Finite Group Databasebeta

← #391972

Subgroups — #391972

Order 512.

1174 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13847C211yesmaximal, characteristic
2#11202C212yesmaximal, complemented
3#11147C212yesmaximal, complemented
4#11147C212yesmaximal, complemented
5#11042C212yesmaximal, complemented
6#11042C212yesmaximal, complemented
7#10964C211yesmaximal, characteristic
8#10964C211yesmaximal, characteristic
9#10796C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11C22×C4⋊C8C2212yes—
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C22⋊C4C2212yes—
20C2×C4×C22⋊C4C2212yes—
21C2×#376C2211yescharacteristic
22C2×#336C2212yes—
23C2×#336C2212yes—
24C2×#336C2212yes—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31#1445C412yes—
32#1439C411yescharacteristic
33#1439C411yescharacteristic
34#1439C411yescharacteristic
35#1439C411yescharacteristic
36#1438C412yes—
37C23×D4C2×C411yescharacteristic
38C24×C4C2311yescharacteristic
39C24×C4C2311yescharacteristic
40C23×C8C2312yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×C4×D4—24no—
45C2×C4×D4—42no—
46C2×C4×D4—41no—
47C2×C4×D4—41no—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2×C412yes—
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C412yes—
56C22×C22⋊C4C2×C412yes—
57C22×C42C2312yes—
58C22×C42C2312yes—
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—42no—
61C2×C4⋊C8—42no—
62C2×C4⋊C8—42no—
63C2×C22⋊C8—42no—
64C2×C22⋊C8—42no—
65C2×C22⋊C8—41no—
66C2×C22⋊C8—41no—
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—41no—
76C2×C22⋊C8—41no—
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—42no—
84C2×#95C2×C412yes—
85C2×#95C2×C412yes—
86C2×#95C2×C411yescharacteristic
87C2×#95C2×C411yescharacteristic
88C2×#95C2×C412yes—
89C2×#95C2×C412yes—
90C2×#95C2×C412yes—
91C2×#95C2×C412yes—
92#336—24no—
93#336—24no—
94#336—42no—
95#336—42no—
96#336—42no—
97#336—42no—
98#336—42no—
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—22no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4C22×C412yes—
127C23×C4C22×C412yes—
128C23×C4C22×C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C412yes—
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—22no—
162C22×C8—22no—
163C4×D4—48no—
164C4×D4—48no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—21no—
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—22no—
204C2×C42—22no—
205C2×C42#4114yes—
206C4⋊C8—48no—
207C22⋊C8—48no—
208C22⋊C8—48no—
209C22⋊C8—44no—
210C22⋊C8—44no—
211C22⋊C8—44no—
212C22⋊C8—44no—
213#95—41no—
214#95—41no—
215#95—42no—
216#95—42no—
217#95—42no—
218#95—42no—
219#95—42no—
220#95—42no—
221C24—21no—
222C24—21no—
223C24—22no—
224C24—22no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—22no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—22no—
238C24—22no—
239C24—21no—
240C24—21no—
241C24#13112yes—
242C24#13112yes—
243C24C23×C411yescharacteristic
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4—44no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
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—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—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—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4#11712yes—
331C22×C4#11712yes—
332C22×C4#13114yes—
333C22×C4C2×#4112yes—
334C22×C4H(5,𝔽2)11yescharacteristic
335C22×C4H(5,𝔽2)12yes—
336C22×C4H(5,𝔽2)11yescharacteristic
337C22×C4H(5,𝔽2)12yes—
338C22×C4H(5,𝔽2)12yes—
339C22×C4H(5,𝔽2)12yes—
340C22×C42−1+412yes—
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—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C22⋊C4—48no—
373C22⋊C4—48no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—48no—
377C22⋊C4—48no—
378C42—24no—
379C42—24no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—42no—
388C23—22no—
389C23—22no—
390C23—24no—
391C23—22no—
392C23—22no—
393C23—24no—
394C23—22no—
395C23—22no—
396C23—24no—
397C23—24no—
398C23—41no—
399C23—41no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—41no—
414C23—41no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—41no—
419C23—41no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23C2×#11711yescharacteristic
435C23#51811yescharacteristic
436C23#51811yescharacteristic
437C23#51812yes—
438C23#52012yes—
439C23C2×#13111yescharacteristic
440C23C2×#13111yescharacteristic
441C23C2×#13112yes—
442C23#56811yescharacteristic
443C23#56811yescharacteristic
444C23#56812yes—
445D4—416no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
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—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4#53812yes—
505C2×C4#53812yes—
506C2×C4#54012yes—
507C2×C4#54012yes—
508C2×C4#54212yes—
509C2×C4#54212yes—
510C2×C4#56012yes—
511C2×C4#56012yes—
512C2×C4#56012yes—
513C2×C4#56011yescharacteristic
514C2×C4#56011yescharacteristic
515C2×C4H(3,𝔽4)11yescharacteristic
516C2×C4#56212yes—
517C2×C4#56212yes—
518C2×C4#56212yes—
519C2×C4#56311yescharacteristic
520C8—82no—
521C8—82no—
522C8—82no—
523C8—82no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22#231211yescharacteristic
547C22#231912yes—
548C22#237712yes—
549C22#237911yescharacteristic
550C22#238011yescharacteristic
551C22#292411yescharacteristic
552C22#292411yescharacteristic
553C22#292512yes—
554C22#297612yes—
555C22#297612yes—
556C22#297611yescharacteristic
557C22#297611yescharacteristic
558C22#297712yes—
559C22#297711yescharacteristic
560C22#297711yescharacteristic
561C22#297812yes—
562C22#297811yescharacteristic
563C22#297811yescharacteristic
564C22#298112yes—
565C22#299212yes—
566C22#299312yes—
567C22#299412yes—
568C22#299512yes—
569C4—44no—
570C4—42no—
571C4—42no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—81no—
576C4—81no—
577C4#282812yes—
578C4#282912yes—
579C4#282912yes—
580C4#283012yes—
581C2—82no—
582C2—82no—
583C2#1466711yescharacteristic
584C2#2191011yescharacteristic
585C2#2191111yescharacteristic
586C2#2201612yes—
587C2#2201712yes—
588C2#2333012yes—
589C2#2333212yes—
590C2#2334811yescharacteristic
591C2#2335411yescharacteristic
592C2#2335811yescharacteristic
593C2#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. #11147C2semidirect
4. #11147C2semidirect
5. #11042C2semidirect
6. #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