Subgroups — #11807113

Order 1152.

744 conjugacy classes of subgroups.

The subgroup lattice has 291 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63431	C₂	1	1	yes	maximal, characteristic
2	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₉	C₂	1	2	yes	maximal
4	#10588652	C₂	1	2	yes	maximal
5	C₄×C₄⋊Dic₉	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
7	#10588608	C₂	1	2	yes	maximal
8	#10588326	C₂	1	4	yes	maximal
9	#73656	—	3	1	no	maximal
10	C₂²×Dic₁₈	C₂²	1	2	yes	—
11	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₉⋊C₄	C₂²	1	4	yes	—
20	C₂×C₄×Dic₉	C₂²	1	2	yes	—
21	C₂×C₄×Dic₉	C₂²	1	4	yes	—
22	#63431	C₄	1	8	yes	complemented
23	C₉×#95	C₂²	1	2	yes	—
24	C₉⋊#95	C₂²	1	2	yes	—
25	C₉⋊#95	C₂²	1	4	yes	—
26	C₉⋊#95	C₂²	1	4	yes	—
27	C₂×#869	—	3	1	no	—
28	C₃×C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
29	Dic₃×C₄⋊C₄	—	3	2	no	—
30	#4972	—	3	2	no	—
31	C₄×C₄⋊Dic₃	—	3	1	no	—
32	C₄×C₄⋊Dic₃	—	3	2	no	—
33	#4928	—	3	2	no	—
34	#4646	—	3	4	no	—
35	C₂²×C₄×C₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂²×Dic₉	C₂³	1	2	yes	—
41	C₂²×Dic₉	C₂³	1	4	yes	—
42	C₂²×Dic₉	C₂³	1	2	yes	—
43	C₂×Dic₁₈	—	4	2	no	—
44	C₂×Dic₁₈	—	4	2	no	—
45	C₂×Dic₁₈	C₂×C₄	1	8	yes	—
46	C₉×C₄⋊C₄	—	2	2	no	—
47	C₉×C₄⋊C₄	Q₈	1	4	yes	complemented
48	C₄²×C₉	—	2	4	no	—
49	C₄²×C₉	C₂×C₄	1	4	yes	—
50	C₄⋊Dic₉	—	2	2	no	—
51	C₄⋊Dic₉	C₂×C₄	1	8	yes	—
52	C₄⋊Dic₉	C₂×C₄	1	8	yes	—
53	C₄⋊Dic₉	Q₈	1	4	yes	complemented
54	Dic₉⋊C₄	—	4	4	no	—
55	C₄×Dic₉	—	2	4	no	—
56	C₄×Dic₉	—	4	4	no	—
57	#2369	—	9	1	no	Sylow
58	C₂²×Dic₆	—	3	2	no	—
59	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
60	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
61	C₂×C₃×C₄²	C₂×D₃	1	2	yes	—
62	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
63	C₂×C₄⋊Dic₃	—	3	2	no	—
64	C₂×C₄⋊Dic₃	—	3	1	no	—
65	C₂×C₄⋊Dic₃	—	3	2	no	—
66	C₂×C₄⋊Dic₃	—	3	1	no	—
67	C₂×Dic₃⋊C₄	—	3	4	no	—
68	C₂×C₄×Dic₃	—	3	2	no	—
69	C₂×C₄×Dic₃	—	3	4	no	—
70	#869	—	3	8	no	—
71	C₃×#95	C₂×D₃	1	2	yes	—
72	C₃⋊#95	—	3	2	no	—
73	C₃⋊#95	—	3	4	no	—
74	C₃⋊#95	—	3	4	no	—
75	C₂³×C₉	C₂⁴	1	1	yes	characteristic
76	C₂×C₄×C₉	—	2	2	no	—
77	C₂×C₄×C₉	—	2	2	no	—
78	C₂×C₄×C₉	—	2	1	no	—
79	C₂×C₄×C₉	—	2	1	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	2	2	no	—
83	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₉	C₂×Q₈	1	4	yes	—
87	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
88	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
89	C₂×C₄×C₉	#41	1	2	yes	—
90	C₂×C₄×C₉	#41	1	2	yes	—
91	C₂×Dic₉	—	2	2	no	—
92	C₂×Dic₉	—	2	2	no	—
93	C₂×Dic₉	—	2	4	no	—
94	C₂×Dic₉	—	2	4	no	—
95	C₂×Dic₉	—	2	2	no	—
96	C₂×Dic₉	—	2	2	no	—
97	C₂×Dic₉	—	2	2	no	—
98	C₂×Dic₉	C₂²×C₄	1	8	yes	—
99	C₂×Dic₉	C₂×Q₈	1	4	yes	—
100	Dic₁₈	—	4	8	no	—
101	C₂×C₄⋊Q₈	—	9	1	no	—
102	#391	—	9	4	no	—
103	#389	—	9	2	no	—
104	#389	—	9	2	no	—
105	C₄×C₄⋊C₄	—	9	2	no	—
106	C₄×C₄⋊C₄	—	9	2	no	—
107	C₄×C₄⋊C₄	—	9	1	no	—
108	C₄×C₄⋊C₄	D₉	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
110	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
111	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
114	C₂²×Dic₃	—	3	2	no	—
115	C₂²×Dic₃	—	3	4	no	—
116	C₂²×Dic₃	—	3	2	no	—
117	C₂×Dic₆	—	3	8	no	—
118	C₂×Dic₆	—	12	2	no	—
119	C₂×Dic₆	—	12	2	no	—
120	C₃×C₄⋊C₄	—	2	2	no	—
121	C₃×C₄⋊C₄	Dic₆	1	4	yes	—
122	C₃×C₄²	—	2	4	no	—
123	C₃×C₄²	C₄×D₃	1	4	yes	—
124	C₄⋊Dic₃	—	3	8	no	—
125	C₄⋊Dic₃	—	3	8	no	—
126	C₄⋊Dic₃	—	3	4	no	—
127	C₄⋊Dic₃	—	6	2	no	—
128	Dic₃⋊C₄	—	12	4	no	—
129	C₄×Dic₃	—	6	4	no	—
130	C₄×Dic₃	—	12	4	no	—
131	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
132	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
133	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
134	C₂²×C₉	C₂×#41	1	1	yes	characteristic
135	C₂²×C₉	C₂×#41	1	1	yes	characteristic
136	C₂²×C₉	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂²×C₉	H(5,𝔽₂)	1	1	yes	characteristic
138	C₄×C₉	—	2	2	no	—
139	C₄×C₉	—	2	4	no	—
140	C₄×C₉	—	4	2	no	—
141	C₄×C₉	C₄×Q₈	1	4	yes	—
142	C₄×C₉	C₄×Q₈	1	4	yes	—
143	Dic₉	—	2	8	no	—
144	Dic₉	—	2	4	no	—
145	Dic₉	—	4	2	no	—
146	C₂²×Q₈	—	9	2	no	—
147	C₄⋊Q₈	—	9	8	no	—
148	C₂×C₄⋊C₄	—	9	2	no	—
149	C₂×C₄⋊C₄	—	9	2	no	—
150	C₂×C₄⋊C₄	—	9	1	no	—
151	C₂×C₄⋊C₄	—	9	1	no	—
152	C₂×C₄⋊C₄	—	9	4	no	—
153	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
155	C₂×C₄²	—	9	2	no	—
156	C₂×C₄²	—	9	4	no	—
157	C₂×C₄²	C₂×D₉	1	2	yes	—
158	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
159	#95	—	9	4	no	—
160	#95	—	9	4	no	—
161	#95	—	9	2	no	—
162	#95	C₂×D₉	1	2	yes	—
163	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
172	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
173	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
174	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
175	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
176	C₂×C₃×C₄	#235	1	2	yes	—
177	C₂×C₃×C₄	#237	1	2	yes	—
178	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
179	C₂×Dic₃	—	3	8	no	—
180	C₂×Dic₃	—	3	4	no	—
181	C₂×Dic₃	—	6	2	no	—
182	C₂×Dic₃	—	6	2	no	—
183	C₂×Dic₃	—	6	4	no	—
184	C₂×Dic₃	—	6	4	no	—
185	C₂×Dic₃	—	6	2	no	—
186	C₂×Dic₃	—	6	2	no	—
187	C₂×Dic₃	—	6	2	no	—
188	Dic₆	—	12	8	no	—
189	C₂×C₉	C₂×C₄×Q₈	1	1	yes	characteristic
190	C₂×C₉	C₂×C₄×Q₈	1	1	yes	characteristic
191	C₂×C₉	#518	1	1	yes	characteristic
192	C₂×C₉	#554	1	1	yes	characteristic
193	C₂×C₉	#554	1	1	yes	characteristic
194	C₂×C₉	#555	1	1	yes	characteristic
195	C₂×C₉	Q₈²	1	1	yes	characteristic
196	C₂×Q₈	—	9	8	no	—
197	C₂×Q₈	—	36	2	no	—
198	C₂×Q₈	—	36	2	no	—
199	C₂²×C₄	—	9	2	no	—
200	C₂²×C₄	—	9	4	no	—
201	C₂²×C₄	—	9	2	no	—
202	C₂²×C₄	C₂²×D₉	1	2	yes	—
203	C₂²×C₄	C₂²×D₉	1	2	yes	—
204	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	9	8	no	—
209	C₄⋊C₄	—	9	8	no	—
210	C₄⋊C₄	—	9	4	no	—
211	C₄⋊C₄	—	18	2	no	—
212	C₄⋊C₄	—	36	4	no	—
213	C₄⋊C₄	Dic₁₈	1	4	yes	complemented
214	C₄²	—	2	4	no	—
215	C₄²	—	18	4	no	—
216	C₄²	—	36	4	no	—
217	C₄²	C₄×D₉	1	4	yes	—
218	C₂²×C₃	C₂²×Dic₆	1	1	yes	characteristic
219	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
220	C₂²×C₃	C₂×#235	1	1	yes	characteristic
221	C₂²×C₃	C₂×#237	1	1	yes	characteristic
222	C₂²×C₃	C₂×D₃×Q₈	1	1	yes	characteristic
223	C₂²×C₃	#1009	1	1	yes	characteristic
224	C₂²×C₃	#1009	1	1	yes	characteristic
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	4	no	—
227	C₃×C₄	—	4	2	no	—
228	C₃×C₄	C₄×Dic₆	1	4	yes	—
229	C₃×C₄	#887	1	4	yes	—
230	Dic₃	—	6	8	no	—
231	Dic₃	—	6	4	no	—
232	Dic₃	—	12	2	no	—
233	C₉	#2369	1	1	yes	characteristic, complemented, Sylow
234	C₂³	C₂³×D₉	1	1	yes	characteristic
235	Q₈	—	36	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	9	8	no	—
244	C₂×C₄	—	9	4	no	—
245	C₂×C₄	—	18	2	no	—
246	C₂×C₄	—	18	2	no	—
247	C₂×C₄	—	18	4	no	—
248	C₂×C₄	—	18	4	no	—
249	C₂×C₄	—	18	2	no	—
250	C₂×C₄	—	18	2	no	—
251	C₂×C₄	—	18	2	no	—
252	C₂×C₄	C₂×Dic₁₈	1	4	yes	—
253	C₂×C₄	C₂×Dic₁₈	1	2	yes	—
254	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
255	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
256	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
257	C₂×C₄	#3697	1	2	yes	—
258	C₂×C₄	#3699	1	2	yes	—
259	C₂×C₄	Q₈×D₉	1	2	yes	—
260	C₂×C₃	C₂×C₄×Dic₆	1	1	yes	characteristic
261	C₂×C₃	C₂×#887	1	1	yes	characteristic
262	C₂×C₃	#5521	1	1	yes	characteristic
263	C₂×C₃	#5543	1	1	yes	characteristic
264	C₂×C₃	#5563	1	1	yes	characteristic
265	C₂×C₃	Q₈×Dic₆	1	1	yes	characteristic
266	C₂×C₃	#5579	1	1	yes	characteristic
267	C₂²	C₂²×Dic₁₈	1	1	yes	characteristic
268	C₂²	C₂²×C₄×D₉	1	1	yes	characteristic
269	C₂²	C₂×#3697	1	1	yes	characteristic
270	C₂²	C₂×#3699	1	1	yes	characteristic
271	C₂²	C₂×Q₈×D₉	1	1	yes	characteristic
272	C₂²	#63715	1	1	yes	characteristic
273	C₂²	#63715	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	4	2	no	—
277	C₄	—	18	8	no	—
278	C₄	—	18	4	no	—
279	C₄	—	36	2	no	—
280	C₄	C₄×Dic₁₈	1	4	yes	—
281	C₄	#63449	1	4	yes	—
282	C₃	#73656	1	1	yes	characteristic
283	C₂	C₂×C₄×Dic₁₈	1	1	yes	characteristic
284	C₂	C₂×#63449	1	1	yes	characteristic
285	C₂	#10589623	1	1	yes	characteristic
286	C₂	#10589645	1	1	yes	characteristic
287	C₂	#10589665	1	1	yes	characteristic
288	C₂	Q₈×Dic₁₈	1	1	yes	characteristic
289	C₂	#10589681	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini, socle, prime radical	C₂³×C₃	C₂³×D₃
Fitting	C₄×C₉×C₄⋊C₄	C₂
cosocle	C₂³×C₉	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₉
hypocenter	C₉	#2369
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
22. #63431	C₄	semidirect
47. C₉×C₄⋊C₄	Q₈	semidirect
53. C₄⋊Dic₉	Q₈	semidirect
213. C₄⋊C₄	Dic₁₈	semidirect
233. C₉	#2369	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63431	2
C₄×C₉×C₄⋊C₄	2
C₄⋊C₄×Dic₉	2
#10588652	2
C₄×C₄⋊Dic₉	2
#10588608	2
#10588326	2
#73656	3