Finite Group Databasebeta

← #385137

Subgroups — #385137

Order 512.

1536 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1954C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3C2×#1850C212yesmaximal, complemented
4C2×#1798C212yesmaximal, complemented
5#13839C211yesmaximal, characteristic
6#11322C212yesmaximal, complemented
7#11042C212yesmaximal, complemented
8#11042C212yesmaximal, complemented
9#10876C212yesmaximal, complemented
10C24×C8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1954—24no—
25#1869—24no—
26#1850—28no—
27#1798—28no—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1441C412yes—
35#1440C411yescharacteristic
36#1440C411yescharacteristic
37#1439C412yes—
38#1438C411yescharacteristic
39#1438C411yescharacteristic
40C4×#95C2212yes—
41C23×D4C2×C411yescharacteristic
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8—22no—
45C23×C8—24no—
46C23×C8—42no—
47C23×C8—41no—
48C23×C8—41no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4×D4—41no—
54C2×C4×D4—41no—
55C2×C4×D4—41no—
56C2×C4×D4—41no—
57C2×C4×D4D414yes—
58C2×C4×D4Q814yes—
59C2×#117—24no—
60C2×#117—24no—
61C2×#117—41no—
62C2×#117—41no—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75C22×C42C2312yes—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—42no—
83C2×C4⋊C8—42no—
84C2×C4⋊C8—41no—
85C2×C4⋊C8—41no—
86C2×C4⋊C8—42no—
87C2×C4⋊C8—42no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—42no—
97C2×C22⋊C8—42no—
98C2×C22⋊C8—41no—
99C2×C22⋊C8—41no—
100C2×C4×C8—22no—
101C2×C4×C8—41no—
102C2×C4×C8—41no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4—41no—
106C4×C22⋊C4—41no—
107C2×#95C2×C412yes—
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114#336—24no—
115#336—24no—
116#336—24no—
117#336—42no—
118#336—42no—
119#336D418yescomplemented
120C25C22×C411yescharacteristic
121C25C22×C411yescharacteristic
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—41no—
125C22×D4—41no—
126C22×D4C4⋊C414yes—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C22×C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C412yes—
148C23×C4C22×C412yes—
149C23×C4C22×C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×Q812yes—
154C23×C4C2×Q812yes—
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
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—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—44no—
183C22×C8—48no—
184C22×C8—48no—
185C22×C8—44no—
186C22×C8—44no—
187C22×C8C2×D414yes—
188C22×C8C2×D414yes—
189C4×D4—44no—
190C4×D4—44no—
191C4×D4—44no—
192C4×D4—44no—
193#117—44no—
194#117—48no—
195#117—44no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4C4⋊C414yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4C4⋊C414yes—
221C2×C22⋊C4C4⋊C414yes—
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—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42C2×D412yes—
237C2×C42C2×D414yes—
238C2×C42C2×Q812yes—
239C2×C42#4114yes—
240C4⋊C8—48no—
241C4⋊C8—48no—
242C4⋊C8—44no—
243C4⋊C8—48no—
244C22⋊C8—48no—
245C22⋊C8—48no—
246C22⋊C8—48no—
247C22⋊C8—44no—
248C4×C8—44no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—42no—
255#95—42no—
256C24—22no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—21no—
265C24—21no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24C2×C4⋊C412yes—
272C24C2×C4⋊C412yes—
273C24#13114yes—
274C24C23×C411yescharacteristic
275C2×D4—28no—
276C2×D4—28no—
277C2×D4—44no—
278C2×D4—48no—
279C2×D4—48no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
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—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4C2×C4⋊C412yes—
357C22×C4C2×C4⋊C412yes—
358C22×C4C2×C4⋊C412yes—
359C22×C4C2×C4⋊C412yes—
360C22×C4C4×D414yes—
361C22×C4C4×D414yes—
362C22×C4#13114yes—
363C22×C4#13114yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D412yes—
366C22×C4C22×Q811yescharacteristic
367C22×C4C2×#4112yes—
368C22×C4H(5,𝔽2)11yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C4H(5,𝔽2)12yes—
373C22×C42−1+412yes—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—416no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C4⋊C4—44no—
404C4⋊C4—48no—
405C22⋊C4—216no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—48no—
412C22⋊C4—44no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—24no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—42no—
425C23—22no—
426C23—24no—
427C23—22no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—24no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—42no—
444C23—42no—
445C23—41no—
446C23—41no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23#42414yes—
467C23C22×C4⋊C411yescharacteristic
468C23C2×C4×D412yes—
469C23#51811yescharacteristic
470C23#51811yescharacteristic
471C23#52012yes—
472C23C2×#13112yes—
473C23C2×#13112yes—
474C23C2×#13112yes—
475C23#56812yes—
476D4—416no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4#42414yes—
537C2×C4#53411yescharacteristic
538C2×C4#53411yescharacteristic
539C2×C4#53512yes—
540C2×C4#53712yes—
541C2×C4#54012yes—
542C2×C4#54212yes—
543C2×C4#54311yescharacteristic
544C2×C4#54311yescharacteristic
545C2×C4D4212yes—
546C2×C4D4211yescharacteristic
547C2×C4#54612yes—
548C2×C4#54712yes—
549C2×C4D4×Q812yes—
550C2×C4#55111yescharacteristic
551C2×C4#55412yes—
552C2×C4#56011yescharacteristic
553C2×C4#56212yes—
554C2×C4#56311yescharacteristic
555C8—44no—
556C8—44no—
557C8—82no—
558C8—82no—
559C22—28no—
560C22—24no—
561C22—44no—
562C22—44no—
563C22—42no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—42no—
568C22—42no—
569C22—44no—
570C22—44no—
571C22—42no—
572C22—42no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22#231511yescharacteristic
578C22#232412yes—
579C22D4×C4⋊C412yes—
580C22#235811yescharacteristic
581C22#237611yescharacteristic
582C22C2×#42411yescharacteristic
583C22C2×#42411yescharacteristic
584C22#290612yes—
585C22#292912yes—
586C22#293112yes—
587C22#293312yes—
588C22#293512yes—
589C22#297712yes—
590C22#297711yescharacteristic
591C22#297712yes—
592C22#297811yescharacteristic
593C22#298112yes—
594C22#299212yes—
595C22#299312yes—
596C22#299412yes—
597C22#299412yes—
598C4—24no—
599C4—44no—
600C4—44no—
601C4—42no—
602C4—42no—
603C4—44no—
604C4—42no—
605C4—82no—
606C4#266911yescharacteristic
607C4#267112yes—
608C4#268311yescharacteristic
609C4#269212yes—
610C4#271711yescharacteristic
611C4#274811yescharacteristic
612C2—44no—
613C2—82no—
614C2#1440111yescharacteristic
615C2#2193111yescharacteristic
616C2#2193711yescharacteristic
617C2#2213412yes—
618C2#2214412yes—
619C2#2264712yes—
620C2#2265112yes—
621C2#2284712yes—
622C2#2331912yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×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
3. C2×#1850C2semidirect
4. C2×#1798C2semidirect
6. #11322C2semidirect
7. #11042C2semidirect
8. #11042C2semidirect
9. #10876C2semidirect
119. #336D4semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#18692
C2×#18502
C2×#17982
#138392
#113222
#110422
#108762