Finite Group Databasebeta

← #391971

Subgroups — #391971

Order 512.

1174 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13847C211yesmaximal, characteristic
2#11202C211yesmaximal, characteristic, complemented
3#11202C211yesmaximal, characteristic, complemented
4#11147C212yesmaximal, complemented
5#11147C212yesmaximal, complemented
6#11042C212yesmaximal, complemented
7#11042C212yesmaximal, complemented
8#10964C212yesmaximal
9#10796C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C22⋊C4C2212yes—
19C2×C4×C22⋊C4C2212yes—
20C2×#376C2211yescharacteristic
21C2×#336C2212yes—
22C2×#336C2212yes—
23C2×#336C2212yes—
24C2×#336C2212yes—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31#1445C411yescharacteristic
32#1445C411yescharacteristic
33#1439C412yes—
34#1439C412yes—
35#1438C411yescharacteristic
36#1438C411yescharacteristic
37C23×D4C2×C411yescharacteristic
38C24×C4C2312yes—
39C23×C8C2312yes—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—41no—
46C2×C4×D4—41no—
47C2×C4×D4—41no—
48C2×C4×D4—41no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C412yes—
53C22×C22⋊C4C2×C412yes—
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C412yes—
56C22×C22⋊C4C2×C412yes—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61C2×C4⋊C8—42no—
62C2×C4⋊C8—42no—
63C2×C4⋊C8—41no—
64C2×C4⋊C8—41no—
65C2×C22⋊C8—42no—
66C2×C22⋊C8—42no—
67C2×C22⋊C8—42no—
68C2×C22⋊C8—42no—
69C2×C22⋊C8—42no—
70C2×C22⋊C8—42no—
71C2×C22⋊C8—42no—
72C2×C22⋊C8—42no—
73C2×C22⋊C8—42no—
74C2×C22⋊C8—42no—
75C2×C22⋊C8—42no—
76C2×C22⋊C8—42no—
77C4×C4⋊C4—41no—
78C4×C4⋊C4—41no—
79C4×C22⋊C4—42no—
80C4×C22⋊C4—42no—
81C4×C22⋊C4—42no—
82C4×C22⋊C4—42no—
83#376—41no—
84#376—41no—
85C2×#95C2×C411yescharacteristic
86C2×#95C2×C411yescharacteristic
87C2×#95C2×C411yescharacteristic
88C2×#95C2×C411yescharacteristic
89C2×#95C2×C412yes—
90C2×#95C2×C412yes—
91C2×#95C2×C412yes—
92C2×#95C2×C412yes—
93C2×#95C2×C412yes—
94#336—24no—
95#336—24no—
96#336—42no—
97#336—42no—
98#336—42no—
99#336—42no—
100#336—42no—
101#336—42no—
102#336—41no—
103#336—41no—
104C25C22×C412yes—
105C22×D4—24no—
106C22×D4—22no—
107C22×D4—41no—
108C22×D4—41no—
109C22×D4—42no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C412yes—
127C23×C4C22×C412yes—
128C23×C4C22×C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—22no—
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—22no—
162C4×D4—44no—
163C4×D4—44no—
164C4×D4—44no—
165C4×D4—44no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
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—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42#4112yes—
209C2×C42#4112yes—
210C4⋊C8—44no—
211C4⋊C8—44no—
212C22⋊C8—48no—
213C22⋊C8—48no—
214C22⋊C8—48no—
215C22⋊C8—48no—
216#95—42no—
217#95—41no—
218#95—41no—
219#95—42no—
220#95—42no—
221#95—42no—
222#95—42no—
223#95—41no—
224#95—41no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24#13114yes—
240C24C23×C411yescharacteristic
241C2×D4—48no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—48no—
245C2×D4—44no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4#11712yes—
324C22×C4#11712yes—
325C22×C4#13112yes—
326C22×C4#13112yes—
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331C22×C4H(5,𝔽2)11yescharacteristic
332C22×C4H(5,𝔽2)11yescharacteristic
333C22×C4H(5,𝔽2)12yes—
334C22×C4H(5,𝔽2)12yes—
335C22×C4H(5,𝔽2)12yes—
336C22×C42−1+412yes—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
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—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C22⋊C4—48no—
369C22⋊C4—48no—
370C22⋊C4—48no—
371C22⋊C4—48no—
372C22⋊C4—48no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—42no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—24no—
392C23—24no—
393C23—24no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23C2×#11711yescharacteristic
423C23#51811yescharacteristic
424C23#51811yescharacteristic
425C23#51812yes—
426C23#52012yes—
427C23C2×#13112yes—
428C23C2×#13111yescharacteristic
429C23C2×#13111yescharacteristic
430C23#56812yes—
431C23#56811yescharacteristic
432C23#56811yescharacteristic
433D4—416no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4#53811yescharacteristic
496C2×C4#53811yescharacteristic
497C2×C4#53811yescharacteristic
498C2×C4#53811yescharacteristic
499C2×C4#54012yes—
500C2×C4#54012yes—
501C2×C4#54212yes—
502C2×C4#54212yes—
503C2×C4#56012yes—
504C2×C4#56012yes—
505C2×C4#56011yescharacteristic
506C2×C4#56011yescharacteristic
507C2×C4#56012yes—
508C2×C4H(3,𝔽4)11yescharacteristic
509C2×C4#56212yes—
510C2×C4#56212yes—
511C2×C4#56212yes—
512C2×C4#56311yescharacteristic
513C8—82no—
514C8—82no—
515C8—82no—
516C8—82no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22#231211yescharacteristic
533C22#231912yes—
534C22#237712yes—
535C22#237911yescharacteristic
536C22#238011yescharacteristic
537C22#292412yes—
538C22#292511yescharacteristic
539C22#292511yescharacteristic
540C22#297612yes—
541C22#297612yes—
542C22#297612yes—
543C22#297611yescharacteristic
544C22#297611yescharacteristic
545C22#297712yes—
546C22#297711yescharacteristic
547C22#297812yes—
548C22#297811yescharacteristic
549C22#297911yescharacteristic
550C22#298011yescharacteristic
551C22#299212yes—
552C22#299312yes—
553C22#299412yes—
554C22#299512yes—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—82no—
561C4—81no—
562C4—81no—
563C4—82no—
564C4#282811yescharacteristic
565C4#282811yescharacteristic
566C4#282912yes—
567C4#282912yes—
568C4#283011yescharacteristic
569C4#283011yescharacteristic
570C2—84no—
571C2#1466711yescharacteristic
572C2#2190811yescharacteristic
573C2#2190911yescharacteristic
574C2#2201612yes—
575C2#2201712yes—
576C2#2332812yes—
577C2#2333512yes—
578C2#2335011yescharacteristic
579C2#2335111yescharacteristic
580C2#2335811yescharacteristic
581C2#2335911yescharacteristic
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. #11202C2semidirect
3. #11202C2semidirect
4. #11147C2semidirect
5. #11147C2semidirect
6. #11042C2semidirect
7. #11042C2semidirect
9. #10796C2semidirect

Maximal subgroups

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

subgroupindex
#138472
#112022
#111472
#110422
#109642
#107962