Finite Group Databasebeta

← #382628

Subgroups — #382628

Order 512.

1788 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2290C211yesmaximal, characteristic
2C2×#1799C212yesmaximal
3#14686C211yesmaximal, characteristic, complemented
4#14682C211yesmaximal, characteristic, complemented
5#11203C214yesmaximal, complemented
6#11043C214yesmaximal
7#10846C212yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9C22×#117C411yescharacteristic, complemented
10C23×C42C2211yescharacteristic
11C22×#99C2214yes—
12#2290—21no—
13#2290—21no—
14#2290—22no—
15#2290—21no—
16#2290—21no—
17#2290—22no—
18C2×#388C412yescomplemented
19C2×#387C412yescomplemented
20C2×#386C2212yes—
21C2×#385C2212yes—
22C2×#384—21no—
23C2×#382—21no—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2214yes—
27C2×C4×C4⋊C4C411yescharacteristic, complemented
28C2×C4×C22⋊C4C411yescharacteristic, complemented
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×#376C411yescharacteristic, complemented
31#1799—28no—
32C2×#342C2214yes—
33C2×#342C2218yes—
34C4⋊#95—21no—
35C4⋊#95—22no—
36C4⋊#95—21no—
37#1434—22no—
38C22⋊#95C2214yes—
39C23×D4C2311yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2312yes—
42C24×C4C2311yescharacteristic
43C2×C4×D4—21no—
44C2×C4×D4—21no—
45C2×C4×D4—21no—
46C2×C4×D4—21no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4D414yes—
50C2×C4×D4Q814yes—
51C2×#117—22no—
52C2×#117—21no—
53C2×#117—21no—
54C2×#117—44no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2314yes—
65C22×C22⋊C4C2314yes—
66C22×C22⋊C4C2312yes—
67C22×C22⋊C4C2311yescharacteristic
68C22×C42—44no—
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C412yes—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D412yes—
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D412yes—
81C2×#99—28no—
82C2×#99—48no—
83#388—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#386—22no—
92#386—22no—
93#386—22no—
94#386—22no—
95#385—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#384—44no—
100#382—44no—
101#380—24no—
102#380—24no—
103#380—24no—
104#380—24no—
105#380—48no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—22no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—22no—
116#376—21no—
117#376—22no—
118#376—21no—
119C2×#95—24no—
120C2×#95—24no—
121C2×#95—22no—
122C2×#95—24no—
123C2×#95—22no—
124C2×#95C2×C412yes—
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129#342—28no—
130#342—48no—
131C25C22×C411yescharacteristic
132C25C2411yescharacteristic
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4C2×D412yes—
140C22×D4C2×Q812yes—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—42no—
149C23×C4—44no—
150C23×C4—42no—
151C23×C4—44no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4C4⋊C414yescomplemented
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C412yes—
160C23×C4C22×C412yes—
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×Q812yes—
174C23×C4#4112yes—
175C23×C4#4112yes—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—24no—
184#117—48no—
185#117—44no—
186#117—44no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4C4⋊C414yescomplemented
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4C4⋊C414yescomplemented
232C2×C22⋊C4C2×D412yes—
233C2×C22⋊C4C2×Q812yes—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—42no—
245C2×C42—44no—
246C2×C42—42no—
247C2×C42—44no—
248C2×C42—42no—
249C2×C42—44no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C412yes—
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C412yes—
257C2×C42C22⋊C412yes—
258C2×C42C22⋊C412yes—
259C2×C42C22⋊C412yes—
260C2×C42C22⋊C412yes—
261C2×C42C22⋊C411yescharacteristic
262C2×C42C4⋊C414yescomplemented
263#99—416no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—48no—
275#95—48no—
276#95—48no—
277#95—44no—
278#95—44no—
279C24—21no—
280C24—21no—
281C24—24no—
282C24—41no—
283C24—41no—
284C24—42no—
285C24—41no—
286C24—42no—
287C24—41no—
288C24C2×C4⋊C412yes—
289C24C23×C411yescharacteristic
290C24C22×D411yescharacteristic
291C24C22×D411yescharacteristic
292C24C22×D411yescharacteristic
293C24C22×Q811yescharacteristic
294C24C2×#4111yescharacteristic
295C24C2×#4111yescharacteristic
296C24H(5,𝔽2)14yes—
297C242−1+414yes—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—28no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—44no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—44no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—416no—
362C22×C4—416no—
363C22×C4—42no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—48no—
377C22×C4—44no—
378C22×C4—48no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C4⋊C412yes—
394C22×C4C2×C4⋊C412yes—
395C22×C4C2×C4⋊C412yes—
396C22×C4C2×C4⋊C412yes—
397C22×C4C2×C4⋊C412yes—
398C22×C4#11712yes—
399C22×C4#11712yes—
400C22×C4#12112yes—
401C22×C4#12112yes—
402C22×C4#12112yes—
403C22×C4#12112yes—
404C22×C4C22⋊Q812yes—
405C22×C4C22⋊Q812yes—
406C22×C4C22⋊Q812yes—
407C22×C4C22⋊Q812yes—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—816no—
415C4⋊C4—88no—
416C4⋊C4—88no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C42—24no—
437C42—24no—
438C42—24no—
439C42—22no—
440C42—24no—
441C42—22no—
442C42—24no—
443C42—22no—
444C42—42no—
445C42—42no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42#9914yescomplemented
452C23—21no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23—21no—
461C23—22no—
462C23—24no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—44no—
468C23—44no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—41no—
482C23—42no—
483C23—41no—
484C23—84no—
485C23—84no—
486C23C22×C22⋊C411yescharacteristic
487C23C22×C4⋊C411yescharacteristic
488C23C2×#11711yescharacteristic
489C23#52014yes—
490C23C2×#12111yescharacteristic
491C23C2×#12111yescharacteristic
492C23C2×C22⋊Q811yescharacteristic
493C23C2×C22⋊Q811yescharacteristic
494C23#53512yes—
495C23#53512yes—
496C23#53612yes—
497C23#53612yes—
498C23#53714yes—
499C23#54012yes—
500C23#54112yes—
501C23#54214yes—
502C23#54414yes—
503D4—42no—
504D4—44no—
505D4—42no—
506D4—44no—
507D4—44no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—88no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—88no—
571C2×C4—88no—
572C2×C4—88no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4#38012yes—
578C2×C4#38012yes—
579C2×C4#38012yes—
580C2×C4#38012yes—
581C2×C4C2×#9914yes—
582C2×C4C2×#9912yes—
583C2×C4#41014yes—
584C2×C4#41012yes—
585C22—22no—
586C22—22no—
587C22—24no—
588C22—24no—
589C22—22no—
590C22—24no—
591C22—24no—
592C22—22no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—44no—
610C22—44no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—84no—
615C22—84no—
616C22—82no—
617C22—82no—
618C22C2×#38011yescharacteristic
619C22#231112yes—
620C22#231412yes—
621C22#231712yes—
622C22#261911yescharacteristic
623C22#262012yes—
624C22#262111yescharacteristic
625C22#262812yes—
626C22#262912yes—
627C22C22×#9911yescharacteristic
628C22C2×#41012yes—
629C22C2×#41011yescharacteristic
630C4—24no—
631C4—24no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—82no—
641C4—82no—
642C4—164no—
643C4—164no—
644C4#179914yes—
645C2—21no—
646C2—22no—
647C2—22no—
648C2—22no—
649C2—21no—
650C2—22no—
651C2—22no—
652C2—82no—
653C2#1434011yescharacteristic
654C2C2×#179911yescharacteristic
655C2#2063211yescharacteristic
656C2#2187412yes—
657C2#2187512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14686C2semidirect
4. #14682C2semidirect
5. #11203C2semidirect
7. #10846C2semidirect
9. C22×#117C4semidirect
18. C2×#388C4semidirect
19. C2×#387C4semidirect
27. C2×C4×C4⋊C4C4semidirect
28. C2×C4×C22⋊C4C4semidirect
30. C2×#376C4semidirect
154. C23×C4C4⋊C4semidirect
205. C2×C4⋊C4C4⋊C4semidirect
231. C2×C22⋊C4C4⋊C4semidirect
262. C2×C42C4⋊C4semidirect
451. C42#99semidirect

Maximal subgroups

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

subgroupindex
C2×#22902
C2×#17992
#146862
#146822
#112032
#110432
#108462