Finite Group Databasebeta

← #7228749

Subgroups — #7228749

Order 512.

1531 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46152C211yesmaximal, characteristic, complemented
2#36303C211yesmaximal, characteristic
3#33072C212yesmaximal, complemented
4#32820C212yesmaximal, complemented
5#32631C212yesmaximal, complemented
6#32269C212yesmaximal, complemented
7#32218C214yesmaximal, complemented
8#30777C214yesmaximal, complemented
9#27533C214yesmaximal, complemented
10#25978C214yesmaximal, complemented
11#25834C212yesmaximal, complemented
12#24055C212yesmaximal
13#23347C211yesmaximal, characteristic, complemented
14C2×D4×Q8C2211yescharacteristic
15C2×#547C2211yescharacteristic
16C2×#498C2212yes—
17C2×#483C2212yes—
18C2×#476C2212yes—
19C2×#417C2212yes—
20#2862C2211yescharacteristic
21#2859C2212yes—
22#2850—21no—
23#2844—21no—
24#2809—21no—
25#2720—24no—
26#2683—21no—
27#2568C2211yescharacteristic
28#2567C2212yes—
29#2547C2211yescharacteristic
30#2503—21no—
31#2497C2214yes—
32#2492C2214yes—
33#2488—24no—
34#2487C2212yes—
35#2440C2214yes—
36#2436—21no—
37#2379—21no—
38#2339C2214yes—
39C4×#124C2211yescharacteristic
40C2×#391—21no—
41C42×D4C2211yescharacteristic
42#2096C2214yes—
43#2077C2214yes—
44#2075C2214yes—
45#2072C2214yes—
46#2068C2214yes—
47#2066C2214yes—
48#2066C2214yes—
49#2058C2214yes—
50#2058C2214yes—
51#2043C2214yes—
52#2035C2214yes—
53#2024C2214yes—
54#1988C2212yes—
55#1984C2212yes—
56#1948C2212yes—
57#1947C2212yes—
58#1933C2214yes—
59#1928C2214yes—
60#1923C2214yes—
61#1921C2214yes—
62#1919C2214yes—
63#1894C2212yes—
64#1870C2212yes—
65#1864C2214yes—
66#1864C2214yes—
67#1862C2214yes—
68#1853C2214yes—
69#1851C2214yes—
70#1831C2212yes—
71#1809C2214yes—
72#1802C2212yes—
73#1802C2212yes—
74#1788C2214yes—
75#1788C2212yes—
76C4×#102C2212yes—
77C22×#41C2312yes—
78C23×Q8—21no—
79C22×M16C2312yes—
80D4×Q8—42no—
81D4×Q8—42no—
82#547—42no—
83#547—42no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×#125—22no—
89C2×#124—21no—
90C2×#124D411yescharacteristic
91C2×#123—22no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8C2312yes—
96C2×#121—21no—
97C2×C4×Q8C2311yescharacteristic
98C2×C4×D4C2311yescharacteristic
99C2×C4×D4C2312yes—
100C2×C4×D4C2312yes—
101C2×C4×D4C2311yescharacteristic
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2312yes—
104C22×C42—21no—
105#498—44no—
106#483—44no—
107#476—44no—
108C2×#107C2314yes—
109C2×#106C2314yes—
110C2×#106C2314yes—
111C2×#106C2314yes—
112C2×C4⋊C8C2312yes—
113C2×C4⋊C8C2312yes—
114C2×C4⋊C8C2312yes—
115#417—44no—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×C22⋊C8C2312yes—
123C2×C22⋊C8C2314yes—
124C2×C4×C8C2312yes—
125#400—22no—
126#400—22no—
127#400—24no—
128#400—24no—
129#399—22no—
130#398—24no—
131#397—22no—
132#396—24no—
133#395—22no—
134#395—22no—
135#395—24no—
136#393—24no—
137#391—22no—
138#391—21no—
139#391—21no—
140#391—21no—
141#391—21no—
142#391—24no—
143#391D411yescharacteristic
144#391C2314yes—
145#391C2314yes—
146#390D411yescharacteristic
147#389—22no—
148#389—22no—
149#389C2314yes—
150#389C2314yes—
151#389C2314yes—
152#388—24no—
153#388—24no—
154#388D414yes—
155#387—24no—
156#386—22no—
157#386—24no—
158#386—24no—
159#385—24no—
160#385—22no—
161#384—22no—
162#384C2312yes—
163#384C2312yes—
164#382—21no—
165#382—22no—
166#382—22no—
167#382D411yescharacteristic
168#381—24no—
169#380C2314yes—
170#380C2314yes—
171#380C2312yes—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2311yescharacteristic
174C4×C4⋊C4C2312yes—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—22no—
177C4×C22⋊C4C2314yes—
178C2×#95—24no—
179C43C2311yescharacteristic
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#336C2314yes—
187#336C2314yes—
188#336C2314yes—
189C2×#41—24no—
190C2×#41—28no—
191C2×#41—42no—
192C2×#41—44no—
193C22×Q8—21no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—44no—
197C22×Q8—44no—
198C22×Q8—82no—
199C22×Q8C2×D411yescharacteristic
200C22×Q8C2412yes—
201C22×D4—22no—
202C22×D4C2×D411yescharacteristic
203C22×D4C2411yescharacteristic
204C23×C4—21no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4C2412yes—
210C2×M16—44no—
211C2×M16—42no—
212C22×C8C2414yes—
213C22×C8C2414yes—
214C4⋊Q8—24no—
215C4⋊Q8—42no—
216C4⋊Q8—41no—
217C4⋊Q8—41no—
218C4⋊Q8—42no—
219C4⋊Q8—42no—
220C4⋊Q8—41no—
221C4⋊Q8—41no—
222#125—84no—
223#124—24no—
224#124—82no—
225#123—48no—
226C22⋊Q8—24no—
227C22⋊Q8—48no—
228C22⋊Q8—42no—
229C22⋊Q8—44no—
230C22⋊Q8—88no—
231#121—44no—
232C4×Q8—41no—
233C4×Q8—41no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4—41no—
241C4×D4—41no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2411yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4C2×D414yes—
275C2×C22⋊C4C2×D414yes—
276C2×C22⋊C4C2×D412yes—
277C2×C42—21no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—82no—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2411yescharacteristic
285C2×C42C2412yes—
286C2×C42C2412yes—
287#107—44no—
288#106—28no—
289#106—44no—
290#106—44no—
291C4⋊C8—42no—
292C4⋊C8—42no—
293C4⋊C8—42no—
294#103—44no—
295#103—44no—
296#103—44no—
297#103—44no—
298#102—28no—
299#102—28no—
300C22⋊C8—44no—
301C22⋊C8—42no—
302C4×C8—42no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—21no—
308#95—22no—
309#95—24no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95C2×D414yes—
317#95C2×D414yes—
318C24—21no—
319C24C22×D412yes—
320#41—48no—
321#41—84no—
322#41—88no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—44no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—44no—
333C2×Q8—42no—
334C2×Q8—82no—
335C2×Q8—84no—
336C2×Q8—82no—
337C2×Q8—816no—
338C2×Q8#13714yes—
339C2×D4—22no—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—44no—
345C2×D4—48no—
346C2×D4H(5,𝔽2)12yes—
347C2×D42−1+414yes—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—48no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4—84no—
371C22×C4C22×D412yes—
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D412yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D412yes—
378C22×C4C2511yescharacteristic
379M16—84no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C4⋊C4—21no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—48no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—84no—
408C4⋊C4—82no—
409C4⋊C4—84no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4H(5,𝔽2)12yes—
413C22⋊C4—24no—
414C22⋊C4—28no—
415C22⋊C4—28no—
416C22⋊C4—42no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—82no—
421C22⋊C4—84no—
422C22⋊C4—84no—
423C42—22no—
424C42—24no—
425C42—22no—
426C42—22no—
427C42—21no—
428C42—21no—
429C42—22no—
430C42—22no—
431C42—41no—
432C42—42no—
433C42—82no—
434C42—82no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—24no—
440C23—22no—
441C23—24no—
442C23—41no—
443C23—41no—
444C23—42no—
445C23C23×D411yescharacteristic
446Q8—44no—
447Q8—42no—
448Q8—42no—
449Q8—44no—
450Q8—88no—
451Q8—84no—
452Q8—164no—
453D4—22no—
454D4—24no—
455D4—82no—
456D4—88no—
457D4#56314yes—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—48no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—88no—
495C2×C4—88no—
496C2×C4—88no—
497C2×C4—164no—
498C2×C4C2×#13712yes—
499C2×C4C2×#13714yes—
500C2×C4#57714yes—
501C2×C4C2×H(5,𝔽2)11yescharacteristic
502C2×C4C2×H(5,𝔽2)11yescharacteristic
503C2×C4C2×2−1+412yes—
504C2×C4#58512yes—
505C8—44no—
506C8—44no—
507C22—24no—
508C22—42no—
509C22—44no—
510C22—44no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—44no—
515C22—44no—
516C22—42no—
517C22#348411yescharacteristic
518C22#348912yes—
519C22C22×#13711yescharacteristic
520C22C2×#57712yes—
521C22#358511yescharacteristic
522C4—22no—
523C4—24no—
524C4—24no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—41no—
529C4—42no—
530C4—82no—
531C4—82no—
532C4—84no—
533C4—82no—
534C4—84no—
535C4C2×#56311yescharacteristic
536C4#354511yescharacteristic
537C4#356112yes—
538C2—44no—
539C2—82no—
540C2#4613611yescharacteristic
541C2#6157011yescharacteristic
542C2#6166411yescharacteristic
543C2#6214712yes—
544C2#6216612yes—
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. #33072C2semidirect
4. #32820C2semidirect
5. #32631C2semidirect
6. #32269C2semidirect
7. #32218C2semidirect
8. #30777C2semidirect
9. #27533C2semidirect
10. #25978C2semidirect
11. #25834C2semidirect
13. #23347C2semidirect

Maximal subgroups

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

subgroupindex
#461522
#363032
#330722
#328202
#326312
#322692
#322182
#307772
#275332
#259782
#258342
#240552
#233472