Finite Group Databasebeta

← #209382

Subgroups — #209382

Order 512.

950 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2994C211yesmaximal, characteristic, complemented
2#23569C211yesmaximal, characteristic
3#22137C211yesmaximal, characteristic, complemented
4C2×#1589C211yesmaximal, characteristic, complemented
5C2×#1588C211yesmaximal, characteristic, complemented
6#8952C211yesmaximal, characteristic, complemented
7#8947C211yesmaximal, characteristic, complemented
8#8939C211yesmaximal, characteristic, complemented
9#8848C211yesmaximal, characteristic, complemented
10#8843C211yesmaximal, characteristic, complemented
11#8829C211yesmaximal, characteristic, complemented
12#8315C211yesmaximal, characteristic, complemented
13#8212C211yesmaximal, characteristic, complemented
14C8×#103C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17#2994—24no—
18#2994—24no—
19C2×#446C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×#436C2211yescharacteristic
24C2×#436C2211yescharacteristic
25C2×#435—21no—
26C2×C8×D4—21no—
27C2×C8×D4C2211yescharacteristic
28C2×#433—21no—
29C2×#431C2211yescharacteristic
30C2×#424—21no—
31C22×C4⋊C8C2211yescharacteristic
32C4×#121C2211yescharacteristic
33#2059C411yescharacteristic
34#2042C411yescharacteristic
35#2036C411yescharacteristic
36#2034C411yescharacteristic
37#2025C411yescharacteristic
38#2012C411yescharacteristic
39#2000C411yescharacteristic
40#1999C411yescharacteristic
41#1954—21no—
42#1954C2211yescharacteristic
43#1939C2211yescharacteristic
44#1923—21no—
45#1870—21no—
46#1851—21no—
47#1797C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C4×#103C2211yescharacteristic
50C4×#102C2211yescharacteristic
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1589—24no—
54#1588—24no—
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×C82C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×SD16C2×C411yescharacteristic
72C22×D8C2×C411yescharacteristic
73C22×M16—21no—
74C22×M16—21no—
75C23×C8—21no—
76C23×C8—21no—
77#529—22no—
78#529—22no—
79#529—22no—
80#529—22no—
81C2×#126—21no—
82C2×#125C2×C411yescharacteristic
83C2×#124C2×C411yescharacteristic
84C2×#123—21no—
85C2×C22⋊Q8C2×C411yescharacteristic
86C2×#121C2×C411yescharacteristic
87C2×#121C2×C411yescharacteristic
88C2×C4×Q8C2311yescharacteristic
89C2×C4×D4—21no—
90C2×C4×D4—21no—
91C2×C4×D4C2311yescharacteristic
92C2×C4×D4C2311yescharacteristic
93C2×#117—21no—
94C22×C42C2311yescharacteristic
95#446—22no—
96#446—22no—
97C8×Q8—22no—
98C8×Q8D414yescomplemented
99C4×SD16—22no—
100C4×SD16—22no—
101C4×D8—22no—
102C4×D8—22no—
103#436—22no—
104#436—22no—
105#436—22no—
106#436D414yescomplemented
107#435—28no—
108C8×D4—28no—
109C8×D4—22no—
110C8×D4—22no—
111#433—28no—
112#431—22no—
113#431—22no—
114C2×#107C2×C411yescharacteristic
115C2×#106C2×C411yescharacteristic
116#424—28no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—22no—
121C2×C4⋊C8—22no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8D411yescharacteristic
124C2×C4⋊C8D411yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×#103C2×C411yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#102C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135C2×#102C2×C411yescharacteristic
136C2×#102C2×C411yescharacteristic
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8—21no—
143C2×C22⋊C8—21no—
144C2×C22⋊C8C2×C411yescharacteristic
145C2×C22⋊C8C2×C411yescharacteristic
146C2×C22⋊C8D411yescharacteristic
147C2×C22⋊C8D411yescharacteristic
148C2×#97—21no—
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157#390C2×C411yescharacteristic
158#389C2×C411yescharacteristic
159#388—21no—
160#386—21no—
161#380C2×C411yescharacteristic
162C4×C4⋊C4C2311yescharacteristic
163C4×C22⋊C4—21no—
164#340C2×C411yescharacteristic
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336C2×C411yescharacteristic
173#336C2×C411yescharacteristic
174#335—22no—
175#335—22no—
176Q8⋊C8—22no—
177Q8⋊C8—22no—
178D4⋊C8—22no—
179D4⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182#322—22no—
183C82—22no—
184C22×Q8C22×C411yescharacteristic
185C22×D4—21no—
186C22×D4—21no—
187C22×D4C22×C411yescharacteristic
188C22×D4C22×C411yescharacteristic
189C23×C4—21no—
190C23×C4—21no—
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4#4111yescharacteristic
194C2×SD16—22no—
195C2×SD16—41no—
196C2×SD16—41no—
197C2×D8—22no—
198C2×D8—22no—
199C2×D8—41no—
200C2×M16—24no—
201C2×M16—42no—
202C2×M16—81no—
203C2×M16—81no—
204C2×M16—81no—
205C2×M16—81no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—24no—
211C22×C8—24no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—81no—
215C22×C8—81no—
216C22×C8—81no—
217C22×C8—81no—
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C22×C411yescharacteristic
221C22×C8C22×C411yescharacteristic
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D411yescharacteristic
226#126—24no—
227#126—24no—
228#125—22no—
229#125—22no—
230#124—22no—
231#124—22no—
232#123—24no—
233#123—24no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236#121—22no—
237#121—22no—
238#121—22no—
239#121—41no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8C2×D412yes—
243C4×Q8#4112yes—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—81no—
253C4×D4—81no—
254C4×D4C2×D412yes—
255C4×D4#4112yes—
256#117—24no—
257#117—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42C22×C411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42#4111yescharacteristic
287C2×C42C2411yescharacteristic
288#107—22no—
289#106—22no—
290C4⋊C8—24no—
291C4⋊C8—24no—
292C4⋊C8—24no—
293C4⋊C8—22no—
294C4⋊C8—22no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—81no—
302C4⋊C8C2×D412yes—
303C4⋊C8C2×D412yes—
304#103—22no—
305#103—22no—
306#103—22no—
307#102—22no—
308#102—22no—
309#102—22no—
310#102—22no—
311#102—41no—
312C22⋊C8—24no—
313C22⋊C8—24no—
314C22⋊C8—24no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318C22⋊C8—41no—
319C22⋊C8—42no—
320C22⋊C8—81no—
321C22⋊C8—81no—
322C22⋊C8—81no—
323#97—24no—
324C4×C8—24no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—21no—
334C4×C8—21no—
335C4×C8—81no—
336C4×C8C2×D412yes—
337C4×C8C2×D412yes—
338#95—21no—
339#95—21no—
340#95—21no—
341C24—21no—
342C24—21no—
343C24C4×D411yescharacteristic
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8C4×D412yes—
350C2×D4—24no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—81no—
367C2×D4C4×D412yes—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4C4×D411yescharacteristic
403C22×C4C4×D411yescharacteristic
404C22×C4C4×D411yescharacteristic
405C22×C4C23×C411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C2×#4111yescharacteristic
408SD16—44no—
409D8—44no—
410M16—84no—
411M16—84no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—84no—
452C2×C8—84no—
453C2×C8#12012yes—
454C2×C8#12012yes—
455C2×C8#13112yes—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—81no—
469C4⋊C4C4×D412yes—
470C4⋊C4C4×D412yes—
471C4⋊C4#13112yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C42—21no—
485C42—21no—
486C42—22no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—22no—
492C42—41no—
493C42—41no—
494C42C22×D411yescharacteristic
495C42C22×D411yescharacteristic
496C42C2×#4111yescharacteristic
497C42C2×#4111yescharacteristic
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23#44312yes—
519C23C2×C4×D411yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524D4—22no—
525D4—22no—
526D4—48no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—41no—
533D4—84no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4C2×C4×D411yescharacteristic
582C2×C4C2×C4×D411yescharacteristic
583C2×C4C2×#12011yescharacteristic
584C2×C4#52511yescharacteristic
585C2×C4#52511yescharacteristic
586C2×C4#52511yescharacteristic
587C2×C4C2×#13111yescharacteristic
588C2×C4C2×#13111yescharacteristic
589C2×C4#56811yescharacteristic
590C2×C4#56811yescharacteristic
591C2×C4#57611yescharacteristic
592C2×C4#57611yescharacteristic
593C2×C4#57711yescharacteristic
594C2×C4#57811yescharacteristic
595C8—22no—
596C8—22no—
597C8—22no—
598C8—42no—
599C8—42no—
600C8—41no—
601C8—41no—
602C8—41no—
603C8—41no—
604C8—42no—
605C8—42no—
606C8—41no—
607C8—81no—
608C8—81no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22C4×#12011yescharacteristic
631C22#293311yescharacteristic
632C22#293511yescharacteristic
633C22#294611yescharacteristic
634C22#295111yescharacteristic
635C22C2×#44311yescharacteristic
636C22#295511yescharacteristic
637C4—21no—
638C4—21no—
639C4—42no—
640C4—41no—
641C4—42no—
642C4—41no—
643C4—42no—
644C4—41no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4#293011yescharacteristic
649C4#293111yescharacteristic
650C4#293311yescharacteristic
651C4#293511yescharacteristic
652C4#300811yescharacteristic
653C4#301311yescharacteristic
654C4#311911yescharacteristic
655C4#312411yescharacteristic
656C2—42no—
657C2—82no—
658C2—161no—
659C2#1801411yescharacteristic
660C2#1803311yescharacteristic
661C2#1808211yescharacteristic
662C2#1808711yescharacteristic
663C2#2144511yescharacteristic
664C2#2146011yescharacteristic
665C2#2146611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radical#1999C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2994C2semidirect
3. #22137C2semidirect
4. C2×#1589C2semidirect
5. C2×#1588C2semidirect
6. #8952C2semidirect
7. #8947C2semidirect
8. #8939C2semidirect
9. #8848C2semidirect
10. #8843C2semidirect
11. #8829C2semidirect
12. #8315C2semidirect
13. #8212C2semidirect
14. C8×#103C2semidirect
15. #8148C2semidirect
98. C8×Q8D4semidirect
106. #436D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29942
#235692
#221372
C2×#15892
C2×#15882
#89522
#89472
#89392
#88482
#88432
#88292
#83152
#82122
C8×#1032
#81482