Subgroups — C₂×Q₈×#6099 #16727307

Order 3328.

2154 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#6099	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11718685	C₂	1	6	yes	maximal, complemented
3	C₂×Q₈×#1055	C₂	1	2	yes	maximal, complemented
4	C₂×#11718585	C₂	1	3	yes	maximal
5	C₂×C₄×#6099	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₁₃	C₄	1	3	yes	—
9	C₂×C₄×Q₈×C₁₃	C₄	1	8	yes	—
10	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₃	C₂²	1	3	yes	—
13	C₂×C₄×Q₈×C₁₃	C₂²	1	8	yes	—
14	C₂²×C₁₃×C₄⋊C₄	C₄	1	3	yes	—
15	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
16	C₂²×#6099	C₂²	1	3	yes	—
17	C₂²×#6099	C₂²	1	1	yes	characteristic
18	#11718685	C₂²	1	48	yes	complemented
19	Q₈×#1055	C₂²	1	16	yes	complemented
20	#11718585	C₂²	1	24	yes	—
21	C₂×C₄⋊#1055	C₂²	1	6	yes	—
22	C₂×C₄⋊#1055	C₂²	1	6	yes	—
23	C₄×#6099	C₂²	1	24	yes	—
24	C₂×#88043	C₂²	1	6	yes	—
25	C₂×C₄×#1055	C₂²	1	6	yes	—
26	C₂²×Q₈×C₁₃	—	2	8	no	—
27	C₂²×Q₈×C₁₃	C₂×C₄	1	8	yes	—
28	C₂²×Q₈×C₁₃	C₂×C₄	1	3	yes	—
29	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
31	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₄×Q₈×C₁₃	C₂×C₄	1	12	yes	—
34	C₄×Q₈×C₁₃	C₂×C₄	1	24	yes	—
35	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
36	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
37	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
38	C₄×Q₈×C₁₃	C₂³	1	12	yes	—
39	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	6	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
42	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	24	yes	—
43	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
44	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
45	C₂×C₄²×C₁₃	C₂×C₄	1	12	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
49	C₂×#6099	—	2	12	no	—
50	C₂×#6099	Q₈	1	8	yes	direct product factor
51	C₂×#6099	C₂³	1	12	yes	—
52	C₂×#6099	C₂³	1	4	yes	—
53	C₂²×#1055	C₂³	1	6	yes	—
54	C₂²×#1055	C₂³	1	2	yes	—
55	C₄⋊#1055	C₂³	1	24	yes	—
56	C₄⋊#1055	C₂³	1	24	yes	—
57	#88043	C₂³	1	24	yes	—
58	C₄×#1055	C₂³	1	24	yes	—
59	C₂×Q₈×M₁₆	—	13	1	no	maximal, Sylow
60	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×Q₈×C₁₃	—	2	32	no	—
62	C₂×Q₈×C₁₃	—	2	12	no	—
63	C₂×Q₈×C₁₃	M₁₆	1	8	yes	complemented
64	C₂×Q₈×C₁₃	C₂²×C₄	1	12	yes	—
65	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃	—	2	12	no	—
67	C₂²×C₄×C₁₃	—	4	6	no	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₁₃	#41	1	4	yes	—
78	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₁₃×C₄⋊C₄	C₂²×C₄	1	24	yes	—
81	C₁₃×C₄⋊C₄	C₂²×C₄	1	12	yes	—
82	C₁₃×C₄⋊C₄	C₂²×C₄	1	12	yes	—
83	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
84	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
85	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
86	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
87	C₄²×C₁₃	C₂⁴	1	6	yes	—
88	C₄²×C₁₃	C₂⁴	1	6	yes	—
89	#6099	—	2	24	no	—
90	#6099	C₂×Q₈	1	16	yes	direct product factor
91	C₂×#1055	—	2	12	no	—
92	C₂×#1055	C₂×Q₈	1	8	yes	complemented
93	C₂×#1055	C₂⁴	1	12	yes	—
94	C₂×#1055	C₂⁴	1	4	yes	—
95	C₂²×C₄×Q₈	D₁₃	1	1	yes	characteristic
96	Q₈×M₁₆	—	13	16	no	—
97	C₂×#446	—	13	6	no	—
98	C₂×C₈×Q₈	—	13	2	no	—
99	C₂×#423	—	13	3	no	—
100	C₂×C₄×M₁₆	—	13	3	no	—
101	C₂³×C₁₃	—	2	2	no	—
102	C₂³×C₁₃	—	2	2	no	—
103	C₂³×C₁₃	C₄×Q₈	1	4	yes	—
104	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
105	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
106	Q₈×C₁₃	—	2	24	no	—
107	Q₈×C₁₃	C₂×M₁₆	1	16	yes	complemented
108	C₂×C₄×C₁₃	—	2	6	no	—
109	C₂×C₄×C₁₃	—	2	6	no	—
110	C₂×C₄×C₁₃	—	2	6	no	—
111	C₂×C₄×C₁₃	—	2	24	no	—
112	C₂×C₄×C₁₃	—	4	12	no	—
113	C₂×C₄×C₁₃	C₄×Q₈	1	8	yes	—
114	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
115	C₂×C₄×C₁₃	C₂×M₁₆	1	12	yes	—
116	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
117	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
118	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
119	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
120	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
122	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
123	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
124	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
126	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
127	#1055	—	2	12	no	—
128	#1055	C₂²×Q₈	1	8	yes	complemented
129	C₂³×Q₈	Dic₁₃	1	1	yes	characteristic
130	C₂²×M₁₆	—	13	1	no	—
131	C₂²×M₁₆	—	13	3	no	—
132	C₂×C₄×Q₈	Dic₁₃	1	3	yes	—
133	C₂×C₄×Q₈	Dic₁₃	1	8	yes	—
134	C₂×C₄×Q₈	Dic₁₃	1	1	yes	characteristic
135	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
136	C₂×C₄×Q₈	C₂×D₁₃	1	3	yes	—
137	C₂×C₄×Q₈	C₂×D₁₃	1	8	yes	—
138	C₂²×C₄⋊C₄	Dic₁₃	1	3	yes	—
139	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
140	#446	—	13	48	no	—
141	C₈×Q₈	—	13	16	no	—
142	#423	—	13	24	no	—
143	C₂×C₄⋊C₈	—	13	6	no	—
144	C₂×C₄⋊C₈	—	13	6	no	—
145	C₄×M₁₆	—	13	24	no	—
146	C₂×#97	—	13	6	no	—
147	C₂×C₄×C₈	—	13	6	no	—
148	C₂²×C₁₃	—	2	8	no	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	C₂×C₄×Q₈	1	4	yes	—
152	C₂²×C₁₃	C₂×C₄×Q₈	1	2	yes	—
153	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
154	C₂²×C₁₃	#568	1	2	yes	—
155	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
156	C₄×C₁₃	—	2	6	no	—
157	C₄×C₁₃	—	2	6	no	—
158	C₄×C₁₃	—	2	6	no	—
159	C₄×C₁₃	C₂×C₄×Q₈	1	2	yes	—
160	C₄×C₁₃	C₂×C₄×Q₈	1	4	yes	—
161	C₄×C₁₃	C₂²×M₁₆	1	12	yes	—
162	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
163	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
164	C₂²×Q₈	—	2	8	no	—
165	C₂²×Q₈	C₂×Dic₁₃	1	8	yes	—
166	C₂²×Q₈	C₂×Dic₁₃	1	3	yes	—
167	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
168	C₂³×C₄	C₂×Dic₁₃	1	3	yes	—
169	C₂³×C₄	C₂×Dic₁₃	1	3	yes	—
170	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
171	C₂×M₁₆	—	13	8	no	—
172	C₂×M₁₆	—	13	4	no	—
173	C₂×M₁₆	—	13	12	no	—
174	C₂×M₁₆	—	26	12	no	—
175	C₂²×C₈	—	13	2	no	—
176	C₂²×C₈	—	13	6	no	—
177	C₄×Q₈	C₂×Dic₁₃	1	12	yes	—
178	C₄×Q₈	C₂×Dic₁₃	1	24	yes	—
179	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
180	C₄×Q₈	C₂×Dic₁₃	1	8	yes	—
181	C₄×Q₈	C₂²×D₁₃	1	4	yes	—
182	C₄×Q₈	C₂²×D₁₃	1	12	yes	—
183	C₂×C₄⋊C₄	C₂×Dic₁₃	1	6	yes	—
184	C₂×C₄⋊C₄	C₂×Dic₁₃	1	3	yes	—
185	C₂×C₄⋊C₄	C₂×Dic₁₃	1	3	yes	—
186	C₂×C₄⋊C₄	C₂×Dic₁₃	1	24	yes	—
187	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
188	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
189	C₂×C₄²	C₂×Dic₁₃	1	12	yes	—
190	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
191	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
192	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
193	C₄⋊C₈	—	13	24	no	—
194	C₄⋊C₈	—	13	24	no	—
195	#97	—	13	24	no	—
196	C₄×C₈	—	13	24	no	—
197	C₂×C₁₃	—	2	4	no	—
198	C₂×C₁₃	Q₈×M₁₆	1	4	yes	complemented
199	C₂×C₁₃	C₂²×C₄×Q₈	1	1	yes	characteristic
200	C₂×C₁₃	C₂³×M₁₆	1	1	yes	characteristic
201	C₂×C₁₃	C₂×#568	1	1	yes	characteristic
202	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
203	C₂×Q₈	—	2	32	no	—
204	C₂×Q₈	—	2	12	no	—
205	C₂×Q₈	#6099	1	8	yes	direct product factor
206	C₂×Q₈	C₂²×Dic₁₃	1	12	yes	—
207	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
208	C₂²×C₄	—	2	12	no	—
209	C₂²×C₄	—	4	6	no	—
210	C₂²×C₄	Q₈×D₁₃	1	4	yes	—
211	C₂²×C₄	#6131	1	4	yes	—
212	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
213	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
214	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
215	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
216	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
217	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
218	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
220	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
221	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
222	M₁₆	—	13	16	no	—
223	M₁₆	—	26	24	no	—
224	C₂×C₈	—	13	8	no	—
225	C₂×C₈	—	13	4	no	—
226	C₂×C₈	—	13	12	no	—
227	C₂×C₈	—	26	12	no	—
228	C₄⋊C₄	C₂²×Dic₁₃	1	24	yes	—
229	C₄⋊C₄	C₂²×Dic₁₃	1	12	yes	—
230	C₄⋊C₄	C₂²×Dic₁₃	1	12	yes	—
231	C₄²	C₂²×Dic₁₃	1	12	yes	—
232	C₄²	C₂²×Dic₁₃	1	6	yes	—
233	C₄²	C₂²×Dic₁₃	1	12	yes	—
234	C₄²	C₂²×Dic₁₃	1	6	yes	—
235	C₄²	C₂³×D₁₃	1	6	yes	—
236	C₄²	C₂³×D₁₃	1	6	yes	—
237	C₁₃	C₂×Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	Q₈×Dic₁₃	1	4	yes	—
241	C₂³	C₂³×Dic₁₃	1	2	yes	—
242	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
243	Q₈	—	2	24	no	—
244	Q₈	C₂×#6099	1	16	yes	direct product factor
245	C₂×C₄	—	2	6	no	—
246	C₂×C₄	—	2	6	no	—
247	C₂×C₄	—	2	6	no	—
248	C₂×C₄	—	2	24	no	—
249	C₂×C₄	—	4	12	no	—
250	C₂×C₄	C₂×#6099	1	12	yes	—
251	C₂×C₄	Q₈×Dic₁₃	1	8	yes	—
252	C₂×C₄	Q₈×Dic₁₃	1	4	yes	—
253	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
254	C₂×C₄	C₂×Q₈×D₁₃	1	4	yes	—
255	C₂×C₄	C₂×#6131	1	2	yes	—
256	C₂×C₄	C₂×#6131	1	4	yes	—
257	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
258	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
259	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
260	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
261	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
262	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
263	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
264	C₈	—	13	8	no	—
265	C₈	—	26	12	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	8	no	—
269	C₂²	C₂²×#6099	1	2	yes	—
270	C₂²	C₂×Q₈×Dic₁₃	1	4	yes	—
271	C₂²	C₂×Q₈×Dic₁₃	1	2	yes	—
272	C₂²	#11719524	1	2	yes	—
273	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
274	C₄	—	2	6	no	—
275	C₄	—	2	6	no	—
276	C₄	—	2	6	no	—
277	C₄	C₂²×#6099	1	12	yes	—
278	C₄	C₂×Q₈×Dic₁₃	1	2	yes	—
279	C₄	C₂×Q₈×Dic₁₃	1	4	yes	—
280	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
281	C₄	C₂²×#6131	1	1	yes	characteristic
282	C₂	—	2	4	no	—
283	C₂	Q₈×#6099	1	4	yes	direct product factor
284	C₂	C₂²×Q₈×Dic₁₃	1	1	yes	characteristic
285	C₂	C₂³×#6099	1	1	yes	characteristic
286	C₂	C₂×#11719524	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×Q₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₁₃
hypocenter	C₁₃	C₂×Q₈×M₁₆
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#6099	C₂	direct (N × H)
2. C₂×#11718685	C₂	semidirect
3. C₂×Q₈×#1055	C₂	semidirect
18. #11718685	C₂²	semidirect
19. Q₈×#1055	C₂²	semidirect
50. C₂×#6099	Q₈	direct (N × H)
63. C₂×Q₈×C₁₃	M₁₆	semidirect
90. #6099	C₂×Q₈	direct (N × H)
92. C₂×#1055	C₂×Q₈	semidirect
107. Q₈×C₁₃	C₂×M₁₆	semidirect
128. #1055	C₂²×Q₈	semidirect
198. C₂×C₁₃	Q₈×M₁₆	semidirect
205. C₂×Q₈	#6099	direct (N × H)
237. C₁₃	C₂×Q₈×M₁₆	semidirect
244. Q₈	C₂×#6099	direct (N × H)
283. C₂	Q₈×#6099	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#6099	2
C₂×#11718685	2
C₂×Q₈×#1055	2
C₂×#11718585	2
C₂×C₄×#6099	2
C₂²×C₄×Q₈×C₁₃	2
C₂×Q₈×M₁₆	13

Subgroups — C2×Q8×#6099 #16727307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#6099 #16727307