Subgroups — #10814890

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#79000	C₂	1	2	yes	maximal, complemented
2	C₄×#904	C₂	1	1	yes	maximal, characteristic
3	C₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
4	#69845	C₂	1	2	yes	maximal
5	#69829	C₂	1	2	yes	maximal, complemented
6	#69792	C₂	1	1	yes	maximal, characteristic, complemented
7	#69494	C₂	1	2	yes	maximal, complemented
8	#68646	C₂	1	4	yes	maximal, complemented
9	#22803	—	3	1	no	maximal, Sylow
10	C₂×#904	C₂²	1	1	yes	characteristic
11	C₄×#235	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×#97	C₂²	1	2	yes	—
14	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	#5113	C₂²	1	2	yes	—
17	#5110	C₂²	1	4	yes	—
18	C₂×#819	C₂²	1	2	yes	—
19	C₂×#819	C₂²	1	2	yes	—
20	#5106	C₂²	1	1	yes	characteristic
21	#5101	C₂²	1	4	yes	—
22	C₂×#805	C₂²	1	4	yes	—
23	#4924	C₂²	1	2	yes	—
24	#4840	—	2	4	no	—
25	#4835	—	2	2	no	—
26	#4833	—	2	2	no	—
27	C₄×Dic₁₂	C₄	1	1	yes	characteristic, complemented
28	C₄×Dic₁₂	C₄	1	1	yes	characteristic, complemented
29	C₄×D₂₄	C₄	1	1	yes	characteristic, complemented
30	C₄×D₂₄	C₄	1	1	yes	characteristic, complemented
31	C₄×#204	C₄	1	2	yes	complemented
32	C₄×#204	C₄	1	2	yes	complemented
33	C₃×#336	C₂²	1	2	yes	—
34	#4523	C₂²	1	4	yes	—
35	C₄×#135	—	3	1	no	—
36	#2925	—	3	2	no	—
37	#1975	—	3	2	no	—
38	#1929	—	3	4	no	—
39	#1848	—	3	1	no	—
40	#1806	—	3	2	no	—
41	#1773	—	3	2	no	—
42	#1770	D₃	1	1	yes	characteristic, complemented
43	C₂×#235	C₂³	1	2	yes	—
44	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₃×C₈	C₂³	1	2	yes	—
46	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₃×C₄⋊C₄	—	2	1	no	—
48	C₂×C₃×C₄²	C₂³	1	2	yes	—
49	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
50	C₄×#67	—	2	2	no	—
51	C₄×#67	—	2	2	no	—
52	#926	C₂³	1	2	yes	—
53	C₂×#208	C₂³	1	4	yes	—
54	C₂×Dic₁₂	C₂×C₄	1	1	yes	characteristic
55	#904	—	2	2	no	—
56	#904	—	2	2	no	—
57	C₂×D₂₄	C₂×C₄	1	1	yes	characteristic
58	C₂×#204	C₂×C₄	1	2	yes	—
59	C₄×D₁₂	C₂×C₄	1	2	yes	—
60	C₄×D₁₂	C₂×C₄	1	2	yes	—
61	#872	—	2	2	no	—
62	C₄²×D₃	—	2	2	no	—
63	C₄×Dic₆	C₂×C₄	1	2	yes	—
64	C₄×Dic₆	C₂×C₄	1	2	yes	—
65	C₃×C₄⋊C₈	D₄	1	2	yes	—
66	C₃×C₄⋊C₈	D₄	1	2	yes	—
67	C₃×#97	—	2	4	no	—
68	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
69	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
70	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
71	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
72	C₃×#95	—	2	1	no	—
73	#821	C₂×C₄	1	2	yes	—
74	#821	C₂×C₄	1	2	yes	—
75	(C₂×D₃)⋊C₈	—	2	4	no	—
76	(C₂×D₃)⋊C₈	—	2	4	no	—
77	#819	—	2	4	no	—
78	#819	—	4	2	no	—
79	#818	C₂×C₄	1	1	yes	characteristic
80	#818	C₂×C₄	1	1	yes	characteristic
81	#817	C₂×C₄	1	2	yes	—
82	#816	C₂×C₄	1	2	yes	—
83	#816	C₂×C₄	1	2	yes	—
84	#815	—	2	4	no	—
85	C₈×Dic₃	—	2	4	no	—
86	#805	—	4	4	no	—
87	C₄⋊(C₃⋊C₈)	—	2	2	no	—
88	C₄⋊(C₃⋊C₈)	—	2	2	no	—
89	C₂×#135	—	3	1	no	—
90	C₄×#41	—	3	2	no	—
91	C₄×Q₁₆	—	3	1	no	—
92	C₄×Q₁₆	—	3	1	no	—
93	C₄×SD₁₆	—	3	2	no	—
94	C₄×SD₁₆	—	3	2	no	—
95	C₄×D₈	—	3	1	no	—
96	C₄×D₈	—	3	1	no	—
97	#433	—	6	4	no	—
98	#432	—	6	2	no	—
99	#431	—	6	2	no	—
100	C₂×#108	—	3	2	no	—
101	C₂×#108	—	3	2	no	—
102	#427	—	3	1	no	—
103	#423	—	3	2	no	—
104	C₂×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
105	C₂×#104	—	3	4	no	—
106	#416	—	3	2	no	—
107	#408	—	3	4	no	—
108	#405	—	3	4	no	—
109	C₂×#97	C₂×D₃	1	2	yes	—
110	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
111	C₄×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
112	#339	—	3	4	no	—
113	#336	C₂×D₃	1	2	yes	—
114	C₂²×C₃×C₄	—	2	2	no	—
115	C₂²×C₃×C₄	#41	1	1	yes	characteristic
116	C₂²×C₃×C₄	#41	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
118	C₂×#67	—	2	2	no	—
119	#235	—	2	2	no	—
120	#235	—	2	2	no	—
121	#235	—	2	2	no	—
122	#235	—	2	2	no	—
123	C₂×D₁₂	C₂²×C₄	1	2	yes	—
124	C₂×C₄×D₃	—	2	2	no	—
125	C₂×C₄×D₃	—	2	2	no	—
126	C₂×C₄×D₃	—	2	2	no	—
127	C₂×Dic₆	C₂²×C₄	1	2	yes	—
128	C₂×C₃×C₈	—	2	2	no	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
137	C₂×C₃×C₈	#41	1	2	yes	—
138	C₂×C₃×C₈	#41	1	2	yes	—
139	C₃×C₄⋊C₄	—	4	2	no	—
140	C₃×C₄²	—	2	2	no	—
141	C₃×C₄²	—	2	2	no	—
142	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
143	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
144	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
145	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
146	#217	—	2	2	no	—
147	(C₂×D₃)⋊C₄	—	2	2	no	—
148	(C₂×D₃)⋊C₄	—	2	2	no	—
149	C₄⋊Dic₃	C₂²×C₄	1	2	yes	—
150	C₄⋊Dic₃	C₂²×C₄	1	2	yes	—
151	Dic₃⋊C₄	—	2	2	no	—
152	Dic₃⋊C₄	—	2	2	no	—
153	C₄×Dic₃	—	2	2	no	—
154	C₄×Dic₃	—	2	2	no	—
155	C₄×Dic₃	—	2	2	no	—
156	#208	—	2	4	no	—
157	#208	—	2	4	no	—
158	C₂×C₃⋊C₈	—	2	4	no	—
159	Dic₁₂	—	2	2	no	—
160	D₂₄	—	2	2	no	—
161	#204	—	2	4	no	—
162	C₂×#41	—	3	2	no	—
163	#135	—	6	2	no	—
164	#135	—	6	2	no	—
165	C₂×Q₁₆	—	3	1	no	—
166	C₂×SD₁₆	—	3	2	no	—
167	C₂×D₈	—	3	1	no	—
168	C₂×M₁₆	—	3	4	no	—
169	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₃	1	2	yes	—
171	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
172	C₄×Q₈	—	3	2	no	—
173	C₄×Q₈	—	3	2	no	—
174	C₄×D₄	—	3	2	no	—
175	C₄×D₄	—	3	2	no	—
176	C₄×D₄	—	6	2	no	—
177	C₄×D₄	—	6	2	no	—
178	#117	—	3	2	no	—
179	#117	—	6	2	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄²	—	6	2	no	—
182	C₂×C₄²	C₂²×D₃	1	2	yes	—
183	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
184	#108	—	6	4	no	—
185	#108	—	12	2	no	—
186	#107	—	3	1	no	—
187	#107	—	3	1	no	—
188	#106	—	3	2	no	—
189	C₄⋊C₈	—	6	2	no	—
190	C₄⋊C₈	—	6	2	no	—
191	C₄⋊C₈	D₁₂	1	2	yes	—
192	C₄⋊C₈	D₁₂	1	2	yes	—
193	#104	—	12	4	no	—
194	#103	—	3	2	no	—
195	#103	—	3	2	no	—
196	#102	—	3	2	no	—
197	#102	—	3	2	no	—
198	C₂²⋊C₈	—	6	4	no	—
199	C₂²⋊C₈	—	6	4	no	—
200	#97	—	2	4	no	—
201	#97	—	6	4	no	—
202	C₄×C₈	—	6	4	no	—
203	C₄×C₈	C₄×D₃	1	1	yes	characteristic
204	C₄×C₈	C₄×D₃	1	1	yes	characteristic
205	C₄×C₈	C₄×D₃	1	1	yes	characteristic
206	C₄×C₈	C₄×D₃	1	1	yes	characteristic
207	#95	—	2	1	no	—
208	C₂³×C₃	C₂×#41	1	1	yes	characteristic
209	C₂²×D₃	—	2	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	—	4	2	no	—
213	C₂×C₃×C₄	—	4	2	no	—
214	C₂×C₃×C₄	—	4	2	no	—
215	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
218	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
223	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
225	#67	—	4	2	no	—
226	#67	—	4	2	no	—
227	C₂×Dic₃	—	2	2	no	—
228	C₂×Dic₃	—	2	2	no	—
229	C₂×Dic₃	—	2	2	no	—
230	D₁₂	—	2	2	no	—
231	D₁₂	—	2	2	no	—
232	C₄×D₃	—	4	2	no	—
233	C₄×D₃	—	4	2	no	—
234	C₄×D₃	—	4	4	no	—
235	C₄×D₃	—	8	2	no	—
236	Dic₆	—	2	2	no	—
237	Dic₆	—	2	2	no	—
238	C₃×C₈	—	2	1	no	—
239	C₃×C₈	—	2	1	no	—
240	C₃×C₈	—	2	4	no	—
241	C₃×C₈	#117	1	2	yes	—
242	C₃×C₈	#117	1	2	yes	—
243	C₃⋊C₈	—	4	4	no	—
244	#41	—	6	2	no	—
245	#41	—	6	2	no	—
246	#41	—	6	2	no	—
247	#41	—	6	2	no	—
248	C₂×Q₈	—	3	2	no	—
249	C₂×D₄	—	3	2	no	—
250	C₂×D₄	—	6	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	6	2	no	—
253	C₂²×C₄	—	6	2	no	—
254	C₂²×C₄	—	6	2	no	—
255	C₂²×C₄	#235	1	1	yes	characteristic
256	C₂²×C₄	#235	1	1	yes	characteristic
257	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
258	Q₁₆	—	6	2	no	—
259	SD₁₆	—	6	4	no	—
260	D₈	—	6	2	no	—
261	M₁₆	—	6	4	no	—
262	M₁₆	—	6	4	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	6	4	no	—
268	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
272	C₂×C₈	C₂×D₁₂	1	4	yes	—
273	C₂×C₈	#237	1	2	yes	—
274	C₂×C₈	#239	1	2	yes	—
275	C₄⋊C₄	—	3	2	no	—
276	C₄⋊C₄	—	3	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	6	2	no	—
279	C₄⋊C₄	—	6	2	no	—
280	C₂²⋊C₄	—	6	2	no	—
281	C₂²⋊C₄	—	6	2	no	—
282	C₂²⋊C₄	—	6	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	6	2	no	—
286	C₄²	—	6	2	no	—
287	C₄²	—	6	2	no	—
288	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
289	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
290	C₄²	C₂×D₁₂	1	1	yes	characteristic
291	C₄²	C₂×D₁₂	1	1	yes	characteristic
292	C₂²×C₃	—	2	1	no	—
293	C₂²×C₃	—	2	1	no	—
294	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
295	C₂²×C₃	#555	1	1	yes	characteristic
296	C₂²×C₃	#556	1	1	yes	characteristic
297	C₂×D₃	—	4	2	no	—
298	C₂×D₃	—	4	2	no	—
299	C₂×D₃	—	4	2	no	—
300	C₃×C₄	—	2	1	no	—
301	C₃×C₄	—	2	1	no	—
302	C₃×C₄	—	2	1	no	—
303	C₃×C₄	—	8	2	no	—
304	C₃×C₄	#444	1	2	yes	—
305	C₃×C₄	C₂×#117	1	1	yes	characteristic
306	C₃×C₄	#520	1	1	yes	characteristic
307	C₃×C₄	#547	1	1	yes	characteristic
308	C₃×C₄	#550	1	1	yes	characteristic
309	Dic₃	—	4	2	no	—
310	Dic₃	—	4	2	no	—
311	Dic₃	—	4	2	no	—
312	C₂³	—	6	2	no	—
313	C₂³	C₂×#235	1	1	yes	characteristic
314	Q₈	—	6	2	no	—
315	Q₈	—	6	2	no	—
316	D₄	—	6	2	no	—
317	D₄	—	6	2	no	—
318	D₄	—	12	2	no	—
319	D₄	—	12	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	6	2	no	—
326	C₂×C₄	—	6	2	no	—
327	C₂×C₄	—	6	2	no	—
328	C₂×C₄	—	12	4	no	—
329	C₂×C₄	—	12	2	no	—
330	C₂×C₄	—	12	2	no	—
331	C₂×C₄	—	24	2	no	—
332	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
333	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
334	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
335	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
336	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
337	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
338	C₂×C₄	C₂×#237	1	1	yes	characteristic
339	C₂×C₄	C₂×#239	1	1	yes	characteristic
340	C₂×C₄	#1009	1	1	yes	characteristic
341	C₂×C₄	#1010	1	1	yes	characteristic
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	2	4	no	—
345	C₈	—	12	4	no	—
346	C₈	#892	1	2	yes	—
347	C₈	#892	1	2	yes	—
348	C₂×C₃	—	2	1	no	—
349	C₂×C₃	—	2	1	no	—
350	C₂×C₃	#2354	1	1	yes	characteristic
351	C₂×C₃	C₂×#444	1	1	yes	characteristic
352	C₂×C₃	#2957	1	1	yes	characteristic
353	D₃	—	8	2	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	12	2	no	—
357	C₂²	—	12	2	no	—
358	C₂²	—	12	2	no	—
359	C₂²	C₂×C₄×D₁₂	1	1	yes	characteristic
360	C₂²	#5563	1	1	yes	characteristic
361	C₂²	#5584	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	8	2	no	—
366	C₄	—	12	2	no	—
367	C₄	—	12	2	no	—
368	C₄	—	12	2	no	—
369	C₄	#4717	1	2	yes	—
370	C₄	C₂×#892	1	1	yes	characteristic
371	C₄	#5523	1	1	yes	characteristic
372	C₄	#5555	1	1	yes	characteristic
373	C₄	#5577	1	1	yes	characteristic
374	C₃	#22803	1	1	yes	characteristic, complemented, Sylow
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	24	2	no	—
378	C₂	#73638	1	1	yes	characteristic
379	C₂	C₂×#4717	1	1	yes	characteristic
380	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	#22803
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. #79000	C₂	semidirect
3. C₃×#1770	C₂	semidirect
5. #69829	C₂	semidirect
6. #69792	C₂	semidirect
7. #69494	C₂	semidirect
8. #68646	C₂	semidirect
27. C₄×Dic₁₂	C₄	semidirect
28. C₄×Dic₁₂	C₄	semidirect
29. C₄×D₂₄	C₄	semidirect
30. C₄×D₂₄	C₄	semidirect
31. C₄×#204	C₄	semidirect
32. C₄×#204	C₄	semidirect
42. #1770	D₃	semidirect
374. C₃	#22803	semidirect

Maximal subgroups

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

subgroup	index
#79000	2
C₄×#904	2
C₃×#1770	2
#69845	2
#69829	2
#69792	2
#69494	2
#68646	2
#22803	3