Subgroups — C₂×Q₈×#11761612 #31720219

Order 17152.

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×#11761612	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16863527	C₂	1	6	yes	maximal, complemented
3	C₂×Q₈×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂×#16863420	C₂	1	3	yes	maximal
5	C₂×C₄×#11761612	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	3	yes	—
8	C₂²×#11761612	C₂²	1	1	yes	characteristic
9	#16863527	C₂²	1	48	yes	complemented
10	Q₈×C₆₇⋊C₈	C₂²	1	16	yes	complemented
11	#16863420	C₂²	1	24	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
14	C₄×#11761612	C₂²	1	24	yes	—
15	C₂×#14523954	C₂²	1	6	yes	—
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	6	yes	—
17	C₂³×Q₈×C₆₇	C₄	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₆₇	C₄	1	3	yes	—
19	C₂×C₄×Q₈×C₆₇	C₄	1	8	yes	—
20	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₆₇	C₂²	1	3	yes	—
23	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	—
24	C₂²×C₄⋊C₄×C₆₇	C₄	1	3	yes	—
25	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
26	C₂×#11761612	—	2	12	no	—
27	C₂×#11761612	Q₈	1	8	yes	direct product factor
28	C₂×#11761612	C₂³	1	12	yes	—
29	C₂×#11761612	C₂³	1	4	yes	—
30	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
31	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
32	C₄⋊(C₆₇⋊C₈)	C₂³	1	24	yes	—
33	C₄⋊(C₆₇⋊C₈)	C₂³	1	24	yes	—
34	#14523954	C₂³	1	24	yes	—
35	C₄×C₆₇⋊C₈	C₂³	1	24	yes	—
36	C₂²×Q₈×C₆₇	—	2	8	no	—
37	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	—
38	C₂²×Q₈×C₆₇	C₂×C₄	1	3	yes	—
39	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
41	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₆₇	C₂×C₄	1	12	yes	—
44	C₄×Q₈×C₆₇	C₂×C₄	1	24	yes	—
45	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
46	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
47	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
48	C₄×Q₈×C₆₇	C₂³	1	12	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
51	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
52	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	24	yes	—
53	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
54	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
55	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
56	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
57	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
58	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
59	#11761612	—	2	24	no	—
60	#11761612	C₂×Q₈	1	16	yes	direct product factor
61	C₂×C₆₇⋊C₈	—	2	12	no	—
62	C₂×C₆₇⋊C₈	C₂×Q₈	1	8	yes	complemented
63	C₂×C₆₇⋊C₈	C₂⁴	1	12	yes	—
64	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
65	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×Q₈×C₆₇	—	2	32	no	—
67	C₂×Q₈×C₆₇	—	2	12	no	—
68	C₂×Q₈×C₆₇	M₁₆	1	8	yes	complemented
69	C₂×Q₈×C₆₇	C₂²×C₄	1	12	yes	—
70	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₆₇	—	2	12	no	—
72	C₂²×C₄×C₆₇	—	4	6	no	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
81	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
82	C₂²×C₄×C₆₇	#41	1	4	yes	—
83	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₆₇	C₂²×C₄	1	24	yes	—
86	C₄⋊C₄×C₆₇	C₂²×C₄	1	12	yes	—
87	C₄⋊C₄×C₆₇	C₂²×C₄	1	12	yes	—
88	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
89	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
90	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
91	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
92	C₄²×C₆₇	C₂⁴	1	6	yes	—
93	C₄²×C₆₇	C₂⁴	1	6	yes	—
94	C₂³×C₆₇	—	2	2	no	—
95	C₂³×C₆₇	—	2	2	no	—
96	C₂³×C₆₇	C₄×Q₈	1	4	yes	—
97	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
98	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
99	Q₈×C₆₇	—	2	24	no	—
100	Q₈×C₆₇	C₂×M₁₆	1	16	yes	complemented
101	C₂×C₄×C₆₇	—	2	6	no	—
102	C₂×C₄×C₆₇	—	2	6	no	—
103	C₂×C₄×C₆₇	—	2	6	no	—
104	C₂×C₄×C₆₇	—	2	24	no	—
105	C₂×C₄×C₆₇	—	4	12	no	—
106	C₂×C₄×C₆₇	C₄×Q₈	1	8	yes	—
107	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
108	C₂×C₄×C₆₇	C₂×M₁₆	1	12	yes	—
109	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
110	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
111	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
112	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
115	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
116	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
117	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
118	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
119	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
120	C₆₇⋊C₈	—	2	12	no	—
121	C₆₇⋊C₈	C₂²×Q₈	1	8	yes	complemented
122	C₂²×C₆₇	—	2	8	no	—
123	C₂²×C₆₇	—	2	2	no	—
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	C₂×C₄×Q₈	1	4	yes	—
126	C₂²×C₆₇	C₂×C₄×Q₈	1	2	yes	—
127	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
128	C₂²×C₆₇	#568	1	2	yes	—
129	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
130	C₄×C₆₇	—	2	6	no	—
131	C₄×C₆₇	—	2	6	no	—
132	C₄×C₆₇	—	2	6	no	—
133	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
134	C₄×C₆₇	C₂×C₄×Q₈	1	4	yes	—
135	C₄×C₆₇	C₂²×M₁₆	1	12	yes	—
136	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
137	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
138	C₂×Q₈×M₁₆	—	67	1	no	maximal, Sylow
139	C₂×C₆₇	—	2	4	no	—
140	C₂×C₆₇	Q₈×M₁₆	1	4	yes	complemented
141	C₂×C₆₇	C₂²×C₄×Q₈	1	1	yes	characteristic
142	C₂×C₆₇	C₂³×M₁₆	1	1	yes	characteristic
143	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
144	C₂²×C₄×Q₈	D₆₇	1	1	yes	characteristic
145	Q₈×M₁₆	—	67	16	no	—
146	C₂×#446	—	67	6	no	—
147	C₂×C₈×Q₈	—	67	2	no	—
148	C₂×#423	—	67	3	no	—
149	C₂×C₄×M₁₆	—	67	3	no	—
150	C₆₇	C₂×Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
151	C₂³×Q₈	Dic₆₇	1	1	yes	characteristic
152	C₂²×M₁₆	—	67	1	no	—
153	C₂²×M₁₆	—	67	3	no	—
154	C₂×C₄×Q₈	Dic₆₇	1	3	yes	—
155	C₂×C₄×Q₈	Dic₆₇	1	8	yes	—
156	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
157	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
158	C₂×C₄×Q₈	C₂×D₆₇	1	3	yes	—
159	C₂×C₄×Q₈	C₂×D₆₇	1	8	yes	—
160	C₂²×C₄⋊C₄	Dic₆₇	1	3	yes	—
161	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
162	#446	—	67	48	no	—
163	C₈×Q₈	—	67	16	no	—
164	#423	—	67	24	no	—
165	C₂×C₄⋊C₈	—	67	6	no	—
166	C₂×C₄⋊C₈	—	67	6	no	—
167	C₄×M₁₆	—	67	24	no	—
168	C₂×#97	—	67	6	no	—
169	C₂×C₄×C₈	—	67	6	no	—
170	C₂²×Q₈	—	2	8	no	—
171	C₂²×Q₈	C₂×Dic₆₇	1	8	yes	—
172	C₂²×Q₈	C₂×Dic₆₇	1	3	yes	—
173	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
174	C₂³×C₄	C₂×Dic₆₇	1	3	yes	—
175	C₂³×C₄	C₂×Dic₆₇	1	3	yes	—
176	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
177	C₂×M₁₆	—	67	8	no	—
178	C₂×M₁₆	—	67	4	no	—
179	C₂×M₁₆	—	67	12	no	—
180	C₂×M₁₆	—	134	12	no	—
181	C₂²×C₈	—	67	2	no	—
182	C₂²×C₈	—	67	6	no	—
183	C₄×Q₈	C₂×Dic₆₇	1	12	yes	—
184	C₄×Q₈	C₂×Dic₆₇	1	24	yes	—
185	C₄×Q₈	C₂×Dic₆₇	1	4	yes	—
186	C₄×Q₈	C₂×Dic₆₇	1	8	yes	—
187	C₄×Q₈	C₂²×D₆₇	1	4	yes	—
188	C₄×Q₈	C₂²×D₆₇	1	12	yes	—
189	C₂×C₄⋊C₄	C₂×Dic₆₇	1	6	yes	—
190	C₂×C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
191	C₂×C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
192	C₂×C₄⋊C₄	C₂×Dic₆₇	1	24	yes	—
193	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
194	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
195	C₂×C₄²	C₂×Dic₆₇	1	12	yes	—
196	C₂×C₄²	C₂²×D₆₇	1	12	yes	—
197	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
198	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
199	C₄⋊C₈	—	67	24	no	—
200	C₄⋊C₈	—	67	24	no	—
201	#97	—	67	24	no	—
202	C₄×C₈	—	67	24	no	—
203	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
204	C₂×Q₈	—	2	32	no	—
205	C₂×Q₈	—	2	12	no	—
206	C₂×Q₈	#11761612	1	8	yes	direct product factor
207	C₂×Q₈	C₂²×Dic₆₇	1	12	yes	—
208	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
209	C₂²×C₄	—	2	12	no	—
210	C₂²×C₄	—	4	6	no	—
211	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
212	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
213	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
214	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
215	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
216	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
217	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
219	C₂²×C₄	Q₈×D₆₇	1	4	yes	—
220	C₂²×C₄	#11761629	1	4	yes	—
221	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
222	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
223	M₁₆	—	67	16	no	—
224	M₁₆	—	134	24	no	—
225	C₂×C₈	—	67	8	no	—
226	C₂×C₈	—	67	4	no	—
227	C₂×C₈	—	67	12	no	—
228	C₂×C₈	—	134	12	no	—
229	C₄⋊C₄	C₂²×Dic₆₇	1	24	yes	—
230	C₄⋊C₄	C₂²×Dic₆₇	1	12	yes	—
231	C₄⋊C₄	C₂²×Dic₆₇	1	12	yes	—
232	C₄²	C₂²×Dic₆₇	1	12	yes	—
233	C₄²	C₂²×Dic₆₇	1	6	yes	—
234	C₄²	C₂²×Dic₆₇	1	12	yes	—
235	C₄²	C₂²×Dic₆₇	1	6	yes	—
236	C₄²	C₂³×D₆₇	1	6	yes	—
237	C₄²	C₂³×D₆₇	1	6	yes	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	Q₈×Dic₆₇	1	4	yes	—
241	C₂³	C₂³×Dic₆₇	1	2	yes	—
242	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
243	Q₈	—	2	24	no	—
244	Q₈	C₂×#11761612	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	Q₈×Dic₆₇	1	8	yes	—
251	C₂×C₄	Q₈×Dic₆₇	1	4	yes	—
252	C₂×C₄	C₂×#11761612	1	12	yes	—
253	C₂×C₄	C₂³×Dic₆₇	1	6	yes	—
254	C₂×C₄	C₂³×Dic₆₇	1	6	yes	—
255	C₂×C₄	C₂³×Dic₆₇	1	6	yes	—
256	C₂×C₄	C₂³×Dic₆₇	1	2	yes	—
257	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
258	C₂×C₄	C₂³×Dic₆₇	1	6	yes	—
259	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
260	C₂×C₄	C₂×Q₈×D₆₇	1	4	yes	—
261	C₂×C₄	C₂×#11761629	1	2	yes	—
262	C₂×C₄	C₂×#11761629	1	4	yes	—
263	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
264	C₈	—	67	8	no	—
265	C₈	—	134	12	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	8	no	—
269	C₂²	C₂×Q₈×Dic₆₇	1	4	yes	—
270	C₂²	C₂×Q₈×Dic₆₇	1	2	yes	—
271	C₂²	C₂²×#11761612	1	2	yes	—
272	C₂²	#16864163	1	2	yes	—
273	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
274	C₄	—	2	6	no	—
275	C₄	—	2	6	no	—
276	C₄	—	2	6	no	—
277	C₄	C₂×Q₈×Dic₆₇	1	4	yes	—
278	C₄	C₂×Q₈×Dic₆₇	1	2	yes	—
279	C₄	C₂²×#11761612	1	12	yes	—
280	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
281	C₄	C₂²×#11761629	1	1	yes	characteristic
282	C₂	—	2	4	no	—
283	C₂	Q₈×#11761612	1	4	yes	direct product factor
284	C₂	C₂²×Q₈×Dic₆₇	1	1	yes	characteristic
285	C₂	C₂³×#11761612	1	1	yes	characteristic
286	C₂	C₂×#16864163	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂²×C₄×Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₆₇
hypocenter	C₆₇	C₂×Q₈×M₁₆
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
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×#11761612	C₂	direct (N × H)
2. C₂×#16863527	C₂	semidirect
3. C₂×Q₈×C₆₇⋊C₈	C₂	semidirect
9. #16863527	C₂²	semidirect
10. Q₈×C₆₇⋊C₈	C₂²	semidirect
27. C₂×#11761612	Q₈	direct (N × H)
60. #11761612	C₂×Q₈	direct (N × H)
62. C₂×C₆₇⋊C₈	C₂×Q₈	semidirect
68. C₂×Q₈×C₆₇	M₁₆	semidirect
100. Q₈×C₆₇	C₂×M₁₆	semidirect
121. C₆₇⋊C₈	C₂²×Q₈	semidirect
140. C₂×C₆₇	Q₈×M₁₆	semidirect
150. C₆₇	C₂×Q₈×M₁₆	semidirect
206. C₂×Q₈	#11761612	direct (N × H)
244. Q₈	C₂×#11761612	direct (N × H)
283. C₂	Q₈×#11761612	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×#11761612	2
C₂×#16863527	2
C₂×Q₈×C₆₇⋊C₈	2
C₂×#16863420	2
C₂×C₄×#11761612	2
C₂²×C₄×Q₈×C₆₇	2
C₂×Q₈×M₁₆	67

Subgroups — C2×Q8×#11761612 #31720219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#11761612 #31720219