Finite Group Databasebeta

← #7459825

Subgroups — #7459825

Order 512.

1605 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#46152C211yesmaximal, characteristic, complemented
2#39360C211yesmaximal, characteristic
3#32683C212yesmaximal, complemented
4#32260C212yesmaximal, complemented
5#31690C214yesmaximal, complemented
6#31086C214yesmaximal, complemented
7#30305C212yesmaximal, complemented
8#29574C212yesmaximal, complemented
9#28845C214yesmaximal, complemented
10#26060C214yesmaximal, complemented
11#25834C212yesmaximal, complemented
12#24089C212yesmaximal
13#23269C211yesmaximal, characteristic, complemented
14C2×D4×Q8C2211yescharacteristic
15C2×#547C2211yescharacteristic
16C2×#536—21no—
17C2×#498C2212yes—
18C2×#481C2212yes—
19C2×#476C2212yes—
20C22×#106C2212yes—
21#2868C2211yescharacteristic
22#2859C2212yes—
23#2844—21no—
24#2809—21no—
25#2720—24no—
26#2683—21no—
27#2680—24no—
28#2621—21no—
29#2567C2212yes—
30#2547C2211yescharacteristic
31#2546C2211yescharacteristic
32#2497C2214yes—
33#2487C2212yes—
34#2440C2214yes—
35#2440C2214yes—
36#2434—21no—
37#2432—21no—
38#2422C2214yes—
39#2379—21no—
40C4×#124C2211yescharacteristic
41#2293C2211yescharacteristic
42#2100C2212yes—
43#2094C2212yes—
44#2088C2214yes—
45#2084C2214yes—
46#2084C2214yes—
47#2084C2212yes—
48#2077C2214yes—
49#2077C2214yes—
50#2071C2212yes—
51#2058C2214yes—
52#2058C2214yes—
53#2058C2214yes—
54#2058C2214yes—
55#2034C2214yes—
56#1988C2212yes—
57#1984C2212yes—
58#1941C2214yes—
59#1941C2214yes—
60#1934C2214yes—
61#1923C2214yes—
62#1920C2214yes—
63#1919C2214yes—
64#1893C2212yes—
65#1869C2212yes—
66#1864C2214yes—
67#1864C2214yes—
68#1853C2214yes—
69#1853C2214yes—
70#1844C2214yes—
71#1834C2214yes—
72#1823C2212yes—
73#1807C2214yes—
74C4×#102C2212yes—
75C8×C22⋊C4C2214yes—
76#1744C2212yes—
77C22×#41—21no—
78C23×Q8—21no—
79C23×C8C2312yes—
80D4×Q8—42no—
81D4×Q8—42no—
82#547—42no—
83#547—42no—
84#536—48no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#125—22no—
93C2×#124—21no—
94C2×#124D411yescharacteristic
95C2×#123—24no—
96C2×#123—22no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×#121C2312yes—
103C2×C4×Q8C2311yescharacteristic
104C2×C4×D4C2311yescharacteristic
105C2×C4×D4C2312yes—
106C2×C4×D4C2311yescharacteristic
107C2×#117—21no—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110#498—44no—
111#481—44no—
112#476—44no—
113C2×#106—44no—
114C2×#106C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2312yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×C22⋊C8C2314yes—
124C2×C22⋊C8C2312yes—
125C2×C4×C8C2314yes—
126C2×C4×C8C2312yes—
127#400—22no—
128#400—24no—
129#399—22no—
130#398—22no—
131#398—24no—
132#398—24no—
133#397—22no—
134#397—24no—
135#396—24no—
136#395—22no—
137#395—22no—
138#395—24no—
139#395—24no—
140#394—24no—
141#391—22no—
142#391—21no—
143#391—21no—
144#391D411yescharacteristic
145#390D411yescharacteristic
146#390C2312yes—
147#389—22no—
148#389—22no—
149#389C2314yes—
150#389C2314yes—
151#389C2314yes—
152#389C2314yes—
153#389C2312yes—
154#388—24no—
155#388—22no—
156#388D414yes—
157#387—24no—
158#386—22no—
159#385—24no—
160#385—24no—
161#385—22no—
162#385—24no—
163#384—22no—
164#384—22no—
165#384C2312yes—
166#384C2314yes—
167#384C2312yes—
168#382—22no—
169#382D411yescharacteristic
170#380C2314yes—
171#380C2314yes—
172#380C2312yes—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2311yescharacteristic
175C4×C22⋊C4—21no—
176C4×C22⋊C4C2314yes—
177C43C2311yescharacteristic
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#336C2314yes—
187#336C2314yes—
188#336C2312yes—
189#336C2312yes—
190C2×#41—44no—
191C2×#41—44no—
192C2×#41—82no—
193C22×Q8—21no—
194C22×Q8—21no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×Q8—82no—
200C22×Q8C2×D411yescharacteristic
201C22×D4—22no—
202C22×D4C2×D411yescharacteristic
203C22×D4C2411yescharacteristic
204C23×C4—21no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4C2412yes—
209C22×C8—28no—
210C22×C8—42no—
211C22×C8C2414yes—
212C22×C8C2414yes—
213C4⋊Q8—24no—
214C4⋊Q8—41no—
215C4⋊Q8—41no—
216C4⋊Q8—42no—
217C4⋊Q8—41no—
218C4⋊Q8—41no—
219C4⋊Q8—42no—
220C4⋊Q8—42no—
221C4⋊Q8—44no—
222C4⋊Q8—44no—
223C4⋊Q8—82no—
224#125—84no—
225#124—44no—
226#124SD1618yes—
227#123—48no—
228#123—416no—
229C22⋊Q8—48no—
230C22⋊Q8—44no—
231C22⋊Q8—44no—
232C22⋊Q8—416no—
233C22⋊Q8—88no—
234#121—24no—
235#121—42no—
236C4×Q8—41no—
237C4×Q8—41no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—24no—
241C4×D4—42no—
242C4×D4—41no—
243C4×D4—41no—
244#117—44no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2414yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2411yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4C2×D414yes—
284C2×C22⋊C4C2×D412yes—
285C2×C22⋊C4C2×D414yes—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42C2×D412yes—
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2412yes—
293C2×C42C2412yes—
294#106—28no—
295#106—28no—
296#106—44no—
297#106—44no—
298#106—44no—
299#106—44no—
300C4⋊C8—42no—
301#102—28no—
302#102—28no—
303C22⋊C8—28no—
304C22⋊C8—42no—
305C4×C8—28no—
306C4×C8—42no—
307#95—22no—
308#95—21no—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95C2×D414yes—
318#95C2×D414yes—
319C24—21no—
320C24C22×D412yes—
321#41—816no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—44no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—44no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—82no—
333C2×Q8—84no—
334C2×Q8—82no—
335C2×Q8—816no—
336C2×Q8C2×SD1614yes—
337C2×D4—21no—
338C2×D4—24no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—82no—
344C2×D4C2×SD1614yes—
345C2×D4H(5,𝔽2)12yes—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—84no—
367C22×C4C22×D412yes—
368C22×C4C22×D412yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D412yes—
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C2511yescharacteristic
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—48no—
381C2×C82−1+418yes—
382C4⋊C4—21no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—44no—
403C4⋊C4—48no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4—84no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4—84no—
410C4⋊C4—82no—
411C4⋊C4—84no—
412C4⋊C4H(5,𝔽2)12yes—
413C22⋊C4—24no—
414C22⋊C4—28no—
415C22⋊C4—42no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—48no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—82no—
422C22⋊C4—84no—
423C22⋊C4C2×SD16116yes—
424C42—24no—
425C42—22no—
426C42—22no—
427C42—21no—
428C42—21no—
429C42—24no—
430C42—41no—
431C42—42no—
432C42—82no—
433C42C2×SD1614yes—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—41no—
440C23—42no—
441C23—41no—
442C23—42no—
443C23C22×SD1614yes—
444C23C23×D411yescharacteristic
445Q8—44no—
446Q8—88no—
447Q8—84no—
448Q8—84no—
449Q8—164no—
450D4—24no—
451D4—44no—
452D4—42no—
453D4—88no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—48no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—88no—
491C2×C4—88no—
492C2×C4—164no—
493C2×C4C22×SD1612yes—
494C2×C4C22×SD1614yes—
495C2×C4C22×SD1614yes—
496C2×C4#57714yes—
497C2×C4C2×H(5,𝔽2)11yescharacteristic
498C2×C4C2×H(5,𝔽2)11yescharacteristic
499C2×C4C2×2−1+414yes—
500C8—28no—
501C8—44no—
502C22—28no—
503C22—24no—
504C22—44no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—82no—
511C22—82no—
512C22#348411yescharacteristic
513C22#349912yes—
514C22C23×SD1611yescharacteristic
515C22C2×#57712yes—
516C22#358511yescharacteristic
517C4—24no—
518C4—24no—
519C4—22no—
520C4—24no—
521C4—42no—
522C4—42no—
523C4—41no—
524C4—42no—
525C4—82no—
526C4—84no—
527C4—82no—
528C4—82no—
529C4—84no—
530C4#353812yes—
531C4#354412yes—
532C2—44no—
533C2—82no—
534C2#5154211yescharacteristic
535C2#6156011yescharacteristic
536C2#6166411yescharacteristic
537C2#6246912yes—
538C2#6251412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #46152C2semidirect
3. #32683C2semidirect
4. #32260C2semidirect
5. #31690C2semidirect
6. #31086C2semidirect
7. #30305C2semidirect
8. #29574C2semidirect
9. #28845C2semidirect
10. #26060C2semidirect
11. #25834C2semidirect
13. #23269C2semidirect

Maximal subgroups

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

subgroupindex
#461522
#393602
#326832
#322602
#316902
#310862
#303052
#295742
#288452
#260602
#258342
#240892
#232692