Finite Group Databasebeta

← #448419

Subgroups — #448419

Order 512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2496C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4#15125C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic, complemented
6#11292C212yesmaximal, complemented
7#11199C212yesmaximal
8#11096C212yesmaximal
9#10921C211yesmaximal, characteristic
10#10907C212yesmaximal, complemented
11#10880C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#126—21no—
14C22×#107C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C4×C8C2211yescharacteristic
17#2496—24no—
18#2496—24no—
19C2×#400—21no—
20C2×#397—21no—
21C2×#393—21no—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#387—21no—
25C2×#384C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#2057—24no—
32#1990—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1446—21no—
43#1442—21no—
44#1439—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×#126—28no—
56C2×#126—44no—
57C2×#126—44no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#107—24no—
78C2×#107—24no—
79C2×#107—42no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—42no—
83C2×C4×C8—22no—
84C2×C4×C8—42no—
85#400—28no—
86#397—28no—
87#393—44no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389D414yes—
92#387—28no—
93#384—22no—
94#384D414yes—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—24no—
100#380—42no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—42no—
124#336—24no—
125#336—22no—
126C25C2×D411yescharacteristic
127C22×Q8—44no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—82no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8#4114yes—
165C22×C8#4114yes—
166C4⋊Q8—28no—
167C4⋊Q8—48no—
168C4⋊Q8—44no—
169#126—432no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
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—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4#4114yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4D814yes—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238#107—48no—
239C22⋊C8—48no—
240C4×C8—44no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—22no—
247#95—22no—
248#95—44no—
249#95—44no—
250#95—42no—
251#95—42no—
252#95—82no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—22no—
257C24—22no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—22no—
262C24—22no—
263C24C2×D812yes—
264C24C22×D411yescharacteristic
265C2×Q8—48no—
266C2×Q8—88no—
267C2×Q8—88no—
268C2×Q8—84no—
269C2×Q8—84no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4#12012yes—
343C22×C4#12012yes—
344C22×C4C2×D812yes—
345C22×C4C2×D812yes—
346C22×C4#13512yes—
347C22×C4AGL(1,ℤ/8)12yes—
348C22×C4#13712yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C2×#4112yes—
352C22×C4C2×#4112yes—
353C22×C4C2×#4112yes—
354C22×C4H(5,𝔽2)12yes—
355C22×C42−1+412yes—
356C22×C42−1+412yes—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C22⋊C4—28no—
395C22⋊C4—48no—
396C22⋊C4—48no—
397C22⋊C4—44no—
398C22⋊C4—88no—
399C42—22no—
400C42—22no—
401C42—24no—
402C42—48no—
403C42—41no—
404C42—41no—
405C42—42no—
406C23—24no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—24no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—41no—
415C23—42no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—41no—
421C23—42no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23C2×#12011yescharacteristic
430C23#53211yescharacteristic
431C23#53611yescharacteristic
432C23#54612yes—
433C23#54712yes—
434C23C22×D811yescharacteristic
435C23C2×#13511yescharacteristic
436C23C2×AGL(1,ℤ/8)11yescharacteristic
437C23C2×#13711yescharacteristic
438C23#57712yes—
439C23#57812yes—
440Q8—816no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4#44714yes—
507C2×C4#45214yes—
508C2×C4#52511yescharacteristic
509C2×C4#52912yes—
510C2×C4#53211yescharacteristic
511C2×C4#53511yescharacteristic
512C2×C4#53611yescharacteristic
513C2×C4#53912yes—
514C2×C4#54112yes—
515C2×C4#54212yes—
516C2×C4#54612yes—
517C2×C4#54712yes—
518C2×C4#54812yes—
519C2×C4#54812yes—
520C2×C4#55112yes—
521C2×C4#55612yes—
522C2×C4#55612yes—
523C2×C4#56312yes—
524C8—44no—
525C8—44no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—44no—
530C22—44no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—82no—
536C22—82no—
537C22#240411yescharacteristic
538C22#243211yescharacteristic
539C22#243211yescharacteristic
540C22#252112yes—
541C22#268912yes—
542C22C2×#44711yescharacteristic
543C22C2×#45211yescharacteristic
544C22#301011yescharacteristic
545C22#301111yescharacteristic
546C22#316911yescharacteristic
547C22#317812yes—
548C22#318211yescharacteristic
549C22#322611yescharacteristic
550C22#322811yescharacteristic
551C22#324212yes—
552C22#329412yes—
553C22#330012yes—
554C22#330612yes—
555C22#331212yes—
556C22#333412yes—
557C22#334012yes—
558C22#335312yes—
559C22#335612yes—
560C4—24no—
561C4—24no—
562C4—42no—
563C4—81no—
564C4—82no—
565C4—82no—
566C4—81no—
567C4—82no—
568C4—82no—
569C4—84no—
570C4—162no—
571C4#246912yes—
572C4#256012yes—
573C4#271612yes—
574C4#272312yes—
575C2—82no—
576C2#1591011yescharacteristic
577C2#2396011yescharacteristic
578C2#2396711yescharacteristic
579C2#2567311yescharacteristic
580C2#2568611yescharacteristic
581C2#2570111yescharacteristic
582C2#2570311yescharacteristic
583C2#2806512yes—
584C2#2808712yes—
585C2#3142212yes—
586C2#3142312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2057C2semidirect
3. C2×#1990C2semidirect
5. #11373C2semidirect
6. #11292C2semidirect
10. #10907C2semidirect
11. #10880C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24962
C2×#20572
C2×#19902
#151252
#113732
#112922
#111992
#110962
#109212
#109072
#108802