Subgroups — #11878886

Order 1152.

576 conjugacy classes of subgroups.

The subgroup lattice has 279 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#10589572	C₂	1	1	yes	maximal, characteristic
3	#10588907	C₂	1	4	yes	maximal
4	#10588765	C₂	1	2	yes	maximal, complemented
5	#10588763	C₂	1	4	yes	maximal, complemented
6	#10588759	C₂	1	2	yes	maximal, complemented
7	#10588605	C₂	1	1	yes	maximal, characteristic
8	#83222	—	3	1	no	maximal
9	C₂²×Q₈×C₉	C₂²	1	2	yes	—
10	C₉×C₄⋊Q₈	—	2	1	no	—
11	C₉×C₄⋊Q₈	—	2	1	no	—
12	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₂×C₉×C₄⋊C₄	—	2	1	no	—
15	C₂×C₉×C₄⋊C₄	—	2	1	no	—
16	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₂×#3674	C₂²	1	8	yes	—
18	#63488	—	2	1	no	—
19	#63486	—	2	1	no	—
20	#63486	C₂²	1	2	yes	—
21	#63432	—	2	1	no	—
22	#63431	C₂²	1	1	yes	characteristic
23	#63431	C₂²	1	1	yes	characteristic
24	C₄×Dic₁₈	—	2	1	no	—
25	C₄×Dic₁₈	C₂²	1	2	yes	—
26	#63395	C₂²	1	8	yes	—
27	#63389	C₂²	1	2	yes	—
28	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
29	C₄×C₉⋊C₈	C₂²	1	2	yes	—
30	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
31	#5470	—	3	1	no	—
32	#5228	—	3	4	no	—
33	#5092	—	3	2	no	—
34	#5090	—	3	4	no	—
35	#5079	—	3	2	no	—
36	#4925	—	3	1	no	—
37	C₂×Q₈×C₉	—	2	2	no	—
38	C₂×Q₈×C₉	—	4	4	no	—
39	C₂×Q₈×C₉	C₂³	1	4	yes	—
40	C₂²×C₄×C₉	—	2	2	no	—
41	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₂×Dic₁₈	—	2	1	no	—
45	C₂×Dic₁₈	C₂³	1	2	yes	—
46	C₉×C₄⋊C₄	—	2	2	no	—
47	C₉×C₄⋊C₄	—	2	2	no	—
48	C₉×C₄⋊C₄	—	2	2	no	—
49	C₉×C₄⋊C₄	—	2	2	no	—
50	C₄²×C₉	D₄	1	1	yes	characteristic
51	C₄²×C₉	D₄	1	1	yes	characteristic
52	C₄²×C₉	C₂³	1	1	yes	characteristic
53	C₄²×C₉	C₂³	1	1	yes	characteristic
54	#3676	—	2	2	no	—
55	#3674	—	4	8	no	—
56	C₄⋊Dic₉	—	2	1	no	—
57	C₄⋊Dic₉	—	2	1	no	—
58	C₄⋊Dic₉	—	2	1	no	—
59	C₄⋊Dic₉	C₂³	1	2	yes	—
60	Dic₉⋊C₄	—	2	2	no	—
61	Dic₉⋊C₄	—	2	2	no	—
62	C₄×Dic₉	—	2	2	no	—
63	C₂×C₉⋊C₈	C₂³	1	4	yes	—
64	#3209	—	9	1	no	Sylow
65	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	—
66	C₃×C₄⋊Q₈	—	2	1	no	—
67	C₃×C₄⋊Q₈	—	2	1	no	—
68	C₃×C₄⋊Q₈	C₂×D₃	1	2	yes	—
69	C₃×C₄⋊Q₈	C₂×D₃	1	2	yes	—
70	C₂×C₃×C₄⋊C₄	—	2	1	no	—
71	C₂×C₃×C₄⋊C₄	—	2	1	no	—
72	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
73	C₂×#216	—	3	8	no	—
74	#926	—	6	1	no	—
75	#924	—	3	2	no	—
76	#924	—	6	1	no	—
77	#870	—	6	1	no	—
78	#869	—	3	1	no	—
79	#869	—	3	1	no	—
80	C₄×Dic₆	—	3	2	no	—
81	C₄×Dic₆	—	6	1	no	—
82	#835	—	3	8	no	—
83	#830	—	3	2	no	—
84	C₄⋊(C₃⋊C₈)	—	3	2	no	—
85	C₄×C₃⋊C₈	—	3	2	no	—
86	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
87	Q₈×C₉	—	2	8	no	—
88	Q₈×C₉	—	4	4	no	—
89	C₂×C₄×C₉	—	2	2	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	4	no	—
92	C₂×C₄×C₉	—	4	4	no	—
93	C₂×C₄×C₉	Q₁₆	1	4	yes	complemented
94	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
100	C₂×Dic₉	—	2	2	no	—
101	C₂×Dic₉	—	2	2	no	—
102	Dic₁₈	—	2	4	no	—
103	Dic₁₈	—	4	2	no	—
104	C₉⋊C₈	—	2	4	no	—
105	#533	—	9	1	no	—
106	C₂×C₄⋊Q₈	D₉	1	1	yes	characteristic
107	#494	—	9	2	no	—
108	#487	—	9	2	no	—
109	#462	—	9	4	no	—
110	#451	—	9	4	no	—
111	#431	—	9	1	no	—
112	C₂×C₃×Q₈	—	2	2	no	—
113	C₂×C₃×Q₈	—	4	4	no	—
114	C₂×C₃×Q₈	C₂²×D₃	1	4	yes	—
115	C₂²×C₃×C₄	—	2	2	no	—
116	C₂²×C₃×C₄	#67	1	1	yes	characteristic
117	C₂²×C₃×C₄	#67	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
119	C₂×Dic₆	—	3	2	no	—
120	C₂×Dic₆	—	6	1	no	—
121	C₃×C₄⋊C₄	—	2	2	no	—
122	C₃×C₄⋊C₄	—	2	2	no	—
123	C₃×C₄⋊C₄	—	2	2	no	—
124	C₃×C₄⋊C₄	—	2	2	no	—
125	C₃×C₄²	#67	1	1	yes	characteristic
126	C₃×C₄²	#67	1	1	yes	characteristic
127	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
128	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
129	#217	—	6	2	no	—
130	#216	—	12	8	no	—
131	C₄⋊Dic₃	—	3	2	no	—
132	C₄⋊Dic₃	—	6	1	no	—
133	C₄⋊Dic₃	—	6	1	no	—
134	C₄⋊Dic₃	—	6	1	no	—
135	Dic₃⋊C₄	—	6	2	no	—
136	Dic₃⋊C₄	—	6	2	no	—
137	C₄×Dic₃	—	6	2	no	—
138	C₂×C₃⋊C₈	—	3	4	no	—
139	C₂²×C₉	—	2	1	no	—
140	C₂²×C₉	—	2	1	no	—
141	C₂²×C₉	C₂×Q₁₆	1	2	yes	—
142	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₉	—	2	1	no	—
144	C₄×C₉	—	2	1	no	—
145	C₄×C₉	—	4	4	no	—
146	C₄×C₉	C₂×Q₁₆	1	2	yes	—
147	C₄×C₉	C₂×Q₁₆	1	2	yes	—
148	C₄×C₉	#137	1	2	yes	—
149	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
150	Dic₉	—	4	2	no	—
151	Dic₉	—	4	2	no	—
152	C₂²×Q₈	C₂×D₉	1	2	yes	—
153	C₂×Q₁₆	—	9	8	no	—
154	C₄⋊Q₈	—	2	1	no	—
155	C₄⋊Q₈	—	2	1	no	—
156	C₄⋊Q₈	—	9	1	no	—
157	C₄⋊Q₈	—	9	1	no	—
158	C₄⋊Q₈	C₂×D₉	1	2	yes	—
159	C₄⋊Q₈	C₂×D₉	1	2	yes	—
160	#125	—	18	1	no	—
161	C₂²⋊Q₈	—	9	2	no	—
162	C₂²⋊Q₈	—	18	1	no	—
163	C₄×Q₈	—	9	2	no	—
164	C₄×Q₈	—	18	1	no	—
165	#117	—	18	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
169	C₄⋊C₈	—	9	2	no	—
170	#103	—	9	8	no	—
171	C₂²⋊C₈	—	9	2	no	—
172	C₄×C₈	—	9	2	no	—
173	C₂³×C₃	C₂×#67	1	1	yes	characteristic
174	C₃×Q₈	—	2	8	no	—
175	C₃×Q₈	—	4	4	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	4	no	—
179	C₂×C₃×C₄	—	4	4	no	—
180	C₂×C₃×C₄	#216	1	4	yes	—
181	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
187	C₂×Dic₃	—	6	2	no	—
188	C₂×Dic₃	—	6	2	no	—
189	Dic₆	—	6	4	no	—
190	Dic₆	—	12	2	no	—
191	C₃⋊C₈	—	6	4	no	—
192	C₂×C₉	—	2	2	no	—
193	C₂×C₉	#535	1	1	yes	characteristic
194	C₂×C₉	C₂²×Q₁₆	1	1	yes	characteristic
195	C₂×C₉	C₂×#137	1	1	yes	characteristic
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂×Q₈	—	9	2	no	—
199	C₂×Q₈	—	18	1	no	—
200	C₂×Q₈	C₂²×D₉	1	4	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	#611	1	1	yes	characteristic
203	C₂²×C₄	#611	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
205	Q₁₆	—	36	8	no	—
206	C₂×C₈	—	9	4	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	9	2	no	—
212	C₄⋊C₄	—	18	1	no	—
213	C₄⋊C₄	—	18	1	no	—
214	C₄⋊C₄	—	18	2	no	—
215	C₄⋊C₄	—	18	2	no	—
216	C₄⋊C₄	—	18	1	no	—
217	C₂²⋊C₄	—	18	2	no	—
218	C₄²	—	18	2	no	—
219	C₄²	#611	1	1	yes	characteristic
220	C₄²	#611	1	1	yes	characteristic
221	C₄²	C₂²×D₉	1	1	yes	characteristic
222	C₄²	C₂²×D₉	1	1	yes	characteristic
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	C₂×#216	1	2	yes	—
226	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	4	4	no	—
230	C₃×C₄	#942	1	2	yes	—
231	C₃×C₄	C₂×#216	1	2	yes	—
232	C₃×C₄	C₂×#216	1	2	yes	—
233	C₃×C₄	#1004	1	2	yes	—
234	Dic₃	—	12	2	no	—
235	Dic₃	—	12	2	no	—
236	C₉	#3209	1	1	yes	characteristic, complemented, Sylow
237	C₂³	C₂×#611	1	1	yes	characteristic
238	Q₈	—	2	8	no	—
239	Q₈	—	4	4	no	—
240	Q₈	—	18	4	no	—
241	Q₈	—	36	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	18	2	no	—
247	C₂×C₄	—	18	2	no	—
248	C₂×C₄	#3674	1	4	yes	complemented
249	C₂×C₄	C₂×#611	1	1	yes	characteristic
250	C₂×C₄	C₂×#611	1	1	yes	characteristic
251	C₂×C₄	C₂×#611	1	1	yes	characteristic
252	C₂×C₄	C₂×#611	1	1	yes	characteristic
253	C₂×C₄	C₂×#611	1	1	yes	characteristic
254	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
255	C₈	—	18	4	no	—
256	C₂×C₃	—	2	2	no	—
257	C₂×C₃	#5805	1	1	yes	characteristic
258	C₂×C₃	C₂×#942	1	1	yes	characteristic
259	C₂×C₃	C₂²×#216	1	1	yes	characteristic
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	C₂×#3674	1	2	yes	—
263	C₂²	C₂²×#611	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	4	4	no	—
267	C₄	—	36	2	no	—
268	C₄	—	36	2	no	—
269	C₄	C₂×#3674	1	2	yes	—
270	C₄	C₂×#3674	1	2	yes	—
271	C₄	#63505	1	2	yes	—
272	C₄	#63710	1	2	yes	—
273	C₃	#83222	1	1	yes	characteristic
274	C₂	—	2	2	no	—
275	C₂	#10589907	1	1	yes	characteristic
276	C₂	C₂²×#3674	1	1	yes	characteristic
277	C₂	C₂×#63505	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#611
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₉×C₄⋊Q₈	C₂
socle	C₂²×C₃	C₂²×#67
prime radical	C₂³×C₃	C₂×#67
hypercenter	C₂×C₄⋊Q₈	D₉
hypocenter	C₉	#3209
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
4. #10588765	C₂	semidirect
5. #10588763	C₂	semidirect
6. #10588759	C₂	semidirect
93. C₂×C₄×C₉	Q₁₆	semidirect
236. C₉	#3209	semidirect
248. C₂×C₄	#3674	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×C₄⋊Q₈	2
#10589572	2
#10588907	2
#10588765	2
#10588763	2
#10588759	2
#10588605	2
#83222	3