Finite Group Databasebeta

← #188114

Subgroups — #188114

Order 512.

2438 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#539C211yesmaximal, characteristic
2C2×#1821C212yesmaximal, complemented
3#17392C212yesmaximal, complemented
4#17392C212yesmaximal, complemented
5#8556C212yesmaximal, complemented
6#8555C212yesmaximal, complemented
7#8554C212yesmaximal, complemented
8#8553C212yesmaximal, complemented
9C2×#539—28no—
10C2×#539—28no—
11C2×#539—24no—
12C2×#539—24no—
13C22×#126C411yescharacteristic
14C22×#126C2211yescharacteristic
15C22×#125C411yescharacteristic
16C22×#125C2211yescharacteristic
17C22×#123C411yescharacteristic
18C22×#123C2211yescharacteristic
19C22×#123C2211yescharacteristic
20C22×#123C2211yescharacteristic
21C22×C22⋊Q8C411yescharacteristic
22C22×C22⋊Q8C411yescharacteristic
23C22×C22⋊Q8C2211yescharacteristic
24C22×#121C411yescharacteristic
25C22×C4×D4C411yescharacteristic
26C22×C4×D4C411yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28C2×#407C2214yes—
29#1821—24no—
30#1821C2214yes—
31#1821C2214yes—
32C2×#343—24no—
33#1468—28no—
34#1468—28no—
35#1463C2214yes—
36#1463C2214yes—
37#1462C2214yes—
38#1462C2214yes—
39C23×Q8C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41C23×D4C2×C411yescharacteristic
42C23×D4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—24no—
49#539—232no—
50#539—232no—
51#539—232no—
52#539—232no—
53C2×#126—24no—
54C2×#126—24no—
55C2×#126—24no—
56C2×#126—24no—
57C2×#126—24no—
58C2×#126—24no—
59C2×#125—24no—
60C2×#125—24no—
61C2×#125—24no—
62C2×#125—24no—
63C2×#125—24no—
64C2×#125—24no—
65C2×#123—22no—
66C2×#123—24no—
67C2×#123—22no—
68C2×#123—22no—
69C2×#123—24no—
70C2×#123—22no—
71C2×#123—24no—
72C2×#123—24no—
73C2×#123—24no—
74C2×#123—24no—
75C2×#123—24no—
76C2×#123—24no—
77C2×#123D418yes—
78C2×#123D418yes—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×#121—24no—
90C2×#121—24no—
91C2×#121—22no—
92C2×#121—22no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—22no—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4C2×C411yescharacteristic
104C22×C4⋊C4C2×C411yescharacteristic
105C22×C4⋊C4C2×C411yescharacteristic
106C22×C4⋊C4C2×C411yescharacteristic
107C22×C4⋊C4C2×C411yescharacteristic
108C22×C4⋊C4C2×C411yescharacteristic
109C22×C4⋊C4C2×C411yescharacteristic
110C22×C4⋊C4C2×C411yescharacteristic
111C22×C4⋊C4C2×C411yescharacteristic
112C22×C4⋊C4C2×C411yescharacteristic
113C22×C4⋊C4D414yes—
114C22×C4⋊C4D414yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4C2×C411yescharacteristic
120C22×C22⋊C4C2×C411yescharacteristic
121C22×C22⋊C4C2×C411yescharacteristic
122C22×C22⋊C4C2×C411yescharacteristic
123C22×C22⋊C4C2×C411yescharacteristic
124C22×C22⋊C4C2×C411yescharacteristic
125C22×C22⋊C4C2×C411yescharacteristic
126C22×C22⋊C4C2×C411yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2311yescharacteristic
129C22×C42C2×C411yescharacteristic
130C22×C42C2×C411yescharacteristic
131#407—24no—
132#407—24no—
133#407—24no—
134#407C2318yes—
135C2×C22⋊C8—24no—
136C2×C22⋊C8—24no—
137#343—28no—
138#343—28no—
139#324—28no—
140#324—28no—
141#323—28no—
142#323—28no—
143C25C22×C411yescharacteristic
144C25C22×C411yescharacteristic
145C25C22×C411yescharacteristic
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—82no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—44no—
160C22×D4—44no—
161C22×D4—82no—
162C22×D4C22⋊C414yes—
163C22×D4C22⋊C414yes—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—44no—
175C23×C4—82no—
176C23×C4—82no—
177C23×C4—82no—
178C23×C4—82no—
179C23×C4C22⋊C412yes—
180C23×C4C22⋊C412yes—
181C23×C4C22⋊C412yes—
182C23×C4C22⋊C412yes—
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D414yes—
196C23×C4C2×D414yes—
197C23×C4C2×D412yes—
198C23×C4C2411yescharacteristic
199C2×M16—28no—
200C2×M16—44no—
201C2×M16—416no—
202C22×C8—44no—
203#126—232no—
204#126—232no—
205#125—232no—
206#125—232no—
207#123—232no—
208#123—232no—
209#123—28no—
210#123—28no—
211#123—28no—
212#123—28no—
213#123—28no—
214#123—28no—
215#123—28no—
216#123—28no—
217C22⋊Q8—232no—
218C22⋊Q8—232no—
219C22⋊Q8—216no—
220C22⋊Q8—216no—
221#121—216no—
222#121—216no—
223C4×D4—216no—
224C4×D4—216no—
225C4×D4—216no—
226C4×D4—216no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4C22⋊C414yes—
266C2×C4⋊C4C22⋊C414yes—
267C2×C4⋊C4C22⋊C414yes—
268C2×C4⋊C4C22⋊C414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D414yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4C22⋊C414yes—
300C2×C22⋊C4C22⋊C414yes—
301C2×C22⋊C4C22⋊C414yes—
302C2×C22⋊C4C22⋊C414yes—
303C2×C22⋊C4C2×D414yes—
304C2×C22⋊C4C2×D414yes—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C22⋊C8—28no—
312C22⋊C8—28no—
313C22⋊C8—44no—
314C22⋊C8—44no—
315C24—24no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24—42no—
328C24C2×C22⋊C412yes—
329C24C2×C22⋊C412yes—
330C24C2×C22⋊C412yes—
331C24C2×C22⋊C412yes—
332C24C2×C22⋊C411yescharacteristic
333C24C2×C22⋊C411yescharacteristic
334C24C23×C411yescharacteristic
335C2×Q8—28no—
336C2×Q8—84no—
337C2×Q8—84no—
338C2×Q8—84no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—28no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—84no—
353C2×D4—84no—
354C2×D4—84no—
355C22×C4—28no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C22⋊C412yes—
414C22×C4C2×C22⋊C412yes—
415C22×C4C2×C22⋊C412yes—
416C22×C4C2×C22⋊C412yes—
417C22×C4C2×C22⋊C412yes—
418C22×C4C2×C22⋊C412yes—
419C22×C4C2×C22⋊C412yes—
420C22×C4C2×C22⋊C412yes—
421C22×C4C2×C22⋊C412yes—
422C22×C4C2×C22⋊C412yes—
423C22×C4C2×C22⋊C412yes—
424C22×C4#12014yes—
425C22×C4#12014yes—
426C22×C4#12014yes—
427C22×C4#12014yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431M16—416no—
432M16—88no—
433C2×C8—48no—
434C2×C8—88no—
435C4⋊C4—28no—
436C4⋊C4—28no—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C22⋊C4—28no—
450C22⋊C4—28no—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—28no—
460C22⋊C4—28no—
461C42—28no—
462C42—28no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—41no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—416no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—88no—
499C23—88no—
500C23#37912yes—
501C23#37912yes—
502C23#41014yes—
503C23C22×C22⋊C411yescharacteristic
504C23C22×C22⋊C411yescharacteristic
505C23C22×C22⋊C411yescharacteristic
506Q8—88no—
507D4—48no—
508D4—48no—
509D4—48no—
510D4—48no—
511D4—88no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—88no—
567C2×C4—88no—
568C2×C4#37914yes—
569C2×C4C2×#12014yes—
570C8—88no—
571C22—28no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—22no—
577C22—21no—
578C22—21no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—44no—
584C22—48no—
585C22—48no—
586C22—44no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22#178214yes—
600C22C2×#37911yescharacteristic
601C22C2×#41012yes—
602C22#288812yes—
603C22#288912yes—
604C4—24no—
605C4—24no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—42no—
616C2—22no—
617C2—22no—
618C2—22no—
619C2—24no—
620C2—42no—
621C2—82no—
622C2—82no—
623C2C2×#178211yescharacteristic
624C2#2059211yescharacteristic
625C2#2059311yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1821C2semidirect
3. #17392C2semidirect
4. #17392C2semidirect
5. #8556C2semidirect
6. #8555C2semidirect
7. #8554C2semidirect
8. #8553C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5392
C2×#18212
#173922
#85562
#85552
#85542
#85532