Finite Group Databasebeta

← #204366

Subgroups — #204366

Order 512.

1160 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#137C211yesmaximal, characteristic
2#21849C211yesmaximal, characteristic
3#21820C211yesmaximal, characteristic, complemented
4C2×#1569C211yesmaximal, characteristic, complemented
5C2×#1488C211yesmaximal, characteristic
6C2×#1487C211yesmaximal, characteristic, complemented
7C2×#1453C211yesmaximal, characteristic, complemented
8#9666C211yesmaximal, characteristic, complemented
9#9664C211yesmaximal, characteristic, complemented
10#9662C211yesmaximal, characteristic, complemented
11#9661C211yesmaximal, characteristic, complemented
12#9660C211yesmaximal, characteristic, complemented
13#9658C211yesmaximal, characteristic, complemented
14#9656C211yesmaximal, characteristic, complemented
15#9655C211yesmaximal, characteristic, complemented
16C22×#137C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C4×#137—24no—
20C4×#137—24no—
21C2×#441C411yescharacteristic
22C2×#441C411yescharacteristic
23C2×#440C411yescharacteristic
24C2×#440C411yescharacteristic
25C2×C4×Q16C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27C2×C4×SD16C2211yescharacteristic
28C2×C4×SD16C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×#428C411yescharacteristic
33C2×#423C2211yescharacteristic
34C2×#417C411yescharacteristic
35C2×#419C411yescharacteristic
36C2×C4×M16C2211yescharacteristic
37#1985—22no—
38#1984C2211yescharacteristic
39#1984C2211yescharacteristic
40#1984C2211yescharacteristic
41#1984C2211yescharacteristic
42#1984C2211yescharacteristic
43#1984C2211yescharacteristic
44#1954C2211yescharacteristic
45#1954C2211yescharacteristic
46#1870—22no—
47#1817—22no—
48#1797—22no—
49#1569—24no—
50C2×#335C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1488—24no—
55#1487—24no—
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64#1453—24no—
65C2×#322C2211yescharacteristic
66C2×#322C2211yescharacteristic
67C2×#322C2211yescharacteristic
68C2×C82C2211yescharacteristic
69C22×#41C2×C411yescharacteristic
70C23×Q8C2×C411yescharacteristic
71C2×#137—24no—
72C2×#137—24no—
73C2×#137—24no—
74C22×Q16C2×C411yescharacteristic
75C22×Q16C2×C411yescharacteristic
76C22×SD16C2×C411yescharacteristic
77C22×SD16C2×C411yescharacteristic
78C22×M16—21no—
79C22×M16C2×C411yescharacteristic
80C22×M16C2×C411yescharacteristic
81C4×#41—22no—
82C4×#41—22no—
83C4×#41—22no—
84C4×#41—22no—
85C2×C4×Q8—21no—
86C2×C4×Q8—21no—
87C2×C4×Q8—22no—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8C2×C411yescharacteristic
92C2×C4×Q8C2×C411yescharacteristic
93C2×C4×Q8C2×C411yescharacteristic
94C2×C4×Q8C2311yescharacteristic
95C2×C4×Q8C2311yescharacteristic
96C2×C4×Q8C2311yescharacteristic
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4C2×C411yescharacteristic
100C2×C4×D4C2311yescharacteristic
101C2×#117—21no—
102C2×#117C2×C411yescharacteristic
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2×C411yescharacteristic
105C22×C42—21no—
106C22×C42—21no—
107C22×C42C2311yescharacteristic
108#441—24no—
109#441—24no—
110#440—24no—
111#440—24no—
112C4×Q16—22no—
113C4×Q16—22no—
114C4×Q16—22no—
115C4×Q16—22no—
116C4×SD16—22no—
117C4×SD16—22no—
118C4×SD16—22no—
119C4×SD16—22no—
120#432—22no—
121#432—22no—
122#431—22no—
123#431—22no—
124#431—22no—
125#431D414yes—
126#428—24no—
127C2×#107C2×C411yescharacteristic
128C2×#107C2×C411yescharacteristic
129C2×#106C2×C411yescharacteristic
130C2×#106C2×C411yescharacteristic
131#423—22no—
132#423—22no—
133C2×C4⋊C8—21no—
134C2×C4⋊C8—21no—
135C2×C4⋊C8D412yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140#419—24no—
141#417—24no—
142C2×#103C2×C411yescharacteristic
143C2×#103C2×C411yescharacteristic
144C2×#103C2×C411yescharacteristic
145C2×#103C2×C411yescharacteristic
146C2×#103C2×C411yescharacteristic
147C2×#103C2×C411yescharacteristic
148C2×#102C2×C411yescharacteristic
149C2×#102C2×C411yescharacteristic
150C2×C22⋊C8—21no—
151C2×C22⋊C8—21no—
152C2×C22⋊C8—22no—
153C2×C22⋊C8—22no—
154C2×C22⋊C8D412yes—
155C4×M16D414yes—
156C4×M16D414yes—
157C2×#97—21no—
158C2×#97C2×C411yescharacteristic
159C2×#97C2×C411yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166#336—22no—
167#336—22no—
168#336—22no—
169#336—22no—
170#336—22no—
171#336—22no—
172#335—22no—
173#335—22no—
174#334—22no—
175#334—22no—
176Q8⋊C8—22no—
177Q8⋊C8—22no—
178Q8⋊C8—22no—
179Q8⋊C8—22no—
180Q8⋊C8—22no—
181Q8⋊C8—22no—
182D4⋊C8—22no—
183D4⋊C8—22no—
184#322—22no—
185#322—22no—
186#322—22no—
187C82—22no—
188C2×#41—22no—
189C2×#41—22no—
190C2×#41—22no—
191C2×#41—22no—
192C2×#41—41no—
193C2×#41—41no—
194C22×Q8—21no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—41no—
200C22×Q8—41no—
201C22×Q8C22×C411yescharacteristic
202C22×Q8C22×C411yescharacteristic
203C22×Q8C22×C411yescharacteristic
204C22×D4—21no—
205C22×D4C22×C411yescharacteristic
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4C22×C411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4#4111yescharacteristic
214#137—232no—
215C2×Q16—22no—
216C2×Q16—22no—
217C2×Q16—22no—
218C2×Q16—22no—
219C2×Q16—22no—
220C2×Q16M1614yescomplemented
221C2×SD16—22no—
222C2×SD16—22no—
223C2×SD16—22no—
224C2×SD16—22no—
225C2×SD16—22no—
226C2×SD16M1614yescomplemented
227C2×M16—22no—
228C2×M16—22no—
229C2×M16—41no—
230C2×M16—41no—
231C2×M16—42no—
232C2×M16—82no—
233C2×M16C22⋊C414yes—
234C2×M16C22⋊C414yes—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8C22×C411yescharacteristic
242C22×C8C22×C411yescharacteristic
243C22×C8C22×C411yescharacteristic
244C22×C8C22×C411yescharacteristic
245C22×C8C2×D412yes—
246C22×C8C2×D412yes—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—42no—
266C4×Q8—42no—
267C4×Q8—42no—
268C4×Q8—42no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—42no—
276C4×D4—42no—
277C4×D4—42no—
278C4×D4—42no—
279#117—22no—
280#117—22no—
281#117—42no—
282#117—42no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C4⋊C4C22×C411yescharacteristic
296C2×C4⋊C4C22×C411yescharacteristic
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42C22×C411yescharacteristic
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42#4112yes—
318C2×C42#4111yescharacteristic
319C2×C42C2411yescharacteristic
320#107—22no—
321#107—22no—
322#106—22no—
323#106—22no—
324C4⋊C8—24no—
325C4⋊C8—24no—
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—42no—
335#103—22no—
336#103—22no—
337#103—22no—
338#103—22no—
339#103—22no—
340#103—22no—
341#102—22no—
342#102—22no—
343C22⋊C8—24no—
344C22⋊C8—24no—
345C22⋊C8—42no—
346C22⋊C8—82no—
347C22⋊C8—82no—
348#97—42no—
349#97C22⋊C414yes—
350#97C22⋊C414yes—
351C4×C8—21no—
352C4×C8—21no—
353C4×C8—21no—
354C4×C8—21no—
355C4×C8—21no—
356C4×C8—21no—
357C4×C8C2×D412yes—
358C4×C8C2×D412yes—
359C4×C8C2×D412yes—
360C4×C8C2×D412yes—
361C4×C8C2×D412yes—
362C4×C8C2×D412yes—
363C24—21no—
364C24C4×D411yescharacteristic
365#41—28no—
366#41—44no—
367#41—44no—
368#41—44no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—28no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8C2×M1612yes—
392C2×Q8C2×M1612yes—
393C2×Q8C2×M1612yes—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4C2×M1612yes—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4C2×C22⋊C412yes—
432C22×C4C4×D412yes—
433C22×C4C4×D411yescharacteristic
434C22×C4C4×D411yescharacteristic
435C22×C4C4×D411yescharacteristic
436C22×C4#12112yes—
437C22×C4#12112yes—
438C22×C4#12412yes—
439C22×C4#12712yes—
440C22×C4C23×C411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4111yescharacteristic
443Q16—28no—
444Q16—28no—
445SD16—28no—
446SD16—28no—
447M16—28no—
448M16—44no—
449M16—84no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8C2×C22⋊C412yes—
475C2×C8C2×C22⋊C412yes—
476C2×C8C2×C22⋊C412yes—
477C2×C8C2×C22⋊C412yes—
478C2×C8C2×M1612yes—
479C2×C8C2×M1612yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—42no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42C2×C22⋊C412yes—
513C42C4×D412yes—
514C42#12112yes—
515C42#12112yes—
516C42#12412yes—
517C42#12712yes—
518C42C22×D411yescharacteristic
519C42C22×D411yescharacteristic
520C42C22×D411yescharacteristic
521C42C2×#4111yescharacteristic
522C23—21no—
523C23—21no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23#39012yes—
532C23C2×C4×D411yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—22no—
536Q8—22no—
537Q8—22no—
538Q8—22no—
539Q8—48no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546D4—22no—
547D4—22no—
548D4—48no—
549D4—41no—
550D4—41no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—82no—
589C2×C4#39012yes—
590C2×C4#39012yes—
591C2×C4#39012yes—
592C2×C4C22×C22⋊C411yescharacteristic
593C2×C4C2×C4×D411yescharacteristic
594C2×C4C2×#12111yescharacteristic
595C2×C4C2×#12111yescharacteristic
596C2×C4C2×#12411yescharacteristic
597C2×C4C2×#12711yescharacteristic
598C2×C4C22×M1611yescharacteristic
599C2×C4C22×M1611yescharacteristic
600C2×C4#56811yescharacteristic
601C2×C4#56811yescharacteristic
602C2×C4#57712yes—
603C2×C4#57812yes—
604C8—22no—
605C8—22no—
606C8—22no—
607C8—42no—
608C8—42no—
609C8—42no—
610C8—42no—
611C8—81no—
612C8—81no—
613C8—81no—
614C8—81no—
615C8#40714yes—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—42no—
623C22—82no—
624C22—81no—
625C22—81no—
626C22C2×#39011yescharacteristic
627C22D4×M1611yescharacteristic
628C22D4×M1611yescharacteristic
629C22#294712yes—
630C22#295512yes—
631C4—22no—
632C4—22no—
633C4—21no—
634C4—22no—
635C4—21no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—81no—
643C4C2×#40711yescharacteristic
644C4#287911yescharacteristic
645C4D4×M1611yescharacteristic
646C4D4×M1611yescharacteristic
647C4#306312yes—
648C4#315411yescharacteristic
649C4#315611yescharacteristic
650C2—42no—
651C2—82no—
652C2#1805112yes—
653C2#1805312yes—
654C2#2119011yescharacteristic
655C2#2123411yescharacteristic
656C2#2124211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×#41C2×C4
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
3. #21820C2semidirect
4. C2×#1569C2semidirect
6. C2×#1487C2semidirect
7. C2×#1453C2semidirect
8. #9666C2semidirect
9. #9664C2semidirect
10. #9662C2semidirect
11. #9661C2semidirect
12. #9660C2semidirect
13. #9658C2semidirect
14. #9656C2semidirect
15. #9655C2semidirect
220. C2×Q16M16semidirect
226. C2×SD16M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1372
#218492
#218202
C2×#15692
C2×#14882
C2×#14872
C2×#14532
#96662
#96642
#96622
#96612
#96602
#96582
#96562
#96552