Finite Group Databasebeta

← #6453173

Subgroups — #6453173

Order 512.

2144 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3094C211yesmaximal, characteristic
2C2×#2924C211yesmaximal, characteristic, complemented
3#35262C211yesmaximal, characteristic
4#34052C211yesmaximal, characteristic, complemented
5#24618C214yesmaximal, complemented
6#24612C214yesmaximal, complemented
7#24610C214yesmaximal, complemented
8#24605C214yesmaximal, complemented
9#23671C212yesmaximal, complemented
10#23614C212yesmaximal, complemented
11#21160C212yesmaximal, complemented
12#21089C212yesmaximal, complemented
13C2×#1875C211yesmaximal, characteristic
14#20845C211yesmaximal, characteristic, complemented
15#20699C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#3094—28no—
21C2×#481C2212yes—
22C2×#482C2214yes—
23C2×#480C2212yes—
24#2924—28no—
25C2×#433C2214yescomplemented
26C2×#432C2212yescomplemented
27C2×#431C2212yescomplemented
28C2×#428C2212yes—
29C2×#428C2212yes—
30C2×#424C2212yes—
31C2×#418C2212yes—
32C22×#102C2212yes—
33C2×#407C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35C2×#405C2212yes—
36#2854C2214yes—
37#2624C2214yes—
38#2504—21no—
39#2501—21no—
40#2500—21no—
41#2500—21no—
42#2482—22no—
43#2405—22no—
44#2317—21no—
45#2317—21no—
46#2317C2211yescharacteristic
47#2317C2211yescharacteristic
48#2317C2212yescomplemented
49#2317C2212yescomplemented
50#2316C2212yes—
51#2316C2212yes—
52#2315—22no—
53#2309C2211yescharacteristic
54#2309C2211yescharacteristic
55#2296—21no—
56#2296—21no—
57#2296C2214yescomplemented
58#2285—22no—
59#2280C2211yescharacteristic
60#2088C2214yescomplemented
61#2088C2214yescomplemented
62#2087C2214yescomplemented
63#2086C2214yescomplemented
64#2067C2214yescomplemented
65#2067C2214yescomplemented
66#2065C2214yescomplemented
67#2065C2214yescomplemented
68#1944C2214yescomplemented
69#1943C2214yescomplemented
70#1942C2214yescomplemented
71#1941C2214yescomplemented
72#1936C2214yescomplemented
73#1936C2214yescomplemented
74#1934C2214yescomplemented
75#1934C2214yescomplemented
76#1875C2212yes—
77#1875C2212yes—
78#1875C2214yes—
79#1854C2214yescomplemented
80#1854C2214yescomplemented
81#1825C2212yescomplemented
82#1825C2212yescomplemented
83#1824C2212yescomplemented
84#1823C2212yescomplemented
85#1736C2212yes—
86#1735C2212yes—
87#1735C2212yes—
88C2×#340C2212yes—
89C23×D4—21no—
90C23×D4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×M16C2312yes—
93C23×C8C2312yes—
94C2×#123—24no—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121—24no—
98C2×#121—24no—
99C2×#121C2312yes—
100C2×#121C2312yes—
101C2×#117—21no—
102C2×#117—21no—
103C2×#117—24no—
104C2×#117—24no—
105C2×#117—21no—
106C2×#117—21no—
107C2×#117—28no—
108C2×#117—24no—
109C2×#117—24no—
110C2×#117D412yes—
111C2×#117D411yescharacteristic
112C2×#117D411yescharacteristic
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2312yes—
116C2×#117C2312yes—
117C2×#117C2312yes—
118C2×#117C2312yes—
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2312yes—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4D412yes—
132C22×C42—21no—
133C22×C42—21no—
134C22×C42C2312yes—
135#482—216no—
136#481—28no—
137#480—28no—
138#433—216no—
139#432—28no—
140#431—28no—
141#428—28no—
142#428—28no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147#424—28no—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2314yes—
150#418—28no—
151C2×#102—28no—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156#407D418yes—
157C2×C22⋊C8—24no—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2311yescharacteristic
160C2×C22⋊C8C2311yescharacteristic
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163#405—28no—
164C2×#97C2314yes—
165C2×C4×C8C2314yes—
166#399—24no—
167#397—24no—
168#394—24no—
169#394—24no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390C2314yes—
175#390C2314yes—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389C2314yes—
181#389C2314yes—
182#389C2314yes—
183#389C2314yes—
184#389C2314yes—
185#389C2314yes—
186#388—24no—
187#388—24no—
188#386—24no—
189#385—24no—
190#385—24no—
191#385—22no—
192#385—22no—
193#385—22no—
194#385—22no—
195#385—22no—
196#385—22no—
197#384—22no—
198#384—22no—
199#384—21no—
200#384—21no—
201#382—22no—
202#382—22no—
203#382—21no—
204#382—21no—
205#381—21no—
206#380—24no—
207#380—24no—
208#380C2312yes—
209#380C2312yes—
210#379—21no—
211#379—22no—
212#379—22no—
213C4×C4⋊C4—24no—
214C4×C4⋊C4—24no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229C25—21no—
230C25C2×D411yescharacteristic
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—21no—
235C22×D4—22no—
236C22×D4—24no—
237C22×D4—24no—
238C22×D4—44no—
239C22×D4—84no—
240C22×D4C2412yes—
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D414yes—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4#4114yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×M16—24no—
258C2×M16—24no—
259C2×M16C2×D418yes—
260C22×C8—24no—
261C22×C8—24no—
262C22×C8—24no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2414yes—
266#123—88no—
267#121—24no—
268#121—24no—
269#121—24no—
270#121—24no—
271#121—44no—
272#121—44no—
273#117—28no—
274#117—28no—
275#117—24no—
276#117—24no—
277#117—28no—
278#117—22no—
279#117—22no—
280#117—24no—
281#117—24no—
282#117—22no—
283#117—22no—
284#117—28no—
285#117—28no—
286#117—24no—
287#117—24no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—24no—
292#117—44no—
293#117—44no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4#4118yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C22⋊C4—28no—
327C2×C22⋊C4—28no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—28no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—44no—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362#107—28no—
363#107—28no—
364#106—28no—
365#106—28no—
366C4⋊C8—28no—
367C4⋊C8—28no—
368#102—28no—
369#102—28no—
370#102—28no—
371#102—28no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374C22⋊C8—24no—
375C22⋊C8C2×D418yes—
376C22⋊C8C2×D418yes—
377#97—28no—
378C4×C8—28no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—82no—
396C24—82no—
397C24#12114yes—
398C24C22×D411yescharacteristic
399C24C22×D411yescharacteristic
400C24C22×D411yescharacteristic
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—48no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—84no—
414C2×D4—816no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—216no—
422C22×C4—216no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4#12114yes—
447C22×C4#12114yes—
448C22×C4#12114yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4114yes—
463C22×C42−1+414yes—
464C22×C4C2511yescharacteristic
465M16—216no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—48no—
475C2×C8C22×D418yes—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4C2×#4118yes—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—82no—
521C42—28no—
522C42—28no—
523C42—24no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—44no—
530C42—44no—
531C42—42no—
532C42—42no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—28no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—164no—
557C23—164no—
558C23C2×#12112yes—
559C23C2×#12112yes—
560C23C2×#12112yes—
561C23#52512yes—
562C23#53612yes—
563C23C23×D411yescharacteristic
564D4—44no—
565D4—44no—
566D4—88no—
567D4—88no—
568D4—84no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—28no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—28no—
589C2×C4—48no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—84no—
607C2×C4—88no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4C2×#12112yes—
612C2×C4C2×#12112yes—
613C2×C4C2×#12112yes—
614C2×C4#52512yes—
615C2×C4#53612yes—
616C2×C4#54718yes—
617C2×C4#55218yes—
618C2×C4C23×D411yescharacteristic
619C2×C4C22×#4112yes—
620C2×C4C22×#4111yescharacteristic
621C2×C4C2×2−1+412yes—
622C8—28no—
623C8—42no—
624C8—42no—
625C22—22no—
626C22—22no—
627C22—24no—
628C22—24no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—88no—
640C22—82no—
641C22—82no—
642C22—84no—
643C22—82no—
644C22—84no—
645C22—168no—
646C22#241314yes—
647C22C22×#12111yescharacteristic
648C22C2×#52511yescharacteristic
649C22C2×#53611yescharacteristic
650C22#358412yes—
651C22#358512yes—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—48no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4#241314yes—
665C4C2×#54712yes—
666C4C2×#55212yes—
667C2—24no—
668C2—42no—
669C2—42no—
670C2—162no—
671C2C2×#241311yescharacteristic
672C2#6045311yescharacteristic
673C2#6045411yescharacteristic
674C2#6058012yes—
675C2#6087512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2924C2semidirect
4. #34052C2semidirect
5. #24618C2semidirect
6. #24612C2semidirect
7. #24610C2semidirect
8. #24605C2semidirect
9. #23671C2semidirect
10. #23614C2semidirect
11. #21160C2semidirect
12. #21089C2semidirect
14. #20845C2semidirect
15. #20699C2semidirect
25. C2×#433C22semidirect
26. C2×#432C22semidirect
27. C2×#431C22semidirect
48. #2317C22semidirect
49. #2317C22semidirect
57. #2296C22semidirect
60. #2088C22semidirect
61. #2088C22semidirect
62. #2087C22semidirect
63. #2086C22semidirect
64. #2067C22semidirect
65. #2067C22semidirect
66. #2065C22semidirect
67. #2065C22semidirect
68. #1944C22semidirect
69. #1943C22semidirect
70. #1942C22semidirect
71. #1941C22semidirect
72. #1936C22semidirect
73. #1936C22semidirect
74. #1934C22semidirect
75. #1934C22semidirect
79. #1854C22semidirect
80. #1854C22semidirect
81. #1825C22semidirect
82. #1825C22semidirect
83. #1824C22semidirect
84. #1823C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30942
C2×#29242
#352622
#340522
#246182
#246122
#246102
#246052
#236712
#236142
#211602
#210892
C2×#18752
#208452
#206992