteristic
95	#208	—	3	4	no	—
96	#208	—	6	2	no	—
97	#208	—	6	4	no	—
98	C₂×C₃⋊C₈	—	6	2	no	—
99	C₂×C₃⋊C₈	—	6	2	no	—
100	C₂²×C₉	—	2	1	no	—
101	C₂²×C₉	—	2	1	no	—
102	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₉	C₄×Q₈	1	2	yes	—
104	C₄×C₉	—	2	1	no	—
105	C₄×C₉	—	2	1	no	—
106	C₄×C₉	—	4	4	no	—
107	C₄×C₉	#100	1	2	yes	—
108	C₄×C₉	#101	1	2	yes	—
109	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
110	C₄×C₉	#124	1	1	yes	characteristic
111	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
112	C₂²×Q₈	Dic₉	1	2	yes	—
113	C₂×M₁₆	—	9	2	no	—
114	C₂×M₁₆	—	9	2	no	—
115	C₄⋊Q₈	—	2	1	no	—
116	C₄⋊Q₈	—	2	1	no	—
117	C₄⋊Q₈	—	2	1	no	—
118	C₄⋊Q₈	—	2	1	no	—
119	C₂×C₄⋊C₄	Dic₉	1	2	yes	—
120	C₂×C₄⋊C₄	C₂×D₉	1	2	yes	—
121	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
122	C₄⋊C₈	—	18	1	no	—
123	C₄⋊C₈	—	18	1	no	—
124	#97	—	18	1	no	—
125	C₄×C₈	—	18	1	no	—
126	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
127	C₃×Q₈	—	4	4	no	—
128	C₃×Q₈	—	4	4	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	—	4	4	no	—
133	C₂×C₃×C₄	#217	1	1	yes	characteristic
134	C₂×C₃×C₄	#217	1	1	yes	characteristic
135	C₂×C₃×C₄	#217	1	1	yes	characteristic
136	C₂×C₃×C₄	#217	1	1	yes	characteristic
137	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
138	C₂×C₃×C₄	#239	1	2	yes	—
139	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
141	C₃⋊C₈	—	6	4	no	—
142	C₃⋊C₈	—	12	2	no	—
143	C₂×C₉	—	2	2	no	—
144	C₂×C₉	#391	1	1	yes	characteristic
145	C₂×C₉	C₂×#100	1	1	yes	characteristic
146	C₂×C₉	C₂×#101	1	1	yes	characteristic
147	C₂×Q₈	—	2	2	no	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	4	4	no	—
150	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
151	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
152	C₂²×C₄	C₂×Dic₉	1	4	yes	—
153	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
154	M₁₆	—	9	4	no	—
155	M₁₆	—	18	2	no	—
156	M₁₆	—	18	4	no	—
157	C₂×C₈	—	18	2	no	—
158	C₂×C₈	—	18	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄²	#611	1	1	yes	characteristic
164	C₄²	#611	1	1	yes	characteristic
165	C₄²	#611	1	1	yes	characteristic
166	C₄²	#611	1	1	yes	characteristic
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	Q₈×Dic₃	1	2	yes	—
170	C₂²×C₃	C₂×#217	1	1	yes	characteristic
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	4	4	no	—
174	C₃×C₄	#833	1	2	yes	—
175	C₃×C₄	#836	1	2	yes	—
176	C₃×C₄	#944	1	1	yes	characteristic
177	C₃×C₄	#946	1	2	yes	—
178	C₃×C₄	#947	1	1	yes	characteristic
179	C₉	#1997	1	1	yes	characteristic, complemented, Sylow
180	C₂³	C₂²×Dic₉	1	1	yes	characteristic
181	Q₈	—	4	4	no	—
182	Q₈	—	4	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	#3676	1	1	yes	characteristic
188	C₂×C₄	#3676	1	1	yes	characteristic
189	C₂×C₄	#3676	1	1	yes	characteristic
190	C₂×C₄	#3676	1	1	yes	characteristic
191	C₂×C₄	Q₈×D₉	1	2	yes	—
192	C₂×C₄	#3701	1	2	yes	—
193	C₂×C₄	C₂×#611	1	1	yes	characteristic
194	C₂×C₄	C₂×#611	1	1	yes	characteristic
195	C₈	—	18	4	no	—
196	C₈	—	36	2	no	—
197	C₂×C₃	—	2	2	no	—
198	C₂×C₃	C₂×#833	1	1	yes	characteristic
199	C₂×C₃	C₂×#836	1	1	yes	characteristic
200	C₂×C₃	#5229	1	1	yes	characteristic
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	Q₈×Dic₉	1	2	yes	—
204	C₂²	C₂×#3676	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	4	4	no	—
208	C₄	#63391	1	2	yes	—
209	C₄	#63394	1	2	yes	—
210	C₄	#63506	1	1	yes	characteristic
211	C₄	#63508	1	2	yes	—
212	C₄	#63509	1	1	yes	characteristic
213	C₃	#71790	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	C₂×#63391	1	1	yes	characteristic
216	C₂	C₂×#63394	1	1	yes	characteristic
217	C₂	#10588909	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#3676
commutator	C₂³×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×C₄⋊Q₈	C₂
socle	C₂²×C₃	C₂×#217
prime radical	C₂³×C₃	C₂²×Dic₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₉
hypocenter	C₉	#1997
LCS	G ▷ C₂³×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
41. #3667	Q₈	semidirect
179. C₉	#1997	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×C₄⋊Q₈	2
#10588604	2
C₄×#3667	2
#10588207	2
#71790	3