Finite Group Databasebeta

← #110425

Subgroups — #110425

Order 512.

1204 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1964C211yesmaximal, characteristic
2C2×#1893C211yesmaximal, characteristic
3C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
4C2×#1519C211yesmaximal, characteristic
5#10854C211yesmaximal, characteristic, complemented
6#10721C211yesmaximal, characteristic, complemented
7#8023C211yesmaximal, characteristic, complemented
8C24×C8C2211yescharacteristic
9C22×#121C2211yescharacteristic
10C22×#102C412yescomplemented
11C22×#99—22no—
12C22×C22⋊C8C2211yescharacteristic
13C22×C4×C8—21no—
14C2×#380C2211yescharacteristic
15C2×#380C2211yescharacteristic
16C2×#380C2211yescharacteristic
17C2×C4×C22⋊C4C2211yescharacteristic
18#1964—22no—
19#1964C414yescomplemented
20#1893—21no—
21#1893—22no—
22#1893—21no—
23C8×C22⋊C4—22no—
24C8×C22⋊C4—22no—
25C2×#342—22no—
26C2×#340—22no—
27C2×#340C412yescomplemented
28C2×#336—21no—
29#1519—22no—
30#1519C414yescomplemented
31C2×#327—22no—
32#1294—22no—
33C23×D4—21no—
34C23×D4C2×C411yescharacteristic
35C24×C4C2311yescharacteristic
36C23×C8—21no—
37C23×C8—21no—
38C23×C8—21no—
39C23×C8—21no—
40C23×C8—21no—
41C23×C8—44no—
42C23×C8C2×C412yes—
43C23×C8D411yescharacteristic
44C23×C8D412yes—
45C23×C8D411yescharacteristic
46C2×#121—21no—
47C2×#121—21no—
48C2×#121—21no—
49C2×#121—21no—
50C2×#121—21no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121C2×C412yes—
57C2×#121C2×C412yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4D412yes—
67C22×C42—21no—
68C2×#102—22no—
69C2×#102—22no—
70C2×#102—44no—
71C2×#99—28no—
72C2×#99—44no—
73C2×#99—44no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8C2×C412yes—
78C2×C22⋊C8D412yes—
79C2×C22⋊C8D412yes—
80C2×C4×C8—24no—
81C2×C4×C8—44no—
82#380—21no—
83#380—21no—
84#380—21no—
85#380—21no—
86#380—21no—
87#380—21no—
88#380—21no—
89#380—21no—
90#380—21no—
91#380—21no—
92#380C2×C412yes—
93#380C2×C412yes—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—21no—
96C4×C22⋊C4C2×C412yes—
97C4×C22⋊C4D412yes—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102#342—44no—
103#340—22no—
104#340—22no—
105#340—44no—
106#336—24no—
107#327—28no—
108C25—21no—
109C25C2×D411yescharacteristic
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—21no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—42no—
118C22×D4—42no—
119C22×D4—41no—
120C22×D4—41no—
121C22×D4—41no—
122C22×D4—41no—
123C22×D4—42no—
124C22×D4—42no—
125C22×D4—42no—
126C22×D4—42no—
127C22×D4C4212yes—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—41no—
134C23×C4—41no—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4#4111yescharacteristic
144C23×C4#4111yescharacteristic
145C23×C4#4111yescharacteristic
146C23×C4#4111yescharacteristic
147C23×C4#4111yescharacteristic
148C23×C4#4111yescharacteristic
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—24no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—24no—
162C22×C8—24no—
163C22×C8—42no—
164C22×C8—42no—
165C22×C8—42no—
166C22×C8—44no—
167C22×C8—44no—
168C22×C8—44no—
169C22×C8—44no—
170C22×C8—42no—
171C22×C8—42no—
172C22×C8C22⋊C412yes—
173C22×C8C22⋊C412yes—
174#121—22no—
175#121—22no—
176#121—22no—
177#121—22no—
178#121—22no—
179#121—22no—
180#121—22no—
181#121—22no—
182#121—41no—
183#121—41no—
184#121—41no—
185#121—41no—
186#121—41no—
187#121—41no—
188#121—41no—
189#121—41no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4C4212yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4C22⋊C414yes—
230C2×C22⋊C4D814yes—
231C2×C22⋊C4SD1614yes—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—42no—
235C2×C42—42no—
236#102—44no—
237#102—44no—
238#99—416no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C22⋊C8C22⋊C414yes—
243C4×C8—48no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252C24—22no—
253C24—42no—
254C24—42no—
255C24—41no—
256C24—41no—
257C24—42no—
258C24—42no—
259C24—82no—
260C24—82no—
261C24C2×C22⋊C411yescharacteristic
262C24C4×D411yescharacteristic
263C24C4×D411yescharacteristic
264C24#12111yescharacteristic
265C24#12111yescharacteristic
266C24#12411yescharacteristic
267C24#12711yescharacteristic
268C24C2×D812yes—
269C24C2×SD1612yes—
270C24#13512yes—
271C24#13512yes—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—82no—
289C2×D4—81no—
290C2×D4—81no—
291C2×D4—81no—
292C2×D4—81no—
293C2×D4—82no—
294C2×D4—82no—
295C2×D4—82no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4—82no—
299C2×D4—81no—
300C2×D4—81no—
301C2×D4—81no—
302C2×D4—81no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—81no—
343C22×C4—81no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4C2×C4211yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4#11711yescharacteristic
352C22×C4#11711yescharacteristic
353C22×C4#11711yescharacteristic
354C22×C4C4×D411yescharacteristic
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4#12011yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4#12111yescharacteristic
367C22×C4#12111yescharacteristic
368C22×C4#12111yescharacteristic
369C22×C4#12311yescharacteristic
370C22×C4#12311yescharacteristic
371C22×C4#12311yescharacteristic
372C22×C4#12411yescharacteristic
373C22×C4#12411yescharacteristic
374C22×C4#12611yescharacteristic
375C22×C4#12611yescharacteristic
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—44no—
385C2×C8—44no—
386C2×C8—44no—
387C2×C8—44no—
388C2×C8—42no—
389C2×C8—44no—
390C2×C8—42no—
391C2×C8—44no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8#9914yescomplemented
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—82no—
404C4⋊C4—81no—
405C4⋊C4—81no—
406C4⋊C4—81no—
407C4⋊C4—81no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C22⋊C4—81no—
421C22⋊C4—81no—
422C22⋊C4#10218yescomplemented
423C42—42no—
424C42—42no—
425C42—44no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—42no—
445C23—44no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—81no—
453C23—81no—
454C23—82no—
455C23—164no—
456C23—164no—
457C23C4×C22⋊C411yescharacteristic
458C23#38611yescharacteristic
459C23#38811yescharacteristic
460C23#38811yescharacteristic
461C23#39011yescharacteristic
462C23#39011yescharacteristic
463C23#39011yescharacteristic
464C23C2×#10212yes—
465C23#41612yes—
466C23C4×D812yes—
467C23C4×D812yes—
468C23C4×SD1612yes—
469C23C4×SD1612yes—
470C23#46512yes—
471C23#46512yes—
472C23#46612yes—
473C23#46612yes—
474C23#48612yes—
475C23#48812yes—
476C23#49211yescharacteristic
477C23#49311yescharacteristic
478C23#49511yescharacteristic
479C23#49511yescharacteristic
480D4—44no—
481D4—82no—
482D4—82no—
483D4—84no—
484D4—84no—
485D4—82no—
486D4—82no—
487D4—82no—
488D4—82no—
489D4—81no—
490D4—81no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—48no—
527C2×C4—48no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—162no—
544C2×C4—162no—
545C2×C4—162no—
546C2×C4—162no—
547C2×C4—164no—
548C2×C4C4×C22⋊C411yescharacteristic
549C2×C4C4×C22⋊C411yescharacteristic
550C2×C4#38611yescharacteristic
551C2×C4#38611yescharacteristic
552C2×C4#38811yescharacteristic
553C2×C4#38811yescharacteristic
554C2×C4#38811yescharacteristic
555C2×C4#38811yescharacteristic
556C2×C4C2×#9912yes—
557C2×C4#41012yes—
558C8—24no—
559C8—44no—
560C8—42no—
561C22—24no—
562C22—22no—
563C22—22no—
564C22—24no—
565C22—24no—
566C22—22no—
567C22—24no—
568C22—22no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—164no—
593C22—164no—
594C22—162no—
595C22—162no—
596C22#143711yescharacteristic
597C22C4×#10212yes—
598C22#189312yes—
599C22#193412yes—
600C22#193412yes—
601C22#195711yescharacteristic
602C22#195811yescharacteristic
603C22#196412yes—
604C22#196811yescharacteristic
605C22#196811yescharacteristic
606C22#197011yescharacteristic
607C22#197011yescharacteristic
608C22#197612yes—
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—48no—
616C4—82no—
617C4—162no—
618C4—162no—
619C4—162no—
620C4C4×#9912yes—
621C4#189811yescharacteristic
622C4#189911yescharacteristic
623C2—22no—
624C2—21no—
625C2—21no—
626C2—22no—
627C2—22no—
628C2—22no—
629C2—21no—
630C2—21no—
631C2—162no—
632C2#978112yes—
633C2#978312yes—
634C2#1092511yescharacteristic
635C2#1094411yescharacteristic
636C2#1096011yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C8×C22⋊C4C2semidirect
5. #10854C2semidirect
6. #10721C2semidirect
7. #8023C2semidirect
10. C22×#102C4semidirect
19. #1964C4semidirect
27. C2×#340C4semidirect
30. #1519C4semidirect
394. C2×C8#99semidirect
422. C22⋊C4#102semidirect

Maximal subgroups

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

subgroupindex
C2×#19642
C2×#18932
C2×C8×C22⋊C42
C2×#15192
#108542
#107212
#80232