Finite Group Databasebeta

← #368519

Subgroups — #368519

Order 512.

1604 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1892C211yesmaximal, characteristic
2C2×#1872C211yesmaximal, characteristic
3C22×#340C211yesmaximal, characteristic
4#14139C211yesmaximal, characteristic
5C22⋊C4×C4⋊C4C211yesmaximal, characteristic
6#11315C212yesmaximal, complemented
7#11190C212yesmaximal, complemented
8#10932C212yesmaximal, complemented
9#10806C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C2×#391C2212yes—
21C2×#380C2211yescharacteristic
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C22⋊C4—21no—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1892C418yescomplemented
26#1872—24no—
27C2×#340—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1441—22no—
38#1436—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C22⋊#95—21no—
42C22⋊#95—21no—
43C22⋊#95—21no—
44C23×Q8C2311yescharacteristic
45C24×C4—21no—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—42no—
51C23×C8—42no—
52C23×C8C2×C414yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8C2×C414yes—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4—42no—
72C22×C4⋊C4D414yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C42—22no—
83C22×C42C2312yes—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—24no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103C2×C418yescomplemented
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94#391—24no—
95#391—42no—
96#380—22no—
97#380C2×C414yes—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—42no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—44no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95D412yes—
112#340—22no—
113#340—22no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340C2×C418yescomplemented
119#336—42no—
120#336—42no—
121C25C2×D411yescharacteristic
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8C22×C412yes—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—82no—
151C23×C4—82no—
152C23×C4C22×C412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4C2411yescharacteristic
162C22×C8—28no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—48no—
180C22×C8—44no—
181C22×C8—44no—
182C22×C8—48no—
183C22×C8C22×C414yes—
184C22⋊Q8—28no—
185C22⋊Q8—28no—
186C22⋊Q8—44no—
187C22⋊Q8—44no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
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—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C414yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4#4114yes—
226C2×C4⋊C4#4114yes—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4SD1614yes—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42—82no—
248#103—48no—
249#103—44no—
250#103—44no—
251C22⋊C8—44no—
252#95—24no—
253#95—24no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—82no—
259#95—82no—
260#95—82no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24C4×D412yes—
266C24#12112yes—
267C24#12312yes—
268C24C2×SD1612yes—
269C24#13512yes—
270C24#13712yes—
271C24#13712yes—
272C24C22×D411yescharacteristic
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—42no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—48no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—48no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—48no—
357C22×C4—84no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4#11714yes—
365C22×C4C4×D414yes—
366C22×C4C4×D412yes—
367C22×C4#12112yes—
368C22×C4#12312yes—
369C22×C4#12414yes—
370C22×C4#12614yes—
371C22×C4C2×SD1614yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4111yescharacteristic
380C22×C4H(5,𝔽2)12yes—
381C22×C4H(5,𝔽2)12yes—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—416no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—48no—
420C4⋊C4#12314yes—
421C22⋊C4—28no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—44no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—88no—
430C23—24no—
431C23—24no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—22no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23#38814yes—
447C23C4×SD1614yes—
448C23#44114yes—
449C23#46114yes—
450C23#46414yes—
451C23#48314yes—
452C23#48514yes—
453C23C2×C4×D411yescharacteristic
454C23C2×#12111yescharacteristic
455C23C2×#12311yescharacteristic
456C23#54612yes—
457C23#54612yes—
458C23C22×SD1611yescharacteristic
459C23C2×#13511yescharacteristic
460C23C2×#13711yescharacteristic
461C23C2×#13711yescharacteristic
462C23#57612yes—
463C23#57812yes—
464Q8—42no—
465Q8—44no—
466Q8—42no—
467Q8—81no—
468Q8—81no—
469Q8—82no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—816no—
513C2×C4—816no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4#38814yes—
531C2×C4C2×#11711yescharacteristic
532C2×C4C2×C4×D411yescharacteristic
533C2×C4C4×#4112yes—
534C2×C4#51812yes—
535C2×C4C2×#12311yescharacteristic
536C2×C4#52511yescharacteristic
537C2×C4C2×#12411yescharacteristic
538C2×C4C2×#12611yescharacteristic
539C2×C4#52912yes—
540C2×C4#53411yescharacteristic
541C2×C4#53511yescharacteristic
542C2×C4#53812yes—
543C2×C4#55112yes—
544C2×C4#55112yes—
545C2×C4#55312yes—
546C2×C4#55312yes—
547C2×C4#55312yes—
548C2×C4#55312yes—
549C8—44no—
550C8—81no—
551C8—81no—
552C22—24no—
553C22—28no—
554C22—28no—
555C22—28no—
556C22—28no—
557C22—24no—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22#192718yes—
563C22C2×#38811yescharacteristic
564C22#236412yes—
565C22#241111yescharacteristic
566C22#241211yescharacteristic
567C22#244412yes—
568C22C2×C4×SD1611yescharacteristic
569C22C2×#44111yescharacteristic
570C22#295112yes—
571C22C2×#46111yescharacteristic
572C22C2×#46411yescharacteristic
573C22#304111yescharacteristic
574C22#304411yescharacteristic
575C22C2×#48311yescharacteristic
576C22C2×#48511yescharacteristic
577C22#309812yes—
578C22#329512yes—
579C22#329712yes—
580C22#329812yes—
581C22#330012yes—
582C22#331212yes—
583C22#331212yes—
584C22#331412yes—
585C22#331412yes—
586C4—24no—
587C4—44no—
588C4—44no—
589C4—82no—
590C4—84no—
591C4—82no—
592C4—84no—
593C4—82no—
594C4—82no—
595C4C4×#12311yescharacteristic
596C4#232111yescharacteristic
597C4#235312yes—
598C4#250411yescharacteristic
599C4#251012yes—
600C4#259111yescharacteristic
601C2—28no—
602C2#1402011yescharacteristic
603C2C2×#192711yescharacteristic
604C2#2106711yescharacteristic
605C2#2312612yes—
606C2#2313312yes—
607C2#2447911yescharacteristic
608C2#2448011yescharacteristic
609C2#2452911yescharacteristic
610C2#2453011yescharacteristic
611C2#2620012yes—
612C2#2621012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11315C2semidirect
7. #11190C2semidirect
8. #10932C2semidirect
9. #10806C2semidirect
25. #1892C4semidirect
90. C2×#103C2×C4semidirect
118. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18922
C2×#18722
C22×#3402
#141392
C22⋊C4×C4⋊C42
#113152
#111902
#109322
#108062
#107302
#107142