Finite Group Databasebeta

← #190358

Subgroups — #190358

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34126C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3#17362C211yesmaximal, characteristic
4#8779C212yesmaximal, complemented
5#8778C212yesmaximal, complemented
6#8777C212yesmaximal, complemented
7#8776C212yesmaximal, complemented
8C2×#1297C212yesmaximal, complemented
9#7942C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11C22×#99—21no—
12C2×#407C2212yes—
13#2357C2211yescharacteristic
14#2357C2211yescharacteristic
15#2353C411yescharacteristic, complemented
16#2353C411yescharacteristic, complemented
17#2353C2211yescharacteristic
18#2353C2211yescharacteristic
19#2349C411yescharacteristic, complemented
20#2349C411yescharacteristic, complemented
21#2345—21no—
22#2342—21no—
23#2342—21no—
24D4×C22⋊C4—21no—
25#2293—21no—
26#2290—21no—
27#2290—21no—
28#2283C411yescharacteristic, complemented
29#2280C411yescharacteristic, complemented
30#2280C2211yescharacteristic
31C42×D4—21no—
32C4×#117C411yescharacteristic, complemented
33C2×C4×C22⋊C4C411yescharacteristic, complemented
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36C4×#99—21no—
37C4×#99—21no—
38C4×#99C2212yes—
39C4×#99C2212yes—
40C2×#342—21no—
41#1465C2212yes—
42#1465C2212yes—
43#1463—21no—
44#1462—21no—
45#1459—21no—
46#1459—21no—
47#1297C2214yescomplemented
48#1297C2214yescomplemented
49#1284C2214yescomplemented
50#1284C2214yescomplemented
51C23×D4D411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4C2311yescharacteristic
54C22×M16—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117—21no—
71C2×#117—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C22⋊C4—22no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4D411yescharacteristic
77C22×C22⋊C4D411yescharacteristic
78C22×C42—21no—
79C22×C42—21no—
80C22×C42—22no—
81C22×C42C2×C411yescharacteristic
82C2×#99—28no—
83C2×#99—22no—
84C2×#99—22no—
85#407—22no—
86#407—22no—
87#407—22no—
88#407C2314yes—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91#390—21no—
92#390—21no—
93#389—21no—
94#389—21no—
95#388—21no—
96#388—21no—
97#388—21no—
98#388—21no—
99#388—21no—
100#388—21no—
101#388—21no—
102#388—21no—
103#387—21no—
104#387—21no—
105#387—21no—
106#387—21no—
107#386—21no—
108#386—21no—
109#386—21no—
110#386—21no—
111#385—21no—
112#385—21no—
113#385—21no—
114#385—21no—
115#383C2×C411yescharacteristic, complemented
116#383C2×C411yescharacteristic, complemented
117#382C2×C411yescharacteristic, complemented
118#382C2×C411yescharacteristic, complemented
119#381—21no—
120#381C2×C411yescharacteristic, complemented
121#381C2×C411yescharacteristic, complemented
122#381C2311yescharacteristic
123#381C2311yescharacteristic
124#380—21no—
125#380—21no—
126#380—21no—
127#380C2×C411yescharacteristic
128#380C2×C411yescharacteristic
129#380C2×C411yescharacteristic, complemented
130#380C2×C411yescharacteristic, complemented
131#379—21no—
132#379—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2×C411yescharacteristic, complemented
138C4×C4⋊C4C2×C411yescharacteristic, complemented
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—22no—
151C4×C22⋊C4—22no—
152C4×C22⋊C4—22no—
153C4×C22⋊C4C2×C411yescharacteristic
154C4×C22⋊C4C2×C411yescharacteristic
155C4×C22⋊C4C2×C411yescharacteristic
156C4×C22⋊C4C2×C411yescharacteristic
157C4×C22⋊C4C2×C411yescharacteristic, complemented
158C4×C22⋊C4C2×C411yescharacteristic, complemented
159C4×C22⋊C4C2314yes—
160C4×C22⋊C4C2311yescharacteristic
161C4×C22⋊C4C2311yescharacteristic
162#376—21no—
163#376C2×C411yescharacteristic, complemented
164#376C2×C411yescharacteristic, complemented
165C2×#95—22no—
166C2×#95C2×C411yescharacteristic
167C2×#95C2×C411yescharacteristic
168C2×#95C2×C411yescharacteristic, complemented
169C2×#95C2×C411yescharacteristic, complemented
170C43—21no—
171#342—22no—
172#342—22no—
173#324—22no—
174#324—22no—
175#323—22no—
176#323—22no—
177C25C22⋊C411yescharacteristic
178C25C2×D411yescharacteristic
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—24no—
182C22×D4—42no—
183C22×D4C22⋊C411yescharacteristic
184C22×D4C22⋊C411yescharacteristic
185C22×D4C2×D411yescharacteristic
186C22×D4C2×D411yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—42no—
196C23×C4—82no—
197C23×C4—82no—
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22×C411yescharacteristic
204C23×C4C22×C411yescharacteristic
205C23×C4C22×C411yescharacteristic
206C23×C4C22×C411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4#4112yes—
211C23×C4#4112yes—
212C23×C4#4112yes—
213C23×C4#4112yes—
214C23×C4C2411yescharacteristic
215C2×M16—24no—
216C2×M16—42no—
217C2×M16—84no—
218C22×C8—42no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—44no—
236C4×D4—44no—
237#117—44no—
238#117—82no—
239#117—82no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C22⋊C411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4C22⋊C411yescharacteristic
284C2×C22⋊C4C2×D411yescharacteristic
285C2×C22⋊C4C2×D411yescharacteristic
286C2×C22⋊C4C2×D411yescharacteristic
287C2×C22⋊C4C2×D411yescharacteristic
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—24no—
305C2×C42—42no—
306C2×C42—82no—
307C2×C42—82no—
308C2×C42—84no—
309C2×C42C22⋊C411yescharacteristic
310C2×C42C22⋊C411yescharacteristic
311C2×C42C22×C411yescharacteristic
312C2×C42C22×C411yescharacteristic
313#99—28no—
314#99—28no—
315C22⋊C8—24no—
316C22⋊C8—24no—
317C22⋊C8—42no—
318C22⋊C8—42no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—24no—
336#95—42no—
337#95C22×C411yescharacteristic
338#95C22×C411yescharacteristic
339#95C22×C411yescharacteristic
340#95C22×C411yescharacteristic
341#95C22×C411yescharacteristic
342#95C22×C411yescharacteristic
343#95C22×C411yescharacteristic
344#95C22×C411yescharacteristic
345C24—22no—
346C24—22no—
347C24—21no—
348C24—21no—
349C24—22no—
350C24—42no—
351C24—41no—
352C24—42no—
353C24—81no—
354C24—81no—
355C24C2×C22⋊C411yescharacteristic
356C24C2×C22⋊C411yescharacteristic
357C24C2×C22⋊C411yescharacteristic
358C24C2×C22⋊C411yescharacteristic
359C24#12312yes—
360C24#12312yes—
361C24C22×D411yescharacteristic
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4#10414yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—48no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4—81no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4#11711yescharacteristic
445C22×C4#11711yescharacteristic
446C22×C4#11711yescharacteristic
447C22×C4#11711yescharacteristic
448C22×C4#11712yes—
449C22×C4#11712yes—
450C22×C4#12312yes—
451C22×C4#12312yes—
452C22×C4#12312yes—
453C22×C4#12312yes—
454C22×C4#12312yes—
455C22×C4#12312yes—
456C22×C4C23×C411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460M16—84no—
461M16—84no—
462C2×C8—44no—
463C2×C8—84no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C22⋊C4—24no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—28no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—81no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C22⋊C4—81no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—42no—
512C42—42no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—84no—
520C42—81no—
521C42—81no—
522C42—81no—
523C42—81no—
524C42—84no—
525C23—24no—
526C23—22no—
527C23—24no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—24no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—81no—
555C23—81no—
556C23—84no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23#38111yescharacteristic
561C23#38112yes—
562C23#38111yescharacteristic
563C23C2×#10412yes—
564C23C2×#10412yes—
565C23#42112yes—
566C23C22×C22⋊C411yescharacteristic
567C23C2×#12312yes—
568D4—28no—
569D4—44no—
570D4—44no—
571D4—44no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—48no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—816no—
649C2×C4#38111yescharacteristic
650C2×C4#38112yes—
651C2×C4#38111yescharacteristic
652C2×C4C2×#10412yes—
653C2×C4C2×#11711yescharacteristic
654C2×C4C2×#11711yescharacteristic
655C2×C4C2×#12312yes—
656C8—84no—
657C22—21no—
658C22—21no—
659C22—22no—
660C22—21no—
661C22—21no—
662C22—22no—
663C22—22no—
664C22—21no—
665C22—22no—
666C22—21no—
667C22—22no—
668C22—22no—
669C22—21no—
670C22—42no—
671C22—42no—
672C22—48no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—81no—
683C22—81no—
684C22—84no—
685C22—84no—
686C22—81no—
687C22—81no—
688C22—84no—
689C22#182614yes—
690C22#182712yes—
691C22C2×#38111yescharacteristic
692C22C22×#10411yescharacteristic
693C22C2×#42111yescharacteristic
694C4—22no—
695C4—22no—
696C4—21no—
697C4—21no—
698C4—21no—
699C4—22no—
700C4—42no—
701C4—42no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—41no—
707C4—81no—
708C4—81no—
709C4—81no—
710C4—81no—
711C4—81no—
712C4—81no—
713C4—88no—
714C4#182812yes—
715C2—21no—
716C2—21no—
717C2—22no—
718C2—24no—
719C2—42no—
720C2—41no—
721C2—42no—
722C2—81no—
723C2C2×#182611yescharacteristic
724C2C2×#182711yescharacteristic
725C2C2×#182811yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#381
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8779C2semidirect
5. #8778C2semidirect
6. #8777C2semidirect
7. #8776C2semidirect
8. C2×#1297C2semidirect
9. #7942C2semidirect
15. #2353C4semidirect
16. #2353C4semidirect
19. #2349C4semidirect
20. #2349C4semidirect
28. #2283C4semidirect
29. #2280C4semidirect
32. C4×#117C4semidirect
33. C2×C4×C22⋊C4C4semidirect
47. #1297C22semidirect
48. #1297C22semidirect
49. #1284C22semidirect
50. #1284C22semidirect
115. #383C2×C4semidirect
116. #383C2×C4semidirect
117. #382C2×C4semidirect
118. #382C2×C4semidirect
120. #381C2×C4semidirect
121. #381C2×C4semidirect
129. #380C2×C4semidirect
130. #380C2×C4semidirect
137. C4×C4⋊C4C2×C4semidirect
138. C4×C4⋊C4C2×C4semidirect
157. C4×C22⋊C4C2×C4semidirect
158. C4×C22⋊C4C2×C4semidirect
163. #376C2×C4semidirect
164. #376C2×C4semidirect
168. C2×#95C2×C4semidirect
169. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341262
C2×C4×#992
#173622
#87792
#87782
#87772
#87762
C2×#12972
#79422