Finite Group Databasebeta

← #6470495

Subgroups — #6470495

Order 512.

2258 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37307C211yesmaximal, characteristic, complemented
2#34257C211yesmaximal, characteristic
3#32967C212yesmaximal, complemented
4#32954C212yesmaximal, complemented
5#29965C212yesmaximal, complemented
6#29954C212yesmaximal, complemented
7#25981C214yesmaximal, complemented
8#25967C214yesmaximal, complemented
9#25955C214yesmaximal, complemented
10#25953C214yesmaximal, complemented
11#21946C212yesmaximal, complemented
12#21937C211yesmaximal, characteristic, complemented
13#20603C212yesmaximal
14C22×2−1+4C2211yescharacteristic
15C2×#519C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C2×#479C2214yes—
18C2×#475C2214yes—
19C2×#417C2212yes—
20C2×#419C2212yes—
21C2×#416C2212yes—
22C22×#103C2212yes—
23C2×#408C2212yes—
24#2857C2212yes—
25#2806—22no—
26#2748—22no—
27#2732—22no—
28#2683—22no—
29#2629C2212yes—
30#2441C2214yes—
31#2440C2214yes—
32#2341—22no—
33#2341C2214yes—
34Q8×C22⋊C4—22no—
35Q8×C22⋊C4C2214yes—
36#2311—22no—
37#2311C2211yescharacteristic
38#2311C2212yes—
39#2311C2211yescharacteristic
40#2297—22no—
41#2297C2211yescharacteristic
42#2297C2211yescharacteristic
43#2096C2212yes—
44#2095C2212yes—
45#2094C2212yes—
46#2094C2212yes—
47#2067C2214yes—
48#2066C2214yes—
49#2066C2214yes—
50#2065C2214yes—
51#2026C2212yes—
52#2025C2212yes—
53#2024C2212yes—
54#2023C2212yes—
55#1865C2214yes—
56#1865C2214yes—
57#1865C2214yes—
58#1863C2214yes—
59#1854C2214yes—
60#1854C2214yes—
61#1853C2214yes—
62#1853C2214yes—
63#1851C2214yes—
64#1850C2214yes—
65#1809C2214yes—
66#1807C2214yes—
67#1803C2214yes—
68#1803C2214yes—
69#1798C2212yes—
70#1797C2212yes—
71#1788C2214yes—
72#1788C2214yes—
73#1787C2214yes—
74#1787C2214yes—
75C2×2−1+4—24no—
76C2×2−1+4—44no—
77C2×2−1+4—44no—
78C2×2−1+4Q818yescomplemented
79C22×#41—22no—
80C22×#41—21no—
81C22×#41C2312yes—
82C22×#41C2311yescharacteristic
83C22×#41C2311yescharacteristic
84C23×Q8—21no—
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C22×M16C2312yes—
88C23×C8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×#125C2312yes—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—24no—
93#519—44no—
94C2×C4×Q8—22no—
95C2×C4×Q8—22no—
96C2×C4×Q8—44no—
97C2×C4×Q8C2314yes—
98C2×C4×Q8C2314yes—
99C2×#117—22no—
100C2×#117C2311yescharacteristic
101C2×#117C2312yes—
102C2×#117C2311yescharacteristic
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C42—21no—
107C22×C42C2311yescharacteristic
108#479—48no—
109#475—48no—
110C2×#107C2314yes—
111C2×#106C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2314yes—
115C2×C4⋊C8C2314yes—
116#419—44no—
117#417—44no—
118#416—44no—
119C2×#103—44no—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128#408—28no—
129C2×C22⋊C8C2314yes—
130C2×C22⋊C8C2314yes—
131#399—24no—
132#398—24no—
133#398—24no—
134#396—24no—
135#396—24no—
136#395—24no—
137#391—21no—
138#391—22no—
139#391—22no—
140#391—24no—
141#391—24no—
142#391D412yes—
143#391C2312yes—
144#391C2312yes—
145#390—21no—
146#390—22no—
147#390—24no—
148#390—24no—
149#390D412yes—
150#390C2312yes—
151#390C2312yes—
152#389—22no—
153#389—24no—
154#389C2314yes—
155#389C2314yes—
156#387—24no—
157#386—22no—
158#386—22no—
159#386—22no—
160#386—21no—
161#386—21no—
162#386—22no—
163#386D412yes—
164#386D412yes—
165#385—24no—
166#384C2312yes—
167#384C2312yes—
168#380—22no—
169#380—22no—
170#380—24no—
171#380—24no—
172#380C2314yes—
173#380C2314yes—
174#380C2314yes—
175C4×C22⋊C4—24no—
176C4×C22⋊C4—22no—
177C4×C22⋊C4—24no—
178C4×C22⋊C4C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#336C2312yes—
184#336C2312yes—
185#336C2312yes—
186#336C2312yes—
1872−1+4—432no—
1882−1+4—432no—
189C2×#41—216no—
190C2×#41—28no—
191C2×#41—22no—
192C2×#41—24no—
193C2×#41—24no—
194C2×#41—24no—
195C2×#41—42no—
196C2×#41—42no—
197C2×#41—42no—
198C2×#41—42no—
199C2×#41—42no—
200C2×#41—42no—
201C2×#41—48no—
202C2×#41—48no—
203C2×#41—44no—
204C2×#41—44no—
205C2×#41C2×Q818yes—
206C2×#41C2×Q814yes—
207C2×#41C2×Q814yes—
208C2×#41#4114yes—
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—28no—
212C22×Q8—21no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—24no—
217C22×Q8—24no—
218C22×Q8—24no—
219C22×Q8—42no—
220C22×Q8—42no—
221C22×Q8—44no—
222C22×Q8—44no—
223C22×Q8C2×D411yescharacteristic
224C22×Q8C2×Q814yes—
225C22×Q8C2×Q818yes—
226C22×Q8C2411yescharacteristic
227C22×Q8C2412yes—
228C22×D4—21no—
229C22×D4—22no—
230C22×D4C2×D412yes—
231C22×D4C2×D411yescharacteristic
232C22×D4C2411yescharacteristic
233C23×C4—21no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243C2×M16—24no—
244C2×M16—42no—
245C2×M16—42no—
246C22×C8—24no—
247C22×C8—42no—
248C22×C8—42no—
249C22×C8C2414yes—
250C22×C8C2414yes—
251C4⋊Q8—44no—
252#125—44no—
253C22⋊Q8—88no—
254C22⋊Q8—84no—
255C4×Q8—28no—
256C4×Q8—28no—
257C4×Q8—44no—
258C4×Q8—44no—
259C4×Q8—48no—
260C4×Q8—48no—
261#117—42no—
262#117—42no—
263#117—48no—
264#117—42no—
265#117—42no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2414yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2414yes—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4C2×D414yes—
295C2×C22⋊C4C2×D412yes—
296C2×C22⋊C4C2×D414yes—
297C2×C42—24no—
298C2×C42—21no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—42no—
302C2×C42—44no—
303C2×C42C2×D412yes—
304C2×C42C2412yes—
305C2×C42C2412yes—
306#107—44no—
307#106—44no—
308#106—44no—
309#106—44no—
310C4⋊C8—44no—
311C4⋊C8—44no—
312#103—28no—
313#103—28no—
314#103—28no—
315#103—28no—
316#103—44no—
317#103—44no—
318#102—44no—
319#102—44no—
320C22⋊C8—28no—
321C22⋊C8—28no—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95C2×D414yes—
331#95C2×D414yes—
332C24—21no—
333C24C22×D412yes—
334C24C22×D411yescharacteristic
335#41—216no—
336#41—48no—
337#41—48no—
338#41—48no—
339#41—48no—
340#41—48no—
341#41—48no—
342#41—464no—
343#41—48no—
344#41—432no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—216no—
348C2×Q8—216no—
349C2×Q8—24no—
350C2×Q8—28no—
351C2×Q8—22no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—216no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—432no—
366C2×Q8—44no—
367C2×Q8—82no—
368C2×Q8—82no—
369C2×Q8C22⋊Q814yes—
370C2×Q8C22×Q812yes—
371C2×Q8C22×Q814yes—
372C2×Q8H(5,𝔽2)14yes—
373C2×Q8H(5,𝔽2)14yes—
374C2×D4—28no—
375C2×D4—24no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4C22⋊Q818yes—
388C2×D4C22⋊Q814yes—
389C2×D4C22×Q812yes—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—21no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—41no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4—84no—
422C22×C4C22×D412yes—
423C22×C4C22×D414yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×Q812yes—
428C22×C4C22×Q814yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4114yes—
431C22×C4C2511yescharacteristic
432M16—84no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—84no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—42no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—48no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4—84no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C4⋊C4—82no—
458C22⋊C4—28no—
459C22⋊C4—44no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—44no—
463C22⋊C4—84no—
464C22⋊C4—82no—
465C22⋊C4—84no—
466C22⋊C4—82no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C42—24no—
470C42—24no—
471C42—42no—
472C42—48no—
473C42—48no—
474C42—42no—
475C42—82no—
476C23—24no—
477C23—22no—
478C23—21no—
479C23—22no—
480C23—21no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—42no—
486C23—41no—
487C23—41no—
488C23—42no—
489C23C2×C22⋊Q812yes—
490C23C2×C22⋊Q814yes—
491C23C23×D411yescharacteristic
492Q8—216no—
493Q8—24no—
494Q8—24no—
495Q8—24no—
496Q8—24no—
497Q8—44no—
498Q8—416no—
499Q8—42no—
500Q8—164no—
501Q8#54318yes—
502D4—24no—
503D4—44no—
504D4—48no—
505D4—416no—
506D4—48no—
507D4—42no—
508C2×C4—216no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—44no—
532C2×C4—416no—
533C2×C4—416no—
534C2×C4—48no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—82no—
549C2×C4—84no—
550C2×C4—164no—
551C2×C4C2×C22⋊Q812yes—
552C2×C4C2×C22⋊Q814yes—
553C2×C4#57714yes—
554C2×C4C23×Q811yescharacteristic
555C2×C4C22×#4111yescharacteristic
556C2×C4C2×H(5,𝔽2)12yes—
557C2×C4C2×H(5,𝔽2)12yes—
558C8—44no—
559C8—44no—
560C22—28no—
561C22—24no—
562C22—22no—
563C22—24no—
564C22—41no—
565C22—42no—
566C22—41no—
567C22—42no—
568C22—44no—
569C22—42no—
570C22—44no—
571C22—42no—
572C22—81no—
573C22—81no—
574C22—82no—
575C22C22×C22⋊Q811yescharacteristic
576C22#348411yescharacteristic
577C22#348411yescharacteristic
578C22C2×#57712yes—
579C22#358612yes—
580C4—28no—
581C4—22no—
582C4—24no—
583C4—24no—
584C4—42no—
585C4—42no—
586C4—44no—
587C4—84no—
588C4—82no—
589C4—82no—
590C4—84no—
591C4C2×#54312yes—
592C4#349012yes—
593C2—42no—
594C2—44no—
595C2—82no—
596C2#3498211yescharacteristic
597C2#6049712yes—
598C2#6163311yescharacteristic
599C2#6165211yescharacteristic
600C2#6168612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×2−1+4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37307C2semidirect
3. #32967C2semidirect
4. #32954C2semidirect
5. #29965C2semidirect
6. #29954C2semidirect
7. #25981C2semidirect
8. #25967C2semidirect
9. #25955C2semidirect
10. #25953C2semidirect
11. #21946C2semidirect
12. #21937C2semidirect
78. C2×2−1+4Q8semidirect

Maximal subgroups

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

subgroupindex
#373072
#342572
#329672
#329542
#299652
#299542
#259812
#259672
#259552
#259532
#219462
#219372
#206032