Subgroups — C₂×Q₈⋊#63216 #17958755

Order 4352.

382 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#63216	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#63216	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#63216	C₂	1	4	yes	maximal, direct product factor
4	C₂×Q₈×#3622	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63216	—	2	1	no	—
6	C₂²×#63216	—	2	1	no	—
7	C₄⋊#63216	—	2	1	no	—
8	C₄⋊#63216	C₂²	1	2	yes	—
9	C₄×#63216	—	2	1	no	—
10	C₄×#63216	C₂²	1	2	yes	—
11	Q₈×#3622	—	2	1	no	—
12	Q₈×#3622	—	2	1	no	—
13	Q₈×#3622	C₄	1	2	yes	—
14	Q₈×#3622	C₂²	1	2	yes	—
15	C₂×C₄⋊#3622	—	2	1	no	—
16	C₂×C₄⋊#3622	C₄	1	1	yes	characteristic
17	C₂×C₄×#3622	—	2	1	no	—
18	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
20	C₂²×Q₈×C₁₇	C₈	1	1	yes	characteristic
21	C₄×Q₈×C₁₇	—	2	1	no	—
22	C₄×Q₈×C₁₇	—	2	1	no	—
23	C₄×Q₈×C₁₇	C₈	1	2	yes	—
24	C₄×Q₈×C₁₇	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₇	C₈	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	—	2	1	no	—
28	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×#3622	—	2	1	no	—
30	C₂²×#3622	D₄	1	1	yes	characteristic
31	C₂²×#3622	D₄	1	1	yes	characteristic
32	C₂×#63216	—	2	4	no	—
33	C₂×#63216	—	2	2	no	—
34	C₂×#63216	—	2	2	no	—
35	C₂×#63216	—	4	2	no	—
36	C₄⋊#3622	—	2	2	no	—
37	C₄⋊#3622	—	2	2	no	—
38	C₄⋊#3622	—	2	1	no	—
39	C₄⋊#3622	C₂×C₄	1	2	yes	—
40	C₄×#3622	—	2	2	no	—
41	C₄×#3622	—	2	2	no	—
42	C₄×#3622	C₂×C₄	1	1	yes	characteristic
43	C₄×#3622	D₄	1	1	yes	characteristic
44	C₄×#3622	D₄	1	1	yes	characteristic
45	C₄×#3622	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₁₇	—	2	1	no	—
47	C₂×Q₈×C₁₇	—	2	2	no	—
48	C₂×Q₈×C₁₇	C₁₆	1	4	yes	complemented
49	C₂×Q₈×C₁₇	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₇	—	2	1	no	—
51	C₂²×C₄×C₁₇	—	2	1	no	—
52	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
55	C₄⋊C₄×C₁₇	—	2	2	no	—
56	C₄⋊C₄×C₁₇	—	2	1	no	—
57	C₄⋊C₄×C₁₇	—	2	2	no	—
58	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
59	C₄²×C₁₇	—	2	2	no	—
60	C₄²×C₁₇	—	2	2	no	—
61	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
64	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×#3622	—	2	2	no	—
66	C₂×#3622	—	2	2	no	—
67	C₂×#3622	—	4	2	no	—
68	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×#3622	SD₁₆	1	2	yes	—
71	C₂×#3622	Q₁₆	1	2	yes	—
72	C₂×#3622	C₂×D₄	1	1	yes	characteristic
73	C₂×#3622	C₂×D₄	1	1	yes	characteristic
74	#63216	—	2	4	no	—
75	#63216	—	4	2	no	—
76	C₂×Q₈⋊C₁₆	—	17	1	no	maximal, Sylow
77	C₂³×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
78	Q₈×C₁₇	—	2	4	no	—
79	Q₈×C₁₇	—	2	2	no	—
80	Q₈×C₁₇	C₂×C₁₆	1	4	yes	complemented
81	C₂×C₄×C₁₇	—	2	4	no	—
82	C₂×C₄×C₁₇	—	2	2	no	—
83	C₂×C₄×C₁₇	—	2	2	no	—
84	C₂×C₄×C₁₇	—	2	2	no	—
85	C₂×C₄×C₁₇	—	4	2	no	—
86	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₇	#103	1	2	yes	—
90	C₂×C₄×C₁₇	#104	1	2	yes	—
91	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
92	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
93	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
96	#3622	—	2	1	no	—
97	#3622	—	2	1	no	—
98	#3622	—	4	2	no	—
99	#3622	#103	1	2	yes	—
100	#3622	C₂×SD₁₆	1	1	yes	characteristic
101	#3622	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₈×Q₈	—	17	1	no	—
103	C₂×C₄⋊C₁₆	—	17	1	no	—
104	C₂×C₄×C₁₆	—	17	1	no	—
105	Q₈⋊C₁₆	—	17	4	no	—
106	C₂²×C₁₇	Q₈⋊C₈	1	2	yes	—
107	C₂²×C₁₇	C₂²⋊C₁₆	1	2	yes	—
108	C₂²×C₁₇	#350	1	2	yes	—
109	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
110	C₄×C₁₇	—	2	4	no	—
111	C₄×C₁₇	—	2	1	no	—
112	C₄×C₁₇	—	2	1	no	—
113	C₄×C₁₇	—	4	2	no	—
114	C₄×C₁₇	Q₈⋊C₈	1	2	yes	—
115	C₄×C₁₇	C₂²⋊C₁₆	1	2	yes	—
116	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
117	C₄×C₁₇	C₂×#104	1	1	yes	characteristic
118	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
119	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
120	C₂×C₄×Q₈	#595	1	1	yes	characteristic
121	C₂²×C₁₆	—	34	1	no	—
122	C₂²×C₁₆	—	34	1	no	—
123	C₈×Q₈	—	17	2	no	—
124	C₈×Q₈	—	17	2	no	—
125	C₈×Q₈	—	34	1	no	—
126	C₈×Q₈	—	34	1	no	—
127	C₂×C₄⋊C₈	—	17	1	no	—
128	C₂×C₄⋊C₈	—	34	1	no	—
129	C₂×C₄×C₈	—	17	1	no	—
130	C₂×C₄×C₈	—	34	1	no	—
131	C₄⋊C₁₆	—	17	2	no	—
132	C₄⋊C₁₆	—	34	1	no	—
133	C₄×C₁₆	—	17	2	no	—
134	C₄×C₁₆	—	34	1	no	—
135	C₂×C₁₇	Q₈⋊C₁₆	1	4	yes	complemented
136	C₂×C₁₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
137	C₂×C₁₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
138	C₂×C₁₇	C₂×#350	1	1	yes	characteristic
139	C₂²×Q₈	#3624	1	1	yes	characteristic
140	C₂²×C₈	—	17	1	no	—
141	C₂²×C₈	—	17	1	no	—
142	C₂²×C₈	—	34	1	no	—
143	C₄×Q₈	—	2	1	no	—
144	C₄×Q₈	—	2	1	no	—
145	C₄×Q₈	#3624	1	2	yes	—
146	C₄×Q₈	C₂×#595	1	2	yes	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	#3624	1	1	yes	characteristic
149	C₂×C₄²	—	2	1	no	—
150	C₂×C₄²	C₂×#595	1	1	yes	characteristic
151	C₂×C₁₆	—	34	2	no	—
152	C₂×C₁₆	—	34	4	no	—
153	C₂×C₁₆	—	34	2	no	—
154	C₂×C₁₆	—	68	2	no	—
155	C₄⋊C₈	—	17	2	no	—
156	C₄⋊C₈	—	34	2	no	—
157	C₄⋊C₈	—	34	2	no	—
158	C₄⋊C₈	—	34	1	no	—
159	C₄×C₈	—	17	1	no	—
160	C₄×C₈	—	17	1	no	—
161	C₄×C₈	—	17	1	no	—
162	C₄×C₈	—	17	1	no	—
163	C₄×C₈	—	34	2	no	—
164	C₄×C₈	—	34	2	no	—
165	C₁₇	C₂×Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	1	no	—
168	C₂×Q₈	#63216	1	4	yes	complemented
169	C₂×Q₈	C₂×#3624	1	2	yes	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
173	C₂²×C₄	#63247	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
175	C₂×C₈	—	17	1	no	—
176	C₂×C₈	—	17	1	no	—
177	C₂×C₈	—	17	2	no	—
178	C₂×C₈	—	17	2	no	—
179	C₂×C₈	—	17	1	no	—
180	C₂×C₈	—	17	1	no	—
181	C₂×C₈	—	34	2	no	—
182	C₂×C₈	—	34	2	no	—
183	C₂×C₈	—	68	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	C₂×#3624	1	2	yes	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	C₂×#3624	1	1	yes	characteristic
191	C₄²	#63247	1	1	yes	characteristic
192	C₄²	C₂²⋊#595	1	1	yes	characteristic
193	C₄²	C₂²×#595	1	1	yes	characteristic
194	C₁₆	—	34	4	no	—
195	C₁₆	—	68	2	no	—
196	C₂³	C₂²⋊#3624	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	Q₈	—	2	4	no	—
199	Q₈	C₂×#63216	1	4	yes	complemented
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₂×#63216	1	2	yes	—
206	C₂×C₄	#10587538	1	2	yes	—
207	C₂×C₄	#10587549	1	2	yes	—
208	C₂×C₄	#10587550	1	2	yes	—
209	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
210	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
211	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
212	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
213	C₂×C₄	C₂×#63247	1	1	yes	characteristic
214	C₂×C₄	C₂×C₂²⋊#595	1	1	yes	characteristic
215	C₈	—	17	1	no	—
216	C₈	—	17	2	no	—
217	C₈	—	17	1	no	—
218	C₈	—	34	1	no	—
219	C₈	—	34	1	no	—
220	C₈	—	68	2	no	—
221	C₂²	Q₈⋊#3624	1	2	yes	—
222	C₂²	C₂²⋊#63216	1	2	yes	—
223	C₂²	#11763707	1	2	yes	—
224	C₂²	C₂×C₂²⋊#3624	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	2	no	—
229	C₄	Q₈⋊#3624	1	2	yes	—
230	C₄	C₂²⋊#63216	1	2	yes	—
231	C₄	C₂×#10587549	1	1	yes	characteristic
232	C₄	C₂×#10587550	1	1	yes	characteristic
233	C₄	C₂²×#63216	1	1	yes	characteristic
234	C₄	C₂×#10587538	1	1	yes	characteristic
235	C₂	Q₈⋊#63216	1	4	yes	direct product factor
236	C₂	C₂×Q₈⋊#3624	1	1	yes	characteristic
237	C₂	C₂×C₂²⋊#63216	1	1	yes	characteristic
238	C₂	C₂×#11763707	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#595
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₄²	C₂²×#595
Fitting	C₂×C₄×Q₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²⋊C₈
cosocle	C₄×#3622	C₂³
hypercenter	C₂×C₄×Q₈	#595
hypocenter	C₁₇	C₂×Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
3. Q₈⋊#63216	C₂	direct (N × H)
48. C₂×Q₈×C₁₇	C₁₆	semidirect
80. Q₈×C₁₇	C₂×C₁₆	semidirect
135. C₂×C₁₇	Q₈⋊C₁₆	semidirect
165. C₁₇	C₂×Q₈⋊C₁₆	semidirect
168. C₂×Q₈	#63216	semidirect
199. Q₈	C₂×#63216	semidirect
235. C₂	Q₈⋊#63216	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#63216	2
C₂×C₄×#63216	2
Q₈⋊#63216	2
C₂×Q₈×#3622	2
C₂×Q₈⋊C₁₆	17

Subgroups — C2×Q8⋊#63216 #17958755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈⋊#63216 #17958755