Subgroups — #13179900

Order 1728.

3612 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11724685	C₂	1	1	yes	maximal, characteristic, complemented
2	#11724679	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
4	#11723991	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88861	C₂	1	1	yes	maximal, characteristic, complemented
6	#11723956	C₂	1	2	yes	maximal, complemented
7	SD₁₆×C₃⋊(C₃⋊D₃)	C₂	1	2	yes	maximal, complemented
8	#11723953	C₂	1	2	yes	maximal, complemented
9	#11723952	C₂	1	2	yes	maximal, complemented
10	#11723950	C₂	1	1	yes	maximal, characteristic, complemented
11	#11723948	C₂	1	1	yes	maximal, characteristic, complemented
12	#10596246	—	3	13	no	maximal
13	C₃³×#41	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₃³×D₄	C₂²	1	1	yes	characteristic
15	C₂×#6312	—	2	2	no	—
16	Q₈×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	complemented
17	#89078	—	2	2	no	—
18	#89078	—	2	1	no	—
19	#89078	C₂²	1	2	yes	—
20	#89078	C₂²	1	2	yes	complemented
21	D₄×C₃⋊(C₃⋊D₃)	—	2	1	no	—
22	D₄×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	—
23	#89076	—	2	1	no	—
24	#89076	C₂²	1	1	yes	characteristic
25	#89076	C₂²	1	1	yes	characteristic
26	C₂×#6308	—	2	1	no	—
27	C₂×#6308	C₂²	1	1	yes	characteristic, complemented
28	C₃³×SD₁₆	C₂²	1	2	yes	complemented
29	C₃³×D₈	C₂²	1	2	yes	complemented
30	C₃³×M₁₆	C₂²	1	1	yes	characteristic, complemented
31	#88863	C₂²	1	1	yes	characteristic, complemented
32	#88862	C₂²	1	1	yes	characteristic, complemented
33	#88861	C₂²	1	2	yes	complemented
34	#88861	C₂²	1	1	yes	characteristic, complemented
35	#88861	C₂²	1	2	yes	complemented
36	#88860	C₂²	1	1	yes	characteristic, complemented
37	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic, complemented
38	#88853	C₂²	1	2	yes	complemented
39	#88851	C₂²	1	2	yes	complemented
40	#88850	C₂²	1	2	yes	complemented
41	C₈×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	complemented
42	#64367	—	3	13	no	—
43	#64361	—	3	13	no	—
44	C₃²×AGL(1,ℤ/8)	D₃	1	13	yes	complemented
45	#64159	—	3	13	no	—
46	C₂×#3754	—	3	13	no	—
47	#64124	—	3	26	no	—
48	SD₁₆×C₃⋊D₃	—	3	26	no	—
49	#64121	—	3	26	no	—
50	#64120	—	3	26	no	—
51	#64118	—	3	13	no	—
52	#64116	—	3	13	no	—
53	C₂³×C₃³	—	2	1	no	—
54	C₂²×C₃⋊(C₃⋊D₃)	—	2	2	no	—
55	C₃³×Q₈	C₂³	1	1	yes	characteristic
56	C₃³×D₄	—	2	1	no	—
57	C₃³×D₄	—	2	1	no	—
58	C₃³×D₄	C₂³	1	1	yes	characteristic
59	C₃³×D₄	C₂³	1	2	yes	—
60	C₂×C₃³×C₄	—	2	1	no	—
61	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
62	#6312	—	2	1	no	—
63	#6312	—	2	2	no	—
64	#6312	—	2	2	no	—
65	#6312	—	2	2	no	—
66	#6312	D₄	1	2	yes	—
67	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
68	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
69	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
70	C₄⋊(C₃⋊(C₃⋊D₃))	D₄	1	1	yes	characteristic
71	C₄⋊(C₃⋊(C₃⋊D₃))	C₂³	1	1	yes	characteristic
72	C₄×C₃⋊(C₃⋊D₃)	—	2	2	no	—
73	C₄×C₃⋊(C₃⋊D₃)	—	2	1	no	—
74	C₄×C₃⋊(C₃⋊D₃)	C₂³	1	2	yes	—
75	#6308	—	2	1	no	—
76	#6308	—	2	2	no	—
77	#6308	D₄	1	1	yes	characteristic
78	#6308	C₂³	1	2	yes	—
79	#6308	C₂³	1	1	yes	characteristic
80	C₃³×C₈	C₂³	1	2	yes	—
81	C₃⋊(C₃⋊(C₃⋊C₈))	C₂³	1	1	yes	characteristic
82	C₃⋊(C₃⋊(C₃⋊C₈))	C₂³	1	1	yes	characteristic
83	#5775	—	9	13	no	—
84	C₃²×#41	C₂×D₃	1	13	yes	complemented
85	C₂×C₃²×D₄	C₂×D₃	1	13	yes	—
86	C₂×#638	—	6	26	no	—
87	Q₈×C₃⋊D₃	—	3	26	no	—
88	#3830	—	3	26	no	—
89	#3830	—	3	26	no	—
90	#3830	—	6	13	no	—
91	#3830	—	6	26	no	—
92	D₄×C₃⋊D₃	—	3	26	no	—
93	D₄×C₃⋊D₃	—	6	13	no	—
94	#3828	—	3	13	no	—
95	#3828	—	3	13	no	—
96	#3828	—	6	13	no	—
97	C₂×#634	—	3	13	no	—
98	C₂×#634	—	6	13	no	—
99	C₃²×SD₁₆	C₂×D₃	1	26	yes	complemented
100	C₃²×D₈	C₂×D₃	1	26	yes	complemented
101	C₃²×M₁₆	C₂×D₃	1	13	yes	complemented
102	#3756	—	3	13	no	—
103	#3755	—	3	13	no	—
104	#3754	—	3	26	no	—
105	#3754	—	3	13	no	—
106	#3754	—	3	26	no	—
107	#3753	—	3	13	no	—
108	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
109	#3746	—	3	26	no	—
110	#3744	—	3	26	no	—
111	#3743	—	3	26	no	—
112	C₈×C₃⋊D₃	—	3	26	no	—
113	C₂²×C₃³	—	2	1	no	—
114	C₂²×C₃³	—	2	2	no	—
115	C₂²×C₃³	—	8	1	no	—
116	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
117	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
118	C₂×C₃⋊(C₃⋊D₃)	—	8	2	no	—
119	C₂×C₃⋊(C₃⋊D₃)	C₂×D₄	1	2	yes	—
120	C₃³×C₄	—	2	1	no	—
121	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
122	C₃³×C₄	C₂⁴	1	1	yes	characteristic
123	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
124	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
125	C₃⋊(C₃⋊Dic₃)	C₂×D₄	1	2	yes	—
126	#1010	—	9	13	no	—
127	#1004	—	9	13	no	—
128	C₃×AGL(1,ℤ/8)	C₃⋊D₃	1	13	yes	complemented
129	#950	—	9	13	no	—
130	C₂×#214	—	9	13	no	—
131	#915	—	9	26	no	—
132	D₃×SD₁₆	—	9	26	no	—
133	#912	—	9	26	no	—
134	#911	—	9	26	no	—
135	#909	—	9	13	no	—
136	#907	—	9	13	no	—
137	C₂³×C₃²	—	2	13	no	—
138	C₂²×C₃⋊D₃	—	6	26	no	—
139	C₃²×Q₈	C₂²×D₃	1	13	yes	—
140	C₃²×D₄	—	2	13	no	—
141	C₃²×D₄	—	2	13	no	—
142	C₃²×D₄	C₂²×D₃	1	13	yes	—
143	C₃²×D₄	C₂²×D₃	1	26	yes	—
144	C₂×C₃²×C₄	—	2	13	no	—
145	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
146	#638	—	3	26	no	—
147	#638	—	6	26	no	—
148	#638	—	6	26	no	—
149	#638	—	6	26	no	—
150	#638	—	6	13	no	—
151	C₂×C₃⋊Dic₃	—	6	26	no	—
152	C₂×C₃⋊Dic₃	—	6	26	no	—
153	C₂×C₃⋊Dic₃	—	6	13	no	—
154	C₄⋊(C₃⋊D₃)	—	3	13	no	—
155	C₄⋊(C₃⋊D₃)	—	3	13	no	—
156	C₄×C₃⋊D₃	—	3	26	no	—
157	C₄×C₃⋊D₃	—	6	26	no	—
158	C₄×C₃⋊D₃	—	6	13	no	—
159	#634	—	3	13	no	—
160	#634	—	3	26	no	—
161	#634	—	3	13	no	—
162	#634	—	6	26	no	—
163	#634	—	6	13	no	—
164	C₃²×C₈	C₂²×D₃	1	26	yes	—
165	C₃⋊(C₃⋊C₈)	—	3	13	no	—
166	C₃⋊(C₃⋊C₈)	—	3	13	no	—
167	#577	—	27	1	no	Sylow
168	C₂×C₃³	—	2	1	no	—
169	C₂×C₃³	—	4	2	no	—
170	C₂×C₃³	—	4	1	no	—
171	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
172	C₃⋊(C₃⋊D₃)	—	2	2	no	—
173	C₃⋊(C₃⋊D₃)	—	4	1	no	—
174	C₃×#41	C₂×C₃⋊D₃	1	13	yes	complemented
175	C₂×C₃×D₄	C₂×C₃⋊D₃	1	13	yes	—
176	C₂×#67	—	18	26	no	—
177	D₃×Q₈	—	9	26	no	—
178	#237	—	9	26	no	—
179	#237	—	9	26	no	—
180	#237	—	18	13	no	—
181	#237	—	18	26	no	—
182	D₃×D₄	—	9	26	no	—
183	D₃×D₄	—	18	13	no	—
184	#235	—	9	13	no	—
185	#235	—	9	13	no	—
186	#235	—	18	13	no	—
187	C₂×Dic₆	—	9	13	no	—
188	C₂×Dic₆	—	18	13	no	—
189	C₃×SD₁₆	C₂×C₃⋊D₃	1	26	yes	complemented
190	C₃×D₈	C₂×C₃⋊D₃	1	26	yes	complemented
191	C₃×M₁₆	C₂×C₃⋊D₃	1	13	yes	complemented
192	#216	—	9	13	no	—
193	#215	—	9	13	no	—
194	#214	—	9	26	no	—
195	#214	—	9	13	no	—
196	#214	—	9	26	no	—
197	#213	—	9	13	no	—
198	C₂×C₃⋊C₈	—	9	13	no	—
199	Dic₁₂	—	9	26	no	—
200	#204	—	9	26	no	—
201	ΓL(1,𝔽₂₅)	—	9	26	no	—
202	D₃×C₈	—	9	26	no	—
203	C₂²×C₃²	—	2	13	no	—
204	C₂²×C₃²	—	2	26	no	—
205	C₂²×C₃²	—	8	13	no	—
206	C₂²×C₃²	D₃×D₄	1	13	yes	—
207	C₂×C₃⋊D₃	—	3	26	no	—
208	C₂×C₃⋊D₃	—	6	13	no	—
209	C₂×C₃⋊D₃	—	24	26	no	—
210	C₃²×C₄	—	2	13	no	—
211	C₃²×C₄	D₃×D₄	1	13	yes	—
212	C₃²×C₄	C₂³×D₃	1	13	yes	—
213	C₃⋊Dic₃	—	3	26	no	—
214	C₃⋊Dic₃	—	6	26	no	—
215	C₃⋊Dic₃	—	6	13	no	—
216	2₋¹⁺⁴	—	27	1	no	—
217	H(5,𝔽₂)	—	27	1	no	—
218	#137	—	27	2	no	—
219	#137	—	27	1	no	—
220	AGL(1,ℤ/8)	—	27	2	no	—
221	AGL(1,ℤ/8)	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
222	#135	—	27	1	no	—
223	#135	—	27	2	no	—
224	C₂×SD₁₆	—	27	1	no	—
225	C₂×SD₁₆	—	27	2	no	—
226	#131	—	27	1	no	—
227	C₃³	#577	1	1	yes	characteristic, complemented, Sylow
228	C₂³×C₃	—	2	13	no	—
229	C₂²×D₃	—	18	26	no	—
230	C₃×Q₈	C₂²×C₃⋊D₃	1	13	yes	—
231	C₃×D₄	—	2	13	no	—
232	C₃×D₄	—	2	13	no	—
233	C₃×D₄	C₂²×C₃⋊D₃	1	13	yes	—
234	C₃×D₄	C₂²×C₃⋊D₃	1	26	yes	—
235	C₂×C₃×C₄	—	2	13	no	—
236	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
237	#67	—	9	26	no	—
238	#67	—	18	26	no	—
239	#67	—	18	26	no	—
240	#67	—	18	26	no	—
241	#67	—	18	13	no	—
242	C₂×Dic₃	—	18	26	no	—
243	C₂×Dic₃	—	18	26	no	—
244	C₂×Dic₃	—	18	13	no	—
245	D₁₂	—	9	13	no	—
246	D₁₂	—	9	13	no	—
247	C₄×D₃	—	9	26	no	—
248	C₄×D₃	—	18	26	no	—
249	C₄×D₃	—	18	13	no	—
250	Dic₆	—	9	13	no	—
251	Dic₆	—	9	26	no	—
252	Dic₆	—	9	13	no	—
253	Dic₆	—	18	26	no	—
254	Dic₆	—	18	13	no	—
255	C₃×C₈	C₂²×C₃⋊D₃	1	26	yes	—
256	C₃⋊C₈	—	9	13	no	—
257	C₃⋊C₈	—	9	13	no	—
258	C₂×C₃²	—	2	13	no	—
259	C₂×C₃²	—	4	26	no	—
260	C₂×C₃²	—	4	13	no	—
261	C₂×C₃²	C₂×D₃×D₄	1	13	yes	—
262	C₃⋊D₃	—	6	26	no	—
263	C₃⋊D₃	—	12	13	no	—
264	#41	—	27	1	no	—
265	#41	—	27	2	no	—
266	#41	—	27	2	no	—
267	#41	—	27	1	no	—
268	#41	—	54	1	no	—
269	#41	—	54	2	no	—
270	#41	—	54	1	no	—
271	#41	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
272	C₂×Q₈	—	27	2	no	—
273	C₂×Q₈	—	27	1	no	—
274	C₂×Q₈	—	54	1	no	—
275	C₂×D₄	—	27	2	no	—
276	C₂×D₄	—	54	1	no	—
277	C₂×D₄	—	54	2	no	—
278	C₂×D₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
279	Q₁₆	—	27	1	no	—
280	Q₁₆	—	27	2	no	—
281	SD₁₆	—	27	1	no	—
282	SD₁₆	—	27	2	no	—
283	SD₁₆	—	27	2	no	—
284	SD₁₆	—	27	1	no	—
285	SD₁₆	—	27	2	no	—
286	SD₁₆	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	complemented
287	D₈	—	27	1	no	—
288	D₈	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	complemented
289	M₁₆	—	27	2	no	—
290	M₁₆	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
291	C₂×C₈	—	27	1	no	—
292	C₂×C₈	—	27	2	no	—
293	C₂²×C₃	—	2	13	no	—
294	C₂²×C₃	—	2	26	no	—
295	C₂²×C₃	—	8	13	no	—
296	C₂²×C₃	D₄×C₃⋊D₃	1	13	yes	—
297	C₂×D₃	—	9	26	no	—
298	C₂×D₃	—	18	13	no	—
299	C₂×D₃	—	72	26	no	—
300	C₃×C₄	—	2	13	no	—
301	C₃×C₄	D₄×C₃⋊D₃	1	13	yes	—
302	C₃×C₄	C₂³×C₃⋊D₃	1	13	yes	—
303	Dic₃	—	9	26	no	—
304	Dic₃	—	18	26	no	—
305	Dic₃	—	18	13	no	—
306	C₃²	#5775	1	13	yes	complemented
307	C₂³	—	2	1	no	—
308	C₂³	—	54	2	no	—
309	Q₈	—	27	1	no	—
310	Q₈	—	27	2	no	—
311	Q₈	—	27	1	no	—
312	Q₈	—	54	2	no	—
313	Q₈	—	54	1	no	—
314	Q₈	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
315	D₄	—	2	1	no	—
316	D₄	—	2	1	no	—
317	D₄	—	27	2	no	—
318	D₄	—	27	1	no	—
319	D₄	—	27	1	no	—
320	D₄	—	54	2	no	—
321	D₄	—	54	2	no	—
322	D₄	—	54	2	no	—
323	D₄	—	54	1	no	—
324	D₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
325	D₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	27	2	no	—
328	C₂×C₄	—	54	2	no	—
329	C₂×C₄	—	54	2	no	—
330	C₂×C₄	—	54	2	no	—
331	C₂×C₄	—	54	1	no	—
332	C₂×C₄	—	54	1	no	—
333	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
334	C₈	—	27	1	no	—
335	C₈	—	27	1	no	—
336	C₈	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
337	C₂×C₃	—	2	13	no	—
338	C₂×C₃	—	4	26	no	—
339	C₂×C₃	—	4	13	no	—
340	C₂×C₃	C₂×D₄×C₃⋊D₃	1	13	yes	—
341	D₃	—	18	26	no	—
342	D₃	—	36	13	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	8	1	no	—
346	C₂²	—	27	2	no	—
347	C₂²	—	54	1	no	—
348	C₂²	—	216	2	no	—
349	C₂²	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	27	2	no	—
352	C₄	—	54	1	no	—
353	C₄	—	54	2	no	—
354	C₄	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
355	C₄	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
356	C₃	#10596246	1	13	yes	complemented
357	C₂	—	2	1	no	—
358	C₂	—	4	2	no	—
359	C₂	—	4	1	no	—
360	C₂	—	54	2	no	—
361	C₂	—	108	1	no	—
362	C₂	C₂×D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×C₃⋊(C₃⋊D₃)
commutator, cosocle	C₃³×C₄	C₂⁴
Frattini	C₄	C₂³×C₃⋊(C₃⋊D₃)
Fitting	C₃³×AGL(1,ℤ/8)	C₂
socle	C₂×C₃³	C₂²×D₄
hypercenter	AGL(1,ℤ/8)	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#577
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #11724685	C₂	semidirect
2. #11724679	C₂	semidirect
3. C₃³×AGL(1,ℤ/8)	C₂	semidirect
4. #11723991	C₂	semidirect
5. C₂×#88861	C₂	semidirect
6. #11723956	C₂	semidirect
7. SD₁₆×C₃⋊(C₃⋊D₃)	C₂	semidirect
8. #11723953	C₂	semidirect
9. #11723952	C₂	semidirect
10. #11723950	C₂	semidirect
11. #11723948	C₂	semidirect
13. C₃³×#41	C₂²	semidirect
16. Q₈×C₃⋊(C₃⋊D₃)	C₂²	semidirect
20. #89078	C₂²	semidirect
27. C₂×#6308	C₂²	semidirect
28. C₃³×SD₁₆	C₂²	semidirect
29. C₃³×D₈	C₂²	semidirect
30. C₃³×M₁₆	C₂²	semidirect
31. #88863	C₂²	semidirect
32. #88862	C₂²	semidirect
33. #88861	C₂²	semidirect
34. #88861	C₂²	semidirect
35. #88861	C₂²	semidirect
36. #88860	C₂²	semidirect
37. C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	semidirect
38. #88853	C₂²	semidirect
39. #88851	C₂²	semidirect
40. #88850	C₂²	semidirect
41. C₈×C₃⋊(C₃⋊D₃)	C₂²	semidirect
44. C₃²×AGL(1,ℤ/8)	D₃	semidirect
84. C₃²×#41	C₂×D₃	semidirect
99. C₃²×SD₁₆	C₂×D₃	semidirect
100. C₃²×D₈	C₂×D₃	semidirect
101. C₃²×M₁₆	C₂×D₃	semidirect
128. C₃×AGL(1,ℤ/8)	C₃⋊D₃	semidirect
174. C₃×#41	C₂×C₃⋊D₃	semidirect
189. C₃×SD₁₆	C₂×C₃⋊D₃	semidirect
190. C₃×D₈	C₂×C₃⋊D₃	semidirect
191. C₃×M₁₆	C₂×C₃⋊D₃	semidirect
221. AGL(1,ℤ/8)	C₃⋊(C₃⋊D₃)	semidirect
227. C₃³	#577	semidirect
271. #41	C₂×C₃⋊(C₃⋊D₃)	semidirect
286. SD₁₆	C₂×C₃⋊(C₃⋊D₃)	semidirect
288. D₈	C₂×C₃⋊(C₃⋊D₃)	semidirect
290. M₁₆	C₂×C₃⋊(C₃⋊D₃)	semidirect
306. C₃²	#5775	semidirect
356. C₃	#10596246	semidirect

Maximal subgroups

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

subgroup	index
#11724685	2
#11724679	2
C₃³×AGL(1,ℤ/8)	2
#11723991	2
C₂×#88861	2
#11723956	2
SD₁₆×C₃⋊(C₃⋊D₃)	2
#11723953	2
#11723952	2
#11723950	2
#11723948	2
#10596246	3