Subgroups — #10813636

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#79012	C₂	1	2	yes	maximal, complemented
2	C₂×#4717	C₂	1	2	yes	maximal
3	C₄×#904	C₂	1	1	yes	maximal, characteristic
4	C₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
5	#69844	C₂	1	2	yes	maximal, complemented
6	#69792	C₂	1	1	yes	maximal, characteristic, complemented
7	#69494	C₂	1	2	yes	maximal, complemented
8	#68694	C₂	1	4	yes	maximal, complemented
9	#22686	—	3	1	no	maximal, Sylow
10	C₂×#904	C₂²	1	1	yes	characteristic
11	C₂×#901	C₂²	1	4	yes	—
12	C₄×#235	C₂²	1	2	yes	—
13	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×#97	C₂²	1	2	yes	—
15	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	#5113	C₂²	1	2	yes	—
18	C₂×#819	C₂²	1	2	yes	—
19	C₂×#819	C₂²	1	2	yes	—
20	#5106	C₂²	1	1	yes	characteristic
21	#5102	C₂²	1	4	yes	—
22	C₂×#805	C₂²	1	4	yes	—
23	#4924	C₂²	1	2	yes	—
24	#4839	—	2	4	no	—
25	#4837	—	2	2	no	—
26	D₃×C₄⋊C₈	—	2	2	no	—
27	#4717	—	2	8	no	—
28	C₄×Dic₁₂	C₄	1	1	yes	characteristic, complemented
29	C₄×Dic₁₂	C₄	1	1	yes	characteristic, complemented
30	C₄×D₂₄	C₄	1	1	yes	characteristic, complemented
31	C₄×D₂₄	C₄	1	1	yes	characteristic, complemented
32	C₄×#204	C₄	1	2	yes	complemented
33	C₄×#204	C₄	1	2	yes	complemented
34	C₃×#336	C₂²	1	2	yes	—
35	#4523	C₂²	1	4	yes	—
36	C₂×#444	—	3	2	no	—
37	C₄×#135	—	3	1	no	—
38	#2906	—	3	2	no	—
39	#1946	—	3	2	no	—
40	#1866	—	3	4	no	—
41	#1848	—	3	1	no	—
42	#1806	—	3	2	no	—
43	#1770	D₃	1	1	yes	characteristic, complemented
44	C₂×#235	C₂³	1	2	yes	—
45	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₃×C₈	C₂³	1	2	yes	—
47	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₄⋊C₄	—	2	1	no	—
49	C₂×C₃×C₄²	C₂³	1	2	yes	—
50	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
51	C₄×#67	—	2	2	no	—
52	C₄×#67	—	2	2	no	—
53	#926	C₂³	1	2	yes	—
54	C₂×#208	C₂³	1	4	yes	—
55	C₂×Dic₁₂	C₂×C₄	1	1	yes	characteristic
56	#904	D₄	1	4	yes	—
57	#904	Q₈	1	4	yes	—
58	C₂×D₂₄	C₂×C₄	1	1	yes	characteristic
59	C₂×#204	C₂×C₄	1	2	yes	—
60	#901	—	2	8	no	—
61	#901	—	4	4	no	—
62	C₂×ΓL(1,𝔽₂₅)	—	2	4	no	—
63	C₂×D₃×C₈	—	2	4	no	—
64	C₄×D₁₂	C₂×C₄	1	2	yes	—
65	C₄×D₁₂	C₂×C₄	1	2	yes	—
66	#872	—	2	2	no	—
67	C₄²×D₃	—	2	2	no	—
68	C₄×Dic₆	C₂×C₄	1	2	yes	—
69	C₄×Dic₆	C₂×C₄	1	2	yes	—
70	C₃×C₄⋊C₈	D₄	1	2	yes	—
71	C₃×C₄⋊C₈	D₄	1	2	yes	—
72	C₃×#97	—	2	4	no	—
73	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
74	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
75	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
76	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
77	C₃×#95	—	2	1	no	—
78	#821	C₂×C₄	1	2	yes	—
79	#821	C₂×C₄	1	2	yes	—
80	#819	—	2	4	no	—
81	#819	—	4	2	no	—
82	#818	C₂×C₄	1	1	yes	characteristic
83	#818	C₂×C₄	1	1	yes	characteristic
84	#817	C₂×C₄	1	2	yes	—
85	#816	C₂×C₄	1	2	yes	—
86	#816	C₂×C₄	1	2	yes	—
87	Dic₃⋊C₈	—	2	4	no	—
88	Dic₃⋊C₈	—	2	4	no	—
89	#805	—	2	8	no	—
90	C₄⋊(C₃⋊C₈)	—	2	2	no	—
91	C₄⋊(C₃⋊C₈)	—	2	2	no	—
92	C₂×#135	—	3	1	no	—
93	C₂×#131	—	3	4	no	—
94	C₄×#41	—	3	2	no	—
95	#444	—	6	8	no	—
96	C₄×Q₁₆	—	3	1	no	—
97	C₄×Q₁₆	—	3	1	no	—
98	C₄×SD₁₆	—	3	2	no	—
99	C₄×SD₁₆	—	3	2	no	—
100	C₄×D₈	—	3	1	no	—
101	C₄×D₈	—	3	1	no	—
102	C₂×#108	—	3	2	no	—
103	C₂×#108	—	3	2	no	—
104	#427	—	3	1	no	—
105	#424	—	3	4	no	—
106	#424	—	6	4	no	—
107	#423	—	3	2	no	—
108	#423	—	6	2	no	—
109	C₂×C₄⋊C₈	—	6	2	no	—
110	C₂×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
111	C₂×#104	—	3	4	no	—
112	#416	—	3	2	no	—
113	C₂×#97	C₂×D₃	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
115	C₄×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
116	#339	—	3	4	no	—
117	#336	C₂×D₃	1	2	yes	—
118	C₂²×C₃×C₄	—	2	2	no	—
119	C₂²×C₃×C₄	#41	1	1	yes	characteristic
120	C₂²×C₃×C₄	#41	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
122	C₂×#67	—	2	2	no	—
123	#235	—	2	2	no	—
124	#235	—	2	2	no	—
125	#235	C₂×D₄	1	4	yes	—
126	#235	C₂×Q₈	1	4	yes	—
127	C₂×D₁₂	C₂²×C₄	1	2	yes	—
128	C₂×C₄×D₃	—	2	2	no	—
129	C₂×C₄×D₃	—	2	2	no	—
130	C₂×C₄×D₃	—	2	2	no	—
131	C₂×Dic₆	C₂²×C₄	1	2	yes	—
132	C₂×C₃×C₈	—	2	2	no	—
133	C₂×C₃×C₈	—	2	1	no	—
134	C₂×C₃×C₈	—	2	2	no	—
135	C₂×C₃×C₈	—	2	1	no	—
136	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
141	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
142	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
143	C₃×C₄⋊C₄	—	4	2	no	—
144	C₃×C₄²	—	2	2	no	—
145	C₃×C₄²	—	2	2	no	—
146	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
147	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
148	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
149	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
150	#217	—	2	2	no	—
151	(C₂×D₃)⋊C₄	—	2	2	no	—
152	(C₂×D₃)⋊C₄	—	2	2	no	—
153	C₄⋊Dic₃	C₂²×C₄	1	2	yes	—
154	C₄⋊Dic₃	C₂²×C₄	1	2	yes	—
155	Dic₃⋊C₄	—	2	2	no	—
156	Dic₃⋊C₄	—	2	2	no	—
157	C₄×Dic₃	—	2	2	no	—
158	C₄×Dic₃	—	2	2	no	—
159	C₄×Dic₃	—	2	2	no	—
160	#208	—	2	4	no	—
161	#208	—	2	4	no	—
162	C₂×C₃⋊C₈	—	2	4	no	—
163	Dic₁₂	C₄⋊C₄	1	4	yes	complemented
164	D₂₄	C₄⋊C₄	1	4	yes	complemented
165	#204	C₄⋊C₄	1	8	yes	complemented
166	ΓL(1,𝔽₂₅)	—	4	8	no	—
167	D₃×C₈	—	4	8	no	—
168	C₂×#41	—	3	2	no	—
169	#135	—	3	4	no	—
170	#135	—	3	4	no	—
171	C₂×Q₁₆	—	3	1	no	—
172	C₂×SD₁₆	—	3	2	no	—
173	C₂×D₈	—	3	1	no	—
174	#131	—	6	8	no	—
175	#131	—	12	4	no	—
176	C₂×M₁₆	—	3	4	no	—
177	C₂×M₁₆	—	6	4	no	—
178	C₂²×C₈	—	6	4	no	—
179	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₃	1	2	yes	—
181	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
182	C₄×Q₈	—	3	2	no	—
183	C₄×Q₈	—	3	2	no	—
184	C₄×D₄	—	3	2	no	—
185	C₄×D₄	—	3	2	no	—
186	C₄×D₄	—	6	2	no	—
187	C₄×D₄	—	6	2	no	—
188	#117	—	3	2	no	—
189	#117	—	6	2	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄²	—	6	2	no	—
192	C₂×C₄²	C₂²×D₃	1	2	yes	—
193	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
194	#108	—	6	4	no	—
195	#108	—	12	2	no	—
196	#107	—	3	1	no	—
197	#107	—	3	1	no	—
198	#106	—	3	2	no	—
199	C₄⋊C₈	—	6	4	no	—
200	C₄⋊C₈	—	6	4	no	—
201	C₄⋊C₈	—	6	2	no	—
202	C₄⋊C₈	—	6	2	no	—
203	C₄⋊C₈	D₁₂	1	2	yes	—
204	C₄⋊C₈	D₁₂	1	2	yes	—
205	#104	—	6	8	no	—
206	#103	—	3	2	no	—
207	#103	—	3	2	no	—
208	#102	—	3	2	no	—
209	#102	—	3	2	no	—
210	#97	—	2	4	no	—
211	C₄×C₈	C₄×D₃	1	1	yes	characteristic
212	C₄×C₈	C₄×D₃	1	1	yes	characteristic
213	C₄×C₈	C₄×D₃	1	1	yes	characteristic
214	C₄×C₈	C₄×D₃	1	1	yes	characteristic
215	#95	—	2	1	no	—
216	C₂³×C₃	C₂×#41	1	1	yes	characteristic
217	C₂²×D₃	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	4	1	no	—
220	C₂×C₃×C₄	—	4	2	no	—
221	C₂×C₃×C₄	—	4	2	no	—
222	C₂×C₃×C₄	—	4	2	no	—
223	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
224	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
225	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
226	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
231	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
232	C₂×C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
233	#67	—	2	4	no	—
234	#67	—	2	4	no	—
235	C₂×Dic₃	—	2	2	no	—
236	C₂×Dic₃	—	2	2	no	—
237	C₂×Dic₃	—	2	2	no	—
238	D₁₂	—	2	2	no	—
239	D₁₂	C₂×C₄⋊C₄	1	4	yes	—
240	C₄×D₃	—	2	4	no	—
241	C₄×D₃	—	2	4	no	—
242	C₄×D₃	—	4	4	no	—
243	C₄×D₃	—	8	2	no	—
244	Dic₆	—	2	2	no	—
245	Dic₆	C₂×C₄⋊C₄	1	4	yes	—
246	C₃×C₈	—	2	1	no	—
247	C₃×C₈	—	2	1	no	—
248	C₃×C₈	—	2	4	no	—
249	C₃×C₈	C₂×C₄⋊C₄	1	2	yes	—
250	C₃×C₈	C₂×C₄⋊C₄	1	2	yes	—
251	C₃⋊C₈	—	4	4	no	—
252	#41	—	3	4	no	—
253	#41	—	3	4	no	—
254	#41	—	6	2	no	—
255	#41	—	6	2	no	—
256	C₂×Q₈	—	3	2	no	—
257	C₂×D₄	—	3	2	no	—
258	C₂×D₄	—	6	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	6	2	no	—
261	C₂²×C₄	—	6	2	no	—
262	C₂²×C₄	—	6	2	no	—
263	C₂²×C₄	#235	1	1	yes	characteristic
264	C₂²×C₄	#235	1	1	yes	characteristic
265	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
266	Q₁₆	—	3	4	no	—
267	SD₁₆	—	3	8	no	—
268	D₈	—	3	4	no	—
269	M₁₆	—	6	4	no	—
270	M₁₆	—	6	4	no	—
271	M₁₆	—	12	8	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	6	4	no	—
277	C₂×C₈	—	12	8	no	—
278	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
279	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
280	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
281	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
282	C₂×C₈	C₂×D₁₂	1	4	yes	—
283	C₂×C₈	D₃×D₄	1	2	yes	—
284	C₂×C₈	D₃×Q₈	1	2	yes	—
285	C₄⋊C₄	—	3	2	no	—
286	C₄⋊C₄	—	3	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	6	2	no	—
289	C₄⋊C₄	—	6	2	no	—
290	C₂²⋊C₄	—	6	2	no	—
291	C₂²⋊C₄	—	6	2	no	—
292	C₂²⋊C₄	—	6	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	6	2	no	—
296	C₄²	—	6	2	no	—
297	C₄²	—	6	2	no	—
298	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
299	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
300	C₄²	C₂×D₁₂	1	1	yes	characteristic
301	C₄²	C₂×D₁₂	1	1	yes	characteristic
302	C₂²×C₃	—	2	1	no	—
303	C₂²×C₃	—	2	1	no	—
304	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
305	C₂²×C₃	#547	1	1	yes	characteristic
306	C₂²×C₃	#554	1	1	yes	characteristic
307	C₂×D₃	—	2	4	no	—
308	C₂×D₃	—	4	2	no	—
309	C₂×D₃	—	4	2	no	—
310	C₃×C₄	—	2	1	no	—
311	C₃×C₄	—	2	1	no	—
312	C₃×C₄	—	2	1	no	—
313	C₃×C₄	—	8	2	no	—
314	C₃×C₄	#444	1	2	yes	complemented
315	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
316	C₃×C₄	#520	1	1	yes	characteristic
317	C₃×C₄	D₄²	1	1	yes	characteristic
318	C₃×C₄	D₄×Q₈	1	1	yes	characteristic
319	Dic₃	—	2	4	no	—
320	Dic₃	—	4	2	no	—
321	Dic₃	—	4	2	no	—
322	C₂³	—	6	2	no	—
323	C₂³	C₂×#235	1	1	yes	characteristic
324	Q₈	—	3	4	no	—
325	Q₈	—	6	2	no	—
326	D₄	—	3	4	no	—
327	D₄	—	6	4	no	—
328	D₄	—	6	4	no	—
329	D₄	—	6	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	6	2	no	—
336	C₂×C₄	—	6	4	no	—
337	C₂×C₄	—	6	4	no	—
338	C₂×C₄	—	6	2	no	—
339	C₂×C₄	—	6	2	no	—
340	C₂×C₄	—	12	4	no	—
341	C₂×C₄	—	24	2	no	—
342	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
343	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
344	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
345	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
346	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
347	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
348	C₂×C₄	C₂×D₃×D₄	1	1	yes	characteristic
349	C₂×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
350	C₂×C₄	#1009	1	1	yes	characteristic
351	C₂×C₄	#1010	1	1	yes	characteristic
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	2	4	no	—
355	C₈	—	12	4	no	—
356	C₈	D₃×C₄⋊C₄	1	2	yes	—
357	C₈	D₃×C₄⋊C₄	1	2	yes	—
358	C₂×C₃	—	2	1	no	—
359	C₂×C₃	—	2	1	no	—
360	C₂×C₃	D₄×C₄⋊C₄	1	1	yes	characteristic
361	C₂×C₃	C₂×#444	1	1	yes	characteristic
362	C₂×C₃	#2957	1	1	yes	characteristic
363	D₃	—	4	4	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	6	4	no	—
367	C₂²	—	12	2	no	—
368	C₂²	—	12	2	no	—
369	C₂²	C₂×C₄×D₁₂	1	1	yes	characteristic
370	C₂²	#5551	1	1	yes	characteristic
371	C₂²	#5579	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	6	4	no	—
376	C₄	—	8	2	no	—
377	C₄	—	12	2	no	—
378	C₄	—	12	2	no	—
379	C₄	#4717	1	2	yes	complemented
380	C₄	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic
381	C₄	#5523	1	1	yes	characteristic
382	C₄	D₄×D₁₂	1	1	yes	characteristic
383	C₄	Q₈×D₁₂	1	1	yes	characteristic
384	C₃	#22686	1	1	yes	characteristic, complemented, Sylow
385	C₂	—	2	1	no	—
386	C₂	—	2	1	no	—
387	C₂	—	12	4	no	—
388	C₂	C₄⋊C₄×D₁₂	1	1	yes	characteristic
389	C₂	C₂×#4717	1	1	yes	characteristic
390	C₂	#78580	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₂
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₃×#1770	C₂
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂×#904	C₂²
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	#1770	D₃
hypocenter	C₃	#22686
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
1. #79012	C₂	semidirect
4. C₃×#1770	C₂	semidirect
5. #69844	C₂	semidirect
6. #69792	C₂	semidirect
7. #69494	C₂	semidirect
8. #68694	C₂	semidirect
28. C₄×Dic₁₂	C₄	semidirect
29. C₄×Dic₁₂	C₄	semidirect
30. C₄×D₂₄	C₄	semidirect
31. C₄×D₂₄	C₄	semidirect
32. C₄×#204	C₄	semidirect
33. C₄×#204	C₄	semidirect
43. #1770	D₃	semidirect
163. Dic₁₂	C₄⋊C₄	semidirect
164. D₂₄	C₄⋊C₄	semidirect
165. #204	C₄⋊C₄	semidirect
314. C₃×C₄	#444	semidirect
379. C₄	#4717	semidirect
384. C₃	#22686	semidirect

Maximal subgroups

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

subgroup	index
#79012	2
C₂×#4717	2
C₄×#904	2
C₃×#1770	2
#69844	2
#69792	2
#69494	2
#68694	2
#22686	3