Finite Group Databasebeta

← #851819

Subgroups — #851819

Order 512.

3158 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2454C212yesmaximal, complemented
2C2×#2445C212yesmaximal
3#16467C214yesmaximal, complemented
4#15384C214yesmaximal, complemented
5#15303C214yesmaximal, complemented
6#15218C211yesmaximal, characteristic, complemented
7#15194C212yesmaximal, complemented
8#15147C212yesmaximal, complemented
9#15127C211yesmaximal, characteristic
10#14700C212yesmaximal, complemented
11#14359C214yesmaximal, complemented
12#14018C212yesmaximal, complemented
13C4×#384C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×#123C2214yes—
17C22×C4×D4C2212yes—
18#2454—216no—
19#2445—216no—
20C2×#399C2214yes—
21C2×#397C2214yes—
22C2×#396C2214yes—
23C2×#394C2214yes—
24C2×#394C2214yes—
25C2×#394C2214yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#386C2214yes—
37C2×#385C2214yes—
38C2×#385C2214yes—
39C2×#384C2212yes—
40C2×#380C2214yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C43C2211yescharacteristic
46#1445C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C22⋊#95C2214yes—
69C23×D4C2312yes—
70C23×D4C2314yes—
71C24×C4C2314yes—
72C2×#127—28no—
73C2×#127—42no—
74C2×#125—28no—
75C2×#125—42no—
76C2×#123—216no—
77C2×#123—216no—
78C2×#123—48no—
79C2×C4×D4—28no—
80C2×C4×D4—28no—
81C2×C4×D4—44no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#399—216no—
115#397—48no—
116#396—216no—
117#394—216no—
118#394—216no—
119#394—48no—
120#390—28no—
121#390—28no—
122#390—28no—
123#390—28no—
124#390D4116yes—
125#389—28no—
126#389—28no—
127#389—28no—
128#388—48no—
129#388—48no—
130#386—216no—
131#385—216no—
132#385—48no—
133#384—28no—
134#380—216no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—42no—
138C4×C4⋊C4—42no—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2311yescharacteristic
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C43—24no—
158C25C2414yes—
159C22×D4—28no—
160C22×D4—28no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—28no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4C2×D418yes—
173C23×C4—28no—
174C23×C4—28no—
175C23×C4—28no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2414yes—
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192#127—416no—
193#125—416no—
194#123—464no—
195C4×D4—432no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D418yes—
234C2×C4⋊C4#4118yes—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4C2×D4116yes—
267C2×C22⋊C4C2×D4116yes—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—42no—
286C2×C42C2×D418yes—
287C2×C42#4114yes—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—41no—
298#95—42no—
299#95—41no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24C22×D418yes—
321C24C2511yescharacteristic
322C2×D4—216no—
323C2×D4—216no—
324C2×D4—216no—
325C2×D4—416no—
326C2×D4—416no—
327C2×D4—416no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—88no—
333C2×D4—88no—
334C2×D4—88no—
335C2×D4—88no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—416no—
400C22×C4—416no—
401C22×C4—416no—
402C22×C4—88no—
403C22×C4—88no—
404C22×C4—88no—
405C22×C4—88no—
406C22×C4C22×D414yes—
407C22×C4C22×D418yes—
408C22×C4C22×D414yes—
409C22×C4C22×D414yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4114yes—
412C22×C4C2×#4114yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4114yes—
415C22×C4C2×#4112yes—
416C22×C4H(5,𝔽2)14yes—
417C22×C4H(5,𝔽2)14yes—
418C22×C4H(5,𝔽2)12yes—
419C22×C42−1+412yes—
420C22×C42−1+414yes—
421C4⋊C4—216no—
422C4⋊C4—216no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—416no—
428C4⋊C4—416no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—88no—
433C4⋊C4—88no—
434C22⋊C4—232no—
435C22⋊C4—416no—
436C22⋊C4—416no—
437C22⋊C4—416no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C42—28no—
446C42—28no—
447C42—216no—
448C42—48no—
449C42—44no—
450C42—48no—
451C42—44no—
452C42—44no—
453C23—28no—
454C23—28no—
455C23—28no—
456C23—28no—
457C23—28no—
458C23—28no—
459C23—28no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23C23×D412yes—
489C23C22×#4111yescharacteristic
490C23C22×#4112yes—
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×H(5,𝔽2)12yes—
493C23C2×H(5,𝔽2)12yes—
494C23C2×2−1+411yescharacteristic
495C23C2×2−1+412yes—
496C23#58512yes—
497D4—432no—
498D4—832no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—832no—
571C2×C4#53618yes—
572C2×C4#53914yes—
573C2×C4#54014yes—
574C2×C4D4214yes—
575C2×C4#54618yes—
576C2×C4#54718yes—
577C2×C4#56314yes—
578C22—48no—
579C22—48no—
580C22—48no—
581C22—48no—
582C22—48no—
583C22—48no—
584C22—48no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—84no—
593C22#344411yescharacteristic
594C22C2×#53612yes—
595C22C2×#53911yescharacteristic
596C22C2×#54011yescharacteristic
597C22C2×D4211yescharacteristic
598C22C2×#54612yes—
599C22C2×#54712yes—
600C22D4×#4112yes—
601C22#349412yes—
602C22#349612yes—
603C22#349812yes—
604C22#350712yes—
605C22#351912yes—
606C22#352212yes—
607C22#352212yes—
608C22C2×#56311yescharacteristic
609C22#354612yes—
610C22#354812yes—
611C22#354811yescharacteristic
612C22#355412yes—
613C22#355711yescharacteristic
614C4—24no—
615C4—28no—
616C4—48no—
617C4—44no—
618C4—44no—
619C4—42no—
620C4—44no—
621C4—82no—
622C4—82no—
623C4—84no—
624C4—84no—
625C4#268018yes—
626C2—88no—
627C2#3508211yescharacteristic
628C2#3613912yes—
629C2#3728611yescharacteristic
630C2#3895212yes—
631C2#3935212yes—
632C2#3994912yes—
633C2#4634012yes—
634C2C2×#268011yescharacteristic
635C2#5941212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2454C2semidirect
3. #16467C2semidirect
4. #15384C2semidirect
5. #15303C2semidirect
6. #15218C2semidirect
7. #15194C2semidirect
8. #15147C2semidirect
10. #14700C2semidirect
11. #14359C2semidirect
12. #14018C2semidirect
13. C4×#384C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24542
C2×#24452
#164672
#153842
#153032
#152182
#151942
#151472
#151272
#147002
#143592
#140182
C4×#3842