Finite Group Databasebeta

← #362008

Subgroups — #362008

Order 512.

1490 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#440C211yesmaximal, characteristic
2C2×#1977C212yesmaximal
3C2×#1849C211yesmaximal, characteristic, complemented
4#20647C211yesmaximal, characteristic, complemented
5#20646C211yesmaximal, characteristic
6C2×#1801C211yesmaximal, characteristic, complemented
7C2×#1800C211yesmaximal, characteristic
8#20438C211yesmaximal, characteristic, complemented
9#11128C212yesmaximal, complemented
10#11127C212yesmaximal
11#10870C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#440—24no—
16C2×#440—22no—
17C2×#440—22no—
18C2×#440C418yescomplemented
19C22×#108C2212yes—
20C22×#108C2212yes—
21C22×#107C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×#419C2212yes—
26C2×#418C2212yes—
27C22×#103C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#101C2212yes—
30C22×#100C2212yes—
31C2×#405C2212yes—
32C22×#97C2211yescharacteristic
33#2285C2211yescharacteristic
34#1977—28no—
35#1849—24no—
36#1805—22no—
37#1804—22no—
38#1803—22no—
39#1802—22no—
40#1801—24no—
41#1800—24no—
42#1771—22no—
43#1771—22no—
44C2×#343C2212yes—
45C2×#343C2212yes—
46#1736C2212yes—
47#1736C2212yes—
48#1729C2212yes—
49C23×Q8C2311yescharacteristic
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16—22no—
55C22×SD16—22no—
56C22×SD16C2×C414yes—
57C22×SD16D414yes—
58C22×SD16Q814yes—
59C22×M16C2312yes—
60C22×M16C2312yes—
61C22×M16C2312yes—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C4×Q8—22no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8C2×C414yes—
72C2×C4×D4—22no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4C2×C414yes—
80C2×#117C2312yes—
81C22×C4⋊C4—21no—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C22⋊C4—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#440—216no—
88#440—216no—
89C2×#108—24no—
90C2×#108—24no—
91C2×#108—24no—
92C2×#108—24no—
93C2×#108—42no—
94C2×#108—42no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#107C2×C414yes—
98#423—22no—
99#423—22no—
100#423—22no—
101#423—22no—
102#423—22no—
103#423D414yes—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110#419—24no—
111#419—42no—
112#418—24no—
113#418—42no—
114C2×#103—22no—
115C2×#103—22no—
116C2×#103—24no—
117C2×#103C2×C414yes—
118C2×#102—22no—
119C2×#102—22no—
120C2×#102—24no—
121C2×#102C2×C414yes—
122C2×#101—24no—
123C2×#101—24no—
124C2×#101—42no—
125C2×#100—24no—
126C2×#100—24no—
127C2×#100—42no—
128#405—24no—
129#405—24no—
130C2×#97—22no—
131C2×#97C2×C414yes—
132C2×#97D414yes—
133C2×#97D412yes—
134C2×#97D412yes—
135C2×C4×C8—22no—
136#384D412yes—
137#382D412yes—
138#380—21no—
139#380—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142#343—24no—
143#343—24no—
144#340—24no—
145#340—24no—
146#336—24no—
147C25—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8C22×C412yes—
160C22×Q8C2×D412yes—
161C22×Q8C2×Q812yes—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C22×C412yes—
174C22×D4C2×D412yes—
175C22×D4C2×Q812yes—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—24no—
180C23×C4—41no—
181C23×C4—41no—
182C23×C4—42no—
183C23×C4—84no—
184C23×C4#4111yescharacteristic
185C23×C4#4111yescharacteristic
186C23×C4C2411yescharacteristic
187C2×SD16—28no—
188C2×SD16—28no—
189C2×SD16—28no—
190C2×SD16—28no—
191C2×SD16—28no—
192C2×SD16—28no—
193C2×SD16C4⋊C4116yescomplemented
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16C2×D414yes—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8C22×C412yes—
220C22×C8C22×C412yes—
221C22×C8C2×D412yes—
222C22×C8C2×D412yes—
223C22×C8C2×D412yes—
224C22×C8C2×D412yes—
225C22×C8C2×D412yes—
226C22×C8C2×Q812yes—
227C22×C8#4112yes—
228C22×C8#4112yes—
229C4×Q8—24no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—42no—
234C4×Q8—42no—
235C4×Q8—42no—
236C4×Q8—42no—
237C4×D4—24no—
238C4×D4—24no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—42no—
242C4×D4—42no—
243C4×D4—42no—
244C4×D4—42no—
245#117—22no—
246#117—22no—
247#117—22no—
248#117—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C22×C412yes—
267C2×C4⋊C4C2×D414yes—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—42no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—41no—
276C2×C42—41no—
277C2×C42—42no—
278C2×C42C22×C412yes—
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D412yes—
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D412yes—
285C2×C42#4112yes—
286#108—48no—
287#108—48no—
288#107—28no—
289C4⋊C8—24no—
290C4⋊C8—24no—
291C4⋊C8—42no—
292C4⋊C8—42no—
293C4⋊C8—42no—
294C4⋊C8—42no—
295#103—28no—
296#103—48no—
297#102—28no—
298#102—48no—
299#101—48no—
300#100—48no—
301#97—44no—
302#97C22⋊C418yes—
303#97C22⋊C418yes—
304C4×C8—44no—
305#95—22no—
306#95—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—41no—
311C24—41no—
312C24—41no—
313C24—41no—
314C24—82no—
315C24—82no—
316C24C2×#4111yescharacteristic
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8C2×C4⋊C414yes—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4C2×C4⋊C414yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4—88no—
394C22×C4—88no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4C4×D412yes—
398C22×C4C4×D412yes—
399C22×C4#12111yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12412yes—
404C22×C4#12412yes—
405C22×C4C23×C411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×Q811yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4H(5,𝔽2)11yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C42−1+411yescharacteristic
418C22×C42−1+411yescharacteristic
419SD16—232no—
420M16—24no—
421M16—24no—
422M16—42no—
423M16—42no—
424M16—42no—
425M16—42no—
426M16—42no—
427M16—42no—
428M16—42no—
429M16—42no—
430M16—42no—
431M16—42no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8C2×C22⋊C414yes—
448C2×C8C2×C22⋊C414yes—
449C2×C8C2×C4⋊C414yes—
450C2×C8#11714yes—
451C2×C8#12114yes—
452C2×C8#12114yes—
453C2×C8C22⋊Q814yes—
454C2×C8C22⋊Q814yes—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C42—42no—
474C42—42no—
475C42—44no—
476C42—44no—
477C42C2×C22⋊C412yes—
478C42C2×C22⋊C412yes—
479C42C4×D412yes—
480C42C4×D412yes—
481C42#12112yes—
482C42#12112yes—
483C42#12712yes—
484C42#12712yes—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—82no—
495C23—82no—
496C23—84no—
497C23—84no—
498C23—82no—
499C23—82no—
500C23—84no—
501C23—84no—
502C23—82no—
503C23—82no—
504C23C2×C4×D411yescharacteristic
505C23C2×#12111yescharacteristic
506C23C2×#12411yescharacteristic
507C23#54711yescharacteristic
508C23#55411yescharacteristic
509C23#55511yescharacteristic
510C23#55611yescharacteristic
511Q8—24no—
512Q8—24no—
513Q8—42no—
514Q8—42no—
515Q8—42no—
516Q8—42no—
517Q8—42no—
518Q8—42no—
519D4—24no—
520D4—24no—
521D4—42no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—42no—
526D4—42no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—88no—
548C2×C4—88no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4#39012yes—
556C2×C4#39012yes—
557C2×C4#39012yes—
558C2×C4#39012yes—
559C2×C4C22×C22⋊C411yescharacteristic
560C2×C4C22×C4⋊C411yescharacteristic
561C2×C4C2×#11711yescharacteristic
562C2×C4C2×C4×D411yescharacteristic
563C2×C4#52011yescharacteristic
564C2×C4#52011yescharacteristic
565C2×C4C2×#12111yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4C2×#12111yescharacteristic
568C2×C4C2×C22⋊Q811yescharacteristic
569C2×C4C2×C22⋊Q811yescharacteristic
570C2×C4C2×#12711yescharacteristic
571C2×C4#53511yescharacteristic
572C2×C4#53611yescharacteristic
573C2×C4#53711yescharacteristic
574C2×C4#53711yescharacteristic
575C2×C4D4211yescharacteristic
576C2×C4D4211yescharacteristic
577C2×C4#54711yescharacteristic
578C2×C4#54711yescharacteristic
579C2×C4#54711yescharacteristic
580C2×C4D4×Q811yescharacteristic
581C2×C4D4×Q811yescharacteristic
582C2×C4#55011yescharacteristic
583C2×C4#55011yescharacteristic
584C2×C4#55411yescharacteristic
585C2×C4#55511yescharacteristic
586C2×C4#55611yescharacteristic
587C8—24no—
588C8—44no—
589C8—82no—
590C8—82no—
591C8#38018yes—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—44no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—82no—
606C22—82no—
607C22—84no—
608C22—84no—
609C22C2×#39011yescharacteristic
610C22D4×C4⋊C411yescharacteristic
611C22#235411yescharacteristic
612C22#253711yescharacteristic
613C22#253811yescharacteristic
614C22#254711yescharacteristic
615C22#256311yescharacteristic
616C22#306812yes—
617C22#306912yes—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—44no—
623C4—82no—
624C4—82no—
625C4—84no—
626C4C2×#38011yescharacteristic
627C4#231111yescharacteristic
628C4D4×C4⋊C411yescharacteristic
629C4#235411yescharacteristic
630C4#253711yescharacteristic
631C4#253811yescharacteristic
632C4#254111yescharacteristic
633C4#254211yescharacteristic
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—84no—
639C2#1390611yescharacteristic
640C2#2124911yescharacteristic
641C2#2125011yescharacteristic
642C2#2824212yes—
643C2#2829812yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1849C2semidirect
4. #20647C2semidirect
6. C2×#1801C2semidirect
8. #20438C2semidirect
9. #11128C2semidirect
11. #10870C2semidirect
18. C2×#440C4semidirect
193. C2×SD16C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4402
C2×#19772
C2×#18492
#206472
#206462
C2×#18012
C2×#18002
#204382
#111282
#111272
#108702