Finite Group Databasebeta

← #188531

Subgroups — #188531

Order 512.

1160 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34212C211yesmaximal, characteristic
2#21934C211yesmaximal, characteristic
3C2×#1511C211yesmaximal, characteristic
4C2×#1508C211yesmaximal, characteristic
5#13953C211yesmaximal, characteristic
6#8578C212yesmaximal, complemented
7#8576C212yesmaximal, complemented
8#8574C212yesmaximal, complemented
9#8572C212yesmaximal, complemented
10#8019C211yesmaximal, characteristic
11#8015C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C2×#407C2211yescharacteristic
14C2×#407C2211yescharacteristic
15#2368—22no—
16#2360C412yescomplemented
17#2360C412yescomplemented
18#2345—22no—
19#2345C412yescomplemented
20Q8×C22⋊C4C412yescomplemented
21#2329—21no—
22#2329—21no—
23#2315—21no—
24C4×C22⋊Q8C2212yes—
25C4×C22⋊Q8C2212yes—
26C2×#381C2211yescharacteristic
27C2×#380C2211yescharacteristic
28#2273—21no—
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1851—22no—
32#1851—22no—
33#1797C2212yes—
34#1797C2212yes—
35#1511—24no—
36#1508—24no—
37C2×#328C2212yes—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41#1439—21no—
42#1439—21no—
43#1436—21no—
44#1436—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47#1434C2211yescharacteristic
48#1434C2211yescharacteristic
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C23×Q8C2×C411yescharacteristic
54C24×C4C2311yescharacteristic
55C22×M16—21no—
56C22×M16—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8C2×C412yes—
62C2×C22⋊Q8D414yes—
63C2×C22⋊Q8Q814yes—
64C2×C22⋊Q8C2312yes—
65C2×C4×Q8C2×C412yes—
66C2×C4×Q8C2×C412yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84#407—22no—
85#407—22no—
86#407—22no—
87#407—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8C2312yes—
91C2×C22⋊C8C2312yes—
92#391—22no—
93#391—22no—
94#391C2×C412yes—
95#391C2×C412yes—
96#389—22no—
97#389—22no—
98#389C2×C412yes—
99#389C2×C412yes—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#385—22no—
110#381—22no—
111#381—22no—
112#380—21no—
113#380—21no—
114#380—22no—
115#380—22no—
116#380C2×C412yes—
117#380C2312yes—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C412yes—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4C2×C412yes—
131C4×C22⋊C4C2×C412yes—
132C4×C22⋊C4D412yes—
133C4×C22⋊C4D412yes—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95D411yescharacteristic
143C2×#95D411yescharacteristic
144C2×#95C2311yescharacteristic
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148#336—22no—
149#336—22no—
150#328—24no—
151#328—24no—
152#328—24no—
153#328—24no—
154C25C2×D411yescharacteristic
155C22×Q8—21no—
156C22×Q8—22no—
157C22×Q8—81no—
158C22×Q8—81no—
159C22×Q8C4⋊C414yescomplemented
160C22×Q8C22×C412yes—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—81no—
170C23×C4—81no—
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×Q812yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4C2411yescharacteristic
185C2×M16—42no—
186C2×M16—42no—
187C2×M16—81no—
188C2×M16—81no—
189C2×M16—81no—
190C2×M16—81no—
191C22×C8—22no—
192C22×C8—22no—
193C22⋊Q8—24no—
194C22⋊Q8—24no—
195C22⋊Q8—24no—
196C22⋊Q8—24no—
197C22⋊Q8—24no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8C4⋊C418yescomplemented
201C22⋊Q8C2×D414yes—
202C22⋊Q8C2×Q814yes—
203C4×Q8—82no—
204C4×Q8—82no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—81no—
222C2×C4⋊C4—81no—
223C2×C4⋊C4—81no—
224C2×C4⋊C4—81no—
225C2×C4⋊C4C4⋊C414yescomplemented
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—81no—
246C2×C22⋊C4—81no—
247C2×C22⋊C4C22⋊C412yes—
248C2×C22⋊C4C22⋊C412yes—
249C2×C22⋊C4C22⋊C412yes—
250C2×C22⋊C4C22⋊C412yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—42no—
256C2×C42—81no—
257C2×C42—81no—
258C2×C42—81no—
259C2×C42—81no—
260C2×C42C22×C412yes—
261C2×C42C22×C412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C4⋊C8—82no—
265C4⋊C8—82no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—42no—
271C22⋊C8—42no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—42no—
291#95—42no—
292#95—42no—
293#95—42no—
294#95—42no—
295#95—81no—
296#95—81no—
297#95C22⋊C412yes—
298#95C22⋊C412yes—
299#95C22⋊C412yes—
300#95C22⋊C412yes—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24C2×C22⋊C411yescharacteristic
310C24C2×C22⋊C411yescharacteristic
311C24#12112yes—
312C24C22⋊Q812yes—
313C24C22×D411yescharacteristic
314C2×Q8—22no—
315C2×Q8—24no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—81no—
319C2×Q8—81no—
320C2×Q8—84no—
321C2×Q8—84no—
322C2×Q8C2×C4⋊C414yes—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C412yes—
403C22×C4C2×C4⋊C412yes—
404C22×C4C2×C4⋊C412yes—
405C22×C4#11712yes—
406C22×C4#11712yes—
407C22×C4#12112yes—
408C22×C4#12112yes—
409C22×C4#12112yes—
410C22×C4C22⋊Q812yes—
411C22×C4C22⋊Q812yes—
412C22×C4C22⋊Q812yes—
413C22×C4#13712yes—
414C22×C4#13712yes—
415C22×C4C23×C411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×Q811yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421M16—84no—
422M16—84no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—28no—
433C4⋊C4—24no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—82no—
437C4⋊C4—82no—
438C4⋊C4—81no—
439C4⋊C4—81no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4C2×C4⋊C414yes—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—28no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—44no—
453C22⋊C4—48no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—84no—
457C42—42no—
458C42—42no—
459C42—42no—
460C42—42no—
461C42—82no—
462C42—82no—
463C42—82no—
464C42—82no—
465C42—84no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—81no—
491C23—81no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23—81no—
496C23—81no—
497C23—81no—
498C23#38012yes—
499C23#38012yes—
500C23C22×C22⋊C411yescharacteristic
501C23C2×#12111yescharacteristic
502C23C2×C22⋊Q811yescharacteristic
503C23C2×#13711yescharacteristic
504C23C2×#13711yescharacteristic
505C23#57511yescharacteristic
506C23#57511yescharacteristic
507Q8—44no—
508Q8—44no—
509Q8—84no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
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—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—162no—
594C2×C4—162no—
595C2×C4#38012yes—
596C2×C4#38012yes—
597C2×C4#41914yes—
598C2×C4C22×C4⋊C411yescharacteristic
599C2×C4C2×#11711yescharacteristic
600C2×C4C2×#12111yescharacteristic
601C2×C4C2×C22⋊Q811yescharacteristic
602C8—82no—
603C8—82no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22C2×#38011yescharacteristic
635C22#288311yescharacteristic
636C22#288311yescharacteristic
637C22C2×#41911yescharacteristic
638C22#289611yescharacteristic
639C22#290111yescharacteristic
640C22#290111yescharacteristic
641C22#303711yescharacteristic
642C22#303711yescharacteristic
643C22#307811yescharacteristic
644C22#307811yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—22no—
650C4—22no—
651C4—24no—
652C4—44no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—161no—
662C4—161no—
663C2—21no—
664C2—21no—
665C2—21no—
666C2—21no—
667C2—42no—
668C2—42no—
669C2#1762111yescharacteristic
670C2#1762311yescharacteristic
671C2#1762511yescharacteristic
672C2#1762711yescharacteristic
673C2#2063311yescharacteristic
674C2#2064711yescharacteristic
675C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8578C2semidirect
7. #8576C2semidirect
8. #8574C2semidirect
9. #8572C2semidirect
16. #2360C4semidirect
17. #2360C4semidirect
19. #2345C4semidirect
20. Q8×C22⋊C4C4semidirect
159. C22×Q8C4⋊C4semidirect
200. C22⋊Q8C4⋊C4semidirect
225. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342122
#219342
C2×#15112
C2×#15082
#139532
#85782
#85762
#85742
#85722
#80192
#80152