Finite Group Databasebeta

← #195853

Subgroups — #195853

Order 512.

1694 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#431C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic, complemented
4C2×#1480C212yesmaximal, complemented
5#9076C214yesmaximal, complemented
6#9072C214yesmaximal, complemented
7#7893C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11C2×#431—21no—
12C2×#431—21no—
13C2×#431—28no—
14C2×#431C2212yes—
15C2×#431C2212yes—
16C22×C4⋊C8—21no—
17C22×C4⋊C8C2212yes—
18C22×C22⋊C8—21no—
19C22×C22⋊C8—21no—
20C22×C4×C8C2212yes—
21#2857C411yescharacteristic
22#2857C411yescharacteristic
23#2856C411yescharacteristic
24#2856C411yescharacteristic
25#2854—21no—
26#2854C412yes—
27#2850—21no—
28#2850—21no—
29#2847—21no—
30#2847C412yes—
31C4×#121—21no—
32C4×#121—21no—
33C4×#121C2212yes—
34C4×#121C2212yes—
35#2297—21no—
36#2285C2211yescharacteristic
37#1797—21no—
38#1797—21no—
39#1797C2212yes—
40#1797C2212yes—
41C2×#336—21no—
42#1480—28no—
43C2×D4⋊C8C2218yescomplemented
44#1279C2218yescomplemented
45C22×#41—21no—
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C23×C8—22no—
50C23×C8—22no—
51#532—24no—
52#532—24no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55C2×#124—21no—
56#525—44no—
57#525—44no—
58C2×#123—22no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121C2×C412yes—
67C2×#121C2×C412yes—
68C2×#121C2×C412yes—
69C2×#120—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—21no—
73C2×C4×D4C2312yes—
74C2×#117—21no—
75C22×C42—24no—
76C22×C42—24no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#431—28no—
90#431—28no—
91#431—24no—
92#431—28no—
93#431D418yescomplemented
94C2×C4⋊C8—28no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8C2314yes—
101C2×C22⋊C8—28no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—44no—
107C2×C4×C8—22no—
108C2×C4×C8—28no—
109C2×C4×C8C2314yes—
110#391—21no—
111#391—21no—
112#390—21no—
113#390—21no—
114#390—21no—
115#390—21no—
116#390C2×C412yes—
117#390C2×C412yes—
118#389—21no—
119#389—21no—
120#389C2×C412yes—
121#389C2×C412yes—
122#388—22no—
123#388—22no—
124#388—22no—
125#388—22no—
126#386—22no—
127#386—22no—
128#386—22no—
129#386—22no—
130#386—22no—
131#386—22no—
132#384C2×C411yescharacteristic
133#384C2×C411yescharacteristic
134#382—21no—
135#380—21no—
136#380—21no—
137#380—21no—
138#380C2×C412yes—
139#380C2×C412yes—
140#380C2×C412yes—
141C4×C4⋊C4—21no—
142C4×C4⋊C4C2312yes—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145#336—22no—
146#336—22no—
147D4⋊C8—216no—
148C25C22⋊C411yescharacteristic
149C2×#41—44no—
150C2×#41—42no—
151C2×#41—42no—
152C2×#41—44no—
153C22×Q8—21no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C22×C412yes—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4D814yes—
180C23×C4SD1614yes—
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C22×C8—28no—
191C22×C8—24no—
192C22×C8—24no—
193C22×C8—44no—
194C22×C8—44no—
195C22×C8—44no—
196C22×C8—44no—
197C22×C8—44no—
198C4⋊Q8—24no—
199#127—24no—
200#124—42no—
201#124—42no—
202#123—48no—
203C22⋊Q8—44no—
204C22⋊Q8—44no—
205#121—28no—
206#121—28no—
207#121—44no—
208#121—44no—
209#121—44no—
210#121—44no—
211#121—44no—
212#121—42no—
213#121—42no—
214#120—48no—
215C4×D4—24no—
216C4×D4—24no—
217C4×D4—44no—
218C4×D4—44no—
219C4×D4—44no—
220C4×D4—44no—
221C4×D4—44no—
222#117—42no—
223#117—42no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D414yes—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42#4114yes—
279C2×C42C2411yescharacteristic
280C4⋊C8—28no—
281C4⋊C8—28no—
282C4⋊C8—24no—
283C4⋊C8—24no—
284C4⋊C8—44no—
285C22⋊C8—28no—
286C22⋊C8—28no—
287C22⋊C8—44no—
288C22⋊C8—44no—
289C4×C8—28no—
290C4×C8—24no—
291C4×C8C2×D418yes—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—41no—
304C24—41no—
305C24#10214yes—
306C24C2×C22⋊C411yescharacteristic
307C24C2×C22⋊C411yescharacteristic
308C24C2×C22⋊C411yescharacteristic
309#41—88no—
310#41—88no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—216no—
351C22×C4—216no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
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—416no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4#10214yes—
390C22×C4#10214yes—
391C22×C4#10214yes—
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C4×D414yes—
402C22×C4#12012yes—
403C22×C4#12114yes—
404C22×C4#12312yes—
405C22×C4#13114yes—
406C22×C4C2×D812yes—
407C22×C4C2×D812yes—
408C22×C4C2×D812yes—
409C22×C4C2×SD1612yes—
410C22×C4C2×SD1612yes—
411C22×C4C2×SD1612yes—
412C22×C4#13514yes—
413C22×C4C23×C411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C2×C8—28no—
417C2×C8—416no—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8—44no—
421C2×C8—44no—
422C2×C8—44no—
423C2×C8—88no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C42—22no—
447C42—22no—
448C42—21no—
449C42—21no—
450C42—22no—
451C42—22no—
452C42—21no—
453C42—21no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—42no—
461C42—42no—
462C42C4×D414yes—
463C42#12012yes—
464C42#12114yes—
465C42#12312yes—
466C42C22×D412yes—
467C42C2×#4112yes—
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—28no—
482C23—28no—
483C23—21no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23#38612yes—
497C23#40812yes—
498C23C2×#10212yes—
499C23C2×#10212yes—
500C23C2×#10212yes—
501C23#41612yes—
502C23#42112yes—
503C23C22×C22⋊C411yescharacteristic
504Q8—84no—
505D4—48no—
506D4—84no—
507D4—88no—
508D4—88no—
509D4—88no—
510D4—84no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—28no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
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—84no—
563C2×C4—84no—
564C2×C4#38612yes—
565C2×C4#38612yes—
566C2×C4#38612yes—
567C2×C4#40812yes—
568C2×C4C2×#10212yes—
569C2×C4C2×#10212yes—
570C2×C4C2×#10212yes—
571C2×C4#41612yes—
572C2×C4#42112yes—
573C2×C4#46518yes—
574C2×C4#46618yes—
575C2×C4C2×C4×D412yes—
576C2×C4C2×#12011yescharacteristic
577C2×C4C2×#12112yes—
578C2×C4C2×#12311yescharacteristic
579C2×C4C2×#13112yes—
580C2×C4#56812yes—
581C2×C4C22×D811yescharacteristic
582C2×C4C22×SD1611yescharacteristic
583C2×C4C2×#13512yes—
584C8—48no—
585C8—84no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—416no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22#148014yescomplemented
614C22#189314yes—
615C22C2×#38611yescharacteristic
616C22C2×#40811yescharacteristic
617C22#287911yescharacteristic
618C22C22×#10211yescharacteristic
619C22C2×#41611yescharacteristic
620C22C2×#42111yescharacteristic
621C22C2×#42111yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—21no—
627C4—21no—
628C4—22no—
629C4—22no—
630C4—41no—
631C4—41no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4#189314yes—
636C4#293314yes—
637C4C2×#46512yes—
638C4C2×#46612yes—
639C2—24no—
640C2—24no—
641C2—42no—
642C2—162no—
643C2C2×#148011yescharacteristic
644C2#1745811yescharacteristic
645C2#1748511yescharacteristic
646C2#1748811yescharacteristic
647C2#2089711yescharacteristic
648C2C2×#189311yescharacteristic
649C2#2097911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#431C2semidirect
3. C2×#1797C2semidirect
4. C2×#1480C2semidirect
5. #9076C2semidirect
6. #9072C2semidirect
7. #7893C2semidirect
43. C2×D4⋊C8C22semidirect
44. #1279C22semidirect
93. #431D4semidirect
613. C22#1480semidirect

Maximal subgroups

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

subgroupindex
C22×#4312
#366242
C2×#17972
C2×#14802
#90762
#90722
#78932