Subgroups — C₂²×#11718668 #16730376

Order 3328.

2790 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88123	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718668	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88047	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#88072	C₂	1	2	yes	maximal
6	C₂²×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×D₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₁₃	—	2	1	no	—
9	C₂³×Dic₂₆	—	2	1	no	—
10	C₂×C₄×D₄×C₁₃	—	2	6	no	—
11	C₂×C₄×D₄×C₁₃	C₂²	1	12	yes	—
12	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₂²⋊C₄	—	2	1	no	—
14	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊Dic₁₃	—	2	1	no	—
16	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
17	C₂³×#1055	C₂²	1	2	yes	—
18	C₂×#88123	—	2	6	no	—
19	C₂×#88123	C₂²	1	12	yes	complemented
20	C₂×#88072	C₂²	1	24	yes	—
21	#11718668	C₂²	1	64	yes	direct product factor
22	C₂×#88047	C₂²	1	24	yes	complemented
23	C₂×C₄⋊#1055	C₂²	1	12	yes	complemented
24	C₂⁵×C₁₃	—	2	1	no	—
25	C₂²×D₄×C₁₃	—	2	3	no	—
26	C₂²×D₄×C₁₃	—	2	4	no	—
27	C₂²×D₄×C₁₃	Q₈	1	8	yes	complemented
28	C₂²×D₄×C₁₃	C₂³	1	6	yes	—
29	C₂³×C₄×C₁₃	—	2	1	no	—
30	C₂³×C₄×C₁₃	—	2	12	no	—
31	C₂³×C₄×C₁₃	—	4	8	no	—
32	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂³×Dic₁₃	—	2	2	no	—
36	C₂²×Dic₂₆	—	2	12	no	—
37	C₂²×Dic₂₆	—	4	8	no	—
38	C₄×D₄×C₁₃	—	2	24	no	—
39	C₄×D₄×C₁₃	C₂³	1	16	yes	—
40	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
42	C₂×C₁₃×C₂²⋊C₄	—	2	12	no	—
43	C₂×C₄²×C₁₃	D₄	1	6	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
45	C₂×C₄⋊Dic₁₃	—	2	6	no	—
46	C₂×C₄⋊Dic₁₃	—	2	12	no	—
47	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
48	C₂²×#1055	—	2	8	no	—
49	C₂²×#1055	C₂³	1	12	yes	—
50	#88123	—	2	24	no	—
51	#88123	C₂³	1	16	yes	complemented
52	#88072	C₂³	1	32	yes	—
53	#88047	C₂³	1	32	yes	complemented
54	C₄⋊#1055	C₂³	1	16	yes	complemented
55	C₂²×#474	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	—	2	8	no	—
57	C₂⁴×C₁₃	—	2	6	no	—
58	C₂⁴×C₁₃	—	4	4	no	—
59	C₂⁴×C₁₃	—	4	4	no	—
60	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×D₄×C₁₃	—	2	24	no	—
62	C₂×D₄×C₁₃	—	2	6	no	—
63	C₂×D₄×C₁₃	—	2	12	no	—
64	C₂×D₄×C₁₃	C₂×Q₈	1	24	yes	complemented
65	C₂×D₄×C₁₃	C₂⁴	1	4	yes	—
66	C₂²×C₄×C₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₃	—	2	6	no	—
68	C₂²×C₄×C₁₃	—	2	16	no	—
69	C₂²×C₄×C₁₃	—	4	4	no	—
70	C₂²×C₄×C₁₃	—	4	48	no	—
71	C₂²×C₄×C₁₃	D₈	1	8	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
75	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₁₃	#41	1	4	yes	—
77	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
78	C₂²×Dic₁₃	—	2	12	no	—
79	C₂²×Dic₁₃	—	4	8	no	—
80	C₂×Dic₂₆	—	2	16	no	—
81	C₂×Dic₂₆	—	4	48	no	—
82	C₁₃×C₄⋊C₄	—	2	6	no	—
83	C₁₃×C₄⋊C₄	C₂⁴	1	4	yes	—
84	C₁₃×C₂²⋊C₄	—	2	16	no	—
85	C₄²×C₁₃	C₂×D₄	1	12	yes	—
86	C₄²×C₁₃	C₂⁴	1	4	yes	—
87	C₄⋊Dic₁₃	—	2	12	no	—
88	C₄⋊Dic₁₃	—	2	16	no	—
89	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
90	C₂×#1055	—	2	24	no	—
91	C₂×#1055	C₂⁴	1	8	yes	—
92	C₂²×C₄⋊Q₈	—	13	1	no	—
93	C₂²×C₄×D₄	D₁₃	1	1	yes	characteristic
94	C₂×#474	—	13	24	no	—
95	C₂²×#107	—	13	2	no	—
96	C₂²×C₄⋊C₈	—	13	1	no	—
97	C₂²×#102	—	13	2	no	—
98	C₂³×C₁₃	—	2	24	no	—
99	C₂³×C₁₃	—	2	4	no	—
100	C₂³×C₁₃	—	4	12	no	—
101	C₂³×C₁₃	—	4	12	no	—
102	C₂³×C₁₃	—	4	32	no	—
103	C₂³×C₁₃	C₂²⋊Q₈	1	4	yes	—
104	C₂³×C₁₃	C₂×D₈	1	4	yes	—
105	C₂³×C₁₃	#137	1	4	yes	—
106	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
107	D₄×C₁₃	—	2	24	no	—
108	D₄×C₁₃	—	2	4	no	—
109	D₄×C₁₃	C₂²×Q₈	1	8	yes	complemented
110	C₂×C₄×C₁₃	—	2	12	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	4	12	no	—
113	C₂×C₄×C₁₃	—	4	32	no	—
114	C₂×C₄×C₁₃	C₂²⋊Q₈	1	12	yes	—
115	C₂×C₄×C₁₃	C₂×D₈	1	24	yes	—
116	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
117	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
118	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
119	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
120	C₂×C₄×C₁₃	C₂×#41	1	6	yes	—
121	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
122	C₂×Dic₁₃	—	2	8	no	—
123	C₂×Dic₁₃	—	4	24	no	—
124	Dic₂₆	—	4	32	no	—
125	#1055	—	2	8	no	—
126	C₂³×Q₈	—	26	1	no	—
127	C₂³×D₄	C₂×D₁₃	1	1	yes	characteristic
128	C₂⁴×C₄	—	2	1	no	—
129	C₂³×C₈	—	13	2	no	—
130	C₂×C₄⋊Q₈	—	13	12	no	—
131	C₂×C₄⋊Q₈	—	26	6	no	—
132	C₂×C₄×D₄	—	2	6	no	—
133	C₂×C₄×D₄	C₂×D₁₃	1	12	yes	—
134	C₂²×C₄⋊C₄	—	13	2	no	—
135	C₂²×C₄⋊C₄	—	26	1	no	—
136	C₂²×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
137	C₂²×C₂²⋊C₄	—	2	1	no	—
138	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
139	#474	—	13	64	no	—
140	C₂×#107	—	13	24	no	—
141	C₂×C₄⋊C₈	—	13	12	no	—
142	C₂×#102	—	13	24	no	—
143	C₂²×C₁₃	—	2	8	no	—
144	C₂²×C₁₃	—	4	4	no	—
145	C₂²×C₁₃	—	4	4	no	—
146	C₂²×C₁₃	—	4	48	no	—
147	C₂²×C₁₃	#474	1	16	yes	complemented
148	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	6	yes	—
149	C₂²×C₁₃	C₂²×D₈	1	6	yes	—
150	C₂²×C₁₃	C₂×#137	1	6	yes	—
151	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
152	C₄×C₁₃	—	2	4	no	—
153	C₄×C₁₃	—	4	4	no	—
154	C₄×C₁₃	C₂×C₂²⋊Q₈	1	6	yes	—
155	C₄×C₁₃	C₂²×D₈	1	8	yes	—
156	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
157	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
158	Dic₁₃	—	4	8	no	—
159	C₂⁵	—	2	1	no	—
160	C₂²×Q₈	—	26	12	no	—
161	C₂²×Q₈	—	52	8	no	—
162	C₂²×D₄	—	2	4	no	—
163	C₂²×D₄	—	2	3	no	—
164	C₂²×D₄	Dic₂₆	1	8	yes	complemented
165	C₂²×D₄	C₂²×D₁₃	1	6	yes	—
166	C₂³×C₄	—	2	1	no	—
167	C₂³×C₄	—	2	12	no	—
168	C₂³×C₄	—	4	8	no	—
169	C₂³×C₄	—	26	2	no	—
170	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
171	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
172	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
173	C₂²×C₈	—	13	12	no	—
174	C₂²×C₈	—	26	8	no	—
175	C₄⋊Q₈	—	13	16	no	—
176	C₄⋊Q₈	—	26	24	no	—
177	C₄×D₄	—	2	24	no	—
178	C₄×D₄	C₂²×D₁₃	1	16	yes	—
179	C₂×C₄⋊C₄	—	2	3	no	—
180	C₂×C₄⋊C₄	—	13	12	no	—
181	C₂×C₄⋊C₄	—	26	6	no	—
182	C₂×C₄⋊C₄	—	26	12	no	—
183	C₂×C₄⋊C₄	C₂²×D₁₃	1	6	yes	—
184	C₂×C₂²⋊C₄	—	2	12	no	—
185	C₂×C₄²	C₂²⋊D₁₃	1	6	yes	—
186	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
187	#107	—	13	32	no	—
188	C₄⋊C₈	—	13	16	no	—
189	#102	—	13	32	no	—
190	C₂×C₁₃	—	4	8	no	—
191	C₂×C₁₃	C₂×#474	1	12	yes	complemented
192	C₂×C₁₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
193	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
194	C₂×C₁₃	C₂²×#137	1	1	yes	characteristic
195	C₂⁴	—	2	8	no	—
196	C₂⁴	—	2	6	no	—
197	C₂⁴	—	4	4	no	—
198	C₂⁴	—	4	4	no	—
199	C₂⁴	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
200	C₂×Q₈	—	26	16	no	—
201	C₂×Q₈	—	52	48	no	—
202	C₂×D₄	—	2	24	no	—
203	C₂×D₄	—	2	12	no	—
204	C₂×D₄	—	2	6	no	—
205	C₂×D₄	C₂×Dic₂₆	1	24	yes	complemented
206	C₂×D₄	C₂³×D₁₃	1	4	yes	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	6	no	—
209	C₂²×C₄	—	2	16	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	4	48	no	—
212	C₂²×C₄	—	26	12	no	—
213	C₂²×C₄	—	52	8	no	—
214	C₂²×C₄	#6104	1	8	yes	—
215	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
216	C₂²×C₄	#6127	1	4	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
218	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
219	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
220	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
221	C₂×C₈	—	13	8	no	—
222	C₂×C₈	—	26	24	no	—
223	C₄⋊C₄	—	2	6	no	—
224	C₄⋊C₄	—	13	8	no	—
225	C₄⋊C₄	—	26	12	no	—
226	C₄⋊C₄	—	26	16	no	—
227	C₄⋊C₄	C₂³×D₁₃	1	4	yes	—
228	C₂²⋊C₄	—	2	16	no	—
229	C₄²	C₂×C₂²⋊D₁₃	1	12	yes	—
230	C₄²	C₂³×D₁₃	1	4	yes	—
231	C₁₃	C₂²×#474	1	1	yes	characteristic, complemented, Sylow
232	C₂³	—	2	24	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	4	32	no	—
235	C₂³	—	4	12	no	—
236	C₂³	—	4	12	no	—
237	C₂³	#88178	1	4	yes	—
238	C₂³	C₂×#6104	1	4	yes	—
239	C₂³	#88205	1	4	yes	—
240	C₂³	C₂²×C₂²⋊D₁₃	1	3	yes	—
241	Q₈	—	52	32	no	—
242	D₄	—	2	24	no	—
243	D₄	—	2	4	no	—
244	D₄	C₂²×Dic₂₆	1	8	yes	complemented
245	C₂×C₄	—	2	12	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	12	no	—
248	C₂×C₄	—	4	32	no	—
249	C₂×C₄	—	26	8	no	—
250	C₂×C₄	—	52	24	no	—
251	C₂×C₄	#88178	1	12	yes	—
252	C₂×C₄	C₂×#6104	1	24	yes	—
253	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
254	C₂×C₄	C₂×#6127	1	6	yes	—
255	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
256	C₂×C₄	C₂²×C₂²⋊D₁₃	1	3	yes	—
257	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
258	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
259	C₈	—	26	8	no	—
260	C₂²	—	2	8	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	4	48	no	—
264	C₂²	#11718668	1	16	yes	direct product factor
265	C₂²	C₂×#88178	1	6	yes	—
266	C₂²	C₂²×#6104	1	6	yes	—
267	C₂²	C₂×#88205	1	6	yes	—
268	C₂²	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
269	C₄	—	2	4	no	—
270	C₄	—	4	4	no	—
271	C₄	—	52	8	no	—
272	C₄	C₂×#88178	1	6	yes	—
273	C₄	C₂²×#6104	1	8	yes	—
274	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
275	C₄	C₂²×#6127	1	1	yes	characteristic
276	C₂	—	4	8	no	—
277	C₂	C₂×#11718668	1	12	yes	direct product factor
278	C₂	C₂²×#88178	1	1	yes	characteristic
279	C₂	C₂³×#6104	1	1	yes	characteristic
280	C₂	C₂²×#88205	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×D₄×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂×D₄
prime radical	C₂³×D₄×C₁₃	C₂²
hypercenter	C₂²×C₄×D₄	D₁₃
hypocenter	C₁₃	C₂²×#474
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#88123	C₂	semidirect
2. C₂×#11718668	C₂	direct (N × H)
3. C₂²×#88047	C₂	semidirect
4. C₂²×C₄⋊#1055	C₂	semidirect
19. C₂×#88123	C₂²	semidirect
21. #11718668	C₂²	direct (N × H)
22. C₂×#88047	C₂²	semidirect
23. C₂×C₄⋊#1055	C₂²	semidirect
27. C₂²×D₄×C₁₃	Q₈	semidirect
51. #88123	C₂³	semidirect
53. #88047	C₂³	semidirect
54. C₄⋊#1055	C₂³	semidirect
64. C₂×D₄×C₁₃	C₂×Q₈	semidirect
109. D₄×C₁₃	C₂²×Q₈	semidirect
147. C₂²×C₁₃	#474	semidirect
164. C₂²×D₄	Dic₂₆	semidirect
191. C₂×C₁₃	C₂×#474	semidirect
205. C₂×D₄	C₂×Dic₂₆	semidirect
231. C₁₃	C₂²×#474	semidirect
244. D₄	C₂²×Dic₂₆	semidirect
264. C₂²	#11718668	direct (N × H)
277. C₂	C₂×#11718668	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88123	2
C₂×#11718668	2
C₂²×#88047	2
C₂²×C₄⋊#1055	2
C₂²×#88072	2
C₂²×C₄×D₄×C₁₃	2
C₂²×#474	13

Subgroups — C22×#11718668 #16730376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718668 #16730376