Finite Group Databasebeta

← #7542957

Subgroups — #7542957

Order 512.

1581 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic
2#36534C211yesmaximal, characteristic
3#32268C212yesmaximal, complemented
4#32260C212yesmaximal, complemented
5#29211C212yesmaximal, complemented
6#26315C214yesmaximal, complemented
7#25356C214yesmaximal, complemented
8#25024C214yesmaximal, complemented
9#24108C212yesmaximal, complemented
10#24098C212yesmaximal, complemented
11#23175C211yesmaximal, characteristic
12#22661C214yesmaximal, complemented
13#21615C212yesmaximal, complemented
14C2×#556C2211yescharacteristic
15C2×#554C2211yescharacteristic
16C2×#484C2212yes—
17C2×#483C2212yes—
18C2×#475C2212yes—
19C2×#435C2214yes—
20C2×#428C2212yes—
21C22×#107C2212yes—
22#2862C2211yescharacteristic
23#2861—21no—
24#2804—21no—
25#2804—21no—
26#2580—24no—
27#2571—21no—
28#2565—21no—
29#2563C2214yes—
30#2563C2211yescharacteristic
31#2560—24no—
32#2547C2211yescharacteristic
33#2544—21no—
34#2536C2214yes—
35#2487C2212yes—
36#2487C2212yes—
37#2366—21no—
38#2352C2214yes—
39D4×C4⋊C4C2214yes—
40C4×C4⋊Q8C2212yes—
41C4×#127C2211yescharacteristic
42C4×#124—21no—
43C42×Q8C2211yescharacteristic
44#2077C2214yes—
45#2077C2214yes—
46#2077C2214yes—
47#2075C2214yes—
48#2058C2214yes—
49#2057C2214yes—
50#2036C2214yes—
51#2035C2214yes—
52#1993C2214yes—
53#1988C2212yes—
54#1955C2214yes—
55#1949C2214yes—
56#1949C2214yes—
57#1947C2212yes—
58#1920C2214yes—
59#1894C2212yes—
60#1892C2212yes—
61#1870C2212yes—
62#1869C2212yes—
63#1863C2214yes—
64#1854C2214yes—
65#1853C2214yes—
66#1852C2214yes—
67#1852C2214yes—
68#1847C2214yes—
69#1825C2212yes—
70#1824C2212yes—
71#1809C2214yes—
72#1807C2214yes—
73#1803C2212yes—
74#1776C2214yes—
75#1767C2212yes—
76#1763C2214yes—
77C22×M16C2312yes—
78C23×C8C2312yes—
79#556—42no—
80#556—42no—
81#554—42no—
82#554—42no—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×#127D411yescharacteristic
87C2×#126—22no—
88C2×#125—21no—
89C2×#124—21no—
90C2×#124—21no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8C2312yes—
93C2×C22⋊Q8C2312yes—
94C2×C4×Q8—21no—
95C2×C4×Q8—22no—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2311yescharacteristic
98C2×C4×D4—21no—
99C2×C4×D4—24no—
100C2×C4×D4C2314yes—
101C2×C4×D4C2311yescharacteristic
102C2×C4×D4C2311yescharacteristic
103C2×#117C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107#484—44no—
108#483—44no—
109#475—28no—
110#435—216no—
111#428—44no—
112C2×#107—44no—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×C22⋊C8C2314yes—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2312yes—
127C2×#97C2314yes—
128#400—22no—
129#400—22no—
130#399—22no—
131#399—22no—
132#398—24no—
133#397—22no—
134#394—24no—
135#393—22no—
136#393—24no—
137#393—24no—
138#391—21no—
139#391—22no—
140#391—22no—
141#391—21no—
142#391—21no—
143#390—24no—
144#390—21no—
145#390D414yes—
146#390D411yescharacteristic
147#390D411yescharacteristic
148#389—24no—
149#389—22no—
150#389C2314yes—
151#388—24no—
152#388—24no—
153#388—24no—
154#387—24no—
155#386—24no—
156#386—22no—
157#386—22no—
158#385—22no—
159#385—22no—
160#385—24no—
161#385—24no—
162#384D411yescharacteristic
163#384C2312yes—
164#382—21no—
165#382—24no—
166#382C2314yes—
167#380C2312yes—
168#380C2312yes—
169#380C2314yes—
170#380C2314yes—
171#380C2314yes—
172#380C2314yes—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4C2314yes—
177C4×C4⋊C4C2311yescharacteristic
178C4×C4⋊C4C2312yes—
179C4×C22⋊C4—24no—
180C43—21no—
181C43C2311yescharacteristic
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191C22×Q8—21no—
192C22×Q8—22no—
193C22×Q8—21no—
194C22×Q8C2411yescharacteristic
195C22×D4C2×D414yes—
196C22×D4C2×D411yescharacteristic
197C22×D4C2×D411yescharacteristic
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C2×M16—28no—
204C2×M16—42no—
205C22×C8—28no—
206C22×C8—42no—
207C22×C8C2414yes—
208C22×C8C2414yes—
209C4⋊Q8—24no—
210C4⋊Q8—24no—
211C4⋊Q8—41no—
212C4⋊Q8—41no—
213C4⋊Q8—41no—
214C4⋊Q8—41no—
215#127—24no—
216#126—48no—
217#125—44no—
218#124—44no—
219#124—44no—
220C22⋊Q8—24no—
221C22⋊Q8—24no—
222C22⋊Q8—42no—
223C22⋊Q8—42no—
224C22⋊Q8—48no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—44no—
230C4×Q8—84no—
231C4×D4—28no—
232C4×D4—28no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—44no—
238C4×D4—88no—
239#117—24no—
240#117—42no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2411yescharacteristic
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4C2×D414yes—
279C2×C22⋊C4C2×D414yes—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D412yes—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D412yes—
290C2×C42C2×D411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2412yes—
293#107—28no—
294#107—44no—
295#107—44no—
296#106—28no—
297#106—44no—
298#106—44no—
299C4⋊C8—24no—
300C4⋊C8#4118yes—
301#103—28no—
302#103—28no—
303C22⋊C8—28no—
304C22⋊C8—28no—
305C22⋊C8—42no—
306C22⋊C8—42no—
307#97—28no—
308#95—24no—
309#95—24no—
310#95—22no—
311#95—21no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320C24C22×D412yes—
321C24C22×D412yes—
322C2×Q8—21no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—44no—
329C2×Q8—42no—
330C2×Q8—82no—
331C2×Q8—82no—
332C2×Q8—82no—
333C2×Q8H(5,𝔽2)12yes—
334C2×D4—28no—
335C2×D4—28no—
336C2×D4—22no—
337C2×D4—41no—
338C2×D4—44no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4#13714yes—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—44no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4—84no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4C22×D412yes—
365C22×C4C22×D412yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D412yes—
368C22×C4C22×D412yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C2511yescharacteristic
371M16—48no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—48no—
378C2×C8C2×#4118yes—
379C4⋊C4—24no—
380C4⋊C4—28no—
381C4⋊C4—21no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4—84no—
406C4⋊C4H(5,𝔽2)12yes—
407C22⋊C4—28no—
408C22⋊C4—28no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—48no—
415C22⋊C4—84no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—84no—
419C22⋊C4—84no—
420C42—24no—
421C42—22no—
422C42—21no—
423C42—21no—
424C42—41no—
425C42—42no—
426C42—44no—
427C42—48no—
428C42—44no—
429C42—42no—
430C42—82no—
431C42C2×#4114yes—
432C422−1+414yes—
433C23—24no—
434C23—22no—
435C23—24no—
436C23—22no—
437C23—24no—
438C23—22no—
439C23—24no—
440C23—22no—
441C23—22no—
442C23C2×#13714yes—
443C23C23×D411yescharacteristic
444Q8—24no—
445Q8—41no—
446Q8—41no—
447Q8—84no—
448Q8—88no—
449Q8—162no—
450D4—44no—
451D4—416no—
452D4—82no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—164no—
490C2×C4—164no—
491C2×C4—164no—
492C2×C4#54714yes—
493C2×C4#54714yes—
494C2×C4C2×#13712yes—
495C2×C4C22×#4112yes—
496C2×C4C2×H(5,𝔽2)11yescharacteristic
497C2×C4C2×H(5,𝔽2)11yescharacteristic
498C2×C4C2×2−1+412yes—
499C8—28no—
500C8—44no—
501C22—28no—
502C22—24no—
503C22—44no—
504C22—42no—
505C22—42no—
506C22—44no—
507C22—44no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22C2×#54712yes—
512C22#348411yescharacteristic
513C22C22×#13711yescharacteristic
514C22#358512yes—
515C22#358611yescharacteristic
516C4—24no—
517C4—24no—
518C4—22no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—41no—
523C4—42no—
524C4—82no—
525C4—84no—
526C4—82no—
527C4—84no—
528C4—82no—
529C4#348111yescharacteristic
530C4#352212yes—
531C4#355011yescharacteristic
532C2—44no—
533C2—82no—
534C2#5907011yescharacteristic
535C2#6125412yes—
536C2#6129112yes—
537C2#6157311yescharacteristic
538C2#6167311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32268C2semidirect
4. #32260C2semidirect
5. #29211C2semidirect
6. #26315C2semidirect
7. #25356C2semidirect
8. #25024C2semidirect
9. #24108C2semidirect
10. #24098C2semidirect
12. #22661C2semidirect
13. #21615C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#365342
#322682
#322602
#292112
#263152
#253562
#250242
#241082
#240982
#231752
#226612
#216152