Subgroups — C₄×Q₈×#10596842 #34034546

Order 18688.

1070 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#10596842	C₂	1	2	yes	maximal
2	C₄⋊C₄×#10596842	C₂	1	6	yes	maximal
3	C₄×C₄⋊#10596842	C₂	1	3	yes	maximal
4	C₄²×#10596842	C₂	1	3	yes	maximal
5	C₄×Q₈×#64408	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10596842	C₄	1	16	yes	direct product factor
7	C₂×C₄⋊#10596842	C₂²	1	6	yes	—
8	C₂×C₄×#10596842	C₂²	1	3	yes	—
9	C₂×C₄×#10596842	C₂²	1	6	yes	—
10	C₂×C₄×#10596842	C₂²	1	6	yes	—
11	C₂×C₄×#10596842	C₂²	1	1	yes	characteristic
12	(C₂×#10596842)⋊C₄	C₂²	1	6	yes	—
13	C₂×Q₈×#64408	C₄	1	1	yes	characteristic
14	C₂×Q₈×#64408	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×#64408	C₄	1	3	yes	—
16	C₄⋊C₄×#64408	C₂²	1	3	yes	—
17	C₄×C₄⋊#64408	C₄	1	3	yes	—
18	C₄²×#64408	C₂²	1	3	yes	—
19	C₄×Q₈×D₇₃	C₄	1	1	yes	characteristic
20	C₂²×#10596842	C₂³	1	6	yes	—
21	C₂²×#10596842	C₂³	1	2	yes	—
22	C₄⋊#10596842	C₂×C₄	1	24	yes	—
23	C₄×#10596842	—	2	12	no	—
24	C₄×#10596842	—	2	6	no	—
25	C₄×#10596842	C₂×C₄	1	24	yes	—
26	C₄×#10596842	Q₈	1	4	yes	direct product factor
27	Q₈×#64408	C₈	1	8	yes	complemented
28	Q₈×#64408	C₂×C₄	1	8	yes	—
29	C₂×C₄⋊#64408	C₂×C₄	1	3	yes	—
30	C₂×C₄⋊#64408	C₂×C₄	1	3	yes	—
31	C₂×C₄×#64408	C₂×C₄	1	3	yes	—
32	C₂×C₄×#64408	C₂×C₄	1	3	yes	—
33	C₂×C₄×#64408	C₂×C₄	1	3	yes	—
34	C₂×C₄×#64408	C₂³	1	3	yes	—
35	C₂×C₄×#64408	C₂³	1	3	yes	—
36	C₂×C₄×#64408	C₂³	1	1	yes	characteristic
37	(C₂×#64408)⋊C₄	C₂×C₄	1	6	yes	—
38	C₂×Q₈×D₇₃	C₂×C₄	1	1	yes	characteristic
39	Q₈×Dic₇₃	C₈	1	1	yes	characteristic, complemented
40	C₄×Dic₁₄₆	C₈	1	3	yes	complemented
41	#14556556	C₈	1	3	yes	complemented
42	C₄⋊C₄×D₇₃	C₂×C₄	1	3	yes	—
43	C₄²×D₇₃	C₂×C₄	1	3	yes	—
44	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic, complemented
45	C₂×#10596842	—	2	6	no	—
46	C₂×#10596842	—	2	6	no	—
47	C₂×#10596842	C₂²×C₄	1	12	yes	—
48	C₂×#10596842	C₂²×C₄	1	4	yes	—
49	C₂×#10596842	C₂×Q₈	1	4	yes	—
50	C₂×#10596842	#41	1	4	yes	—
51	C₂²×#64408	C₂²×C₄	1	3	yes	—
52	C₂²×#64408	C₂²×C₄	1	3	yes	—
53	C₂²×#64408	C₂²×C₄	1	1	yes	characteristic
54	C₂²×#64408	C₂⁴	1	1	yes	characteristic
55	C₄⋊#64408	C₄²	1	12	yes	—
56	C₄⋊#64408	C₂×C₈	1	12	yes	—
57	C₄×#64408	—	2	6	no	—
58	C₄×#64408	—	2	6	no	—
59	C₄×#64408	—	2	6	no	—
60	C₄×#64408	C₂×C₈	1	12	yes	—
61	C₄×#64408	C₂²×C₄	1	12	yes	—
62	C₄×#64408	C₂×Q₈	1	2	yes	—
63	C₄×#64408	#41	1	2	yes	—
64	C₂×Dic₁₄₆	C₂×C₈	1	3	yes	—
65	Q₈×D₇₃	C₄²	1	4	yes	—
66	Q₈×D₇₃	C₂×C₈	1	4	yes	—
67	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
68	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
69	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
70	C₄⋊Dic₇₃	C₂×C₈	1	3	yes	—
71	#11924007	C₂×C₈	1	6	yes	—
72	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
73	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
74	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
75	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
76	C₄⋊C₄×C₇₃	C₂×C₈	1	3	yes	—
77	C₄²×C₇₃	C₂×C₈	1	3	yes	—
78	C₂²×D₇₃	C₂³×C₄	1	1	yes	characteristic
79	C₂×#64408	—	2	3	no	—
80	C₂×#64408	—	2	3	no	—
81	C₂×#64408	—	2	3	no	—
82	C₂×#64408	—	2	3	no	—
83	C₂×#64408	C₂×C₄²	1	6	yes	—
84	C₂×#64408	C₄×Q₈	1	2	yes	—
85	C₂×#64408	C₂²×C₈	1	6	yes	—
86	C₂×#64408	C₂²×C₈	1	2	yes	—
87	C₂×#64408	#131	1	2	yes	—
88	C₂×#64408	C₂³×C₄	1	2	yes	—
89	C₂×#64408	C₂²×Q₈	1	1	yes	characteristic
90	C₂×#64408	C₂×#41	1	2	yes	—
91	C₂×#64408	C₂×#41	1	1	yes	characteristic
92	#10596842	—	2	12	no	—
93	#10596842	C₄×Q₈	1	8	yes	direct product factor
94	Q₈×C₇₃	C₄×C₈	1	4	yes	complemented
95	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
96	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
97	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
99	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
100	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
101	C₄×D₇₃	—	2	3	no	—
102	C₄×D₇₃	—	2	3	no	—
103	C₄×D₇₃	C₂×C₄²	1	6	yes	—
104	C₄×D₇₃	C₄×Q₈	1	2	yes	—
105	C₄×D₇₃	C₂²×C₈	1	6	yes	—
106	C₄×D₇₃	#131	1	2	yes	—
107	Dic₁₄₆	C₄×C₈	1	12	yes	complemented
108	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
109	C₂×D₇₃	C₂²×C₄²	1	1	yes	characteristic
110	C₂×D₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
111	C₂×D₇₃	C₄×#41	1	1	yes	characteristic
112	C₂×D₇₃	C₂³×C₈	1	1	yes	characteristic
113	C₂×D₇₃	C₂×#131	1	1	yes	characteristic
114	C₂×D₇₃	C₂×#131	1	1	yes	characteristic
115	#64408	—	2	6	no	—
116	#64408	—	2	6	no	—
117	#64408	C₈×Q₈	1	4	yes	complemented
118	#64408	C₂×C₄×Q₈	1	2	yes	—
119	#64408	C₄×#41	1	2	yes	—
120	C₄×C₇₃	—	2	3	no	—
121	C₄×C₇₃	C₂×C₄×C₈	1	6	yes	—
122	C₄×C₇₃	C₈×Q₈	1	2	yes	complemented
123	Dic₇₃	—	2	3	no	—
124	Dic₇₃	C₂×C₄×C₈	1	6	yes	—
125	Dic₇₃	C₈×Q₈	1	2	yes	complemented
126	C₄×C₈×Q₈	—	73	1	no	maximal, Sylow
127	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
128	C₂×C₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
129	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
130	D₇₃	C₄²×Q₈	1	1	yes	characteristic
131	D₇₃	C₄×#131	1	1	yes	characteristic
132	D₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
133	D₇₃	C₈×#41	1	1	yes	characteristic
134	C₂×C₈×Q₈	—	73	2	no	—
135	C₄²×Q₈	—	73	1	no	—
136	C₈×C₄⋊C₄	—	73	6	no	—
137	C₄×C₄⋊C₈	—	73	3	no	—
138	C₄²×C₈	—	73	3	no	—
139	C₇₃	C₄×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
140	C₂×C₄×Q₈	—	73	1	no	—
141	C₂×C₄×Q₈	—	73	1	no	—
142	C₂×C₄×Q₈	—	73	1	no	—
143	C₈×Q₈	—	73	16	no	—
144	C₂×C₄⋊C₈	—	73	6	no	—
145	C₂×C₄×C₈	—	73	6	no	—
146	C₂×C₄×C₈	—	73	1	no	—
147	C₂×C₄×C₈	—	73	3	no	—
148	C₂×C₄×C₈	—	73	6	no	—
149	C₄×C₄⋊C₄	—	73	3	no	—
150	C₄×C₄⋊C₄	—	73	3	no	—
151	C₄×C₄⋊C₄	—	73	3	no	—
152	C₄³	—	73	3	no	—
153	#336	—	73	6	no	—
154	C₂²×Q₈	—	73	1	no	—
155	C₂²×C₈	—	73	2	no	—
156	C₂²×C₈	—	73	6	no	—
157	C₄×Q₈	—	73	8	no	—
158	C₄×Q₈	—	73	8	no	—
159	C₄×Q₈	—	73	1	no	—
160	C₄×Q₈	—	73	3	no	—
161	C₄×Q₈	—	73	3	no	—
162	C₄×Q₈	#10596842	1	1	yes	characteristic, direct product factor
163	C₂×C₄⋊C₄	—	73	3	no	—
164	C₂×C₄⋊C₄	—	73	3	no	—
165	C₂×C₄⋊C₄	—	73	3	no	—
166	C₂×C₄²	—	73	3	no	—
167	C₂×C₄²	—	73	3	no	—
168	C₂×C₄²	—	73	1	no	—
169	C₂×C₄²	—	73	3	no	—
170	C₂×C₄²	—	73	3	no	—
171	C₂×C₄²	—	73	3	no	—
172	C₂×C₄²	—	73	3	no	—
173	C₄⋊C₈	—	73	24	no	—
174	C₄×C₈	—	73	24	no	—
175	C₄×C₈	—	73	4	no	—
176	C₄×C₈	—	146	6	no	—
177	C₄×C₈	—	146	12	no	—
178	#95	—	73	6	no	—
179	C₂×Q₈	—	73	4	no	—
180	C₂×Q₈	—	73	4	no	—
181	C₂×Q₈	—	73	3	no	—
182	C₂×Q₈	C₂×#10596842	1	1	yes	characteristic
183	C₂²×C₄	—	73	1	no	—
184	C₂²×C₄	—	73	1	no	—
185	C₂²×C₄	—	73	3	no	—
186	C₂²×C₄	—	73	1	no	—
187	C₂²×C₄	—	73	3	no	—
188	C₂²×C₄	—	73	3	no	—
189	C₂²×C₄	—	73	3	no	—
190	C₂×C₈	—	73	4	no	—
191	C₂×C₈	—	73	4	no	—
192	C₂×C₈	—	73	4	no	—
193	C₂×C₈	—	73	12	no	—
194	C₂×C₈	—	146	6	no	—
195	C₂×C₈	—	146	6	no	—
196	C₄⋊C₄	—	73	12	no	—
197	C₄⋊C₄	—	73	12	no	—
198	C₄⋊C₄	—	73	6	no	—
199	C₄⋊C₄	—	73	3	no	—
200	C₄⋊C₄	C₂×#10596842	1	3	yes	—
201	C₄²	—	73	12	no	—
202	C₄²	—	73	12	no	—
203	C₄²	—	73	2	no	—
204	C₄²	—	73	2	no	—
205	C₄²	—	73	3	no	—
206	C₄²	—	73	3	no	—
207	C₄²	—	73	3	no	—
208	C₄²	—	146	6	no	—
209	C₄²	—	146	6	no	—
210	C₄²	—	146	6	no	—
211	C₄²	C₂×#10596842	1	3	yes	—
212	C₂³	—	73	1	no	—
213	Q₈	—	73	12	no	—
214	Q₈	C₄×#10596842	1	4	yes	direct product factor
215	C₂×C₄	—	73	2	no	—
216	C₂×C₄	—	73	2	no	—
217	C₂×C₄	—	73	1	no	—
218	C₂×C₄	—	73	1	no	—
219	C₂×C₄	—	73	2	no	—
220	C₂×C₄	—	73	2	no	—
221	C₂×C₄	—	73	2	no	—
222	C₂×C₄	—	73	1	no	—
223	C₂×C₄	—	73	3	no	—
224	C₂×C₄	—	73	6	no	—
225	C₂×C₄	—	73	6	no	—
226	C₂×C₄	—	73	2	no	—
227	C₂×C₄	—	73	2	no	—
228	C₂×C₄	—	73	6	no	—
229	C₂×C₄	—	73	6	no	—
230	C₂×C₄	—	73	3	no	—
231	C₂×C₄	—	146	3	no	—
232	C₂×C₄	—	146	3	no	—
233	C₂×C₄	—	146	3	no	—
234	C₂×C₄	—	146	3	no	—
235	C₂×C₄	—	146	3	no	—
236	C₂×C₄	—	146	3	no	—
237	C₂×C₄	C₂²×#10596842	1	3	yes	—
238	C₂×C₄	C₂²×#10596842	1	3	yes	—
239	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
240	C₈	—	73	8	no	—
241	C₈	—	146	12	no	—
242	C₂²	—	73	1	no	—
243	C₂²	—	73	1	no	—
244	C₂²	—	73	1	no	—
245	C₂²	—	73	1	no	—
246	C₂²	—	73	1	no	—
247	C₂²	—	73	1	no	—
248	C₂²	C₂³×#10596842	1	1	yes	characteristic
249	C₄	—	2	3	no	—
250	C₄	—	73	2	no	—
251	C₄	—	73	2	no	—
252	C₄	—	73	4	no	—
253	C₄	—	73	6	no	—
254	C₄	—	73	2	no	—
255	C₄	—	146	6	no	—
256	C₄	—	146	6	no	—
257	C₄	—	146	3	no	—
258	C₄	C₂×C₄×#10596842	1	6	yes	—
259	C₄	Q₈×#10596842	1	2	yes	direct product factor
260	C₂	—	73	1	no	—
261	C₂	—	73	1	no	—
262	C₂	—	73	1	no	—
263	C₂	—	73	1	no	—
264	C₂	C₂²×C₄×#10596842	1	1	yes	characteristic
265	C₂	C₂×Q₈×#10596842	1	1	yes	characteristic
266	C₂	#41×#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#10596842
commutator	C₂×C₇₃	C₂²×C₄×C₈
Frattini	C₂²	C₂³×#10596842
Fitting	C₄×Q₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂³×C₈
prime radical	C₂²×D₇₃	C₂³×C₄
cosocle	C₂²×#64408	C₂⁴
hypercenter	C₄×Q₈	#10596842
hypocenter	C₇₃	C₄×C₈×Q₈
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
6. Q₈×#10596842	C₄	direct (N × H)
26. C₄×#10596842	Q₈	direct (N × H)
27. Q₈×#64408	C₈	semidirect
39. Q₈×Dic₇₃	C₈	semidirect
40. C₄×Dic₁₄₆	C₈	semidirect
41. #14556556	C₈	semidirect
44. C₄×Q₈×C₇₃	C₈	semidirect
93. #10596842	C₄×Q₈	direct (N × H)
94. Q₈×C₇₃	C₄×C₈	semidirect
107. Dic₁₄₆	C₄×C₈	semidirect
117. #64408	C₈×Q₈	semidirect
122. C₄×C₇₃	C₈×Q₈	semidirect
125. Dic₇₃	C₈×Q₈	semidirect
139. C₇₃	C₄×C₈×Q₈	semidirect
162. C₄×Q₈	#10596842	direct (N × H)
214. Q₈	C₄×#10596842	direct (N × H)
259. C₄	Q₈×#10596842	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×Q₈×#10596842	2
C₄⋊C₄×#10596842	2
C₄×C₄⋊#10596842	2
C₄²×#10596842	2
C₄×Q₈×#64408	2
C₄×C₈×Q₈	73

Subgroups — C4×Q8×#10596842 #34034546

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₈×#10596842 #34034546