Finite Group Databasebeta

← #388122

Subgroups — #388122

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13839C211yesmaximal, characteristic
2#11298C212yesmaximal, complemented
3#11147C212yesmaximal, complemented
4#10964C211yesmaximal, characteristic
5#10964C211yesmaximal, characteristic
6#10922C212yesmaximal, complemented
7#10922C212yesmaximal, complemented
8#10798C212yesmaximal, complemented
9#10713C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C22×#97C2212yes—
18C2×C4×C22⋊C4C2212yes—
19C2×C4×C22⋊C4C2211yescharacteristic
20C2×#336C2212yes—
21C2×#336C2211yescharacteristic
22C2×#336C2212yes—
23C2×#336C2212yes—
24C2×#336C2212yes—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2211yescharacteristic
28C2×#336C2212yes—
29C2×#336C2212yes—
30#1441C411yescharacteristic
31#1441C411yescharacteristic
32#1440C412yes—
33#1439C411yescharacteristic
34#1439C411yescharacteristic
35#1438C412yes—
36C4×#95C2212yes—
37C23×D4C2×C411yescharacteristic
38C24×C4C2311yescharacteristic
39C24×C4C2311yescharacteristic
40C23×C8C2312yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×C4×D4—24no—
45C2×C4×D4—41no—
46C2×C4×D4—41no—
47C2×C4×D4—42no—
48C2×#117—41no—
49C2×#117—41no—
50C2×#117—42no—
51C2×#117—42no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2×C412yes—
61C22×C42C2312yes—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64C2×C22⋊C8—24no—
65C2×C22⋊C8—24no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—41no—
71C2×C22⋊C8—42no—
72C2×C22⋊C8—42no—
73C2×C22⋊C8—42no—
74C2×C22⋊C8—42no—
75C2×C22⋊C8—41no—
76C2×C22⋊C8—42no—
77C2×#97—24no—
78C2×#97—42no—
79C2×#97—42no—
80C4×C22⋊C4—42no—
81C4×C22⋊C4—42no—
82C4×C22⋊C4—42no—
83C2×#95C2×C412yes—
84C2×#95C2×C412yes—
85C2×#95C2×C412yes—
86C2×#95C2×C412yes—
87C2×#95C2×C411yescharacteristic
88C2×#95C2×C411yescharacteristic
89C2×#95C2×C412yes—
90C2×#95C2×C412yes—
91#336—24no—
92#336—42no—
93#336—42no—
94#336—42no—
95#336—42no—
96#336—41no—
97#336—42no—
98#336—41no—
99#336—42no—
100#336—42no—
101C25C22×C411yescharacteristic
102C25C22×C411yescharacteristic
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—41no—
107C22×D4—41no—
108C22×D4—41no—
109C22×D4—41no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C22×C412yes—
126C23×C4C22×C412yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C412yes—
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4#4114yes—
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8#4114yes—
162C22×C8#4114yes—
163C4×D4—48no—
164C4×D4—48no—
165#117—48no—
166#117—44no—
167#117—44no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C22⋊C8—44no—
210C22⋊C8—48no—
211C22⋊C8—48no—
212C22⋊C8—48no—
213C22⋊C8—44no—
214#97—48no—
215#95—24no—
216#95—42no—
217#95—42no—
218#95—42no—
219#95—41no—
220#95—41no—
221#95—42no—
222#95—42no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—21no—
227C24—21no—
228C24—22no—
229C24—22no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—21no—
241C24—21no—
242C24—21no—
243C24#11712yes—
244C24#13112yes—
245C24C23×C411yescharacteristic
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—44no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
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—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4#11712yes—
334C22×C4#13112yes—
335C22×C4C2×#4112yes—
336C22×C4C2×#4112yes—
337C22×C4C2×#4112yes—
338C22×C4H(5,𝔽2)12yes—
339C22×C4H(5,𝔽2)12yes—
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C4H(5,𝔽2)12yes—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—48no—
376C22⋊C4—44no—
377C22⋊C4—48no—
378C22⋊C4—48no—
379C22⋊C4—48no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—42no—
390C23—22no—
391C23—24no—
392C23—24no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—24no—
397C23—24no—
398C23—22no—
399C23—22no—
400C23—41no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—42no—
409C23—41no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—42no—
421C23—41no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—42no—
430C23—42no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23C2×#11711yescharacteristic
437C23C4×#4112yes—
438C23#51811yescharacteristic
439C23#51811yescharacteristic
440C23#51812yes—
441C23C2×#13111yescharacteristic
442C23C2×#13111yescharacteristic
443C23#56812yes—
444C23#56812yes—
445C23#56811yescharacteristic
446C23#56811yescharacteristic
447D4—416no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4#53812yes—
507C2×C4#53812yes—
508C2×C4#53812yes—
509C2×C4#53812yes—
510C2×C4#54012yes—
511C2×C4#54012yes—
512C2×C4#55111yescharacteristic
513C2×C4#55112yes—
514C2×C4#55112yes—
515C2×C4#55312yes—
516C2×C4#55312yes—
517C2×C4#55312yes—
518C2×C4#55911yescharacteristic
519C2×C4#56012yes—
520C2×C4#56011yescharacteristic
521C2×C4H(3,𝔽4)11yescharacteristic
522C8—44no—
523C8—44no—
524C8—82no—
525C8—82no—
526C22—42no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—42no—
531C22—44no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—81no—
537C22—82no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—81no—
547C22—81no—
548C22#231211yescharacteristic
549C22#233312yes—
550C22#235312yes—
551C22#237411yescharacteristic
552C22#237511yescharacteristic
553C22#292312yes—
554C22#292411yescharacteristic
555C22#292411yescharacteristic
556C22#296712yes—
557C22#296712yes—
558C22#296812yes—
559C22#296912yes—
560C22#297611yescharacteristic
561C22#297611yescharacteristic
562C22#297612yes—
563C22#297711yescharacteristic
564C22#297812yes—
565C22#297812yes—
566C22#297811yescharacteristic
567C22#298011yescharacteristic
568C22#298012yes—
569C22#298011yescharacteristic
570C22#298112yes—
571C4—42no—
572C4—42no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—81no—
578C4—81no—
579C4#274912yes—
580C4#275312yes—
581C4#277812yes—
582C4#279712yes—
583C2—82no—
584C2—82no—
585C2#1451611yescharacteristic
586C2#2190711yescharacteristic
587C2#2191011yescharacteristic
588C2#2233612yes—
589C2#2233812yes—
590C2#2279012yes—
591C2#2279412yes—
592C2#2329711yescharacteristic
593C2#2330311yescharacteristic
594C2#2331111yescharacteristic
595C2#2331311yescharacteristic
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
2. #11298C2semidirect
3. #11147C2semidirect
6. #10922C2semidirect
7. #10922C2semidirect
8. #10798C2semidirect
9. #10713C2semidirect

Maximal subgroups

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

subgroupindex
#138392
#112982
#111472
#109642
#109222
#107982
#107132