Finite Group Databasebeta

← #196121

Subgroups — #196121

Order 512.

1077 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3117C211yesmaximal, characteristic
2#60248C211yesmaximal, characteristic, complemented
3#17465C211yesmaximal, characteristic
4C2×#1479C211yesmaximal, characteristic, complemented
5#17383C211yesmaximal, characteristic
6#9094C212yesmaximal, complemented
7#9092C212yesmaximal, complemented
8#9090C212yesmaximal, complemented
9#9088C212yesmaximal, complemented
10#7922C211yesmaximal, characteristic, complemented
11#7863C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3117—22no—
15#3117—22no—
16#3117—22no—
17#3117—22no—
18C2×#484C411yescharacteristic
19C2×#481C411yescharacteristic
20C2×#479C411yescharacteristic
21C2×#479C411yescharacteristic
22C2×#464C411yescharacteristic
23C2×#464C411yescharacteristic
24C2×#451C411yescharacteristic
25C2×#448C411yescharacteristic
26#2984—22no—
27#2984—22no—
28#2983—22no—
29#2983C2212yes—
30#2982C2212yes—
31Q8×M16—22no—
32Q8×M16—22no—
33C2×#441C2211yescharacteristic
34C2×#441C2211yescharacteristic
35C2×#440C2211yescharacteristic
36C2×#440C2211yescharacteristic
37#2920—22no—
38#2920C2211yescharacteristic
39#2920C2211yescharacteristic
40C2×#432C2211yescharacteristic
41C2×#431C2211yescharacteristic
42C2×#431C2211yescharacteristic
43#1483—22no—
44#1482—22no—
45#1479—22no—
46#1479C2212yes—
47#1479C2212yes—
48#1479C2212yes—
49#1478C2212yes—
50C2×Q8⋊C8—22no—
51#1473—22no—
52#1468—22no—
53#1461C2212yes—
54#1460C2212yes—
55#1281C2212yes—
56#1281C2212yes—
57#1275C2212yes—
58#1275C2212yes—
59C23×Q8C2×C411yescharacteristic
60C22×Q16C2×C411yescharacteristic
61C22×Q16C2×C411yescharacteristic
62C22×SD16C2×C411yescharacteristic
63C22×SD16C2×C411yescharacteristic
64#529—21no—
65#529—21no—
66#529—21no—
67#529—21no—
68#529—21no—
69#529—21no—
70#529—21no—
71#529—21no—
72C2×#126—21no—
73C2×#125C2×C411yescharacteristic
74C2×#124C2×C411yescharacteristic
75C2×#123—21no—
76C2×C22⋊Q8C2×C411yescharacteristic
77C2×#121C2×C411yescharacteristic
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×C4×Q8—21no—
84C2×C4×Q8C2×C411yescharacteristic
85C2×C4×Q8C2×C411yescharacteristic
86C2×C4×Q8D412yes—
87C2×C4×Q8D412yes—
88C2×C4×Q8D412yes—
89C2×C4×Q8D412yes—
90C2×C4×Q8C2311yescharacteristic
91C2×C4×Q8C2311yescharacteristic
92C2×C4×Q8C2311yescharacteristic
93C2×C4×D4—21no—
94C2×C4×D4C2311yescharacteristic
95C2×#117—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4C2×C411yescharacteristic
98C22×C42—21no—
99C22×C42C2311yescharacteristic
100#484—22no—
101#484—22no—
102#481—22no—
103#481—22no—
104#479—22no—
105#479—22no—
106#479—22no—
107#479—22no—
108#464—22no—
109#464—22no—
110#464—22no—
111#464—22no—
112#451—22no—
113#451—22no—
114#448—22no—
115#448—22no—
116#446—28no—
117#446—28no—
118#446—24no—
119#446—24no—
120C8×Q8—24no—
121C8×Q8—24no—
122#441—22no—
123#441—22no—
124#441—22no—
125#441—22no—
126#440—22no—
127#440—22no—
128#440—22no—
129#440—22no—
130#432—24no—
131#432—21no—
132#432—21no—
133#432—21no—
134#432—21no—
135#432—21no—
136#432—21no—
137#432—21no—
138#432—24no—
139#432D412yes—
140#431—24no—
141#431—21no—
142#431—24no—
143#431—21no—
144#431—21no—
145#431—21no—
146#431—21no—
147#431—21no—
148#431—21no—
149#431D412yes—
150#431C2312yes—
151#431C2312yes—
152#431C2312yes—
153#431C2312yes—
154C2×#107C2×C411yescharacteristic
155C2×#107C2×C411yescharacteristic
156C2×#106C2×C411yescharacteristic
157C2×#106C2×C411yescharacteristic
158#423—24no—
159#423—21no—
160#423—21no—
161C2×C4⋊C8—21no—
162C2×C4⋊C8—21no—
163C2×C4⋊C8C2×C411yescharacteristic
164C2×C4⋊C8C2×C411yescharacteristic
165C2×#103C2×C411yescharacteristic
166C2×#103C2×C411yescharacteristic
167C2×#103C2×C411yescharacteristic
168C2×#103C2×C411yescharacteristic
169C2×#103C2×C411yescharacteristic
170C2×#103C2×C411yescharacteristic
171C2×#102C2×C411yescharacteristic
172C2×#102C2×C411yescharacteristic
173#407—21no—
174#407D412yes—
175C2×C22⋊C8—21no—
176C2×C22⋊C8—21no—
177C2×C22⋊C8C2×C411yescharacteristic
178C2×C22⋊C8C2×C411yescharacteristic
179C4×M16—24no—
180C4×M16—21no—
181C4×M16D412yes—
182C2×#97C2311yescharacteristic
183C2×#97C2311yescharacteristic
184C2×C4×C8—21no—
185C2×C4×C8—21no—
186Q8⋊C8—24no—
187Q8⋊C8—24no—
188Q8⋊C8—24no—
189D4⋊C8—24no—
190#324—24no—
191#323—24no—
192C22×Q8—21no—
193C22×Q8—21no—
194C22×Q8—21no—
195C22×Q8—24no—
196C22×Q8—42no—
197C22×Q8C22⋊C412yes—
198C22×Q8C22⋊C412yes—
199C22×Q8C22×C411yescharacteristic
200C22×Q8C22×C411yescharacteristic
201C22×Q8C22×C411yescharacteristic
202C22×D4—21no—
203C22×D4C22×C411yescharacteristic
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4C22×C411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4#4111yescharacteristic
210C2×Q16—22no—
211C2×Q16—22no—
212C2×Q16—24no—
213C2×Q16—42no—
214C2×SD16—22no—
215C2×SD16—22no—
216C2×SD16—24no—
217C2×SD16—42no—
218C2×M16—22no—
219C2×M16—22no—
220C22×C8—41no—
221C22×C8—41no—
222C22×C8C22×C411yescharacteristic
223C22×C8C22×C411yescharacteristic
224C22×C8C22×C411yescharacteristic
225C22×C8C22×C411yescharacteristic
226#126—41no—
227#126—41no—
228#126—41no—
229#126—41no—
230#125—21no—
231#125—21no—
232#125—21no—
233#125—21no—
234#124—21no—
235#124—21no—
236#124—21no—
237#124—21no—
238#123—41no—
239#123—41no—
240#123—41no—
241#123—41no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C22⋊Q8—21no—
246#121—21no—
247#121—21no—
248#121—21no—
249#121—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8C22⋊C412yes—
271C4×Q8C22⋊C412yes—
272C4×Q8C22⋊C412yes—
273C4×Q8C22⋊C412yes—
274C4×Q8C2×D412yes—
275C4×Q8C2×D412yes—
276C4×Q8C2×D412yes—
277C4×Q8C2×D412yes—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—41no—
283C4×D4—41no—
284C4×D4—41no—
285C4×D4—41no—
286#117—41no—
287#117—41no—
288#117—41no—
289#117—41no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4C22⋊C412yes—
306C2×C4⋊C4C22⋊C412yes—
307C2×C4⋊C4C22×C411yescharacteristic
308C2×C4⋊C4C22×C411yescharacteristic
309C2×C4⋊C4C22×C411yescharacteristic
310C2×C4⋊C4C22×C411yescharacteristic
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42C22×C411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42#4111yescharacteristic
334C2×C42#4111yescharacteristic
335C2×C42#4111yescharacteristic
336C2×C42C2411yescharacteristic
337#107—22no—
338#107—22no—
339#106—22no—
340#106—22no—
341C4⋊C8—28no—
342C4⋊C8—28no—
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—22no—
348C4⋊C8—22no—
349C4⋊C8—22no—
350C4⋊C8—22no—
351C4⋊C8—22no—
352C4⋊C8—22no—
353C4⋊C8—41no—
354C4⋊C8—41no—
355#103—22no—
356#103—22no—
357#103—22no—
358#103—22no—
359#103—22no—
360#103—22no—
361#102—22no—
362#102—22no—
363C22⋊C8—21no—
364C22⋊C8—21no—
365C22⋊C8—21no—
366C22⋊C8—21no—
367C22⋊C8—22no—
368C22⋊C8—22no—
369C22⋊C8—22no—
370C22⋊C8—22no—
371C22⋊C8—22no—
372C22⋊C8—41no—
373C22⋊C8—41no—
374C22⋊C8C2×D414yes—
375#97—28no—
376#97—22no—
377#97—22no—
378#97—21no—
379#97—21no—
380#97—21no—
381#97—21no—
382C4×C8—28no—
383C4×C8—22no—
384C4×C8—22no—
385C4×C8—22no—
386C4×C8—41no—
387C4×C8—41no—
388C4×C8C2×D414yes—
389C24—21no—
390C24C4×D411yescharacteristic
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8C2×C22⋊C412yes—
410C2×Q8C2×C22⋊C412yes—
411C2×Q8#13114yes—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4—82no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—81no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C4×D411yescharacteristic
442C22×C4C4×D411yescharacteristic
443C22×C4C4×D411yescharacteristic
444C22×C4C4×D411yescharacteristic
445C22×C4C4×D411yescharacteristic
446C22×C4#12011yescharacteristic
447C22×C4#12011yescharacteristic
448C22×C4#12111yescharacteristic
449C22×C4#12111yescharacteristic
450C22×C4#12111yescharacteristic
451C22×C4#12111yescharacteristic
452C22×C4#12311yescharacteristic
453C22×C4#12311yescharacteristic
454C22×C4C23×C411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C2×#4111yescharacteristic
457Q16—44no—
458Q16—44no—
459SD16—44no—
460SD16—44no—
461M16—44no—
462M16—44no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—82no—
476C2×C8—82no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4C2×C22⋊C412yes—
502C4⋊C4C2×C22⋊C412yes—
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—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—41no—
522C42—41no—
523C42C2×C22⋊C411yescharacteristic
524C42C2×C22⋊C411yescharacteristic
525C42C4×D411yescharacteristic
526C42C4×D411yescharacteristic
527C42#12011yescharacteristic
528C42#12011yescharacteristic
529C42#12111yescharacteristic
530C42#12111yescharacteristic
531C42#12111yescharacteristic
532C42#12111yescharacteristic
533C42#12311yescharacteristic
534C42#12311yescharacteristic
535C42C22×D411yescharacteristic
536C42C22×D411yescharacteristic
537C42C22×D411yescharacteristic
538C42C2×#4111yescharacteristic
539C23—22no—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23—81no—
545C23—81no—
546C23#38611yescharacteristic
547C23#38611yescharacteristic
548C23C2×C4×D411yescharacteristic
549Q8—22no—
550Q8—24no—
551Q8—22no—
552Q8—22no—
553Q8—44no—
554Q8—44no—
555Q8—42no—
556Q8—41no—
557Q8—41no—
558Q8—42no—
559Q8#40814yes—
560D4—41no—
561D4—41no—
562D4—42no—
563D4—82no—
564D4—82no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—82no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4#38611yescharacteristic
605C2×C4#38611yescharacteristic
606C2×C4#38611yescharacteristic
607C2×C4#38611yescharacteristic
608C2×C4#38611yescharacteristic
609C2×C4#38611yescharacteristic
610C2×C4C22×C22⋊C411yescharacteristic
611C2×C4C2×C4×D411yescharacteristic
612C2×C4C2×#12011yescharacteristic
613C2×C4C2×#12111yescharacteristic
614C2×C4C2×#12111yescharacteristic
615C2×C4C2×#12311yescharacteristic
616C2×C4C2×#13112yes—
617C2×C4#56812yes—
618C8—44no—
619C8—44no—
620C8—41no—
621C8—41no—
622C8—41no—
623C8—41no—
624C8—82no—
625C8—82no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—44no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—162no—
638C22#190512yes—
639C22C2×#38611yescharacteristic
640C22#293312yes—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—24no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4C2×#40811yescharacteristic
659C4#287911yescharacteristic
660C4#293312yes—
661C2—22no—
662C2—22no—
663C2—41no—
664C2—161no—
665C2#2089711yescharacteristic
666C2#2099511yescharacteristic
667C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60248C2semidirect
4. C2×#1479C2semidirect
6. #9094C2semidirect
7. #9092C2semidirect
8. #9090C2semidirect
9. #9088C2semidirect
10. #7922C2semidirect
11. #7863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31172
#602482
#174652
C2×#14792
#173832
#90942
#90922
#90902
#90882
#79222
#78632