Finite Group Databasebeta

← #191113

Subgroups — #191113

Order 512.

1068 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34212C211yesmaximal, characteristic
2#21902C211yesmaximal, characteristic
3#13947C211yesmaximal, characteristic
4#8712C211yesmaximal, characteristic
5#8712C211yesmaximal, characteristic
6#8576C212yesmaximal, complemented
7#8574C212yesmaximal, complemented
8#8329C211yesmaximal, characteristic
9#8329C211yesmaximal, characteristic
10#8221C212yesmaximal, complemented
11C4×#328C212yesmaximal, complemented
12C22×C22⋊Q8C2211yescharacteristic
13#2368—22no—
14#2360C412yescomplemented
15#2360C412yescomplemented
16#2345—22no—
17#2345C412yescomplemented
18Q8×C22⋊C4C412yescomplemented
19#2329—21no—
20#2329—21no—
21#2315—21no—
22C4×C22⋊Q8C2212yes—
23C4×C22⋊Q8C2212yes—
24C2×#380C2211yescharacteristic
25#2273—21no—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1923—22no—
28#1923—22no—
29#1797C2212yes—
30#1779C2211yescharacteristic
31#1779C2211yescharacteristic
32C4×C22⋊C8C2212yes—
33C2×#328C2212yes—
34C2×#328C2212yes—
35C2×#328C2212yes—
36C2×#328C2212yes—
37#1443—21no—
38#1440—21no—
39#1437—21no—
40#1434—21no—
41#1434C2211yescharacteristic
42#1434C2211yescharacteristic
43#1434C2211yescharacteristic
44C22⋊#95C2211yescharacteristic
45C22⋊#95C2211yescharacteristic
46C4×#95C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C23×Q8C2×C411yescharacteristic
52C24×C4C2311yescharacteristic
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2312yes—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C412yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×C4⋊C8—22no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8C2312yes—
80C2×C22⋊C8C2312yes—
81C2×C4×C8—22no—
82#391—22no—
83#391—22no—
84#391C2×C412yes—
85#391C2×C412yes—
86#389—22no—
87#389—22no—
88#389C2×C412yes—
89#389C2×C412yes—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#380—21no—
101#380—21no—
102#380—22no—
103#380—22no—
104#380C2×C412yes—
105#380C2312yes—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C412yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4C2×C412yes—
117C4×C22⋊C4C2×C412yes—
118C4×C22⋊C4D412yes—
119C4×C22⋊C4D412yes—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95D411yescharacteristic
131C2×#95D411yescharacteristic
132C2×#95C2311yescharacteristic
133C2×#95C2311yescharacteristic
134C2×#95C2311yescharacteristic
135C2×#95C2311yescharacteristic
136#336—22no—
137#336—22no—
138#328—24no—
139#328—24no—
140#328—24no—
141#328—24no—
142C25C2×D411yescharacteristic
143C22×Q8—24no—
144C22×Q8—21no—
145C22×Q8—81no—
146C22×Q8—81no—
147C22×Q8C22×C412yes—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—81no—
159C23×C4—81no—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8—22no—
171C22×C8—22no—
172C22⋊Q8—28no—
173C22⋊Q8—24no—
174C22⋊Q8—24no—
175C22⋊Q8—24no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—24no—
179C22⋊Q8#4114yes—
180C22⋊Q8#4114yes—
181C4×Q8—82no—
182C4×Q8—82no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—81no—
216C2×C22⋊C4—81no—
217C2×C22⋊C4C22⋊C412yes—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—81no—
228C2×C42—81no—
229C2×C42—81no—
230C2×C42C22×C412yes—
231C2×C42C22×C412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C4⋊C8—82no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—81no—
240C22⋊C8—81no—
241C22⋊C8—81no—
242C22⋊C8—81no—
243C4×C8—82no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—42no—
259#95—42no—
260#95—42no—
261#95—42no—
262#95—81no—
263#95—81no—
264#95—81no—
265#95—81no—
266#95—81no—
267#95C22⋊C412yes—
268#95C22⋊C412yes—
269#95C22⋊C412yes—
270#95C22⋊C412yes—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24C2×C22⋊C411yescharacteristic
280C24C2×C22⋊C411yescharacteristic
281C24#13512yes—
282C24#13512yes—
283C24C22×D411yescharacteristic
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—81no—
289C2×Q8—81no—
290C2×Q8—88no—
291C2×Q8#11714yes—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C412yes—
376C22×C4#13712yes—
377C22×C4#13712yes—
378C22×C4C23×C411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C2×#4112yes—
381C22×C4C2×#4112yes—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C4⋊C4—24no—
389C4⋊C4—28no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4—81no—
398C4⋊C4—81no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4#11714yes—
402C22⋊C4—28no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—44no—
412C22⋊C4—84no—
413C22⋊C4—84no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42—42no—
418C42—82no—
419C42—82no—
420C42—82no—
421C42—82no—
422C42—84no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—81no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23#41612yes—
456C23#41612yes—
457C23C22×C22⋊C411yescharacteristic
458C23#52511yescharacteristic
459C23#52511yescharacteristic
460C23C2×#13511yescharacteristic
461C23C2×#13511yescharacteristic
462C23C2×#13711yescharacteristic
463C23C2×#13711yescharacteristic
464Q8—44no—
465Q8—44no—
466Q8—84no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—162no—
547C2×C4—162no—
548C2×C4#41914yes—
549C2×C4C2×#11712yes—
550C2×C4#52512yes—
551C8—82no—
552C8—82no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22#228011yescharacteristic
584C22#288311yescharacteristic
585C22#288311yescharacteristic
586C22C2×#41611yescharacteristic
587C22C2×#41911yescharacteristic
588C22#290111yescharacteristic
589C22#290111yescharacteristic
590C22#310811yescharacteristic
591C22#310811yescharacteristic
592C22#310811yescharacteristic
593C22#310811yescharacteristic
594C4—24no—
595C4—22no—
596C4—22no—
597C4—44no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—44no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—161no—
609C4—161no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—42no—
615C2—42no—
616C2#1761511yescharacteristic
617C2#1761511yescharacteristic
618C2#1762111yescharacteristic
619C2#1762711yescharacteristic
620C2#2072411yescharacteristic
621C2#2073511yescharacteristic
622C2#2074911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8576C2semidirect
7. #8574C2semidirect
10. #8221C2semidirect
11. C4×#328C2semidirect
14. #2360C4semidirect
15. #2360C4semidirect
17. #2345C4semidirect
18. Q8×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#342122
#219022
#139472
#87122
#85762
#85742
#83292
#82212
C4×#3282