Finite Group Databasebeta

← #380581

Subgroups — #380581

Order 512.

1760 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#465C211yesmaximal, characteristic
2C2×#1895C211yesmaximal, characteristic
3C2×#1894C211yesmaximal, characteristic, complemented
4C2×#1808C211yesmaximal, characteristic, complemented
5C2×#1729C211yesmaximal, characteristic, complemented
6#14258C211yesmaximal, characteristic, complemented
7#14257C211yesmaximal, characteristic
8#11378C212yesmaximal, complemented
9#11124C212yesmaximal, complemented
10#11123C212yesmaximal, complemented
11#11023C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#465—24no—
19C2×#465—24no—
20C2×#465C418yescomplemented
21C22×#107C2212yes—
22C22×#106C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#103C2212yes—
25C22×#102C2211yescharacteristic
26C22×#102C2212yes—
27C2×#391C2212yes—
28C2×#390C2212yes—
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31#1895—24no—
32#1894—24no—
33#1808—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37#1729—24no—
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1441—22no—
41#1440—22no—
42#1439—22no—
43#1438—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×D4—21no—
48C23×D4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C22×SD16—22no—
51C22×SD16—22no—
52C22×SD16—42no—
53C22×SD16—42no—
54C22×SD16C2×C414yes—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—42no—
58C22×M16—42no—
59C22×M16D414yes—
60C23×C8—42no—
61C23×C8—42no—
62C23×C8C2×C414yes—
63C23×C8D414yes—
64C23×C8D414yes—
65C23×C8C2312yes—
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8C2×C414yes—
74C2×#121—22no—
75C2×#121—22no—
76C2×#121—22no—
77C2×#121—22no—
78C2×#121—22no—
79C2×#121C2×C414yes—
80C2×#117—22no—
81C2×#117—22no—
82C2×#117—42no—
83C2×#117—42no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D412yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—24no—
95C22×C22⋊C4D412yes—
96C22×C42C2312yes—
97#465—232no—
98C2×#107—24no—
99C2×#107—42no—
100C2×#107—42no—
101C2×#106—22no—
102C2×#106—22no—
103C2×#106C2×C414yes—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—24no—
107C2×#103—42no—
108C2×#103—42no—
109C2×#103C2×C414yes—
110C2×#102—22no—
111C2×#102—22no—
112C2×#102—24no—
113C2×#102—42no—
114C2×#102—42no—
115C2×#102C2×C414yes—
116#391—42no—
117#391—42no—
118#390—42no—
119#390—42no—
120#380—22no—
121#380—22no—
122#380—22no—
123#380—22no—
124C2×#95—21no—
125C2×#95—24no—
126C2×#95—21no—
127C2×#95—24no—
128C2×#95—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#336—24no—
133#336—24no—
134C25—21no—
135C25C2×D411yescharacteristic
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8C22×C412yes—
146C22×D4—24no—
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×D4—42no—
157C22×D4—42no—
158C22×D4—84no—
159C22×D4C22×C412yes—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4C22×C412yes—
170C23×C4C2×D414yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4#4112yes—
176C23×C4#4112yes—
177C23×C4C2411yescharacteristic
178C2×SD16—28no—
179C2×SD16—48no—
180C2×SD16—44no—
181C2×SD16—44no—
182C2×SD16—48no—
183C2×M16—28no—
184C2×M16—48no—
185C2×M16—44no—
186C2×M16—44no—
187C2×M16—48no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—48no—
199C22×C8—48no—
200C22×C8—44no—
201C22×C8—44no—
202C22×C8C22⋊C4116yescomplemented
203C22×C8C22×C412yes—
204C22×C8C22×C412yes—
205C22×C8C2×D414yes—
206C22×C8C2×D412yes—
207C22×C8C2×D412yes—
208C22×C8C2×D412yes—
209C22×C8C2×D412yes—
210C22⋊Q8—28no—
211C22⋊Q8—28no—
212C22⋊Q8—44no—
213C22⋊Q8—44no—
214#121—28no—
215#121—28no—
216#121—44no—
217#121—44no—
218#117—44no—
219#117—44no—
220#117—48no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C22×C412yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—84no—
262C2×C22⋊C4—84no—
263C2×C22⋊C4—82no—
264C2×C22⋊C4—82no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271#107—48no—
272#106—28no—
273#103—28no—
274#103—48no—
275#102—28no—
276#102—48no—
277#95—48no—
278#95—42no—
279#95—42no—
280#95—84no—
281#95—84no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—24no—
286C24—22no—
287C24—22no—
288C24—41no—
289C24—41no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—44no—
294C24—42no—
295C24C4×D412yes—
296C24#12112yes—
297C24#12712yes—
298C24C22×D411yescharacteristic
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—42no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—48no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—42no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—42no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—88no—
337C2×D4—88no—
338C2×D4—88no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4C2×C22⋊C414yes—
387C22×C4C4×D414yes—
388C22×C4C4×D412yes—
389C22×C4#12114yes—
390C22×C4#12112yes—
391C22×C4#12414yes—
392C22×C4#12712yes—
393C22×C4C23×C411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C4H(5,𝔽2)12yes—
402C22×C42−1+412yes—
403SD16—416no—
404M16—416no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—416no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8C2×C22⋊C414yes—
422C2×C8C2×C22⋊C414yes—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—48no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—48no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—816no—
445C22⋊C4—84no—
446C22⋊C4—84no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C23—24no—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—42no—
470C23—81no—
471C23—81no—
472C23—82no—
473C23—81no—
474C23—81no—
475C23—82no—
476C23—84no—
477C23—84no—
478C23—84no—
479C23—84no—
480C23—82no—
481C23—82no—
482C23#39014yes—
483C23C2×C4×D411yescharacteristic
484C23C2×#12111yescharacteristic
485C23C2×#12711yescharacteristic
486C23#53412yes—
487C23#53712yes—
488C23#57612yes—
489C23#57714yes—
490C23#57812yes—
491Q8—42no—
492Q8—42no—
493Q8—44no—
494Q8—81no—
495Q8—81no—
496Q8—82no—
497D4—42no—
498D4—42no—
499D4—44no—
500D4—816no—
501D4—81no—
502D4—81no—
503D4—82no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4#39014yes—
546C2×C4C22×C22⋊C411yescharacteristic
547C2×C4C2×C4×D411yescharacteristic
548C2×C4#51812yes—
549C2×C4#52012yes—
550C2×C4C2×#12111yescharacteristic
551C2×C4C2×#12411yescharacteristic
552C2×C4#53411yescharacteristic
553C2×C4#53411yescharacteristic
554C2×C4#53412yes—
555C2×C4#53412yes—
556C2×C4#53511yescharacteristic
557C2×C4#53611yescharacteristic
558C2×C4#53712yes—
559C2×C4#53712yes—
560C2×C4#53812yes—
561C2×C4#53812yes—
562C2×C4#53912yes—
563C2×C4#53912yes—
564C8—24no—
565C8—24no—
566C8—44no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—48no—
575C22—88no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—162no—
580C22—162no—
581C22—162no—
582C22—162no—
583C22C2×#39011yescharacteristic
584C22#232512yes—
585C22#261711yescharacteristic
586C22#261811yescharacteristic
587C22#265612yes—
588C22#294912yes—
589C22#295112yes—
590C22#305712yes—
591C22#306112yes—
592C22#315412yes—
593C22#315511yescharacteristic
594C22#315611yescharacteristic
595C22#319814yes—
596C22#320114yes—
597C22#325714yes—
598C22#325814yes—
599C4—24no—
600C4—84no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—162no—
605C4—162no—
606C4#230811yescharacteristic
607C4#231111yescharacteristic
608C4#232512yes—
609C4#261711yescharacteristic
610C4#261811yescharacteristic
611C4#263912yes—
612C2—44no—
613C2—162no—
614C2#1427111yescharacteristic
615C2#2123011yescharacteristic
616C2#2123111yescharacteristic
617C2#2219414yes—
618C2#2974012yes—
619C2#2978012yes—
620C2#3025112yes—
621C2#3025212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1894C2semidirect
4. C2×#1808C2semidirect
5. C2×#1729C2semidirect
6. #14258C2semidirect
8. #11378C2semidirect
9. #11124C2semidirect
10. #11123C2semidirect
11. #11023C2semidirect
20. C2×#465C4semidirect
202. C22×C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4652
C2×#18952
C2×#18942
C2×#18082
C2×#17292
#142582
#142572
#113782
#111242
#111232
#110232