Finite Group Databasebeta

← #366412

Subgroups — #366412

Order 512.

1208 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1940C211yesmaximal, characteristic, complemented
2C2×#1798C211yesmaximal, characteristic
3C2×#1770C211yesmaximal, characteristic, complemented
4C2×#1750C211yesmaximal, characteristic
5#13852C211yesmaximal, characteristic
6#11298C212yesmaximal, complemented
7#11042C212yesmaximal
8#10922C212yesmaximal
9#10877C212yesmaximal, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12C22×#117C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×#97C2211yescharacteristic
20C22×#97C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2212yes—
25#1940—24no—
26#1798—24no—
27#1770—24no—
28#1750—24no—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1441C412yes—
39#1441C411yescharacteristic
40#1441C411yescharacteristic
41#1439C411yescharacteristic
42#1439C412yes—
43#1439C411yescharacteristic
44C4×#95C2212yes—
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C42C2312yes—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—42no—
79C2×C4⋊C8D414yes—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8Q814yes—
89C2×#97—24no—
90C2×#97—22no—
91C2×#97—22no—
92C2×#97—22no—
93C2×#97—42no—
94C2×#97—42no—
95C2×C4×C8—22no—
96C2×C4×C8—22no—
97C2×C4×C8D414yes—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C22⋊C4—42no—
103C4×C22⋊C4—42no—
104C2×#95C2×C412yes—
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C412yes—
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113#336—24no—
114#336—24no—
115#336—22no—
116#336—22no—
117#336—24no—
118#336—24no—
119#336—42no—
120#336—42no—
121#336—42no—
122C25C22×C411yescharacteristic
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C412yes—
138C23×C4C22×C412yes—
139C23×C4C22×C412yes—
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C412yes—
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×Q812yes—
145C23×C4#4112yes—
146C23×C4C2411yescharacteristic
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8C2×D412yes—
183C22×C8C2×D414yes—
184C22×C8C2×D412yes—
185C22×C8C2×Q812yes—
186C22×C8C2×Q812yes—
187C22×C8#4112yes—
188C22×C8#4112yes—
189#117—44no—
190#117—44no—
191#117—44no—
192#117—44no—
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—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42#4112yes—
242C4⋊C8—28no—
243C4⋊C8—48no—
244C4⋊C8—44no—
245C22⋊C8—28no—
246C22⋊C8—28no—
247C22⋊C8—48no—
248#97—48no—
249#97—44no—
250C4×C8—28no—
251#95—22no—
252#95—22no—
253#95—24no—
254#95—24no—
255#95—42no—
256#95—42no—
257#95—42no—
258#95—42no—
259#95—42no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24C4×Q812yes—
271C24C23×C411yescharacteristic
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4#11712yes—
348C22×C4#11712yes—
349C22×C4C4×D412yes—
350C22×C4C4×D412yes—
351C22×C4C4×D412yes—
352C22×C4C4×D412yes—
353C22×C4C4×Q812yes—
354C22×C4#13114yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×Q811yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C42−1+411yescharacteristic
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—24no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—24no—
382C2×C8—22no—
383C2×C8—24no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8C22⋊Q814yes—
411C4⋊C4—48no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—42no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42#12114yes—
433C42#12314yes—
434C42#12414yes—
435C42#12614yes—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—22no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—42no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—41no—
459C23C2×#11711yescharacteristic
460C23C2×C4×D411yescharacteristic
461C23C2×C4×D411yescharacteristic
462C23C2×C4×Q811yescharacteristic
463C23C4×#4111yescharacteristic
464C23#51811yescharacteristic
465C23#52011yescharacteristic
466C23C2×#13112yes—
467C23#56812yes—
468C23#56812yes—
469C23#56812yes—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4#38814yes—
532C2×C4#38814yes—
533C2×C4C2×#12111yescharacteristic
534C2×C4C2×C22⋊Q811yescharacteristic
535C2×C4C2×#12311yescharacteristic
536C2×C4#52511yescharacteristic
537C2×C4C2×#12411yescharacteristic
538C2×C4C2×#12611yescharacteristic
539C2×C4#52911yescharacteristic
540C2×C4#53311yescharacteristic
541C2×C4#53511yescharacteristic
542C2×C4#53711yescharacteristic
543C2×C4#54011yescharacteristic
544C2×C4#54211yescharacteristic
545C2×C4#54611yescharacteristic
546C2×C4#54611yescharacteristic
547C2×C4D4×Q811yescharacteristic
548C2×C4#55011yescharacteristic
549C2×C4#55211yescharacteristic
550C2×C4#55311yescharacteristic
551C2×C4#55311yescharacteristic
552C2×C4#55311yescharacteristic
553C2×C4#55311yescharacteristic
554C2×C4#55311yescharacteristic
555C2×C4#55411yescharacteristic
556C2×C4#55411yescharacteristic
557C2×C4#55411yescharacteristic
558C2×C4#55511yescharacteristic
559C2×C4#55611yescharacteristic
560C2×C4#55611yescharacteristic
561C8—24no—
562C8—42no—
563C8—44no—
564C8—42no—
565C8—42no—
566C8—82no—
567C22—42no—
568C22—42no—
569C22—44no—
570C22—42no—
571C22—44no—
572C22—82no—
573C22—81no—
574C22—81no—
575C22—82no—
576C22—81no—
577C22—81no—
578C22C2×#38811yescharacteristic
579C22C4×C22⋊Q811yescharacteristic
580C22#232411yescharacteristic
581C22#234511yescharacteristic
582C22#234711yescharacteristic
583C22#236011yescharacteristic
584C22#236311yescharacteristic
585C22#292312yes—
586C22#292512yes—
587C22#293011yescharacteristic
588C22#293011yescharacteristic
589C22#293111yescharacteristic
590C22#293212yes—
591C22#293211yescharacteristic
592C22#293312yes—
593C22#296112yes—
594C22#296212yes—
595C22#296712yes—
596C22#296812yes—
597C22#297612yes—
598C22#297812yes—
599C22#299312yes—
600C22#299512yes—
601C4—24no—
602C4—24no—
603C4—44no—
604C4—42no—
605C4—82no—
606C4—81no—
607C4—81no—
608C4—82no—
609C4—82no—
610C4#248511yescharacteristic
611C4#248711yescharacteristic
612C4#249011yescharacteristic
613C4#249211yescharacteristic
614C4#249411yescharacteristic
615C4#249711yescharacteristic
616C4#253511yescharacteristic
617C4#253811yescharacteristic
618C2—82no—
619C2#1399311yescharacteristic
620C2#2104711yescharacteristic
621C2#2105111yescharacteristic
622C2#2160311yescharacteristic
623C2#2160411yescharacteristic
624C2#2213811yescharacteristic
625C2#2214011yescharacteristic
626C2#2263112yes—
627C2#2264512yes—
628C2#2296012yes—
629C2#2309112yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1940C2semidirect
3. C2×#1770C2semidirect
6. #11298C2semidirect
9. #10877C2semidirect
10. #10835C2semidirect
11. #10834C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19402
C2×#17982
C2×#17702
C2×#17502
#138522
#112982
#110422
#109222
#108772
#108352
#108342