Finite Group Databasebeta

← #402842

Subgroups — #402842

Order 512.

1532 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic, complemented
3C2×#1918C211yesmaximal, characteristic, complemented
4#15129C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic, complemented
6#11200C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#10964C211yesmaximal, characteristic
9#10806C211yesmaximal, characteristic, complemented
10#10731C212yesmaximal, complemented
11#10731C212yesmaximal, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2212yes—
14C22×#107C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C2×#394—21no—
18#2354—24no—
19#2354—24no—
20C2×#390—21no—
21C2×#389C2211yescharacteristic
22C2×#388—22no—
23C2×#384C2211yescharacteristic
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2100—24no—
30#1918—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1441—21no—
43#1440—22no—
44C4⋊#95C2211yescharacteristic
45C23×D4D411yescharacteristic
46C24×C4C2312yes—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×#117—24no—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—24no—
58C2×#117—42no—
59C2×#117—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D412yes—
75C22×C42—22no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C2×#107—22no—
79C2×#107—22no—
80C2×#107—22no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—42no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—42no—
87#394—44no—
88#390—28no—
89#389—22no—
90#389—22no—
91#388—216no—
92#384—22no—
93#384D414yes—
94#380—24no—
95#380—24no—
96#380—24no—
97#380—24no—
98#380—24no—
99#380—42no—
100C4×C4⋊C4—28no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4D414yes—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110#340—24no—
111#340—24no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—24no—
116#340—42no—
117#340—42no—
118#340—42no—
119#336—22no—
120#336—22no—
121C25C2×D412yes—
122C22×D4—24no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—42no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4#4114yes—
149C23×C4#4114yes—
150C23×C4C2411yescharacteristic
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8#4112yes—
168C4×D4—28no—
169C4×D4—28no—
170C4×D4—216no—
171#117—48no—
172#117—48no—
173#117—48no—
174#117—48no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—81no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4#4112yes—
216C2×C4⋊C4#4114yes—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4—82no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—28no—
235C2×C42—24no—
236C2×C42—44no—
237C2×C42—44no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242#107—28no—
243C22⋊C8—48no—
244C22⋊C8—48no—
245#95—28no—
246#95—24no—
247#95—44no—
248#95—42no—
249#95—44no—
250#95—42no—
251#95—81no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24#12314yes—
267C24C22×D411yescharacteristic
268C2×D4—28no—
269C2×D4—48no—
270C2×D4—48no—
271C2×D4—44no—
272C2×D4—44no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4#12012yes—
352C22×C4#12012yes—
353C22×C4#12314yes—
354C22×C4#13512yes—
355C22×C4AGL(1,ℤ/8)12yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4112yes—
363C22×C4H(5,𝔽2)12yes—
364C22×C4H(5,𝔽2)11yescharacteristic
365C22×C42−1+411yescharacteristic
366C2×C8—22no—
367C2×C8—24no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C4⋊C4—28no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—48no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—84no—
407C4⋊C4—84no—
408C4⋊C4#12414yes—
409C22⋊C4—216no—
410C22⋊C4—416no—
411C22⋊C4—88no—
412C22⋊C4—88no—
413C22⋊C4—88no—
414C42—22no—
415C42—22no—
416C42—24no—
417C42—416no—
418C42—48no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23C2×#12011yescharacteristic
455C23C2×#12312yes—
456C23#54612yes—
457C23#54611yescharacteristic
458C23#54711yescharacteristic
459C23C2×#13511yescharacteristic
460C23C2×AGL(1,ℤ/8)11yescharacteristic
461C23#57512yes—
462C23#57611yescharacteristic
463C23#57711yescharacteristic
464C23#57812yes—
465D4—416no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4#44914yes—
529C2×C4#52511yescharacteristic
530C2×C4#52511yescharacteristic
531C2×C4#52511yescharacteristic
532C2×C4#52512yes—
533C2×C4C2×#12411yescharacteristic
534C2×C4#52912yes—
535C2×C4#52912yes—
536C2×C4#53511yescharacteristic
537C2×C4#53711yescharacteristic
538C2×C4#53812yes—
539C2×C4#53912yes—
540C2×C4#54612yes—
541C2×C4#54811yescharacteristic
542C2×C4#55011yescharacteristic
543C2×C4#55112yes—
544C2×C4#55212yes—
545C2×C4#55312yes—
546C2×C4#55312yes—
547C8—42no—
548C8—81no—
549C8—82no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22#240711yescharacteristic
566C22#241912yes—
567C22#243511yescharacteristic
568C22#244412yes—
569C22#284911yescharacteristic
570C22C2×#44911yescharacteristic
571C22#300511yescharacteristic
572C22#301111yescharacteristic
573C22#301211yescharacteristic
574C22#309112yes—
575C22#309412yes—
576C22#309712yes—
577C22#309812yes—
578C22#330212yes—
579C22#330311yescharacteristic
580C22#330912yes—
581C22#331312yes—
582C22#331712yes—
583C22#332511yescharacteristic
584C22#333012yes—
585C22#334111yescharacteristic
586C22#334912yes—
587C22#335411yescharacteristic
588C4—24no—
589C4—44no—
590C4—44no—
591C4—42no—
592C4—84no—
593C4—82no—
594C4—82no—
595C4—162no—
596C4—161no—
597C4#239811yescharacteristic
598C4#246012yes—
599C4#246712yes—
600C4#256211yescharacteristic
601C4#259811yescharacteristic
602C4#263611yescharacteristic
603C2—84no—
604C2#1486011yescharacteristic
605C2#2411711yescharacteristic
606C2#2413511yescharacteristic
607C2#2494812yes—
608C2#2495812yes—
609C2#2588211yescharacteristic
610C2#2589311yescharacteristic
611C2#2621112yes—
612C2#2622512yes—
613C2#3298011yescharacteristic
614C2#3298911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2100C2semidirect
3. C2×#1918C2semidirect
5. #11357C2semidirect
6. #11200C2semidirect
7. #11096C2semidirect
9. #10806C2semidirect
10. #10731C2semidirect
11. #10731C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#21002
C2×#19182
#151292
#113572
#112002
#110962
#109642
#108062
#107312