Finite Group Databasebeta

← #193471

Subgroups — #193471

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
4C2×#1492C212yesmaximal, complemented
5#8943C214yesmaximal, complemented
6#8940C214yesmaximal, complemented
7#7897C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11#2920—28no—
12C2×#432—21no—
13C2×#432C2212yes—
14C2×#431—21no—
15C2×#431C2212yes—
16C2×#423C2212yes—
17C2×#407—21no—
18C22×C22⋊C8—21no—
19C2×C4×M16—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
37C4×C22⋊C8—21no—
38C4×C22⋊C8—21no—
39C4×C22⋊C8C2212yes—
40C4×C22⋊C8C2212yes—
41C2×#336—21no—
42#1492—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—28no—
91#431—28no—
92#423—24no—
93#423D418yescomplemented
94C2×C4⋊C8—22no—
95C2×C4⋊C8C2314yes—
96#407—28no—
97C2×C22⋊C8—28no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C4×M16—28no—
103C2×#97—22no—
104C2×C4×C8—22no—
105C2×C4×C8—24no—
106C2×C4×C8—24no—
107C2×C4×C8—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—42no—
149C2×#41—42no—
150C2×#41—44no—
151C2×#41—44no—
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—24no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—22no—
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×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C2×M16—28no—
189C2×M16—44no—
190C2×M16—44no—
191C22×C8—28no—
192C22×C8—24no—
193C22×C8—24no—
194C22×C8—44no—
195C22×C8—44no—
196C4⋊Q8—24no—
197#127—24no—
198#124—44no—
199#123—48no—
200C22⋊Q8—44no—
201C22⋊Q8—44no—
202#121—28no—
203#121—28no—
204#121—44no—
205#121—44no—
206#121—44no—
207#121—44no—
208#121—44no—
209#121—44no—
210#120—48no—
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—48no—
214C4×D4—48no—
215C4×D4—42no—
216C4×D4—42no—
217#117—44no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
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×C42—24no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D414yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42#4114yes—
273C2×C42C2411yescharacteristic
274C4⋊C8—28no—
275C4⋊C8—24no—
276C4⋊C8C2×D418yes—
277C22⋊C8—28no—
278C22⋊C8—28no—
279C22⋊C8—28no—
280C22⋊C8—28no—
281#97—28no—
282C4×C8—28no—
283C4×C8—24no—
284C4×C8—24no—
285C4×C8—44no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—41no—
299C24—41no—
300C24C2×C22⋊C411yescharacteristic
301C24C2×C22⋊C411yescharacteristic
302C24C2×C22⋊C411yescharacteristic
303#41—88no—
304#41—88no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
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—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
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—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—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—416no—
379C22×C4—416no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4#10414yes—
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#12114yes—
399C22×C4#12312yes—
400C22×C4#13114yes—
401C22×C4#13514yes—
402C22×C4AGL(1,ℤ/8)14yes—
403C22×C4C23×C411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406M16—416no—
407C2×C8—28no—
408C2×C8—28no—
409C2×C8—44no—
410C2×C8—416no—
411C2×C8—44no—
412C2×C8—44no—
413C2×C8—44no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C42—22no—
436C42—22no—
437C42—21no—
438C42—21no—
439C42—22no—
440C42—22no—
441C42—21no—
442C42—21no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—44no—
450C42C4×D414yes—
451C42#12012yes—
452C42#12114yes—
453C42#12312yes—
454C42C22×D412yes—
455C42C2×#4112yes—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—28no—
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—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23#38612yes—
487C23#40812yes—
488C23#41612yes—
489C23#41812yes—
490C23C2×#10412yes—
491C23C22×C22⋊C411yescharacteristic
492Q8—84no—
493D4—48no—
494D4—88no—
495D4—88no—
496D4—88no—
497D4—84no—
498D4—84no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—21no—
512C2×C4—21no—
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—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4#38612yes—
550C2×C4#38612yes—
551C2×C4#38612yes—
552C2×C4#40812yes—
553C2×C4#41612yes—
554C2×C4#41812yes—
555C2×C4C2×#10412yes—
556C2×C4C2×#10412yes—
557C2×C4#42112yes—
558C2×C4#44918yes—
559C2×C4#48218yes—
560C2×C4C2×C4×D412yes—
561C2×C4C2×#12011yescharacteristic
562C2×C4C2×#12112yes—
563C2×C4C2×#12311yescharacteristic
564C2×C4C2×#13112yes—
565C2×C4#56812yes—
566C2×C4C2×#13512yes—
567C2×C4C2×AGL(1,ℤ/8)12yes—
568C8—48no—
569C8—48no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—416no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22#187514yes—
599C22C2×#38611yescharacteristic
600C22C2×#40811yescharacteristic
601C22#287911yescharacteristic
602C22C2×#41611yescharacteristic
603C22C2×#41811yescharacteristic
604C22C22×#10411yescharacteristic
605C22C2×#42111yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4#187114yes—
618C4#187514yes—
619C4#293314yes—
620C4C2×#44912yes—
621C4C2×#48212yes—
622C2—24no—
623C2—42no—
624C2—42no—
625C2—162no—
626C2#1747311yescharacteristic
627C2#1747811yescharacteristic
628C2#1748911yescharacteristic
629C2#1750011yescharacteristic
630C2#2089711yescharacteristic
631C2C2×#187111yescharacteristic
632C2C2×#187511yescharacteristic
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×C4×C22⋊C8C2semidirect
4. C2×#1492C2semidirect
5. #8943C2semidirect
6. #8940C2semidirect
7. #7897C2semidirect
43. C2×D4⋊C8C22semidirect
44. #1279C22semidirect
93. #423D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
#366242
C2×C4×C22⋊C82
C2×#14922
#89432
#89402
#78972