Finite Group Databasebeta

← #6388972

Subgroups — #6388972

Order 512.

2250 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#463C211yesmaximal, characteristic
2#34377C211yesmaximal, characteristic, complemented
3#34194C211yesmaximal, characteristic, complemented
4#23109C214yesmaximal, complemented
5#23007C214yesmaximal, complemented
6#22574C212yesmaximal, complemented
7#22567C212yesmaximal, complemented
8C22⋊C4×SD16C212yesmaximal, complemented
9#22436C212yesmaximal
10#22161C214yesmaximal, complemented
11#21959C211yesmaximal, characteristic, complemented
12#21808C211yesmaximal, characteristic, complemented
13#21510C214yesmaximal, complemented
14#20738C211yesmaximal, characteristic, complemented
15#20544C211yesmaximal, characteristic, complemented
16C23×SD16C2211yescharacteristic
17C23×D8C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#463C4116yescomplemented
21C2×#463C2212yes—
22C2×#463C2212yes—
23C2×#463C2212yes—
24C2×#463C2212yes—
25C2×#442C2214yes—
26C2×C4×SD16C2214yescomplemented
27C2×#432C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31#2379—22no—
32#2367C2214yescomplemented
33#2354C2212yescomplemented
34#2352—22no—
35D4×C4⋊C4C2212yescomplemented
36#2345C2212yes—
37#2342—22no—
38#2342—22no—
39Q8×C22⋊C4C2212yes—
40D4×C22⋊C4C2212yes—
41D4×C22⋊C4C2212yes—
42#2337—22no—
43#2326C2214yescomplemented
44#2319C2212yescomplemented
45#2318—22no—
46#2315—21no—
47#2312—21no—
48#2285C2211yescharacteristic
49#2285C2211yescharacteristic
50#2283C2211yescharacteristic
51#2280—21no—
52C2×#380—21no—
53#2275C2212yescomplemented
54#1983C2214yescomplemented
55#1972C2214yes—
56#1968C2214yescomplemented
57#1959C2214yescomplemented
58#1948C2214yescomplemented
59#1948C2214yescomplemented
60#1935C2214yescomplemented
61#1934C2214yescomplemented
62#1927C2214yescomplemented
63#1926C2214yescomplemented
64#1880C2214yescomplemented
65#1879C2214yes—
66#1879C2214yes—
67#1877C2214yescomplemented
68#1865C2214yescomplemented
69#1864C2214yescomplemented
70#1824C2212yes—
71#1807C2212yes—
72#1807C2212yes—
73#1802C2212yescomplemented
74#1802C2212yescomplemented
75#1787C2212yes—
76#1786C2212yes—
77#1786C2212yes—
78#1771C2214yescomplemented
79#1768C2214yescomplemented
80#1763C2214yescomplemented
81#1762C2212yescomplemented
82#1762C2212yescomplemented
83C4×#102C2214yescomplemented
84#1752C2212yes—
85C8×C22⋊C4C2212yes—
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×SD16—44no—
91C22×SD16C2×C418yescomplemented
92C22×SD16C2312yes—
93C22×SD16C2312yes—
94C22×D8—44no—
95C22×D8C2×C418yes—
96C22×D8C2312yes—
97C22×D8C2312yes—
98C23×C8C2311yescharacteristic
99C23×C8C2311yescharacteristic
100C2×#124—21no—
101C2×#124—21no—
102C2×#124—24no—
103C2×#124C2×C418yescomplemented
104C2×#124C2312yes—
105C2×#124C2312yes—
106C2×C4×Q8C2314yes—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—24no—
110C2×C4×D4C2×C418yescomplemented
111C2×C4×D4C2314yes—
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2314yes—
117C2×#117—21no—
118C2×#117—21no—
119C2×#117—22no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—22no—
126C22×C22⋊C4—22no—
127C22×C42C2311yescharacteristic
128#463C2×C4116yescomplemented
129#463C2×C4116yescomplemented
130#463C2×C4116yescomplemented
131#463C2×C4116yescomplemented
132#442—48no—
133C4×SD16—48no—
134#432—24no—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2×C418yescomplemented
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×#103C2×C418yescomplemented
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#102C2×C418yescomplemented
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×C22⋊C8C2312yes—
154C2×#97C2312yes—
155C2×#97C2314yes—
156C2×C4×C8C2314yes—
157#391—24no—
158#391C2314yes—
159#390C2314yes—
160#390C2314yes—
161#389C2314yes—
162#388—24no—
163#388—24no—
164#388—24no—
165#388—24no—
166#388—24no—
167#388—24no—
168#387—24no—
169#386—24no—
170#386—24no—
171#385—24no—
172#385—24no—
173#385—24no—
174#385—24no—
175#384C2312yes—
176#384C2312yes—
177#383D412yes—
178#382—21no—
179#382—21no—
180#382D412yes—
181#381—22no—
182#381D412yes—
183#380—21no—
184#380—21no—
185#380—22no—
186#380—22no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380D411yescharacteristic
191#380D411yescharacteristic
192#380C2312yes—
193#380C2312yes—
194#379—22no—
195#379—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—24no—
206C4×C22⋊C4—24no—
207C4×C22⋊C4C2312yes—
208#376C2312yes—
209C2×#95—22no—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2314yes—
216C25—21no—
217C25—21no—
218C22×Q8—21no—
219C22×Q8—22no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8C22×C414yes—
223C22×Q8C2412yes—
224C22×D4—24no—
225C22×D4—21no—
226C22×D4—21no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—42no—
230C22×D4—42no—
231C22×D4—42no—
232C22×D4C22×C414yes—
233C22×D4C22×C414yes—
234C22×D4C2412yes—
235C22×D4C2411yescharacteristic
236C22×D4C2411yescharacteristic
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—28no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4—24no—
246C23×C4—84no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×SD16—28no—
251C2×SD16—44no—
252C2×SD16—416no—
253C2×SD16C22×C418yes—
254C2×SD16C22×C418yes—
255C2×D8—28no—
256C2×D8—44no—
257C2×D8—416no—
258C2×D8C22×C418yes—
259C2×D8C22×C418yes—
260C22×C8—42no—
261C22×C8—42no—
262C22×C8C22×C414yes—
263C22×C8C22×C414yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267#124—28no—
268#124—28no—
269#124—24no—
270#124—24no—
271#124C22×C418yes—
272#124C22×C418yes—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4C22×C414yes—
286C4×D4C22×C414yes—
287C4×D4C22×C414yes—
288C4×D4C22×C414yes—
289#117—82no—
290#117—82no—
291#117—84no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—84no—
301C2×C4⋊C4C22×C414yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—84no—
328C2×C22⋊C4—84no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42C22×C414yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42#4114yes—
339C2×C42C2414yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#106—44no—
343#106—44no—
344C4⋊C8—24no—
345C4⋊C8C22×C418yes—
346C4⋊C8C22×C418yes—
347#103—44no—
348#103—44no—
349#103C22×C418yes—
350#103C22×C418yes—
351#102—28no—
352#102—44no—
353#102—44no—
354#102—44no—
355#102C22×C418yes—
356#102C22×C418yes—
357C22⋊C8—24no—
358#97—24no—
359#97—44no—
360C4×C8—44no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D414yes—
372#95C2×D414yes—
373#95C2×D414yes—
374C24—21no—
375C24—22no—
376C24—21no—
377C24—24no—
378C24—24no—
379C24—44no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—82no—
385C24—82no—
386C24—82no—
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—82no—
395C2×Q8—84no—
396C2×Q8C23×C414yes—
397C2×D4—28no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—48no—
412C2×D4—44no—
413C2×D4—84no—
414C2×D4—82no—
415C2×D4C23×C414yes—
416C2×D4C23×C412yes—
417C2×D4C23×C412yes—
418C2×D4H(5,𝔽2)14yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—816no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4C4×D414yes—
449C22×C4C4×D414yes—
450C22×C4C23×C412yes—
451C22×C4C22×D414yes—
452C22×C4C22×D414yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4112yes—
459C22×C4C2511yescharacteristic
460SD16—216no—
461SD16—48no—
462D8—216no—
463D8—48no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—44no—
470C2×C8—44no—
471C2×C8C23×C414yes—
472C2×C8C23×C414yes—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—82no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—84no—
488C4⋊C4C23×C412yes—
489C4⋊C4C23×C412yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C42—24no—
509C42—24no—
510C42—82no—
511C42—82no—
512C42—84no—
513C42—84no—
514C42C4×D414yes—
515C42C4×D414yes—
516C42C23×C412yes—
517C42C23×C412yes—
518C42C2×#4112yes—
519C42C2×#4112yes—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—48no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—44no—
532C23—48no—
533C23—44no—
534C23—42no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—48no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23—88no—
544C23—88no—
545C23—164no—
546C23—164no—
547C23C2×C4×D412yes—
548C23#57712yes—
549C23#57712yes—
550C23C23×D411yescharacteristic
551Q8—24no—
552Q8—24no—
553Q8—44no—
554Q8—84no—
555D4—24no—
556D4—22no—
557D4—22no—
558D4—22no—
559D4—24no—
560D4—24no—
561D4—48no—
562D4—42no—
563D4—44no—
564D4—84no—
565D4#51814yes—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—168no—
603C2×C4—168no—
604C2×C4C2×C4×D412yes—
605C2×C4C2×C4×D412yes—
606C2×C4C2×C4×D412yes—
607C2×C4C2×C4×D412yes—
608C2×C4C2×C4×D412yes—
609C2×C4C24×C411yescharacteristic
610C2×C4C22×#4111yescharacteristic
611C2×C4C2×H(5,𝔽2)12yes—
612C2×C4#58512yes—
613C8—24no—
614C8—24no—
615C8—44no—
616C22—22no—
617C22—24no—
618C22—24no—
619C22—22no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—44no—
625C22—42no—
626C22—44no—
627C22—42no—
628C22—44no—
629C22—42no—
630C22—44no—
631C22—816no—
632C22—82no—
633C22—84no—
634C22—84no—
635C22—82no—
636C22—168no—
637C22#295214yes—
638C22C22×C4×D411yescharacteristic
639C22#348812yes—
640C22C2×#57711yescharacteristic
641C22C2×#57711yescharacteristic
642C22#358512yes—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—42no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—84no—
654C4—84no—
655C4C2×#51811yescharacteristic
656C4#342811yescharacteristic
657C4#349412yes—
658C2—24no—
659C2—44no—
660C2—82no—
661C2—84no—
662C2#3420411yescharacteristic
663C2C2×#295211yescharacteristic
664C2#6016811yescharacteristic
665C2#6201612yes—
666C2#6201812yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34377C2semidirect
3. #34194C2semidirect
4. #23109C2semidirect
5. #23007C2semidirect
6. #22574C2semidirect
7. #22567C2semidirect
8. C22⋊C4×SD16C2semidirect
10. #22161C2semidirect
11. #21959C2semidirect
12. #21808C2semidirect
13. #21510C2semidirect
14. #20738C2semidirect
15. #20544C2semidirect
20. C2×#463C4semidirect
26. C2×C4×SD16C22semidirect
32. #2367C22semidirect
33. #2354C22semidirect
35. D4×C4⋊C4C22semidirect
43. #2326C22semidirect
44. #2319C22semidirect
53. #2275C22semidirect
54. #1983C22semidirect
56. #1968C22semidirect
57. #1959C22semidirect
58. #1948C22semidirect
59. #1948C22semidirect
60. #1935C22semidirect
61. #1934C22semidirect
62. #1927C22semidirect
63. #1926C22semidirect
64. #1880C22semidirect
67. #1877C22semidirect
68. #1865C22semidirect
69. #1864C22semidirect
73. #1802C22semidirect
74. #1802C22semidirect
78. #1771C22semidirect
79. #1768C22semidirect
80. #1763C22semidirect
81. #1762C22semidirect
82. #1762C22semidirect
83. C4×#102C22semidirect
91. C22×SD16C2×C4semidirect
103. C2×#124C2×C4semidirect
110. C2×C4×D4C2×C4semidirect
128. #463C2×C4semidirect
129. #463C2×C4semidirect
130. #463C2×C4semidirect
131. #463C2×C4semidirect
137. C2×C4⋊C8C2×C4semidirect
141. C2×#103C2×C4semidirect
146. C2×#102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4632
#343772
#341942
#231092
#230072
#225742
#225672
C22⋊C4×SD162
#224362
#221612
#219592
#218082
#215102
#207382
#205442