Finite Group Databasebeta

← #391959

Subgroups — #391959

Order 512.

1206 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#11255C212yesmaximal, complemented
4#11147C212yesmaximal, complemented
5#11042C212yesmaximal, complemented
6#11042C212yesmaximal, complemented
7#10964C211yesmaximal, characteristic
8#10964C211yesmaximal, characteristic
9#10713C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×C4⋊C8C2212yes—
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×C4×C22⋊C4C2212yes—
20C2×#336C2212yes—
21C2×#336C2212yes—
22C2×#336C2212yes—
23C2×#336C2212yes—
24C2×#336C2212yes—
25C2×#336C2211yescharacteristic
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
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×C412yes—
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2×C412yes—
61C22×C42C2312yes—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64C2×C4⋊C8—42no—
65C2×C4⋊C8—42no—
66C2×C4⋊C8—42no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—42no—
70C2×C22⋊C8—42no—
71C2×C22⋊C8—42no—
72C2×C22⋊C8—42no—
73C2×C22⋊C8—41no—
74C2×C22⋊C8—41no—
75C2×C22⋊C8—42no—
76C2×C22⋊C8—42no—
77C2×C22⋊C8—42no—
78C2×C22⋊C8—42no—
79C2×C22⋊C8—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—42no—
92#336—42no—
93#336—42no—
94#336—42no—
95#336—41no—
96#336—41no—
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—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C22×C412yes—
126C23×C4C22×C412yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C412yes—
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—22no—
162C22×C8—22no—
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—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—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—21no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C4⋊C8—48no—
210C22⋊C8—48no—
211C22⋊C8—48no—
212C22⋊C8—44no—
213C22⋊C8—48no—
214C22⋊C8—44no—
215#95—24no—
216#95—24no—
217#95—42no—
218#95—42no—
219#95—41no—
220#95—41no—
221#95—42no—
222#95—42no—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—22no—
227C24—22no—
228C24—21no—
229C24—21no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—21no—
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—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
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—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4#11712yes—
333C22×C4#13112yes—
334C22×C4#13114yes—
335C22×C4C2×#4112yes—
336C22×C4H(5,𝔽2)12yes—
337C22×C4H(5,𝔽2)11yescharacteristic
338C22×C4H(5,𝔽2)12yes—
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C4H(5,𝔽2)12yes—
341C22×C4H(5,𝔽2)12yes—
342C22×C42−1+412yes—
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—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C22⋊C4—48no—
375C22⋊C4—48no—
376C22⋊C4—44no—
377C22⋊C4—44no—
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—22no—
393C23—22no—
394C23—22no—
395C23—24no—
396C23—22no—
397C23—24no—
398C23—22no—
399C23—24no—
400C23—41no—
401C23—42no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—42no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—42no—
415C23—41no—
416C23—42no—
417C23—42no—
418C23—41no—
419C23—41no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23C2×#11711yescharacteristic
437C23#51811yescharacteristic
438C23#51811yescharacteristic
439C23#51812yes—
440C23#52012yes—
441C23C2×#13111yescharacteristic
442C23C2×#13111yescharacteristic
443C23C2×#13112yes—
444C23#56811yescharacteristic
445C23#56811yescharacteristic
446C23#56812yes—
447D4—416no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
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—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
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#53912yes—
511C2×C4#53912yes—
512C2×C4#56012yes—
513C2×C4#56011yescharacteristic
514C2×C4#56012yes—
515C2×C4#56012yes—
516C2×C4#56011yescharacteristic
517C2×C4H(3,𝔽4)11yescharacteristic
518C2×C4#56212yes—
519C2×C4#56212yes—
520C2×C4#56212yes—
521C2×C4#56311yescharacteristic
522C8—82no—
523C8—82no—
524C8—82no—
525C8—82no—
526C22—42no—
527C22—42no—
528C22—44no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—44no—
534C22—44no—
535C22—44no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—81no—
546C22—81no—
547C22—82no—
548C22#231211yescharacteristic
549C22#231812yes—
550C22#237511yescharacteristic
551C22#237611yescharacteristic
552C22#237712yes—
553C22#292411yescharacteristic
554C22#292411yescharacteristic
555C22#292512yes—
556C22#297611yescharacteristic
557C22#297611yescharacteristic
558C22#297612yes—
559C22#297712yes—
560C22#297711yescharacteristic
561C22#297812yes—
562C22#297811yescharacteristic
563C22#297911yescharacteristic
564C22#298011yescharacteristic
565C22#298112yes—
566C22#298112yes—
567C22#299212yes—
568C22#299312yes—
569C22#299412yes—
570C22#299512yes—
571C4—44no—
572C4—42no—
573C4—42no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—81no—
578C4—81no—
579C4#282412yes—
580C4#282512yes—
581C4#282612yes—
582C4#282712yes—
583C2—82no—
584C2—82no—
585C2#1466611yescharacteristic
586C2#2190811yescharacteristic
587C2#2190911yescharacteristic
588C2#2200612yes—
589C2#2200712yes—
590C2#2331011yescharacteristic
591C2#2331311yescharacteristic
592C2#2331711yescharacteristic
593C2#2332311yescharacteristic
594C2#2333012yes—
595C2#2333212yes—
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. #11255C2semidirect
4. #11147C2semidirect
5. #11042C2semidirect
6. #11042C2semidirect
9. #10713C2semidirect

Maximal subgroups

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

subgroupindex
#138392
#112982
#112552
#111472
#110422
#109642
#107132