Finite Group Databasebeta

← #6489665

Subgroups — #6489665

Order 512.

3036 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic, complemented
2C2×#3234C211yesmaximal, characteristic
3C2×#3103C211yesmaximal, characteristic
4#35249C211yesmaximal, characteristic
5#33005C212yesmaximal, complemented
6#33003C212yesmaximal, complemented
7#29640C212yesmaximal, complemented
8#29639C212yesmaximal, complemented
9#25944C214yesmaximal, complemented
10#25848C214yesmaximal, complemented
11#24334C218yesmaximal, complemented
12#21836C211yesmaximal, characteristic, complemented
13#21820C211yesmaximal, characteristic, complemented
14#20741C211yesmaximal, characteristic, complemented
15C2×#557—28no—
16C2×#557—28no—
17C2×#557—22no—
18C2×#557—22no—
19C2×#557C2214yes—
20C2×#557C2214yes—
21C2×#532C2211yescharacteristic
22C22×C4⋊Q8—21no—
23C22×C4⋊Q8C2211yescharacteristic
24C22×#125—21no—
25C22×#125—21no—
26C22×#125C2211yescharacteristic
27C22×#125C2211yescharacteristic
28C22×C4×Q8—21no—
29C22×C4×Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×C4×Q8C2211yescharacteristic
33#3234—28no—
34C2×#501C2212yes—
35C2×#490C2212yes—
36#3103—28no—
37C2×#481C2212yes—
38C2×#481C2212yes—
39C2×#480C2212yes—
40C2×#475C2212yes—
41C2×#478C2214yes—
42C2×#460C2212yes—
43C2×#448C2212yes—
44C2×C4×SD16C2214yes—
45C2×#432C2211yescharacteristic
46C2×#431C2211yescharacteristic
47#2854C2212yes—
48#2854C2212yes—
49#2632—22no—
50#2580C2214yes—
51#2567C2212yes—
52#2548C2212yes—
53#2501—21no—
54#2436—21no—
55#2406—21no—
56#2406—21no—
57#2326—22no—
58#2283C2211yescharacteristic
59#2059C2218yes—
60#2058C2218yes—
61#2025C2218yes—
62#2023C2218yes—
63#1991C2214yes—
64#1989C2214yes—
65#1985C2214yes—
66#1984C2214yes—
67#1880C2218yes—
68#1880C2218yes—
69#1865C2218yes—
70#1863C2218yes—
71#1824C2212yes—
72#1824C2212yes—
73#1823C2212yes—
74#1817C2212yes—
75#1817C2212yes—
76#1805C2214yes—
77#1787C2212yes—
78#1763C2214yes—
79C22×#41—21no—
80C23×Q8—21no—
81C23×Q8D411yescharacteristic
82C23×Q8C2311yescharacteristic
83C22×SD16C2314yes—
84#557—232no—
85#557—232no—
86#557—216no—
87#557—216no—
88#557—216no—
89#557—216no—
90#532—24no—
91#532—24no—
92C2×C4⋊Q8—216no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2314yes—
100C2×#127C2311yescharacteristic
101C2×#125—216no—
102C2×#125—216no—
103C2×#125—24no—
104C2×#125—24no—
105C2×#125—24no—
106C2×#125—24no—
107C2×#125—24no—
108C2×#125—24no—
109C2×#125—22no—
110C2×#125—21no—
111C2×#125—21no—
112C2×#125—24no—
113C2×#125D418yes—
114C2×#125C2314yes—
115C2×#125C2312yes—
116C2×#125C2312yes—
117C2×#124—22no—
118C2×#124—21no—
119C2×#123—22no—
120C2×#123—22no—
121C2×#121—21no—
122C2×#121C2312yes—
123C2×C4×Q8—216no—
124C2×C4×Q8—24no—
125C2×C4×Q8—24no—
126C2×C4×Q8—24no—
127C2×C4×Q8—24no—
128C2×C4×Q8—24no—
129C2×C4×Q8—24no—
130C2×C4×Q8—24no—
131C2×C4×Q8—22no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8—22no—
135C2×C4×Q8—24no—
136C2×C4×Q8—24no—
137C2×C4×Q8D418yes—
138C2×C4×Q8D412yes—
139C2×C4×Q8D412yes—
140C2×C4×Q8C2314yes—
141C2×C4×Q8C2314yes—
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2312yes—
144C2×C4×Q8C2312yes—
145C2×C4×Q8C2312yes—
146C2×C4×D4—21no—
147C2×C4×D4C2312yes—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4—21no—
156C22×C4⋊C4D411yescharacteristic
157C22×C4⋊C4D411yescharacteristic
158C22×C4⋊C4D411yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2311yescharacteristic
161C22×C4⋊C4C2311yescharacteristic
162C22×C42—21no—
163C22×C42—21no—
164C22×C42—21no—
165C22×C42C2311yescharacteristic
166C22×C42C2311yescharacteristic
167C22×C42C2311yescharacteristic
168#501—28no—
169#490—28no—
170#481—28no—
171#481—28no—
172#480—28no—
173#478—216no—
174#475—28no—
175#460—28no—
176#448—28no—
177C4×SD16—216no—
178#432—24no—
179#431—24no—
180C2×#107C2314yes—
181C2×#106C2314yes—
182C2×#106C2314yes—
183C2×C4⋊C8C2312yes—
184C2×C4⋊C8C2312yes—
185C2×#103C2314yes—
186C2×#103C2318yes—
187C2×#102C2314yes—
188C2×#102C2314yes—
189C2×#102C2314yes—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2312yes—
192C2×#97C2312yes—
193C2×C4×C8C2312yes—
194#399—24no—
195#395—22no—
196#395—22no—
197#394—22no—
198#394—22no—
199#393—24no—
200#391—22no—
201#390—22no—
202#390C2314yes—
203#390C2314yes—
204#389C2314yes—
205#389C2314yes—
206#388—24no—
207#388—24no—
208#386—22no—
209#386—22no—
210#385—24no—
211#385—24no—
212#384—22no—
213#383—21no—
214#382C2312yes—
215#381—21no—
216#380C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219#376—21no—
220#340C2318yes—
221#340C2318yes—
222#340C2318yes—
223#336C2318yes—
224#336C2318yes—
225C2×#41—44no—
226C2×#41—84no—
227C22×Q8—28no—
228C22×Q8—28no—
229C22×Q8—24no—
230C22×Q8—22no—
231C22×Q8—21no—
232C22×Q8—24no—
233C22×Q8—21no—
234C22×Q8—44no—
235C22×Q8—48no—
236C22×Q8C2×D414yes—
237C22×Q8C2×D412yes—
238C22×Q8C2×D412yes—
239C22×Q8C2412yes—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×SD16—28no—
257C2×SD16—216no—
258C22×C8C2414yes—
259C22×C8C2414yes—
260C4⋊Q8—232no—
261C4⋊Q8—232no—
262C4⋊Q8—216no—
263C4⋊Q8—28no—
264C4⋊Q8—28no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267#127—22no—
268#127—22no—
269#125—232no—
270#125—232no—
271#125—232no—
272#125—232no—
273#125—216no—
274#125—28no—
275#125—28no—
276#125—28no—
277#125—28no—
278#125—24no—
279#125—24no—
280#125C2×D418yes—
281#125C2×D418yes—
282#124—48no—
283#124—44no—
284#123—84no—
285#123—84no—
286#121—24no—
287#121—24no—
288#121—44no—
289C4×Q8—216no—
290C4×Q8—232no—
291C4×Q8—232no—
292C4×Q8—216no—
293C4×Q8—28no—
294C4×Q8—28no—
295C4×Q8—28no—
296C4×Q8—28no—
297C4×Q8—28no—
298C4×Q8—28no—
299C4×Q8—28no—
300C4×Q8—28no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8C2×D418yes—
308C4×Q8C2×D418yes—
309C4×D4—24no—
310C4×D4—24no—
311C4×D4—44no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D412yes—
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2414yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C42—28no—
365C2×C42—28no—
366C2×C42—28no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—24no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—24no—
375C2×C42—22no—
376C2×C42C2×D412yes—
377C2×C42C2×D412yes—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D414yes—
381C2×C42C2×D414yes—
382C2×C42C2412yes—
383C2×C42C2412yes—
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386#107—28no—
387#106—28no—
388#106—28no—
389C4⋊C8—24no—
390C4⋊C8—24no—
391#103—28no—
392#103—216no—
393#102—28no—
394#102—28no—
395#102—28no—
396C22⋊C8—24no—
397C22⋊C8—24no—
398#97—24no—
399C4×C8—24no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—24no—
406C24—22no—
407C24C22×D411yescharacteristic
408#41—816no—
409C2×Q8—216no—
410C2×Q8—28no—
411C2×Q8—216no—
412C2×Q8—216no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—416no—
423C2×Q8—432no—
424C2×Q8—48no—
425C2×Q8—42no—
426C2×Q8—48no—
427C2×Q8—82no—
428C2×Q8C22×D414yes—
429C2×Q82−1+418yes—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—28no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—82no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4#12014yes—
469C22×C4#12014yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C2511yescharacteristic
482SD16—416no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—28no—
486C2×C8—28no—
487C4⋊C4—216no—
488C4⋊C4—216no—
489C4⋊C4—216no—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—28no—
513C4⋊C4—42no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4C22×D414yes—
517C4⋊C4C22×D414yes—
518C4⋊C4C22×D414yes—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C42—216no—
527C42—216no—
528C42—28no—
529C42—28no—
530C42—24no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—44no—
537C42#12014yes—
538C42#12014yes—
539C42C22×D412yes—
540C42C22×D412yes—
541C42C22×D412yes—
542C42C22×D412yes—
543C23—22no—
544C23—22no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—42no—
549C23—42no—
550C23C2×#12012yes—
551C23#57714yes—
552C23C23×D411yescharacteristic
553Q8—216no—
554Q8—24no—
555Q8—28no—
556Q8—48no—
557Q8—48no—
558Q8—432no—
559Q8—84no—
560Q8#53618yes—
561D4—44no—
562D4—44no—
563D4—84no—
564D4—84no—
565D4—88no—
566C2×C4—216no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—88no—
607C2×C4C2×#12012yes—
608C2×C4C2×#12012yes—
609C2×C4C2×#12012yes—
610C2×C4C2×#12012yes—
611C2×C4C2×#12012yes—
612C2×C4#57514yes—
613C2×C4C23×D411yescharacteristic
614C2×C4C23×D411yescharacteristic
615C2×C4C2×2−1+414yes—
616C8—44no—
617C8—44no—
618C22—28no—
619C22—22no—
620C22—22no—
621C22—84no—
622C22—84no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22#301014yes—
627C22C22×#12011yescharacteristic
628C22#348612yes—
629C22C2×#57512yes—
630C22C2×#57712yes—
631C4—28no—
632C4—22no—
633C4—24no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—44no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—84no—
645C4—82no—
646C4C2×#53612yes—
647C4#348612yes—
648C2—24no—
649C2—162no—
650C2#3524211yescharacteristic
651C2#6035911yescharacteristic
652C2C2×#301011yescharacteristic
653C2#6183614yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#557C2semidirect
5. #33005C2semidirect
6. #33003C2semidirect
7. #29640C2semidirect
8. #29639C2semidirect
9. #25944C2semidirect
10. #25848C2semidirect
11. #24334C2semidirect
12. #21836C2semidirect
13. #21820C2semidirect
14. #20741C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#32342
C2×#31032
#352492
#330052
#330032
#296402
#296392
#259442
#258482
#243342
#218362
#218202
#207412