Subgroups — C₂×Q₈⋊#11924031 #34010415

Order 18688.

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⋊#11924031	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11924031	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11924031	C₂	1	4	yes	maximal, direct product factor
4	C₂×Q₈×#10596831	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924031	—	2	1	no	—
6	C₂²×#11924031	—	2	1	no	—
7	C₄⋊#11924031	—	2	1	no	—
8	C₄⋊#11924031	C₂²	1	2	yes	—
9	C₄×#11924031	—	2	1	no	—
10	C₄×#11924031	C₂²	1	2	yes	—
11	Q₈×#10596831	—	2	1	no	—
12	Q₈×#10596831	—	2	1	no	—
13	Q₈×#10596831	C₄	1	2	yes	—
14	Q₈×#10596831	C₂²	1	2	yes	—
15	C₂×C₄⋊#10596831	—	2	1	no	—
16	C₂×C₄⋊#10596831	C₄	1	1	yes	characteristic
17	C₂×C₄×#10596831	—	2	1	no	—
18	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
20	C₂×#11924031	—	2	4	no	—
21	C₂×#11924031	—	2	2	no	—
22	C₂×#11924031	—	2	2	no	—
23	C₂×#11924031	—	4	2	no	—
24	C₂²×#10596831	—	2	1	no	—
25	C₂²×#10596831	D₄	1	1	yes	characteristic
26	C₂²×#10596831	D₄	1	1	yes	characteristic
27	C₄⋊#10596831	—	2	2	no	—
28	C₄⋊#10596831	—	2	2	no	—
29	C₄⋊#10596831	—	2	1	no	—
30	C₄⋊#10596831	C₂×C₄	1	2	yes	—
31	C₄×#10596831	—	2	2	no	—
32	C₄×#10596831	—	2	2	no	—
33	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
34	C₄×#10596831	D₄	1	1	yes	characteristic
35	C₄×#10596831	D₄	1	1	yes	characteristic
36	C₄×#10596831	C₂³	1	1	yes	characteristic
37	C₂²×Q₈×C₇₃	C₈	1	1	yes	characteristic
38	C₄×Q₈×C₇₃	—	2	1	no	—
39	C₄×Q₈×C₇₃	—	2	1	no	—
40	C₄×Q₈×C₇₃	C₈	1	2	yes	—
41	C₄×Q₈×C₇₃	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₇₃	C₈	1	1	yes	characteristic
44	C₂×C₄²×C₇₃	—	2	1	no	—
45	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
46	#11924031	—	2	4	no	—
47	#11924031	—	4	2	no	—
48	C₂×#10596831	—	2	2	no	—
49	C₂×#10596831	—	2	2	no	—
50	C₂×#10596831	—	4	2	no	—
51	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×#10596831	SD₁₆	1	2	yes	—
54	C₂×#10596831	Q₁₆	1	2	yes	—
55	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
56	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
57	C₂×Q₈×C₇₃	—	2	1	no	—
58	C₂×Q₈×C₇₃	—	2	2	no	—
59	C₂×Q₈×C₇₃	C₁₆	1	4	yes	complemented
60	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₇₃	—	2	1	no	—
62	C₂²×C₄×C₇₃	—	2	1	no	—
63	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
66	C₄⋊C₄×C₇₃	—	2	2	no	—
67	C₄⋊C₄×C₇₃	—	2	1	no	—
68	C₄⋊C₄×C₇₃	—	2	2	no	—
69	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
70	C₄²×C₇₃	—	2	2	no	—
71	C₄²×C₇₃	—	2	2	no	—
72	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
75	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
76	C₂³×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
77	Q₈×C₇₃	—	2	4	no	—
78	Q₈×C₇₃	—	2	2	no	—
79	Q₈×C₇₃	C₂×C₁₆	1	4	yes	complemented
80	C₂×C₄×C₇₃	—	2	4	no	—
81	C₂×C₄×C₇₃	—	2	2	no	—
82	C₂×C₄×C₇₃	—	2	2	no	—
83	C₂×C₄×C₇₃	—	2	2	no	—
84	C₂×C₄×C₇₃	—	4	2	no	—
85	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₇₃	#103	1	2	yes	—
89	C₂×C₄×C₇₃	#104	1	2	yes	—
90	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
91	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
92	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
95	#10596831	—	2	1	no	—
96	#10596831	—	2	1	no	—
97	#10596831	—	4	2	no	—
98	#10596831	#103	1	2	yes	—
99	#10596831	C₂×SD₁₆	1	1	yes	characteristic
100	#10596831	C₂×Q₁₆	1	1	yes	characteristic
101	C₂²×C₇₃	Q₈⋊C₈	1	2	yes	—
102	C₂²×C₇₃	C₂²⋊C₁₆	1	2	yes	—
103	C₂²×C₇₃	#350	1	2	yes	—
104	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
105	C₄×C₇₃	—	2	4	no	—
106	C₄×C₇₃	—	2	1	no	—
107	C₄×C₇₃	—	2	1	no	—
108	C₄×C₇₃	—	4	2	no	—
109	C₄×C₇₃	Q₈⋊C₈	1	2	yes	—
110	C₄×C₇₃	C₂²⋊C₁₆	1	2	yes	—
111	C₄×C₇₃	C₂×#103	1	1	yes	characteristic
112	C₄×C₇₃	C₂×#104	1	1	yes	characteristic
113	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
114	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
115	C₂×Q₈⋊C₁₆	—	73	1	no	maximal, Sylow
116	C₂×C₇₃	Q₈⋊C₁₆	1	4	yes	complemented
117	C₂×C₇₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
118	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
119	C₂×C₇₃	C₂×#350	1	1	yes	characteristic
120	C₂×C₈×Q₈	—	73	1	no	—
121	C₂×C₄⋊C₁₆	—	73	1	no	—
122	C₂×C₄×C₁₆	—	73	1	no	—
123	Q₈⋊C₁₆	—	73	4	no	—
124	C₇₃	C₂×Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
125	C₂×C₄×Q₈	#64408	1	1	yes	characteristic
126	C₂²×C₁₆	—	146	1	no	—
127	C₂²×C₁₆	—	146	1	no	—
128	C₈×Q₈	—	73	2	no	—
129	C₈×Q₈	—	73	2	no	—
130	C₈×Q₈	—	146	1	no	—
131	C₈×Q₈	—	146	1	no	—
132	C₂×C₄⋊C₈	—	73	1	no	—
133	C₂×C₄⋊C₈	—	146	1	no	—
134	C₂×C₄×C₈	—	73	1	no	—
135	C₂×C₄×C₈	—	146	1	no	—
136	C₄⋊C₁₆	—	73	2	no	—
137	C₄⋊C₁₆	—	146	1	no	—
138	C₄×C₁₆	—	73	2	no	—
139	C₄×C₁₆	—	146	1	no	—
140	C₂²×Q₈	#10596833	1	1	yes	characteristic
141	C₂²×C₈	—	73	1	no	—
142	C₂²×C₈	—	73	1	no	—
143	C₂²×C₈	—	146	1	no	—
144	C₄×Q₈	—	2	1	no	—
145	C₄×Q₈	—	2	1	no	—
146	C₄×Q₈	#10596833	1	2	yes	—
147	C₄×Q₈	C₂×#64408	1	2	yes	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	#10596833	1	1	yes	characteristic
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
152	C₂×C₁₆	—	292	2	no	—
153	C₂×C₁₆	—	146	2	no	—
154	C₂×C₁₆	—	146	2	no	—
155	C₂×C₁₆	—	146	4	no	—
156	C₄⋊C₈	—	73	2	no	—
157	C₄⋊C₈	—	146	2	no	—
158	C₄⋊C₈	—	146	2	no	—
159	C₄⋊C₈	—	146	1	no	—
160	C₄×C₈	—	73	1	no	—
161	C₄×C₈	—	73	1	no	—
162	C₄×C₈	—	73	1	no	—
163	C₄×C₈	—	73	1	no	—
164	C₄×C₈	—	146	2	no	—
165	C₄×C₈	—	146	2	no	—
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	#11924031	1	4	yes	complemented
169	C₂×Q₈	C₂×#10596833	1	2	yes	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
173	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
174	C₂²×C₄	#11924037	1	1	yes	characteristic
175	C₂×C₈	—	292	2	no	—
176	C₂×C₈	—	73	2	no	—
177	C₂×C₈	—	73	2	no	—
178	C₂×C₈	—	73	1	no	—
179	C₂×C₈	—	73	1	no	—
180	C₂×C₈	—	73	1	no	—
181	C₂×C₈	—	73	1	no	—
182	C₂×C₈	—	146	2	no	—
183	C₂×C₈	—	146	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	C₂×#10596833	1	2	yes	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	C₂²⋊#64408	1	1	yes	characteristic
191	C₄²	C₂×#10596833	1	1	yes	characteristic
192	C₄²	#11924037	1	1	yes	characteristic
193	C₄²	C₂²×#64408	1	1	yes	characteristic
194	C₁₆	—	292	2	no	—
195	C₁₆	—	146	4	no	—
196	C₂³	C₂²⋊#10596833	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	Q₈	—	2	4	no	—
199	Q₈	C₂×#11924031	1	4	yes	complemented
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
206	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
207	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
208	C₂×C₄	#14556654	1	2	yes	—
209	C₂×C₄	#14556656	1	2	yes	—
210	C₂×C₄	C₂×#11924031	1	2	yes	—
211	C₂×C₄	#14556665	1	2	yes	—
212	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
213	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
214	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
215	C₈	—	292	2	no	—
216	C₈	—	73	1	no	—
217	C₈	—	73	2	no	—
218	C₈	—	73	1	no	—
219	C₈	—	146	1	no	—
220	C₈	—	146	1	no	—
221	C₂²	Q₈⋊#10596833	1	2	yes	—
222	C₂²	C₂²⋊#11924031	1	2	yes	—
223	C₂²	#17989485	1	2	yes	—
224	C₂²	C₂×C₂²⋊#10596833	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⋊#10596833	1	2	yes	—
230	C₄	C₂²⋊#11924031	1	2	yes	—
231	C₄	C₂×#14556654	1	1	yes	characteristic
232	C₄	C₂×#14556656	1	1	yes	characteristic
233	C₄	C₂²×#11924031	1	1	yes	characteristic
234	C₄	C₂×#14556665	1	1	yes	characteristic
235	C₂	Q₈⋊#11924031	1	4	yes	direct product factor
236	C₂	C₂×Q₈⋊#10596833	1	1	yes	characteristic
237	C₂	C₂×C₂²⋊#11924031	1	1	yes	characteristic
238	C₂	C₂×#17989485	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#64408
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#64408
Fitting	C₂×C₄×Q₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²⋊C₈
cosocle	C₄×#10596831	C₂³
hypercenter	C₂×C₄×Q₈	#64408
hypocenter	C₇₃	C₂×Q₈⋊C₁₆
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
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⋊#11924031	C₂	direct (N × H)
59. C₂×Q₈×C₇₃	C₁₆	semidirect
79. Q₈×C₇₃	C₂×C₁₆	semidirect
116. C₂×C₇₃	Q₈⋊C₁₆	semidirect
124. C₇₃	C₂×Q₈⋊C₁₆	semidirect
168. C₂×Q₈	#11924031	semidirect
199. Q₈	C₂×#11924031	semidirect
235. C₂	Q₈⋊#11924031	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⋊#11924031	2
C₂×C₄×#11924031	2
Q₈⋊#11924031	2
C₂×Q₈×#10596831	2
C₂×Q₈⋊C₁₆	73

Subgroups — C2×Q8⋊#11924031 #34010415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈⋊#11924031 #34010415