Finite Group Databasebeta

← #375250

Subgroups — #375250

Order 512.

1496 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2640C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4C2×#1959C211yesmaximal, characteristic
5C2×#1958C211yesmaximal, characteristic
6C2×#1768C211yesmaximal, characteristic, complemented
7#13779C211yesmaximal, characteristic, complemented
8#11077C212yesmaximal, complemented
9#11076C212yesmaximal
10#10932C212yesmaximal, complemented
11#10931C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#97C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2640—24no—
22#2640C418yescomplemented
23C2×#397—22no—
24C2×#393—22no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2212yes—
32C2×#376C2211yescharacteristic
33#1993—24no—
34#1990—24no—
35#1959—24no—
36#1958—24no—
37#1768—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43#1436—22no—
44#1436—22no—
45C4⋊#95C2212yes—
46C23×Q8—21no—
47C23×Q8C2311yescharacteristic
48C23×D4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×#124—22no—
58C2×#124—22no—
59C2×#124—22no—
60C2×#124—22no—
61C2×#124—22no—
62C2×#124C2×C414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C42—22no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—42no—
92C2×#102—42no—
93C2×#97—22no—
94C2×#97—22no—
95C2×#97—22no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99#397—216no—
100#393—216no—
101#391—22no—
102#391—22no—
103#391C2×C414yes—
104#391C2×C414yes—
105#390—22no—
106#390—22no—
107#390C2×C414yes—
108#390C2×C414yes—
109#384—22no—
110#384D414yes—
111#382—22no—
112#382D414yes—
113C4×C4⋊C4—42no—
114C4×C4⋊C4—42no—
115#376C2×C414yes—
116#376D414yes—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95D412yes—
125C2×#95D412yes—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#336—24no—
131C25—21no—
132C22×Q8—24no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—82no—
145C22×Q8—82no—
146C22×Q8C22×C412yes—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4C22×C412yes—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
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×C8—22no—
178C22×C8—22no—
179C22×C8#4114yes—
180C22×C8#4114yes—
181C4⋊Q8—28no—
182C4⋊Q8—28no—
183C4⋊Q8—44no—
184C4⋊Q8—44no—
185#124—28no—
186#124—28no—
187#124—44no—
188#124—44no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
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—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D414yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4—82no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—44no—
237C2×C42—82no—
238C2×C42—82no—
239C2×C42C22×C412yes—
240C2×C42C22×C412yes—
241C2×C42C22×C412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C4⋊C8—216no—
250#103—48no—
251#103—48no—
252#102—48no—
253#102—48no—
254#97—44no—
255C4×C8—44no—
256#95—28no—
257#95—28no—
258#95—24no—
259#95—24no—
260#95—44no—
261#95—82no—
262#95—82no—
263#95C22⋊C418yes—
264#95C22⋊C418yes—
265C24—22no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24C22×D411yescharacteristic
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—42no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—42no—
285C2×Q8—48no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—88no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8—88no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
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—24no—
334C22×C4—24no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
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×C4C2×C22⋊C414yes—
362C22×C4C2×C22⋊C414yes—
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C412yes—
365C22×C4#12112yes—
366C22×C4#12112yes—
367C22×C4#12112yes—
368C22×C4#12112yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4112yes—
376C22×C4C2×#4112yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C42−1+412yes—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—48no—
410C4⋊C4—84no—
411C4⋊C4—84no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—88no—
415C22⋊C4—88no—
416C42—24no—
417C42—24no—
418C42—24no—
419C42—24no—
420C42—88no—
421C42C4×D414yes—
422C42C4×D414yes—
423C42#12414yes—
424C42#12714yes—
425C23—42no—
426C23—44no—
427C23—44no—
428C23—42no—
429C23—42no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—82no—
443C23—82no—
444C23—81no—
445C23—81no—
446C23—81no—
447C23—81no—
448C23C22×C22⋊C411yescharacteristic
449C23C2×#12111yescharacteristic
450C23C2×#12111yescharacteristic
451C23#53211yescharacteristic
452C23#53211yescharacteristic
453C23#53511yescharacteristic
454C23#53611yescharacteristic
455C23#57612yes—
456C23#57712yes—
457C23#57712yes—
458C23#57812yes—
459Q8—44no—
460Q8—44no—
461Q8—816no—
462Q8—82no—
463Q8—82no—
464D4—44no—
465D4—44no—
466D4—82no—
467D4—82no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4#39014yes—
522C2×C4#39014yes—
523C2×C4C2×C4×D411yescharacteristic
524C2×C4C2×C4×D411yescharacteristic
525C2×C4C4×#4112yes—
526C2×C4#52012yes—
527C2×C4#52512yes—
528C2×C4C2×#12411yescharacteristic
529C2×C4C2×#12711yescharacteristic
530C2×C4#53712yes—
531C2×C4#54612yes—
532C2×C4#54612yes—
533C2×C4#54812yes—
534C2×C4#54812yes—
535C2×C4#55612yes—
536C2×C4#55612yes—
537C2×C4#55912yes—
538C2×C4#55912yes—
539C8—44no—
540C8—44no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—84no—
545C22—84no—
546C22—162no—
547C22—161no—
548C22—161no—
549C22—161no—
550C22—161no—
551C22—162no—
552C22C2×#39011yescharacteristic
553C22#229711yescharacteristic
554C22#231111yescharacteristic
555C22#254411yescharacteristic
556C22#254411yescharacteristic
557C22#254511yescharacteristic
558C22#254511yescharacteristic
559C22#289712yes—
560C22#289812yes—
561C22#304111yescharacteristic
562C22#304211yescharacteristic
563C22#304311yescharacteristic
564C22#304411yescharacteristic
565C22#304512yes—
566C22#304612yes—
567C22#317312yes—
568C22#317512yes—
569C22#317812yes—
570C22#317912yes—
571C22#322412yes—
572C22#322512yes—
573C22#324312yes—
574C22#324412yes—
575C4—24no—
576C4—24no—
577C4—44no—
578C4—44no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—84no—
583C4—84no—
584C4#236712yes—
585C4#236712yes—
586C4#254012yes—
587C4#259212yes—
588C2—162no—
589C2#1415311yescharacteristic
590C2#2126211yescharacteristic
591C2#2126311yescharacteristic
592C2#2141212yes—
593C2#2187112yes—
594C2#2862612yes—
595C2#2862711yescharacteristic
596C2#2862811yescharacteristic
597C2#2866211yescharacteristic
598C2#2866311yescharacteristic
599C2#2866812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1993C2semidirect
3. C2×#1990C2semidirect
6. C2×#1768C2semidirect
7. #13779C2semidirect
8. #11077C2semidirect
10. #10932C2semidirect
22. #2640C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26402
C2×#19932
C2×#19902
C2×#19592
C2×#19582
C2×#17682
#137792
#110772
#110762
#109322
#109312