Finite Group Databasebeta

← #196848

Subgroups — #196848

Order 512.

1542 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2C2×#2353C211yesmaximal, characteristic
3#21738C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic, complemented
5C22×#328C211yesmaximal, characteristic, complemented
6#9154C212yesmaximal, complemented
7#9153C212yesmaximal, complemented
8#9146C212yesmaximal, complemented
9#9145C212yesmaximal, complemented
10#8023C211yesmaximal, characteristic, complemented
11C2×#1294C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16#2855C2212yes—
17#2848C2212yes—
18#2627—21no—
19#2626—21no—
20#2579—22no—
21#2564—22no—
22#2487C412yescomplemented
23#2417C414yescomplemented
24#2409C412yescomplemented
25#2353—24no—
26#2353—24no—
27#2315—21no—
28#2308—21no—
29C2×#388—21no—
30C2×#387—21no—
31C2×#386—21no—
32C2×#383C2211yescharacteristic
33C2×#381C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×C4×C22⋊C4—21no—
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×#376C2211yescharacteristic
39C22×#95C2211yescharacteristic
40#1869—22no—
41#1869—22no—
42#1817C2212yes—
43#1779C2212yes—
44#1520—24no—
45C2×#328—24no—
46C2×#328C2212yes—
47C2×#328C2212yes—
48C2×#328C2212yes—
49C2×#328C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54C23×Q8C2×C411yescharacteristic
55C23×D4D411yescharacteristic
56C24×C4—21no—
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C23×C8—21no—
60C23×C8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8C2×C412yes—
68C2×C22⋊Q8C2×C414yescomplemented
69C2×C22⋊Q8C2312yes—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4C2×C412yes—
75C2×C4×D4C2×C412yes—
76C2×C4×D4D412yes—
77C2×C4×D4D412yes—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C4⋊C4C2×C412yes—
82C22×C4⋊C4D411yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4D411yescharacteristic
89C22×C22⋊C4D411yescharacteristic
90C22×C42—21no—
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8C2312yes—
101C2×C22⋊C8C2312yes—
102#400—22no—
103#399—22no—
104#398—24no—
105#396—24no—
106#395—22no—
107#393—22no—
108#391C2×C414yescomplemented
109#389C2×C414yescomplemented
110#388—28no—
111#387—28no—
112#387—24no—
113#386—28no—
114#386—24no—
115#386—22no—
116#386—22no—
117#385—24no—
118#385—22no—
119#385—22no—
120#385—22no—
121#385—22no—
122#383—22no—
123#383—22no—
124#381—22no—
125#381—22no—
126#381—22no—
127#380—21no—
128#380—22no—
129#380—21no—
130#380—24no—
131#380C2×C412yes—
132#380C2×C414yescomplemented
133#380D414yes—
134#380C2312yes—
135#379—22no—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139#376—22no—
140#376—22no—
141C2×#95—21no—
142C2×#95—22no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—21no—
146C2×#95—21no—
147C2×#95—21no—
148C2×#95D414yes—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153#336—24no—
154#328—24no—
155#328—24no—
156#328—24no—
157#328—24no—
158C25C2×D411yescharacteristic
159C25C2×D411yescharacteristic
160C22×Q8—21no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—82no—
164C22×Q8C22×C412yes—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4C22⋊C412yes—
172C22×D4C22⋊C412yes—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—24no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—44no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—82no—
189C23×C4—82no—
190C23×C4—82no—
191C23×C4C22×C412yes—
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D412yes—
202C23×C4C2×D412yes—
203C23×C4#4112yes—
204C23×C4#4112yes—
205C23×C4C2411yescharacteristic
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8—82no—
211C22×C8—82no—
212C22⋊Q8—24no—
213C22⋊Q8—24no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—42no—
219C22⋊Q8—42no—
220C22⋊Q8—42no—
221C22⋊Q8—42no—
222C22⋊Q8—84no—
223C4×D4—24no—
224C4×D4—24no—
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—82no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4C22⋊C412yes—
247C2×C4⋊C4C22⋊C412yes—
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C22×C414yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—82no—
271C2×C22⋊C4C22⋊C412yes—
272C2×C22⋊C4C22⋊C412yes—
273C2×C22⋊C4C22⋊C412yes—
274C2×C22⋊C4C22⋊C412yes—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—44no—
282C2×C42—82no—
283C2×C42C22×C412yes—
284C2×C42C2×D412yes—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—22no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291C22⋊C8—84no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—22no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—21no—
303#95—21no—
304#95—22no—
305#95—21no—
306#95—21no—
307#95C2×D412yes—
308#95C2×D412yes—
309#95C2×D412yes—
310#95C2×D412yes—
311C24—21no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24C2×C22⋊C412yes—
322C24C2×C22⋊C411yescharacteristic
323C24C2×C22⋊C411yescharacteristic
324C24#12012yes—
325C24#12312yes—
326C24#13512yes—
327C24#13512yes—
328C24#13712yes—
329C24#13712yes—
330C24C22×D411yescharacteristic
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—44no—
334C2×Q8—42no—
335C2×Q8—82no—
336C2×Q8—84no—
337C2×Q8—84no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—48no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—48no—
399C22×C4—44no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4—84no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4C2×C22⋊C412yes—
416C22×C4C2×C22⋊C412yes—
417C22×C4C2×C22⋊C412yes—
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C4×D412yes—
421C22×C4C4×D412yes—
422C22×C4C4×D412yes—
423C22×C4C4×D412yes—
424C22×C4#12012yes—
425C22×C4#12112yes—
426C22×C4#12112yes—
427C22×C4#12112yes—
428C22×C4#12112yes—
429C22×C4#12312yes—
430C22×C4C23×C411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C2×C8—44no—
437C2×C8—44no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—84no—
441C2×C8—84no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—84no—
452C4⋊C4—82no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—48no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—48no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C22⋊C4—84no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—48no—
475C42—84no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—22no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23#38612yes—
508C23#38612yes—
509C23#41612yes—
510C23#41612yes—
511C23#41912yes—
512C23#41912yes—
513C23#45214yes—
514C23#45214yes—
515C23#48514yes—
516C23#48514yes—
517C23C22×C22⋊C411yescharacteristic
518C23C2×#12011yescharacteristic
519C23C2×#12311yescharacteristic
520C23C2×#13511yescharacteristic
521C23C2×#13511yescharacteristic
522C23C2×#13711yescharacteristic
523C23C2×#13711yescharacteristic
524Q8—82no—
525Q8—82no—
526Q8—84no—
527D4—48no—
528D4—48no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—816no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—164no—
597C2×C4#38612yes—
598C2×C4#38612yes—
599C2×C4C2×C4×D411yescharacteristic
600C2×C4C2×C4×D411yescharacteristic
601C2×C4C2×#12111yescharacteristic
602C2×C4C2×#12111yescharacteristic
603C8—82no—
604C8—82no—
605C22—22no—
606C22—22no—
607C22—24no—
608C22—24no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—24no—
613C22—21no—
614C22—21no—
615C22—22no—
616C22—22no—
617C22—21no—
618C22—21no—
619C22—22no—
620C22—22no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—42no—
626C22—44no—
627C22—44no—
628C22—42no—
629C22—42no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22#187314yes—
635C22#187314yes—
636C22C2×#38611yescharacteristic
637C22#288312yes—
638C22C2×#41611yescharacteristic
639C22C2×#41911yescharacteristic
640C22#290112yes—
641C22C2×#45211yescharacteristic
642C22C2×#45211yescharacteristic
643C22C2×#48511yescharacteristic
644C22C2×#48511yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—22no—
650C4—22no—
651C4—44no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—84no—
659C4—82no—
660C4—82no—
661C2—21no—
662C2—21no—
663C2—21no—
664C2—21no—
665C2—24no—
666C2—24no—
667C2—82no—
668C2#1761512yes—
669C2#1762112yes—
670C2C2×#187311yescharacteristic
671C2#2098011yescharacteristic
672C2#2098411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1520C2semidirect
5. C22×#328C2semidirect
6. #9154C2semidirect
7. #9153C2semidirect
8. #9146C2semidirect
9. #9145C2semidirect
10. #8023C2semidirect
11. C2×#1294C2semidirect
22. #2487C4semidirect
23. #2417C4semidirect
24. #2409C4semidirect
68. C2×C22⋊Q8C2×C4semidirect
108. #391C2×C4semidirect
109. #389C2×C4semidirect
132. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350102
C2×#23532
#217382
C2×#15202
C22×#3282
#91542
#91532
#91462
#91452
#80232
C2×#12942