Subgroups — Q₈×#3676 #11804877

Order 1152.

1230 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₉	C₂	1	2	yes	maximal, complemented
3	#10588909	C₂	1	6	yes	maximal, complemented
4	#10588892	C₂	1	3	yes	maximal
5	C₄×#3676	C₂	1	3	yes	maximal
6	Q₈×#217	—	3	1	no	maximal
7	C₂²×Q₈×C₉	—	2	8	no	—
8	C₂²×Q₈×C₉	C₄	1	8	yes	complemented
9	C₂²×Q₈×C₉	C₂²	1	3	yes	—
10	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₉	C₂²	1	3	yes	—
12	C₂×#3676	C₂²	1	3	yes	—
13	C₂×#3676	C₂²	1	1	yes	characteristic
14	Q₈×Dic₉	—	2	8	no	—
15	C₂×C₄⋊Dic₉	C₂²	1	6	yes	—
16	C₂×C₄×Dic₉	C₂²	1	6	yes	—
17	C₉⋊#95	C₂²	1	6	yes	—
18	C₉⋊#95	C₂²	1	6	yes	—
19	C₂³×C₃×Q₈	D₃	1	1	yes	characteristic
20	C₂×Q₈×Dic₃	—	3	2	no	—
21	#5229	—	3	6	no	—
22	#5212	—	3	3	no	—
23	C₄×#217	—	3	3	no	—
24	C₂⁴×C₉	C₂³	1	1	yes	characteristic
25	C₂×Q₈×C₉	—	2	32	no	—
26	C₂×Q₈×C₉	—	2	12	no	—
27	C₂×Q₈×C₉	C₂×C₄	1	12	yes	—
28	C₂×Q₈×C₉	C₂×C₄	1	4	yes	—
29	C₂×Q₈×C₉	D₄	1	8	yes	—
30	C₂²×C₄×C₉	—	2	12	no	—
31	C₂²×C₄×C₉	C₂×C₄	1	12	yes	—
32	C₂²×C₄×C₉	C₂³	1	3	yes	—
33	C₂²×C₄×C₉	C₂³	1	3	yes	—
34	C₂²×Dic₉	C₂³	1	6	yes	—
35	C₂²×Dic₉	C₂³	1	2	yes	—
36	#3676	—	2	6	no	—
37	#3676	Q₈	1	4	yes	direct product factor
38	C₄⋊Dic₉	—	2	12	no	—
39	C₄×Dic₉	—	2	12	no	—
40	Q₈×C₂²⋊C₄	—	9	1	no	Sylow
41	C₂²×C₃×Q₈	—	2	8	no	—
42	C₂²×C₃×Q₈	Dic₃	1	8	yes	—
43	C₂²×C₃×Q₈	C₂×D₃	1	3	yes	—
44	C₂²×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
45	C₂³×C₃×C₄	C₂×D₃	1	3	yes	—
46	C₂×#217	—	3	1	no	—
47	C₂×#217	—	3	3	no	—
48	Q₈×Dic₃	—	6	8	no	—
49	C₂×C₄⋊Dic₃	—	3	6	no	—
50	C₂×C₄×Dic₃	—	3	6	no	—
51	C₃⋊#95	—	3	6	no	—
52	C₃⋊#95	—	3	6	no	—
53	C₂³×C₉	—	2	2	no	—
54	C₂³×C₉	—	2	2	no	—
55	C₂³×C₉	C₂²×C₄	1	2	yes	—
56	C₂³×C₉	C₂×Q₈	1	2	yes	—
57	C₂³×C₉	#41	1	2	yes	—
58	C₂³×C₉	C₂⁴	1	1	yes	characteristic
59	Q₈×C₉	—	2	24	no	—
60	Q₈×C₉	C₂²⋊C₄	1	16	yes	complemented
61	C₂×C₄×C₉	—	2	6	no	—
62	C₂×C₄×C₉	—	2	24	no	—
63	C₂×C₄×C₉	C₂²×C₄	1	6	yes	—
64	C₂×C₄×C₉	C₂²×C₄	1	6	yes	—
65	C₂×C₄×C₉	C₂×D₄	1	12	yes	—
66	C₂×Dic₉	—	2	2	no	—
67	C₂×Dic₉	—	2	2	no	—
68	C₂×Dic₉	—	2	6	no	—
69	C₂×Dic₉	—	2	6	no	—
70	C₂×Dic₉	—	2	6	no	—
71	C₂×Dic₉	C₂×Q₈	1	4	yes	complemented
72	C₂³×Q₈	D₉	1	1	yes	characteristic
73	C₂×C₄×Q₈	—	9	2	no	—
74	#391	—	9	6	no	—
75	#380	—	9	3	no	—
76	C₄×C₂²⋊C₄	—	9	3	no	—
77	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
78	C₂×C₃×Q₈	—	2	32	no	—
79	C₂×C₃×Q₈	—	2	12	no	—
80	C₂×C₃×Q₈	C₂×Dic₃	1	12	yes	—
81	C₂×C₃×Q₈	C₂×Dic₃	1	4	yes	—
82	C₂×C₃×Q₈	#67	1	8	yes	—
83	C₂²×C₃×C₄	—	2	12	no	—
84	C₂²×C₃×C₄	C₂×Dic₃	1	12	yes	—
85	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
86	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
87	C₂²×Dic₃	—	3	2	no	—
88	C₂²×Dic₃	—	3	6	no	—
89	#217	—	3	4	no	—
90	#217	—	6	6	no	—
91	C₄⋊Dic₃	—	6	12	no	—
92	C₄×Dic₃	—	6	12	no	—
93	C₂²×C₉	—	2	2	no	—
94	C₂²×C₉	—	2	2	no	—
95	C₂²×C₉	—	2	4	no	—
96	C₂²×C₉	—	2	4	no	—
97	C₂²×C₉	C₄×Q₈	1	4	yes	complemented
98	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
99	C₂²×C₉	C₂²×D₄	1	2	yes	—
100	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
101	C₂²×C₉	C₂×#41	1	1	yes	characteristic
102	C₂²×C₉	2₋¹⁺⁴	1	2	yes	—
103	C₄×C₉	—	2	6	no	—
104	C₄×C₉	C₂×C₂²⋊C₄	1	12	yes	—
105	Dic₉	—	2	4	no	—
106	Dic₉	—	4	6	no	—
107	C₂²×Q₈	—	2	8	no	—
108	C₂²×Q₈	Dic₉	1	8	yes	complemented
109	C₂²×Q₈	C₂×D₉	1	3	yes	—
110	C₂²×Q₈	C₂×D₉	1	1	yes	characteristic
111	C₂³×C₄	C₂×D₉	1	3	yes	—
112	C₄×Q₈	—	18	8	no	—
113	C₂×C₄⋊C₄	—	9	6	no	—
114	C₂×C₂²⋊C₄	—	9	1	no	—
115	C₂×C₂²⋊C₄	—	9	3	no	—
116	C₂×C₄²	—	9	6	no	—
117	#95	—	9	6	no	—
118	#95	—	9	6	no	—
119	C₂³×C₃	—	2	2	no	—
120	C₂³×C₃	—	2	2	no	—
121	C₂³×C₃	D₃×Q₈	1	2	yes	—
122	C₂³×C₃	#239	1	2	yes	—
123	C₂³×C₃	C₂²×Dic₃	1	2	yes	—
124	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
125	C₃×Q₈	—	2	24	no	—
126	C₃×Q₈	#217	1	16	yes	—
127	C₂×C₃×C₄	—	2	6	no	—
128	C₂×C₃×C₄	—	2	24	no	—
129	C₂×C₃×C₄	C₂²×Dic₃	1	6	yes	—
130	C₂×C₃×C₄	C₂²×Dic₃	1	6	yes	—
131	C₂×C₃×C₄	C₂×#67	1	12	yes	—
132	C₂×Dic₃	—	3	4	no	—
133	C₂×Dic₃	—	6	2	no	—
134	C₂×Dic₃	—	6	2	no	—
135	C₂×Dic₃	—	6	6	no	—
136	C₂×Dic₃	—	6	6	no	—
137	C₂×Dic₃	—	6	6	no	—
138	C₂×C₉	—	2	4	no	—
139	C₂×C₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₉	C₂×C₄×Q₈	1	1	yes	characteristic
141	C₂×C₉	#519	1	1	yes	characteristic
142	C₂×C₉	#548	1	2	yes	—
143	C₂×C₉	D₄×Q₈	1	2	yes	—
144	C₂⁴	C₂²×D₉	1	1	yes	characteristic
145	C₂×Q₈	—	2	32	no	—
146	C₂×Q₈	—	2	12	no	—
147	C₂×Q₈	C₂×Dic₉	1	12	yes	—
148	C₂×Q₈	C₂×Dic₉	1	4	yes	—
149	C₂×Q₈	#611	1	8	yes	—
150	C₂²×C₄	—	2	12	no	—
151	C₂²×C₄	—	9	2	no	—
152	C₂²×C₄	—	9	6	no	—
153	C₂²×C₄	C₂×Dic₉	1	12	yes	—
154	C₂²×C₄	C₂²×D₉	1	3	yes	—
155	C₂²×C₄	C₂²×D₉	1	3	yes	—
156	C₄⋊C₄	—	18	12	no	—
157	C₂²⋊C₄	—	9	4	no	—
158	C₂²⋊C₄	—	18	6	no	—
159	C₄²	—	18	12	no	—
160	C₂²×C₃	—	2	2	no	—
161	C₂²×C₃	—	2	4	no	—
162	C₂²×C₃	—	2	4	no	—
163	C₂²×C₃	—	2	2	no	—
164	C₂²×C₃	Q₈×Dic₃	1	4	yes	—
165	C₂²×C₃	C₂×D₃×Q₈	1	1	yes	characteristic
166	C₂²×C₃	C₂×#239	1	1	yes	characteristic
167	C₂²×C₃	#1007	1	2	yes	—
168	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
169	C₂²×C₃	C₂²×#67	1	2	yes	—
170	C₃×C₄	—	2	6	no	—
171	C₃×C₄	C₂×#217	1	12	yes	—
172	Dic₃	—	6	4	no	—
173	Dic₃	—	12	6	no	—
174	C₉	Q₈×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	Q₈×D₉	1	2	yes	—
178	C₂³	#3701	1	2	yes	—
179	C₂³	C₂²×Dic₉	1	2	yes	—
180	C₂³	C₂³×D₉	1	1	yes	characteristic
181	Q₈	—	2	24	no	—
182	Q₈	#3676	1	16	yes	direct product factor
183	C₂×C₄	—	2	6	no	—
184	C₂×C₄	—	2	24	no	—
185	C₂×C₄	—	9	4	no	—
186	C₂×C₄	—	18	2	no	—
187	C₂×C₄	—	18	2	no	—
188	C₂×C₄	—	18	6	no	—
189	C₂×C₄	—	18	6	no	—
190	C₂×C₄	—	18	6	no	—
191	C₂×C₄	C₂²×Dic₉	1	6	yes	—
192	C₂×C₄	C₂²×Dic₉	1	6	yes	—
193	C₂×C₄	C₂×#611	1	12	yes	—
194	C₂×C₃	—	2	4	no	—
195	C₂×C₃	C₂×Q₈×Dic₃	1	1	yes	characteristic
196	C₂×C₃	#5812	1	1	yes	characteristic
197	C₂×C₃	Q₈×#67	1	2	yes	—
198	C₂×C₃	#5817	1	2	yes	—
199	C₂×C₃	C₂²×#217	1	1	yes	characteristic
200	C₂²	—	2	2	no	—
201	C₂²	—	2	4	no	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	2	no	—
204	C₂²	Q₈×Dic₉	1	4	yes	complemented
205	C₂²	C₂×Q₈×D₉	1	1	yes	characteristic
206	C₂²	C₂×#3701	1	1	yes	characteristic
207	C₂²	#63713	1	2	yes	—
208	C₂²	C₂³×Dic₉	1	1	yes	characteristic
209	C₂²	C₂²×#611	1	2	yes	—
210	C₄	—	2	6	no	—
211	C₄	—	18	4	no	—
212	C₄	—	36	6	no	—
213	C₄	C₂×#3676	1	12	yes	—
214	C₃	Q₈×#217	1	1	yes	characteristic
215	C₂	—	2	4	no	—
216	C₂	C₂×Q₈×Dic₉	1	1	yes	characteristic
217	C₂	#10589914	1	1	yes	characteristic
218	C₂	Q₈×#611	1	2	yes	—
219	C₂	#10589919	1	2	yes	—
220	C₂	C₂²×#3676	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini, socle	C₂³×C₃	C₂³×D₃
Fitting	C₂³×Q₈×C₉	C₂
prime radical	C₂⁴×C₃	C₂²×D₃
cosocle	C₂³×C₉	C₂⁴
hypercenter	C₂³×Q₈	D₉
hypocenter	C₉	Q₈×C₂²⋊C₄
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×Dic₉	C₂	semidirect
3. #10588909	C₂	semidirect
8. C₂²×Q₈×C₉	C₄	semidirect
37. #3676	Q₈	direct (N × H)
60. Q₈×C₉	C₂²⋊C₄	semidirect
71. C₂×Dic₉	C₂×Q₈	semidirect
97. C₂²×C₉	C₄×Q₈	semidirect
108. C₂²×Q₈	Dic₉	semidirect
174. C₉	Q₈×C₂²⋊C₄	semidirect
182. Q₈	#3676	direct (N × H)
204. C₂²	Q₈×Dic₉	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Q₈×C₉	2
C₂×Q₈×Dic₉	2
#10588909	2
#10588892	2
C₄×#3676	2
Q₈×#217	3

Subgroups — Q8×#3676 #11804877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#3676 #11804877