Finite Group Databasebeta

← #195856

Subgroups — #195856

Order 512.

1624 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic, complemented
4C2×#1480C212yesmaximal, complemented
5#9078C214yesmaximal, complemented
6#9074C214yesmaximal, complemented
7#7893C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11#2920—28no—
12C2×#432—21no—
13C2×#432—21no—
14C2×#431C2212yes—
15C2×#431C2212yes—
16C2×#423—21no—
17C22×C4⋊C8C2212yes—
18C2×#407—21no—
19C22×C22⋊C8—21no—
20C2×C4×M16C2212yes—
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
49C22×M16—22no—
50C23×C8—22no—
51#532—24no—
52#532—24no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55C2×#124—21no—
56#525—48no—
57C2×#123—22no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121C2×C412yes—
66C2×#121C2×C412yes—
67C2×#121C2×C412yes—
68C2×#120—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4C2312yes—
73C2×#117—21no—
74C22×C42—24no—
75C22×C42—24no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#432—28no—
89#432—28no—
90#431—24no—
91#431—28no—
92#431D418yescomplemented
93#423—28no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—24no—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8C2314yes—
100#407—28no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—44no—
106C4×M16—28no—
107C2×#97—22no—
108C2×C4×C8C2314yes—
109#391—21no—
110#391—21no—
111#390—21no—
112#390—21no—
113#390—21no—
114#390—21no—
115#390C2×C412yes—
116#390C2×C412yes—
117#389—21no—
118#389—21no—
119#389C2×C412yes—
120#389C2×C412yes—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#386—22no—
126#386—22no—
127#386—22no—
128#386—22no—
129#386—22no—
130#386—22no—
131#384C2×C411yescharacteristic
132#384C2×C411yescharacteristic
133#382—21no—
134#380—21no—
135#380—21no—
136#380—21no—
137#380C2×C412yes—
138#380C2×C412yes—
139#380C2×C412yes—
140C4×C4⋊C4—21no—
141C4×C4⋊C4C2312yes—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144#336—22no—
145#336—22no—
146D4⋊C8—216no—
147C25C22⋊C411yescharacteristic
148C2×#41—44no—
149C2×#41—44no—
150C2×#41—42no—
151C2×#41—42no—
152C22×Q8—21no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4C22×C412yes—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
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⋊C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C2×M16—28no—
190C2×M16—44no—
191C2×M16—44no—
192C22×C8—24no—
193C22×C8—24no—
194C22×C8—44no—
195C22×C8—44no—
196C22×C8—44no—
197C4⋊Q8—24no—
198#127—24no—
199#124—42no—
200#124—42no—
201#123—48no—
202C22⋊Q8—44no—
203C22⋊Q8—44no—
204#121—28no—
205#121—28no—
206#121—44no—
207#121—44no—
208#121—44no—
209#121—44no—
210#121—44no—
211#121—42no—
212#121—42no—
213#120—48no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—48no—
217C4×D4—48no—
218C4×D4—44no—
219#117—44no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
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×C42—24no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—22no—
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×C42C22⋊C411yescharacteristic
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×D414yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42#4114yes—
275C2×C42C2411yescharacteristic
276C4⋊C8—28no—
277C4⋊C8—28no—
278C4⋊C8—24no—
279C4⋊C8—24no—
280C4⋊C8—44no—
281C22⋊C8—28no—
282C22⋊C8—28no—
283C22⋊C8—44no—
284C22⋊C8—44no—
285#97—28no—
286C4×C8—24no—
287C4×C8C2×D418yes—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—41no—
301C24—41no—
302C24C2×C22⋊C411yescharacteristic
303C24C2×C22⋊C411yescharacteristic
304C24C2×C22⋊C411yescharacteristic
305#41—88no—
306#41—88no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
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—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
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—22no—
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—41no—
383C22×C4—416no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—42no—
391C22×C4—42no—
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×C4AGL(1,ℤ/8)12yes—
407C22×C4AGL(1,ℤ/8)14yes—
408C22×C4#13712yes—
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412M16—416no—
413C2×C8—28no—
414C2×C8—44no—
415C2×C8—44no—
416C2×C8—44no—
417C2×C8—44no—
418C2×C8—44no—
419C2×C8—88no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C42—22no—
443C42—22no—
444C42—21no—
445C42—21no—
446C42—22no—
447C42—22no—
448C42—21no—
449C42—21no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—44no—
457C42C4×D414yes—
458C42#12012yes—
459C42#12114yes—
460C42#12312yes—
461C42C22×D412yes—
462C42C2×#4112yes—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—28no—
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—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23#38612yes—
494C23#40812yes—
495C23#41712yes—
496C23#41812yes—
497C23#42112yes—
498C23C22×C22⋊C411yescharacteristic
499Q8—84no—
500D4—48no—
501D4—84no—
502D4—88no—
503D4—88no—
504D4—88no—
505D4—84no—
506C2×C4—24no—
507C2×C4—24no—
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—21no—
517C2×C4—21no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4#38612yes—
562C2×C4#38612yes—
563C2×C4#38612yes—
564C2×C4#40812yes—
565C2×C4#41712yes—
566C2×C4#41812yes—
567C2×C4#42112yes—
568C2×C4#46818yes—
569C2×C4#46918yes—
570C2×C4C2×C4×D412yes—
571C2×C4C2×#12011yescharacteristic
572C2×C4C2×#12112yes—
573C2×C4C2×#12311yescharacteristic
574C2×C4C2×#13112yes—
575C2×C4#56812yes—
576C2×C4C2×AGL(1,ℤ/8)12yes—
577C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
578C2×C4C2×#13711yescharacteristic
579C8—48no—
580C8—84no—
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—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22#189514yes—
610C22C2×#38611yescharacteristic
611C22C2×#40811yescharacteristic
612C22#287911yescharacteristic
613C22C2×#41811yescharacteristic
614C22C2×#41711yescharacteristic
615C22C2×#42111yescharacteristic
616C22C2×#42111yescharacteristic
617C4—22no—
618C4—22no—
619C4—21no—
620C4—21no—
621C4—22no—
622C4—22no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4#189514yes—
631C4#293314yes—
632C4C2×#46812yes—
633C4C2×#46912yes—
634C2—24no—
635C2—42no—
636C2—42no—
637C2—162no—
638C2#1746311yescharacteristic
639C2#1746611yescharacteristic
640C2#1749511yescharacteristic
641C2#1749811yescharacteristic
642C2#2089711yescharacteristic
643C2C2×#189511yescharacteristic
644C2#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. C2×#2920C2semidirect
3. C2×#1797C2semidirect
4. C2×#1480C2semidirect
5. #9078C2semidirect
6. #9074C2semidirect
7. #7893C2semidirect
43. C2×D4⋊C8C22semidirect
44. #1279C22semidirect
92. #431D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
#366242
C2×#17972
C2×#14802
#90782
#90742
#78932