Finite Group Databasebeta

← #195998

Subgroups — #195998

Order 512.

1638 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2C2×#2854C211yesmaximal, characteristic
3C2×#1817C211yesmaximal, characteristic, complemented
4C2×#1471C212yesmaximal
5#9084C214yesmaximal, complemented
6#9080C214yesmaximal, complemented
7#7915C212yesmaximal, complemented
8C22×#123C412yes—
9C22×#121C2211yescharacteristic
10C23×C42C2211yescharacteristic
11#2920—28no—
12C2×#432—21no—
13C2×#432C2212yes—
14C2×#431—21no—
15C2×#431C2212yes—
16C2×#423—21no—
17C2×#407C2212yes—
18C22×C22⋊C8C2212yes—
19C2×C4×M16—21no—
20#2854—21no—
21#2854—21no—
22#2854—21no—
23#2854—21no—
24#2854—21no—
25#2854—21no—
26#2854—21no—
27#2854—21no—
28C2×#390C411yescharacteristic
29C2×#390C411yescharacteristic
30C2×#389C411yescharacteristic
31C2×#389C411yescharacteristic
32C2×#388C2212yes—
33C2×#388C2212yes—
34C2×#386C412yes—
35C2×#380C2211yescharacteristic
36#1817—21no—
37#1817—21no—
38#1817C2212yes—
39#1817C2212yes—
40C2×#336—21no—
41C2×#336—21no—
42#1471—28no—
43C2×#323C2218yes—
44#1280C2218yes—
45C23×D4C2×C412yes—
46C23×D4C2×C411yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C23×C8—22no—
52C2×#123—22no—
53C2×#123—22no—
54C2×#123—22no—
55C2×#123—22no—
56C2×#123—48no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—24no—
62C2×#121—24no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2312yes—
74C22×C42—24no—
75C22×C42—24no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#432—28no—
89#432—28no—
90#431—28no—
91#431—28no—
92#423—28no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95#407—24no—
96#407D418yes—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8C2314yes—
102C2×C22⋊C8C2314yes—
103C4×M16—28no—
104C2×#97—22no—
105C2×C4×C8—22no—
106#390—21no—
107#390—21no—
108#390—21no—
109#390—21no—
110#390—21no—
111#390—21no—
112#390—21no—
113#390—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#388—22no—
123#388—22no—
124#388—22no—
125#388—22no—
126#388—22no—
127#388—22no—
128#388—22no—
129#388—22no—
130#386—22no—
131#386—22no—
132#386—22no—
133#386—22no—
134#380—21no—
135#380—21no—
136#380—21no—
137#380—21no—
138C2×#95C2×C412yes—
139C2×#95C2×C412yes—
140C2×#95C2312yes—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—22no—
145#323—216no—
146C25C22×C411yescharacteristic
147C25C22×C412yes—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—28no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—24no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—44no—
157C22×D4—44no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
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×C4C22×C412yes—
180C23×C4C22×C412yes—
181C23×C4C22×C412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D414yes—
187C23×C4#4114yes—
188C23×C4C2411yescharacteristic
189C2×M16—28no—
190C2×M16—44no—
191C2×M16—44no—
192C22×C8—24no—
193C22×C8—24no—
194C22×C8—44no—
195C22×C8—44no—
196C22×C8—44no—
197#123—48no—
198#123—48no—
199#123—48no—
200#123—48no—
201#121—42no—
202#121—42no—
203#121—44no—
204#121—44no—
205#121—42no—
206#121—42no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C4⋊C8—28no—
276C4⋊C8—28no—
277C22⋊C8—28no—
278C22⋊C8—24no—
279C22⋊C8—24no—
280C22⋊C8—24no—
281C22⋊C8—44no—
282C22⋊C8C2×D418yes—
283#97—28no—
284C4×C8—28no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—41no—
300C24—41no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24C2×C22⋊C411yescharacteristic
306C24C2×C22⋊C411yescharacteristic
307C24C4×D414yes—
308C24C23×C411yescharacteristic
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—88no—
322C2×D4—816no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
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—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—416no—
378C22×C4—416no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4#10014yes—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C4×D414yes—
397C22×C4#12012yes—
398C22×C4#12012yes—
399C22×C4#12114yes—
400C22×C4#12114yes—
401C22×C4#12312yes—
402C22×C4#12312yes—
403C22×C4#13114yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4C2×#4112yes—
408M16—416no—
409C2×C8—28no—
410C2×C8—44no—
411C2×C8—44no—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8—44no—
415C2×C8—88no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—42no—
433C22⋊C4—44no—
434C22⋊C4—42no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—28no—
474C23—28no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—48no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23—84no—
494C23#38612yes—
495C23#38612yes—
496C23C2×#10012yes—
497C23#41312yes—
498C23C22×C22⋊C411yescharacteristic
499C23C2×C4×D412yes—
500C23C2×#13112yes—
501C23#56812yes—
502D4—84no—
503D4—88no—
504D4—88no—
505D4—84no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
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—84no—
565C2×C4#38612yes—
566C2×C4#38612yes—
567C2×C4#40812yes—
568C2×C4#40812yes—
569C2×C4#41012yes—
570C2×C4#41012yes—
571C2×C4C2×#10012yes—
572C2×C4C2×#10012yes—
573C2×C4C2×#12011yescharacteristic
574C2×C4C2×#12112yes—
575C2×C4C2×#12311yescharacteristic
576C8—48no—
577C8—84no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—416no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—168no—
606C22#190014yes—
607C22C2×#38611yescharacteristic
608C22C2×#40811yescharacteristic
609C22#287911yescharacteristic
610C22C2×#41011yescharacteristic
611C22C2×#41011yescharacteristic
612C22C22×#10011yescharacteristic
613C22C2×#41311yescharacteristic
614C22#293314yes—
615C4—22no—
616C4—22no—
617C4—21no—
618C4—22no—
619C4—21no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4#189914yes—
631C2—24no—
632C2—24no—
633C2—42no—
634C2—162no—
635C2#1738011yescharacteristic
636C2#1738411yescharacteristic
637C2#1739911yescharacteristic
638C2#1740311yescharacteristic
639C2#2089711yescharacteristic
640C2C2×#189911yescharacteristic
641C2C2×#190011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2920C2semidirect
3. C2×#1817C2semidirect
5. #9084C2semidirect
6. #9080C2semidirect
7. #7915C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×#28542
C2×#18172
C2×#14712
#90842
#90802
#79152