Finite Group Databasebeta

← #471514

Subgroups — #471514

Order 512.

1246 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2072C211yesmaximal, characteristic
2C2×#2016C211yesmaximal, characteristic
3#14714C211yesmaximal, characteristic
4#14703C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11190C212yesmaximal
7#11067C212yesmaximal
8#11024C212yesmaximal
9#11024C212yesmaximal
10#11010C211yesmaximal, characteristic
11#10795C211yesmaximal, characteristic
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#389C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×C43C2211yescharacteristic
26#2072—24no—
27#2016—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1436—21no—
36#1436—21no—
37C4⋊#95—21no—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43#1434—21no—
44#1434—21no—
45C23×Q8C2311yescharacteristic
46C23×Q8C2312yes—
47C22×Q16—24no—
48C22×Q16—24no—
49C22×Q16—24no—
50C22×Q16—42no—
51C22×Q16—42no—
52C22×Q16—42no—
53C22×Q16—42no—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88#391—22no—
89#391—42no—
90#391—42no—
91#391—42no—
92#391—42no—
93#391—42no—
94#391—42no—
95#391D414yes—
96#389—42no—
97#384—22no—
98#384—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C43—42no—
115#340—24no—
116#340—42no—
117#340—42no—
118#336—24no—
119#336—22no—
120#336—22no—
121#336—42no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8C2×D412yes—
145C22×Q8#4114yes—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C2×Q16—48no—
160C2×Q16—48no—
161C2×Q16—48no—
162C2×Q16—48no—
163C2×Q16—48no—
164C2×Q16—48no—
165C2×Q16—48no—
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×C8—22no—
178C22×C8—22no—
179C22×C8#4114yes—
180C4⋊Q8—48no—
181C4⋊Q8—48no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
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—21no—
200C2×C4⋊C4—21no—
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⋊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⋊C4C2×D412yes—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—41no—
231C2×C42—41no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42Q1614yescomplemented
237C2×C42C2×D412yes—
238C2×C42#4112yes—
239#103—48no—
240#103—48no—
241#103—48no—
242#103—48no—
243#95—24no—
244#95—24no—
245#95—42no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95—42no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—42no—
255#95—42no—
256#95—42no—
257#95—42no—
258C24C22×D411yescharacteristic
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
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—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
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—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4#12112yes—
378C22×C4#12112yes—
379C22×C4C2×Q1612yes—
380C22×C4C2×Q1612yes—
381C22×C4C2×Q1612yes—
382C22×C4#13512yes—
383C22×C4#13712yes—
384C22×C4#13712yes—
385C22×C4#13712yes—
386C22×C4#13712yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C2×#4112yes—
389C22×C4C2×#4112yes—
390C22×C4C2×#4111yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C4H(5,𝔽2)12yes—
396C22×C4H(5,𝔽2)12yes—
397Q16—432no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C4⋊C4—24no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C4⋊C4—84no—
427C42—24no—
428C42—24no—
429C42—44no—
430C42—44no—
431C42—44no—
432C42—44no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42—42no—
437C23C2×#12111yescharacteristic
438C23#52511yescharacteristic
439C23#53211yescharacteristic
440C23#53411yescharacteristic
441C23#53511yescharacteristic
442C23#53511yescharacteristic
443C23#53511yescharacteristic
444C23C22×Q1611yescharacteristic
445C23C2×#13511yescharacteristic
446C23C2×#13711yescharacteristic
447C23C2×#13711yescharacteristic
448C23C2×#13711yescharacteristic
449C23C2×#13711yescharacteristic
450C23#57511yescharacteristic
451C23#57511yescharacteristic
452Q8—44no—
453Q8—44no—
454Q8—44no—
455Q8—44no—
456Q8—44no—
457Q8—44no—
458Q8—82no—
459Q8—82no—
460Q8—82no—
461Q8—82no—
462Q8—82no—
463Q8—82no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
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—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—81no—
546C2×C4#46214yes—
547C2×C4#46414yes—
548C2×C4#53511yescharacteristic
549C2×C4#53511yescharacteristic
550C2×C4#53812yes—
551C2×C4#53812yes—
552C2×C4#53812yes—
553C2×C4#54012yes—
554C2×C4#54011yescharacteristic
555C2×C4#54011yescharacteristic
556C2×C4#54612yes—
557C2×C4#54612yes—
558C2×C4#55112yes—
559C2×C4#55112yes—
560C2×C4#55312yes—
561C2×C4#55912yes—
562C2×C4#56012yes—
563C2×C4#56012yes—
564C8—44no—
565C8—82no—
566C22#253411yescharacteristic
567C22#254411yescharacteristic
568C22#260311yescharacteristic
569C22#261911yescharacteristic
570C22#263111yescharacteristic
571C22#266111yescharacteristic
572C22#281511yescharacteristic
573C22C2×#46211yescharacteristic
574C22C2×#46411yescharacteristic
575C22#303911yescharacteristic
576C22#304011yescharacteristic
577C22#310411yescharacteristic
578C22#310811yescharacteristic
579C22#311311yescharacteristic
580C22#311711yescharacteristic
581C22#317011yescharacteristic
582C22#317211yescharacteristic
583C22#318111yescharacteristic
584C22#318511yescharacteristic
585C22#318911yescharacteristic
586C22#319011yescharacteristic
587C22#319511yescharacteristic
588C22#319611yescharacteristic
589C22#320911yescharacteristic
590C22#320911yescharacteristic
591C22#320911yescharacteristic
592C22#321011yescharacteristic
593C22#321211yescharacteristic
594C22#321211yescharacteristic
595C22#321511yescharacteristic
596C22#321611yescharacteristic
597C22#321611yescharacteristic
598C22#321611yescharacteristic
599C22#321811yescharacteristic
600C22#321811yescharacteristic
601C4—24no—
602C4—24no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—84no—
608C4—82no—
609C4—82no—
610C4—161no—
611C4—162no—
612C4—161no—
613C4#268512yes—
614C4#270312yes—
615C4#275212yes—
616C4#279712yes—
617C2#1679911yescharacteristic
618C2#2812111yescharacteristic
619C2#2812411yescharacteristic
620C2#2861611yescharacteristic
621C2#2862011yescharacteristic
622C2#2949511yescharacteristic
623C2#2950111yescharacteristic
624C2#2979711yescharacteristic
625C2#2979911yescharacteristic
626C2#3004211yescharacteristic
627C2#3004311yescharacteristic
628C2#3031811yescharacteristic
629C2#3032611yescharacteristic
630C2#3276111yescharacteristic
631C2#3276211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
236. C2×C42Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#20722
C2×#20162
#147142
#147032
#113572
#111902
#110672
#110242
#110102
#107952