Finite Group Databasebeta

← #381847

Subgroups — #381847

Order 512.

1396 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2016C211yesmaximal, characteristic
2C2×#1878C212yesmaximal
3#14714C211yesmaximal, characteristic
4C4×#391C211yesmaximal, characteristic
5#11267C211yesmaximal, characteristic
6#11190C212yesmaximal
7#11050C211yesmaximal, characteristic
8#11024C212yesmaximal
9#11010C211yesmaximal, characteristic
10#10932C212yesmaximal
11#10795C211yesmaximal, characteristic
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C2×#391C2211yescharacteristic
20C2×#391C2212yes—
21C2×#391C2211yescharacteristic
22C2×#391C2212yes—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C43C2211yescharacteristic
26#2016C418yescomplemented
27#1878—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35#1436—21no—
36#1436—21no—
37C4⋊#95—21no—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C4×#95—21no—
44C4×#95—21no—
45C23×Q8C2312yes—
46C23×Q8C2311yescharacteristic
47C22×Q16—24no—
48C22×Q16—24no—
49C22×Q16—42no—
50C22×Q16—42no—
51C22×Q16—42no—
52C22×Q16—42no—
53C22×Q16C2×C418yes—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8C2×C414yes—
60C2×C4×Q8—22no—
61C2×C4×Q8—42no—
62C2×C4×Q8—42no—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42—21no—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#103C2×C418yes—
94#391—22no—
95#391—22no—
96#391—22no—
97#391—24no—
98#391—24no—
99#391—42no—
100#391—42no—
101#391C2×C414yes—
102#384—42no—
103C4×C4⋊C4—42no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C43—42no—
118#340—24no—
119#340—42no—
120#340—42no—
121#336—24no—
122#336—22no—
123#336—42no—
124#336C2×C414yes—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8C22×C414yes—
146C22×Q8C22×C412yes—
147C22×Q8C2×D412yes—
148C22×Q8#4112yes—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C2×Q16—216no—
163C2×Q16—48no—
164C2×Q16—48no—
165C2×Q16—48no—
166C2×Q16—48no—
167C2×Q16—48no—
168C2×Q16—48no—
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—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8C22×C414yes—
183C4⋊Q8—48no—
184C4⋊Q8—48no—
185C4×Q8—48no—
186C4×Q8—48no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
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—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4#4112yes—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42Q1614yescomplemented
246C2×C42C22×C412yes—
247C2×C42C2×D412yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250#103—48no—
251#103—48no—
252#103—48no—
253#103—48no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—42no—
259#95—42no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—42no—
265#95—42no—
266#95—42no—
267C24C22×D411yescharacteristic
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
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—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8C4×D414yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4C4×D412yes—
385C22×C4C4×D412yes—
386C22×C4#12112yes—
387C22×C4#12112yes—
388C22×C4C2×Q1612yes—
389C22×C4C2×Q1612yes—
390C22×C4C2×Q1612yes—
391C22×C4#13512yes—
392C22×C4#13712yes—
393C22×C4#13712yes—
394C22×C4#13712yes—
395C22×C4#13712yes—
396C22×C4C23×C411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4H(5,𝔽2)12yes—
403C22×C4H(5,𝔽2)12yes—
404C22×C4H(5,𝔽2)12yes—
405C22×C4H(5,𝔽2)12yes—
406Q16—432no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C42—24no—
437C42—24no—
438C42—24no—
439C42—24no—
440C42—44no—
441C42—44no—
442C42—44no—
443C42—44no—
444C42—48no—
445C42—42no—
446C42—42no—
447C42—84no—
448C23C2×C4×D411yescharacteristic
449C23C2×#12111yescharacteristic
450C23#52511yescharacteristic
451C23#53411yescharacteristic
452C23#53511yescharacteristic
453C23#53511yescharacteristic
454C23#53511yescharacteristic
455C23C22×Q1611yescharacteristic
456C23C2×#13511yescharacteristic
457C23C2×#13711yescharacteristic
458C23C2×#13711yescharacteristic
459C23C2×#13711yescharacteristic
460C23C2×#13711yescharacteristic
461C23#57511yescharacteristic
462C23#57511yescharacteristic
463Q8—28no—
464Q8—44no—
465Q8—44no—
466Q8—44no—
467Q8—44no—
468Q8—44no—
469Q8—44no—
470Q8—44no—
471Q8—82no—
472Q8—82no—
473Q8—82no—
474Q8—82no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4C4×Q1614yes—
557C2×C4#44114yes—
558C2×C4#46214yes—
559C2×C4C2×C4×D411yescharacteristic
560C2×C4C4×#4111yescharacteristic
561C2×C4#51812yes—
562C2×C4#51812yes—
563C2×C4#52911yescharacteristic
564C2×C4#53211yescharacteristic
565C2×C4#53412yes—
566C2×C4#53512yes—
567C2×C4#53812yes—
568C2×C4#53812yes—
569C2×C4#53812yes—
570C2×C4#53812yes—
571C2×C4#53812yes—
572C2×C4#54012yes—
573C2×C4#54012yes—
574C2×C4#54012yes—
575C8—44no—
576C8—82no—
577C22C4×#12111yescharacteristic
578C22#232111yescharacteristic
579C22#232811yescharacteristic
580C22#261711yescharacteristic
581C22#261911yescharacteristic
582C22#263111yescharacteristic
583C22#263511yescharacteristic
584C22C2×C4×Q1611yescharacteristic
585C22C2×#44111yescharacteristic
586C22C4×#13711yescharacteristic
587C22C4×#13711yescharacteristic
588C22C2×#46211yescharacteristic
589C22#303511yescharacteristic
590C22#303711yescharacteristic
591C22#303911yescharacteristic
592C22#310411yescharacteristic
593C22#310711yescharacteristic
594C22#311311yescharacteristic
595C22#311611yescharacteristic
596C22#318911yescharacteristic
597C22#319011yescharacteristic
598C22#319511yescharacteristic
599C22#319611yescharacteristic
600C22#320911yescharacteristic
601C22#320911yescharacteristic
602C22#320911yescharacteristic
603C22#321011yescharacteristic
604C22#321011yescharacteristic
605C22#321111yescharacteristic
606C22#321511yescharacteristic
607C22#321511yescharacteristic
608C22#321611yescharacteristic
609C22#321811yescharacteristic
610C22#321811yescharacteristic
611C22#321911yescharacteristic
612C4—24no—
613C4—24no—
614C4—24no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—44no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—84no—
623C4—162no—
624C4#232112yes—
625C4#233312yes—
626C4#264812yes—
627C4#266112yes—
628C2#1431111yescharacteristic
629C2C4×#46211yescharacteristic
630C2#2150611yescharacteristic
631C2#2204411yescharacteristic
632C2#2205011yescharacteristic
633C2#2227011yescharacteristic
634C2#2227311yescharacteristic
635C2#2971511yescharacteristic
636C2#2971711yescharacteristic
637C2#2979911yescharacteristic
638C2#2980111yescharacteristic
639C2#3004211yescharacteristic
640C2#3004611yescharacteristic
641C2#3010911yescharacteristic
642C2#3011611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #2016C4semidirect
245. C2×C42Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#20162
C2×#18782
#147142
C4×#3912
#112672
#111902
#110502
#110242
#110102
#109322
#107952