Subgroups — #61480

Order 256.

1005 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3519	C₂	1	2	yes	maximal, complemented
2	#3391	C₂	1	1	yes	maximal, characteristic, complemented
3	#3380	C₂	1	1	yes	maximal, characteristic, complemented
4	#3366	C₂	1	2	yes	maximal, complemented
5	#3341	C₂	1	1	yes	maximal, characteristic, complemented
6	#3339	C₂	1	2	yes	maximal, complemented
7	#3325	C₂	1	2	yes	maximal, complemented
8	#3318	C₂	1	4	yes	maximal, complemented
9	D₄×D₈	C₂	1	2	yes	maximal, complemented
10	D₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#3176	C₂	1	2	yes	maximal, complemented
12	#3169	C₂	1	1	yes	maximal, characteristic, complemented
13	#3122	C₂	1	4	yes	maximal, complemented
14	#3109	C₂	1	2	yes	maximal, complemented
15	#2966	C₂	1	1	yes	maximal, characteristic, complemented
16	#2948	C₂	1	2	yes	maximal, complemented
17	C₂×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂²×D₈	C₂²	1	1	yes	characteristic
19	C₂²×D₈	C₂²	1	2	yes	—
20	#555	C₂²	1	2	yes	complemented
21	#554	C₂²	1	2	yes	complemented
22	#553	—	2	2	no	—
23	#553	C₂²	1	4	yes	complemented
24	#550	C₂²	1	2	yes	complemented
25	#546	—	2	2	no	—
26	#546	—	2	4	no	—
27	D₄²	C₂²	1	4	yes	complemented
28	D₄²	C₂²	1	2	yes	complemented
29	#540	C₂²	1	4	yes	complemented
30	#538	—	2	4	no	—
31	#532	C₂²	1	2	yes	complemented
32	#529	C₂²	1	1	yes	characteristic
33	#525	—	2	2	no	—
34	C₂×#121	—	2	2	no	—
35	#518	C₂²	1	4	yes	complemented
36	C₂×C₄×D₄	C₂²	1	2	yes	complemented
37	#500	C₂²	1	1	yes	characteristic, complemented
38	#499	C₂²	1	2	yes	complemented
39	#493	C₂²	1	1	yes	characteristic, complemented
40	#493	C₂²	1	2	yes	complemented
41	#490	C₂²	1	4	yes	complemented
42	#486	C₂²	1	2	yes	complemented
43	#482	C₂²	1	4	yes	complemented
44	#482	C₂²	1	4	yes	complemented
45	#481	C₂²	1	4	yes	complemented
46	#478	C₂²	1	4	yes	complemented
47	#478	C₂²	1	4	yes	complemented
48	#476	C₂²	1	2	yes	complemented
49	#476	C₂²	1	2	yes	complemented
50	#469	C₂²	1	4	yes	complemented
51	#466	C₂²	1	2	yes	complemented
52	#466	C₂²	1	2	yes	complemented
53	#466	C₂²	1	2	yes	complemented
54	#466	C₂²	1	2	yes	complemented
55	#459	C₂²	1	2	yes	complemented
56	#459	C₂²	1	2	yes	complemented
57	#459	C₂²	1	4	yes	complemented
58	#459	C₂²	1	4	yes	complemented
59	#447	C₂²	1	4	yes	complemented
60	#447	C₂²	1	4	yes	complemented
61	#447	C₂²	1	4	yes	complemented
62	#446	C₂²	1	1	yes	characteristic, complemented
63	C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
64	#442	C₂²	1	4	yes	complemented
65	#442	C₂²	1	4	yes	complemented
66	#442	C₂²	1	4	yes	complemented
67	C₄×D₈	C₂²	1	1	yes	characteristic
68	C₄×D₈	C₂²	1	2	yes	—
69	C₄×D₈	C₂²	1	1	yes	characteristic
70	C₄×D₈	C₂²	1	2	yes	complemented
71	C₄×D₈	C₂²	1	1	yes	characteristic, complemented
72	C₄×D₈	C₂²	1	2	yes	complemented
73	C₄×D₈	C₂²	1	2	yes	complemented
74	C₄×D₈	C₂²	1	2	yes	complemented
75	C₄×D₈	C₂²	1	2	yes	complemented
76	C₄×D₈	C₂²	1	1	yes	characteristic, complemented
77	#436	C₂²	1	1	yes	characteristic, complemented
78	#435	C₂²	1	2	yes	complemented
79	C₈×D₄	C₂²	1	2	yes	complemented
80	C₈×D₄	C₂²	1	1	yes	characteristic, complemented
81	#433	C₂²	1	2	yes	complemented
82	#432	C₂²	1	1	yes	characteristic, complemented
83	#427	C₂²	1	2	yes	complemented
84	C₂×#107	C₂²	1	1	yes	characteristic, complemented
85	#418	C₂²	1	4	yes	complemented
86	C₂×#102	C₂²	1	2	yes	complemented
87	#405	C₂²	1	2	yes	complemented
88	C₂×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
89	C₂×#41	—	2	2	no	—
90	C₂×#41	—	2	2	no	—
91	C₂²×D₄	—	2	2	no	—
92	C₂²×D₄	—	2	4	no	—
93	C₂²×D₄	C₂³	1	2	yes	—
94	C₂²×D₄	C₂³	1	4	yes	—
95	C₂³×C₄	—	2	2	no	—
96	C₂×D₈	—	2	4	no	—
97	C₂×D₈	—	2	8	no	—
98	C₂×D₈	C₂³	1	2	yes	—
99	C₂×D₈	C₂³	1	1	yes	characteristic
100	C₂×D₈	C₂³	1	2	yes	—
101	C₂×D₈	C₂³	1	4	yes	—
102	C₂×D₈	C₂³	1	2	yes	—
103	C₂×D₈	C₂³	1	4	yes	—
104	C₂×D₈	C₂³	1	1	yes	characteristic
105	C₂×M₁₆	C₂³	1	2	yes	—
106	C₂²×C₈	C₂³	1	1	yes	characteristic
107	C₂²×C₈	C₂³	1	2	yes	—
108	C₂²×C₈	C₂³	1	1	yes	characteristic
109	C₄⋊Q₈	C₂³	1	2	yes	—
110	#127	C₂³	1	2	yes	—
111	#127	C₂³	1	4	yes	—
112	#126	—	2	4	no	—
113	#126	D₄	1	2	yes	—
114	#125	D₄	1	1	yes	characteristic
115	#125	C₂³	1	4	yes	—
116	#124	—	2	4	no	—
117	#124	—	2	2	no	—
118	#124	D₄	1	1	yes	characteristic
119	#123	—	2	4	no	—
120	#123	—	2	4	no	—
121	#123	—	2	4	no	—
122	#123	D₄	1	2	yes	—
123	C₂²⋊Q₈	—	2	2	no	—
124	C₂²⋊Q₈	—	2	4	no	—
125	C₂²⋊Q₈	D₄	1	1	yes	characteristic
126	#121	—	2	4	no	—
127	#121	—	2	2	no	—
128	#121	—	2	2	no	—
129	#121	—	2	2	no	—
130	#121	—	2	2	no	—
131	#121	—	2	4	no	—
132	#121	—	2	4	no	—
133	#121	—	2	4	no	—
134	#121	—	2	4	no	—
135	#121	D₄	1	1	yes	characteristic
136	#121	C₂³	1	4	yes	—
137	#121	C₂³	1	4	yes	—
138	#121	C₂³	1	4	yes	—
139	#120	—	2	4	no	—
140	#120	—	2	4	no	—
141	#120	—	2	4	no	—
142	C₄×Q₈	C₂³	1	1	yes	characteristic
143	C₄×D₄	—	2	4	no	—
144	C₄×D₄	—	2	4	no	—
145	C₄×D₄	—	2	4	no	—
146	C₄×D₄	—	2	2	no	—
147	C₄×D₄	—	2	2	no	—
148	C₄×D₄	—	2	2	no	—
149	C₄×D₄	C₂³	1	2	yes	—
150	C₄×D₄	C₂³	1	2	yes	—
151	C₄×D₄	C₂³	1	2	yes	—
152	C₄×D₄	C₂³	1	2	yes	—
153	C₄×D₄	C₂³	1	4	yes	—
154	C₄×D₄	C₂³	1	4	yes	—
155	C₄×D₄	C₂³	1	4	yes	—
156	C₄×D₄	C₂³	1	2	yes	—
157	C₄×D₄	C₂³	1	1	yes	characteristic
158	#117	—	2	2	no	—
159	#117	C₂³	1	4	yes	—
160	#117	C₂³	1	2	yes	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂³	1	2	yes	—
163	C₂×C₂²⋊C₄	—	2	2	no	—
164	C₂×C₂²⋊C₄	—	2	4	no	—
165	C₂×C₂²⋊C₄	—	2	4	no	—
166	C₂×C₄²	C₂³	1	1	yes	characteristic
167	#107	C₂³	1	1	yes	characteristic
168	#107	C₂³	1	1	yes	characteristic
169	#107	C₂³	1	2	yes	—
170	#107	C₂³	1	2	yes	—
171	#107	C₂³	1	2	yes	—
172	#106	C₂³	1	4	yes	—
173	#106	C₂³	1	4	yes	—
174	C₄⋊C₈	C₂³	1	1	yes	characteristic
175	C₄⋊C₈	C₂³	1	1	yes	characteristic
176	C₄⋊C₈	C₂³	1	2	yes	—
177	C₄⋊C₈	C₂³	1	2	yes	—
178	#102	C₂³	1	2	yes	—
179	#102	C₂³	1	2	yes	—
180	#102	C₂³	1	2	yes	—
181	#102	C₂³	1	2	yes	—
182	#102	C₂³	1	4	yes	—
183	#102	C₂³	1	4	yes	—
184	#102	C₂³	1	4	yes	—
185	#102	C₂³	1	4	yes	—
186	#102	C₂³	1	4	yes	—
187	#102	C₂³	1	4	yes	—
188	C₂²⋊C₈	C₂³	1	2	yes	—
189	C₂²⋊C₈	C₂³	1	2	yes	—
190	C₂²⋊C₈	C₂³	1	2	yes	—
191	#97	C₂³	1	2	yes	—
192	#97	C₂³	1	2	yes	—
193	C₄×C₈	C₂³	1	1	yes	characteristic
194	C₄×C₈	C₂³	1	1	yes	characteristic
195	C₄×C₈	C₂³	1	1	yes	characteristic
196	C₄×C₈	C₂³	1	1	yes	characteristic
197	C₄×C₈	C₂³	1	2	yes	—
198	C₂⁴	—	2	2	no	—
199	C₂⁴	—	2	4	no	—
200	#41	—	8	4	no	—
201	#41	—	8	4	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	8	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	2	4	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	4	no	—
217	C₂×D₄	—	8	4	no	—
218	C₂×D₄	—	8	8	no	—
219	C₂×D₄	C₂×D₄	1	2	yes	—
220	C₂×D₄	C₂×D₄	1	1	yes	characteristic
221	C₂×D₄	C₂⁴	1	2	yes	—
222	C₂×D₄	C₂⁴	1	4	yes	—
223	C₂×D₄	C₂⁴	1	2	yes	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	8	4	no	—
233	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
235	C₂²×C₄	C₂⁴	1	1	yes	characteristic
236	C₂²×C₄	C₂⁴	1	2	yes	—
237	D₈	—	2	4	no	—
238	D₈	—	2	8	no	—
239	D₈	—	2	2	no	—
240	D₈	—	2	4	no	—
241	D₈	—	4	2	no	—
242	D₈	—	4	4	no	—
243	D₈	#41	1	4	yes	complemented
244	M₁₆	—	4	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	C₂⁴	1	1	yes	characteristic
249	C₂×C₈	C₂⁴	1	1	yes	characteristic
250	C₂×C₈	C₂⁴	1	1	yes	characteristic
251	C₂×C₈	C₂⁴	1	2	yes	—
252	C₂×C₈	C₂⁴	1	2	yes	—
253	C₂×C₈	C₂⁴	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
259	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
260	C₄⋊C₄	C₂×D₄	1	2	yes	—
261	C₄⋊C₄	C₂×D₄	1	2	yes	—
262	C₄⋊C₄	C₂×D₄	1	2	yes	—
263	C₄⋊C₄	C₂×D₄	1	2	yes	—
264	C₄⋊C₄	C₂⁴	1	2	yes	—
265	C₄⋊C₄	C₂⁴	1	2	yes	—
266	C₄⋊C₄	C₂⁴	1	4	yes	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	4	no	—
270	C₂²⋊C₄	—	2	2	no	—
271	C₂²⋊C₄	—	2	4	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	2	4	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₂²⋊C₄	—	2	4	no	—
276	C₂²⋊C₄	—	2	4	no	—
277	C₂²⋊C₄	—	2	4	no	—
278	C₂²⋊C₄	C₂×D₄	1	2	yes	—
279	C₂²⋊C₄	C₂×D₄	1	2	yes	—
280	C₂²⋊C₄	C₂×D₄	1	2	yes	—
281	C₂²⋊C₄	C₂×D₄	1	2	yes	—
282	C₂²⋊C₄	C₂×D₄	1	2	yes	—
283	C₄²	—	2	4	no	—
284	C₄²	C₂×D₄	1	1	yes	characteristic
285	C₄²	C₂×D₄	1	1	yes	characteristic
286	C₄²	C₂⁴	1	1	yes	characteristic
287	C₄²	C₂⁴	1	1	yes	characteristic
288	C₄²	C₂⁴	1	2	yes	—
289	C₂³	—	2	2	no	—
290	C₂³	—	2	2	no	—
291	C₂³	—	2	4	no	—
292	C₂³	—	4	4	no	—
293	C₂³	—	4	8	no	—
294	C₂³	—	4	4	no	—
295	C₂³	—	8	2	no	—
296	C₂³	—	8	4	no	—
297	C₂³	—	8	4	no	—
298	C₂³	C₂²×D₄	1	2	yes	—
299	C₂³	C₂²×D₄	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	8	2	no	—
302	D₄	—	2	4	no	—
303	D₄	—	2	2	no	—
304	D₄	—	2	2	no	—
305	D₄	—	2	4	no	—
306	D₄	—	2	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	2	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	8	no	—
311	D₄	—	4	8	no	—
312	D₄	—	4	4	no	—
313	D₄	—	8	4	no	—
314	D₄	—	8	2	no	—
315	D₄	—	8	4	no	—
316	D₄	—	8	4	no	—
317	D₄	—	8	4	no	—
318	D₄	C₂×#41	1	4	yes	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	4	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	4	4	no	—
330	C₂×C₄	—	8	2	no	—
331	C₂×C₄	—	8	2	no	—
332	C₂×C₄	—	8	4	no	—
333	C₂×C₄	—	8	4	no	—
334	C₂×C₄	—	8	4	no	—
335	C₂×C₄	—	8	2	no	—
336	C₂×C₄	C₂²×D₄	1	2	yes	—
337	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
338	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
339	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
340	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
341	C₂×C₄	C₂²×D₄	1	2	yes	—
342	C₂×C₄	C₂²×D₄	1	2	yes	—
343	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
344	C₂×C₄	C₂⁵	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	2	no	—
347	C₈	—	2	1	no	—
348	C₈	—	2	1	no	—
349	C₈	—	2	2	no	—
350	C₈	C₂×#41	1	2	yes	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	2	4	no	—
355	C₂²	—	2	2	no	—
356	C₂²	—	2	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	4	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	4	4	no	—
363	C₂²	—	4	4	no	—
364	C₂²	—	4	2	no	—
365	C₂²	—	8	8	no	—
366	C₂²	—	16	8	no	—
367	C₂²	#576	1	2	yes	—
368	C₂²	C₂³×D₄	1	1	yes	characteristic
369	C₄	—	2	1	no	—
370	C₄	—	2	2	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	2	no	—
375	C₄	—	2	2	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	4	no	—
379	C₄	#576	1	2	yes	—
380	C₄	C₂²×#41	1	1	yes	characteristic
381	C₄	#585	1	1	yes	characteristic
382	C₂	—	2	2	no	—
383	C₂	—	4	2	no	—
384	C₂	—	4	4	no	—
385	C₂	—	8	2	no	—
386	C₂	—	8	4	no	—
387	C₂	D₄×#41	1	1	yes	characteristic
388	C₂	C₂×#576	1	1	yes	characteristic
389	C₂	C₂×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3519	C₂	semidirect
2. #3391	C₂	semidirect
3. #3380	C₂	semidirect
4. #3366	C₂	semidirect
5. #3341	C₂	semidirect
6. #3339	C₂	semidirect
7. #3325	C₂	semidirect
8. #3318	C₂	semidirect
9. D₄×D₈	C₂	semidirect
10. D₄×D₈	C₂	semidirect
11. #3176	C₂	semidirect
12. #3169	C₂	semidirect
13. #3122	C₂	semidirect
14. #3109	C₂	semidirect
15. #2966	C₂	semidirect
16. #2948	C₂	semidirect
17. C₂×C₄×D₈	C₂	semidirect
20. #555	C₂²	semidirect
21. #554	C₂²	semidirect
23. #553	C₂²	semidirect
24. #550	C₂²	semidirect
27. D₄²	C₂²	semidirect
28. D₄²	C₂²	semidirect
29. #540	C₂²	semidirect
31. #532	C₂²	semidirect
35. #518	C₂²	semidirect
36. C₂×C₄×D₄	C₂²	semidirect
37. #500	C₂²	semidirect
38. #499	C₂²	semidirect
39. #493	C₂²	semidirect
40. #493	C₂²	semidirect
41. #490	C₂²	semidirect
42. #486	C₂²	semidirect
43. #482	C₂²	semidirect
44. #482	C₂²	semidirect
45. #481	C₂²	semidirect
46. #478	C₂²	semidirect
47. #478	C₂²	semidirect
48. #476	C₂²	semidirect
49. #476	C₂²	semidirect
50. #469	C₂²	semidirect
51. #466	C₂²	semidirect
52. #466	C₂²	semidirect
53. #466	C₂²	semidirect
54. #466	C₂²	semidirect
55. #459	C₂²	semidirect
56. #459	C₂²	semidirect
57. #459	C₂²	semidirect
58. #459	C₂²	semidirect
59. #447	C₂²	semidirect
60. #447	C₂²	semidirect
61. #447	C₂²	semidirect
62. #446	C₂²	semidirect
63. C₈×Q₈	C₂²	semidirect
64. #442	C₂²	semidirect
65. #442	C₂²	semidirect
66. #442	C₂²	semidirect
70. C₄×D₈	C₂²	semidirect
71. C₄×D₈	C₂²	semidirect
72. C₄×D₈	C₂²	semidirect
73. C₄×D₈	C₂²	semidirect
74. C₄×D₈	C₂²	semidirect
75. C₄×D₈	C₂²	semidirect
76. C₄×D₈	C₂²	semidirect
77. #436	C₂²	semidirect
78. #435	C₂²	semidirect
79. C₈×D₄	C₂²	semidirect
80. C₈×D₄	C₂²	semidirect
81. #433	C₂²	semidirect
82. #432	C₂²	semidirect
83. #427	C₂²	semidirect
84. C₂×#107	C₂²	semidirect
85. #418	C₂²	semidirect
86. C₂×#102	C₂²	semidirect
87. #405	C₂²	semidirect
88. C₂×C₄×C₈	C₂²	semidirect
243. D₈	#41	semidirect

Maximal subgroups

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

subgroup	index
#3519	2
#3391	2
#3380	2
#3366	2
#3341	2
#3339	2
#3325	2
#3318	2
D₄×D₈	2
#3176	2
#3169	2
#3122	2
#3109	2
#2966	2
#2948	2
C₂×C₄×D₈	2