Finite Group Databasebeta

← #6595138

Subgroups — #6595138

Order 512.

1540 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#36188C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#33049C212yesmaximal, complemented
5#29208C211yesmaximal, characteristic, complemented
6#28512C211yesmaximal, characteristic, complemented
7#28087C218yesmaximal, complemented
8#27892C212yesmaximal, complemented
9#27875C212yesmaximal, complemented
10#25348C214yesmaximal
11#23068C214yesmaximal, complemented
12C4×#487C211yesmaximal, characteristic, complemented
13#20824C212yesmaximal, complemented
14C4×#436C211yesmaximal, characteristic
15C2×#556—21no—
16C2×#487C2211yescharacteristic
17C2×#475C2214yes—
18C2×#436C2212yes—
19C2×C4×M16C2212yes—
20#2868C2211yescharacteristic
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2857C2212yes—
24#2571—21no—
25#2568—21no—
26#2563—21no—
27#2547—21no—
28#2547C2212yes—
29#2538C2212yes—
30#2499—22no—
31#2497—22no—
32#2496—22no—
33#2496C2214yes—
34#2492C2214yes—
35#2439—22no—
36#2369—21no—
37#2365—21no—
38#2365C2212yes—
39#2355C2212yes—
40#2354—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43#2285C2212yes—
44#2285C2212yes—
45C42×D4C2211yescharacteristic
46#2096C2214yes—
47#2077C2218yes—
48#2057C2214yes—
49#2036C2218yes—
50#1990C2214yes—
51#1990C2214yes—
52#1990C2212yes—
53#1990C2211yescharacteristic
54#1956C2212yes—
55#1956C2211yescharacteristic
56#1947C2214yes—
57#1935C2218yes—
58#1933C2214yes—
59#1928C2218yes—
60#1920C2218yes—
61#1894C2218yes—
62#1845C2212yes—
63#1844C2212yes—
64#1834C2212yes—
65#1834C2212yes—
66#1834C2211yescharacteristic
67#1809C2218yes—
68#1776C2218yes—
69#1770C2212yes—
70C4×C4⋊C8C2211yescharacteristic
71C4×#103C2214yes—
72#1753C2214yes—
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75C22×M16C2314yes—
76#556—48no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2312yes—
85C2×#126—24no—
86C2×#124—22no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8C2314yes—
89C2×C4×Q8—22no—
90C2×C4×Q8—21no—
91C2×C4×Q8C2312yes—
92C2×C4×D4D411yescharacteristic
93C2×C4×D4D411yescharacteristic
94C2×C4×D4C2311yescharacteristic
95C2×#117—22no—
96C2×#117—22no—
97C22×C42C2312yes—
98#487—24no—
99#475—216no—
100#436—28no—
101C2×#107C2318yes—
102C2×#106C2318yes—
103C2×C4⋊C8C2312yes—
104C2×#103C2318yes—
105C2×#103C2314yes—
106C2×#103C2314yes—
107C2×C22⋊C8C2314yes—
108C4×M16—28no—
109C2×#97C2314yes—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2312yes—
112C2×C4×C8C2312yes—
113C2×C4×C8C2312yes—
114C2×C4×C8C2311yescharacteristic
115#400—24no—
116#400—24no—
117#397—24no—
118#397—24no—
119#396—24no—
120#393—24no—
121#391—22no—
122#391—22no—
123#391—21no—
124#391—22no—
125#391—24no—
126#391C2312yes—
127#391C2314yes—
128#390—22no—
129#389—22no—
130#389—21no—
131#389—21no—
132#389—21no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389C2312yes—
137#389C2314yes—
138#388—24no—
139#387—24no—
140#387—24no—
141#387—24no—
142#384—21no—
143#384—21no—
144#384—21no—
145#384—21no—
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#384C2312yes—
149#384C2312yes—
150#384C2312yes—
151#384C2312yes—
152#382—22no—
153#382—22no—
154#382—22no—
155#380—22no—
156#380—22no—
157#380—22no—
158#380—22no—
159#380—22no—
160#380C2314yes—
161#380C2314yes—
162#380C2314yes—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4D411yescharacteristic
170C4×C4⋊C4D411yescharacteristic
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2314yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2311yescharacteristic
175C4×C22⋊C4D412yes—
176C4×C22⋊C4D412yes—
177C4×C22⋊C4C2312yes—
178C43C2311yescharacteristic
179#340C2318yes—
180#340C2314yes—
181#340C2318yes—
182#340C2314yes—
183#340C2318yes—
184#336C2312yes—
185#336C2314yes—
186C22×Q8—21no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8C2412yes—
190C22×D4C2×D411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2412yes—
194C2×M16—28no—
195C2×M16—28no—
196C2×M16—44no—
197C22×C8C2412yes—
198C22×C8C2412yes—
199C22×C8C2414yes—
200C4⋊Q8—28no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208C4⋊Q8—44no—
209C4⋊Q8—82no—
210#126—416no—
211#124—48no—
212C22⋊Q8—48no—
213C22⋊Q8—48no—
214C4×Q8—28no—
215C4×Q8—48no—
216C4×Q8—44no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—24no—
220#117—48no—
221#117—84no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2414yes—
250C2×C4⋊C4C2412yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4C2×D412yes—
256C2×C22⋊C4C2×D412yes—
257C2×C22⋊C4C2×D412yes—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2411yescharacteristic
271C2×C42C2412yes—
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274#107—48no—
275#106—216no—
276C4⋊C8#4118yes—
277#103—216no—
278#103—28no—
279#103—44no—
280C22⋊C8—28no—
281#97—28no—
282C4×C8—24no—
283C4×C8—24no—
284C4×C8—22no—
285C4×C8—42no—
286C4×C8#4114yes—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—24no—
294#95—24no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95C2×D412yes—
300#95C2×D412yes—
301#95C2×D412yes—
302C24C22×D412yes—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—44no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—42no—
314C2×Q8—82no—
315C2×Q8—82no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4C22×D412yes—
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C2511yescharacteristic
345M16—416no—
346C2×C8—24no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8C2×#4114yes—
354C2×C8C2×#4118yes—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—28no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—28no—
364C4⋊C4—48no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—84no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—48no—
389C22⋊C4—48no—
390C22⋊C4—84no—
391C22⋊C4—84no—
392C42—24no—
393C42—24no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—48no—
400C42—44no—
401C42—44no—
402C42—84no—
403C42AGL(1,ℤ/8)14yes—
404C42#13714yes—
405C42C2×#4114yes—
406C42C2×#4112yes—
407C422−1+412yes—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23C23×D411yescharacteristic
416Q8—44no—
417Q8—44no—
418Q8—84no—
419Q8—88no—
420Q8—164no—
421D4—44no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—48no—
438C2×C4—48no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4#53212yes—
460C2×C4#53212yes—
461C2×C4#54712yes—
462C2×C4#54712yes—
463C2×C4C2×AGL(1,ℤ/8)12yes—
464C2×C4C2×AGL(1,ℤ/8)12yes—
465C2×C4C2×AGL(1,ℤ/8)12yes—
466C2×C4C2×#13712yes—
467C2×C4C2×#13712yes—
468C2×C4C2×#13712yes—
469C2×C4#57712yes—
470C2×C4#57812yes—
471C2×C4C22×#4111yescharacteristic
472C2×C4C22×#4112yes—
473C2×C4C2×2−1+411yescharacteristic
474C2×C4#58512yes—
475C8—28no—
476C8—24no—
477C8—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22C2×#53211yescharacteristic
486C22C2×#54711yescharacteristic
487C22#350011yescharacteristic
488C22C22×AGL(1,ℤ/8)11yescharacteristic
489C22C22×#13711yescharacteristic
490C22C2×#57711yescharacteristic
491C22C2×#57811yescharacteristic
492C4—22no—
493C4—22no—
494C4—22no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—84no—
499C4—82no—
500C4—82no—
501C4—82no—
502C4—84no—
503C4—82no—
504C4#318214yes—
505C4#335314yes—
506C4#335614yes—
507C4#347512yes—
508C4#351812yes—
509C2—82no—
510C2#3618511yescharacteristic
511C2C2×#318211yescharacteristic
512C2#6072311yescharacteristic
513C2C2×#335311yescharacteristic
514C2C2×#335611yescharacteristic
515C2#6253611yescharacteristic
516C2#6254111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33289C2semidirect
4. #33049C2semidirect
5. #29208C2semidirect
6. #28512C2semidirect
7. #28087C2semidirect
8. #27892C2semidirect
9. #27875C2semidirect
11. #23068C2semidirect
12. C4×#487C2semidirect
13. #20824C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#361882
#332892
#330492
#292082
#285122
#280872
#278922
#278752
#253482
#230682
C4×#4872
#208242
C4×#4362