Finite Group Databasebeta

← #418703

Subgroups — #418703

Order 512.

1594 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic, complemented
2C2×#2454C211yesmaximal, characteristic
3C2×#2023C211yesmaximal, characteristic
4C2×#1862C212yesmaximal
5#15128C211yesmaximal, characteristic, complemented
6#11385C211yesmaximal, characteristic, complemented
7#11327C212yesmaximal, complemented
8#11219C212yesmaximal, complemented
9#11121C212yesmaximal
10#10927C211yesmaximal, characteristic
11#10875C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#501—28no—
16C2×#501—24no—
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#97C2211yescharacteristic
22#2454—24no—
23#2454—24no—
24C2×#399—21no—
25C2×#396—21no—
26C2×#395—21no—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#385—21no—
33C2×#380—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#2023—24no—
36#1862—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44#1446—21no—
45#1444—21no—
46#1441—21no—
47C4⋊#95C2212yes—
48C4⋊#95C2211yescharacteristic
49C23×Q8—21no—
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×#125—24no—
59C2×#125—22no—
60C2×#125—42no—
61C2×#125D414yes—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—24no—
65C2×C4×D4—42no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#501—232no—
88C2×#107—24no—
89C2×#107—24no—
90C2×#107D418yes—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—42no—
99C2×#102Q818yescomplemented
100C2×#97—24no—
101C2×#97—22no—
102#399—28no—
103#396—28no—
104#395—44no—
105#390—22no—
106#390—22no—
107#390—22no—
108#390D414yes—
109#389—22no—
110#389—22no—
111#389—22no—
112#389—22no—
113#389—24no—
114#389—42no—
115#385—28no—
116#380—28no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95D411yescharacteristic
127C2×#95D411yescharacteristic
128#340—24no—
129#340—24no—
130#340—24no—
131#340—42no—
132#336—24no—
133#336—24no—
134#336—22no—
135C25—21no—
136C22×Q8—44no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—82no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4C2×D412yes—
148C22×D4C2×Q814yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—44no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—82no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8C2×D414yes—
178C22×C8C2×Q814yes—
179C22×C8#4114yes—
180C4⋊Q8—28no—
181C4⋊Q8—48no—
182C4⋊Q8—44no—
183#125—28no—
184#125—44no—
185#125—48no—
186C4×D4—48no—
187C4×D4—44no—
188C4×D4—44no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×Q814yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4—82no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255#107—216no—
256#106—216no—
257#102—216no—
258#102—48no—
259#97—28no—
260#95—24no—
261#95—24no—
262#95—22no—
263#95—22no—
264#95—44no—
265#95—42no—
266#95—44no—
267#95—42no—
268#95—82no—
269C24—22no—
270C24—24no—
271C24—41no—
272C24—41no—
273C24—42no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—42no—
279C24—42no—
280C24C22×D411yescharacteristic
281C2×Q8—48no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8—88no—
285C2×Q8—88no—
286C2×D4—24no—
287C2×D4—22no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4C4⋊Q814yes—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—88no—
362C22×C4—84no—
363C22×C4—88no—
364C22×C4#12012yes—
365C22×C4#12012yes—
366C22×C4AGL(1,ℤ/8)12yes—
367C22×C4#13712yes—
368C22×C4C22×D412yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×Q812yes—
372C22×C4C2×#4112yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C42−1+412yes—
375C22×C42−1+412yes—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8C22⋊Q818yes—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—48no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—88no—
411C4⋊C4—84no—
412C4⋊C4—84no—
413C4⋊C4#12014yes—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—84no—
417C22⋊C4—84no—
418C22⋊C4—84no—
419C42—22no—
420C42—22no—
421C42—24no—
422C42—41no—
423C42—41no—
424C42—42no—
425C42—42no—
426C42—42no—
427C23—24no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—44no—
434C23—42no—
435C23—44no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—81no—
441C23—82no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23C2×#12011yescharacteristic
451C23#53211yescharacteristic
452C23#53611yescharacteristic
453C23D4212yes—
454C23D4×Q812yes—
455C23C2×AGL(1,ℤ/8)11yescharacteristic
456C23C2×#13711yescharacteristic
457C23#57511yescharacteristic
458C23#57511yescharacteristic
459C23#57612yes—
460C23#57712yes—
461Q8—816no—
462D4—28no—
463D4—44no—
464D4—44no—
465D4—44no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—816no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4C2×#12011yescharacteristic
529C2×C4C2×C22⋊Q812yes—
530C2×C4C2×C4⋊Q811yescharacteristic
531C2×C4#53411yescharacteristic
532C2×C4#53612yes—
533C2×C4#53611yescharacteristic
534C2×C4#54212yes—
535C2×C4#54412yes—
536C2×C4D4212yes—
537C2×C4#54712yes—
538C2×C4#54812yes—
539C2×C4#54812yes—
540C2×C4D4×Q812yes—
541C2×C4D4×Q812yes—
542C2×C4#55412yes—
543C2×C4#56312yes—
544C8—28no—
545C8—82no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—84no—
554C22—84no—
555C22—84no—
556C22—84no—
557C22#240311yescharacteristic
558C22#243211yescharacteristic
559C22#243411yescharacteristic
560C22#245212yes—
561C22#267712yes—
562C22#300611yescharacteristic
563C22#300711yescharacteristic
564C22#300811yescharacteristic
565C22#300911yescharacteristic
566C22#317111yescharacteristic
567C22#317912yes—
568C22#318111yescharacteristic
569C22#323911yescharacteristic
570C22#324011yescharacteristic
571C22#324112yes—
572C22#326412yes—
573C22#327112yes—
574C22#327312yes—
575C22#327812yes—
576C22#338412yes—
577C22#338512yes—
578C22#338812yes—
579C22#338912yes—
580C4—42no—
581C4—42no—
582C4—44no—
583C4—42no—
584C4—42no—
585C4—84no—
586C4—81no—
587C4—82no—
588C4—81no—
589C4—82no—
590C4—84no—
591C4#240912yes—
592C4#244012yes—
593C4#267412yes—
594C4#272912yes—
595C2—84no—
596C2#1526511yescharacteristic
597C2#2386111yescharacteristic
598C2#2387011yescharacteristic
599C2#2567811yescharacteristic
600C2#2568511yescharacteristic
601C2#2585011yescharacteristic
602C2#2585111yescharacteristic
603C2#2648812yes—
604C2#2649812yes—
605C2#3096212yes—
606C2#3097112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#501C2semidirect
5. #15128C2semidirect
6. #11385C2semidirect
7. #11327C2semidirect
8. #11219C2semidirect
11. #10875C2semidirect
99. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#24542
C2×#20232
C2×#18622
#151282
#113852
#113272
#112192
#111212
#109272
#108752