Finite Group Databasebeta

← #206064

Subgroups — #206064

Order 512.

906 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3248C211yesmaximal, characteristic
2#21969C211yesmaximal, characteristic, complemented
3#20544C211yesmaximal, characteristic, complemented
4#17584C211yesmaximal, characteristic, complemented
5#17579C211yesmaximal, characteristic
6#17404C211yesmaximal, characteristic, complemented
7#17399C211yesmaximal, characteristic
8#9817C212yesmaximal, complemented
9#9816C212yesmaximal
10#9814C212yesmaximal, complemented
11#9813C212yesmaximal
12C2×#533C2211yescharacteristic
13C2×#529C2211yescharacteristic
14#3248—22no—
15#3248—22no—
16#3248—22no—
17#3248—22no—
18C2×#479C411yescharacteristic, complemented
19C2×#479C411yescharacteristic, complemented
20C2×#475C411yescharacteristic, complemented
21C2×#474C411yescharacteristic, complemented
22C2×#467C2211yescharacteristic
23C2×#467C2211yescharacteristic
24C2×#465C2211yescharacteristic
25C2×#465C2211yescharacteristic
26C2×#464C411yescharacteristic, complemented
27C2×#464C411yescharacteristic, complemented
28C2×#462C411yescharacteristic, complemented
29C2×#461C411yescharacteristic, complemented
30#2920C2211yescharacteristic
31#2920C2211yescharacteristic
32C2×#432C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34#2283C2211yescharacteristic
35#1950—22no—
36#1950—22no—
37#1810C2212yes—
38#1810C2212yes—
39#1771—22no—
40#1768—22no—
41#1752C2212yes—
42C8×C22⋊C4C2212yes—
43#1521—22no—
44#1518—22no—
45#1515—22no—
46#1512—22no—
47#1508C2212yes—
48#1508C2212yes—
49#1508C2212yes—
50#1507C2212yes—
51#1473—22no—
52#1472C2212yes—
53#1472C2212yes—
54#1472C2212yes—
55#1471C2212yes—
56#1470—22no—
57#1467—22no—
58#1464—22no—
59C22×Q16C2×C411yescharacteristic
60C22×Q16C2×C411yescharacteristic
61C22×SD16C2×C411yescharacteristic
62C22×SD16C2×C411yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65#533—21no—
66#533—21no—
67#533—21no—
68#533—21no—
69#533—21no—
70#533—21no—
71#533—21no—
72#533—21no—
73C2×C4⋊Q8C2×C411yescharacteristic
74C2×C4⋊Q8C2×C411yescharacteristic
75#529—21no—
76#529—21no—
77#529—21no—
78#529—21no—
79#529—21no—
80#529—21no—
81#529—21no—
82#529—21no—
83C2×#126—21no—
84C2×#125—21no—
85C2×#125C2×C411yescharacteristic
86C2×#124C2×C411yescharacteristic
87C2×#123—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8C2311yescharacteristic
90C2×C22⋊Q8C2311yescharacteristic
91C2×C22⋊Q8C2311yescharacteristic
92C2×#121C2311yescharacteristic
93C2×C4×Q8—21no—
94C2×C4×Q8C2×C411yescharacteristic
95C2×C4×Q8C2×C411yescharacteristic
96C2×C4×Q8C2×C411yescharacteristic
97C2×C4×D4—21no—
98C2×C4×D4C2×C411yescharacteristic
99C2×#117—21no—
100C2×#117—21no—
101C22×C42C2311yescharacteristic
102#479—22no—
103#479—22no—
104#479—22no—
105#479—22no—
106#475—22no—
107#475—22no—
108#474—22no—
109#474—22no—
110#467—22no—
111#467—22no—
112#467—22no—
113#467—22no—
114#465—22no—
115#465—22no—
116#465—22no—
117#465—22no—
118#464—22no—
119#464—22no—
120#464—22no—
121#464—22no—
122#462—22no—
123#462—22no—
124#461—22no—
125#461—22no—
126#432—21no—
127#432—21no—
128#432—21no—
129#432—21no—
130#432—21no—
131#432—21no—
132#432D412yes—
133#432D412yes—
134#431—21no—
135#431—21no—
136#431—21no—
137#431—21no—
138C2×#108—22no—
139C2×#108—22no—
140C2×#107C2×C411yescharacteristic
141C2×#107C2×C411yescharacteristic
142C2×#106C2×C411yescharacteristic
143C2×#106C2×C411yescharacteristic
144#423—21no—
145#423—21no—
146C2×C4⋊C8—21no—
147C2×C4⋊C8C2×C411yescharacteristic
148C2×C4⋊C8C2×C411yescharacteristic
149C2×C4⋊C8C2×C411yescharacteristic
150C2×C4⋊C8C2×C411yescharacteristic
151C2×C4⋊C8D412yes—
152C2×C4⋊C8D412yes—
153C2×C4⋊C8D412yes—
154C2×C4⋊C8D412yes—
155C2×#103C2×C411yescharacteristic
156C2×#103C2×C411yescharacteristic
157C2×#103C2×C411yescharacteristic
158C2×#103C2×C411yescharacteristic
159C2×#103C2×C411yescharacteristic
160C2×#103C2×C411yescharacteristic
161C2×#102C2×C411yescharacteristic
162C2×#102C2×C411yescharacteristic
163#407C2312yes—
164#407C2312yes—
165C2×C22⋊C8C2312yes—
166C4×M16—21no—
167C4×M16—21no—
168C2×#97—21no—
169C2×#97—22no—
170C2×C4×C8—22no—
171#383D412yes—
172#382D412yes—
173#381—21no—
174#380—21no—
175C4×C4⋊C4—21no—
176C4×C22⋊C4C2312yes—
177#376—21no—
178#341—22no—
179#341—22no—
180#336—22no—
181#336—22no—
182#328—24no—
183#328—24no—
184#328—24no—
185#327—24no—
186#324—24no—
187#324—24no—
188#324—24no—
189#323—24no—
190C22×Q8—21no—
191C22×Q8C22×C411yescharacteristic
192C22×Q8C22×C411yescharacteristic
193C22×Q8C22×C411yescharacteristic
194C22×D4—21no—
195C22×D4C22×C411yescharacteristic
196C23×C4—21no—
197C23×C4#4111yescharacteristic
198C23×C4#4111yescharacteristic
199C23×C4C2411yescharacteristic
200C2×Q16—22no—
201C2×Q16—22no—
202C2×Q16—81no—
203C2×Q16—81no—
204C2×SD16—22no—
205C2×SD16—22no—
206C2×SD16—81no—
207C2×SD16—81no—
208C2×M16—22no—
209C2×M16—22no—
210C22×C8—22no—
211C22×C8—81no—
212C22×C8—81no—
213C22×C8C22×C411yescharacteristic
214C22×C8C22×C411yescharacteristic
215C22×C8C22×C411yescharacteristic
216C22×C8C22×C411yescharacteristic
217C22×C8C2×D412yes—
218C22×C8C2×D412yes—
219C22×C8C2×D412yes—
220C22×C8C2×D412yes—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229#126—41no—
230#126—41no—
231#126—41no—
232#126—41no—
233#125—21no—
234#125—21no—
235#125—21no—
236#125—21no—
237#125—41no—
238#125—41no—
239#125—41no—
240#125—41no—
241#124—21no—
242#124—21no—
243#124—21no—
244#124—21no—
245#123—41no—
246#123—41no—
247#123—41no—
248#123—41no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260C22⋊Q8—21no—
261C22⋊Q8—41no—
262C22⋊Q8—41no—
263C22⋊Q8—41no—
264C22⋊Q8—41no—
265#121—21no—
266#121—21no—
267#121—21no—
268#121—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×Q8—21no—
278C4×Q8—21no—
279C4×Q8—21no—
280C4×Q8—21no—
281C4×Q8—41no—
282C4×Q8—41no—
283C4×Q8—41no—
284C4×Q8—41no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—41no—
290C4×D4—41no—
291C4×D4—41no—
292C4×D4—41no—
293#117—41no—
294#117—41no—
295#117—41no—
296#117—41no—
297#117—41no—
298#117—41no—
299#117—41no—
300#117—41no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4C22×C411yescharacteristic
314C2×C4⋊C4C22×C411yescharacteristic
315C2×C4⋊C4C22×C411yescharacteristic
316C2×C4⋊C4C22×C411yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42C22×C411yescharacteristic
330C2×C42C22×C411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42#4111yescharacteristic
340C2×C42#4111yescharacteristic
341#108—82no—
342#108—82no—
343#107—22no—
344#107—22no—
345#106—22no—
346#106—22no—
347C4⋊C8—22no—
348C4⋊C8—22no—
349C4⋊C8—41no—
350C4⋊C8—41no—
351C4⋊C8—41no—
352C4⋊C8—41no—
353C4⋊C8C22⋊C412yes—
354C4⋊C8C22⋊C412yes—
355C4⋊C8C22⋊C412yes—
356C4⋊C8C22⋊C412yes—
357C4⋊C8C22⋊C412yes—
358C4⋊C8C22⋊C412yes—
359C4⋊C8C22⋊C412yes—
360C4⋊C8C22⋊C412yes—
361#103—22no—
362#103—22no—
363#103—22no—
364#103—22no—
365#103—22no—
366#103—22no—
367#102—22no—
368#102—22no—
369C22⋊C8—22no—
370C22⋊C8—22no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374C22⋊C8C2×D414yes—
375#97—22no—
376#97—22no—
377#97—41no—
378#97—41no—
379#97—82no—
380C4×C8—41no—
381C4×C8—41no—
382C4×C8—82no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95C2×D414yes—
387C24—21no—
388C24C2×#4111yescharacteristic
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—41no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8—41no—
400C2×Q8—41no—
401C2×Q8—41no—
402C2×Q8—41no—
403C2×Q8—82no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—82no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—82no—
426C22×C4C4×D411yescharacteristic
427C22×C4C4×D411yescharacteristic
428C22×C4C4×D411yescharacteristic
429C22×C4C4×D411yescharacteristic
430C22×C4#12111yescharacteristic
431C22×C4#12111yescharacteristic
432C22×C4#12111yescharacteristic
433C22×C4#12111yescharacteristic
434C22×C4#12411yescharacteristic
435C22×C4#12411yescharacteristic
436C22×C4#12411yescharacteristic
437C22×C4#12411yescharacteristic
438C22×C4C23×C411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2×#4111yescharacteristic
444Q16—82no—
445Q16—82no—
446SD16—82no—
447SD16—82no—
448M16—42no—
449M16—42no—
450M16—42no—
451M16—42no—
452C2×C8—24no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8C2×C22⋊C412yes—
466C2×C8C2×C22⋊C412yes—
467C2×C8C2×C22⋊C412yes—
468C2×C8C2×C22⋊C412yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—82no—
500C22⋊C4—44no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C42—42no—
516C42—42no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42C2×C22⋊C411yescharacteristic
524C42C2×C22⋊C411yescharacteristic
525C42C2×C22⋊C411yescharacteristic
526C42C2×C22⋊C411yescharacteristic
527C42C4×D411yescharacteristic
528C42C4×D411yescharacteristic
529C42C4×D411yescharacteristic
530C42C4×D411yescharacteristic
531C42#12111yescharacteristic
532C42#12111yescharacteristic
533C42#12111yescharacteristic
534C42#12111yescharacteristic
535C42#12711yescharacteristic
536C42#12711yescharacteristic
537C42#12711yescharacteristic
538C42#12711yescharacteristic
539C23—22no—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23—81no—
545C23—81no—
546C23C2×C4×D411yescharacteristic
547C23C2×#12111yescharacteristic
548C23C2×#12411yescharacteristic
549Q8—41no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—81no—
556Q8—81no—
557Q8—81no—
558Q8—82no—
559Q8—82no—
560D4—41no—
561D4—41no—
562D4—81no—
563D4—82no—
564D4—82no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4#39011yescharacteristic
606C2×C4#39011yescharacteristic
607C2×C4#39011yescharacteristic
608C2×C4#39011yescharacteristic
609C2×C4#39011yescharacteristic
610C2×C4#39011yescharacteristic
611C2×C4#39011yescharacteristic
612C2×C4#39011yescharacteristic
613C2×C4C22×C22⋊C411yescharacteristic
614C2×C4C2×C4×D411yescharacteristic
615C2×C4C2×#12111yescharacteristic
616C2×C4C2×#12711yescharacteristic
617C8—22no—
618C8—22no—
619C8—22no—
620C8—22no—
621C8—44no—
622C8—82no—
623C8—82no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—44no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—162no—
636C22C2×#39011yescharacteristic
637C22#295712yes—
638C22#306912yes—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—84no—
651C4—81no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4#288311yescharacteristic
659C4#288411yescharacteristic
660C2—22no—
661C2—22no—
662C2—41no—
663C2—161no—
664C2#2121411yescharacteristic
665C2#2124011yescharacteristic
666C2#2125011yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21969C2semidirect
3. #20544C2semidirect
4. #17584C2semidirect
6. #17404C2semidirect
8. #9817C2semidirect
10. #9814C2semidirect
18. C2×#479C4semidirect
19. C2×#479C4semidirect
20. C2×#475C4semidirect
21. C2×#474C4semidirect
26. C2×#464C4semidirect
27. C2×#464C4semidirect
28. C2×#462C4semidirect
29. C2×#461C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32482
#219692
#205442
#175842
#175792
#174042
#173992
#98172
#98162
#98142
#98132