Finite Group Databasebeta

← #481451

Subgroups — #481451

Order 512.

1322 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34633C211yesmaximal, characteristic
2C2×#1740C211yesmaximal, characteristic, complemented
3C2×#1458C211yesmaximal, characteristic, complemented
4#11694C212yesmaximal, complemented
5#11693C212yesmaximal, complemented
6#11692C214yesmaximal, complemented
7#11691C214yesmaximal, complemented
8C2×#532C411yescharacteristic, complemented
9C2×#525—21no—
10C22×#117—21no—
11C22×#117C2211yescharacteristic
12C2×#431—21no—
13#2621C411yescharacteristic, complemented
14#2621C2211yescharacteristic
15#2620—21no—
16#2620—21no—
17#2619C411yescharacteristic, complemented
18#2619C2211yescharacteristic
19#2618—21no—
20#2618C412yescomplemented
21#2617—21no—
22#2617C2212yes—
23#2392—21no—
24#2392—21no—
25#2392C412yescomplemented
26#2392C2212yes—
27#2297—21no—
28#2285C411yescharacteristic, complemented
29#1740—21no—
30#1740—21no—
31#1740C2212yes—
32#1740C2212yes—
33C2×#342—21no—
34C2×#342C2212yes—
35#1458—21no—
36#1458—21no—
37#1458C2212yes—
38#1458C2212yes—
39C2×#324—21no—
40C2×#323C2212yes—
41#1344C2218yes—
42#1343C2218yes—
43C22×#41—21no—
44C24×C4D411yescharacteristic
45#532—41no—
46#532—41no—
47#532—41no—
48#532—41no—
49C2×C4⋊Q8C2×C411yescharacteristic, complemented
50C2×#127C2×C411yescharacteristic, complemented
51C2×#124—21no—
52#525—44no—
53#525—44no—
54C2×#123—22no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121C2×C412yes—
61C2×#120—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C412yescomplemented
66C2×#117—21no—
67C2×#117—28no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—41no—
71C2×#117—41no—
72C2×#117—41no—
73C2×#117—41no—
74C2×#117—41no—
75C2×#117—41no—
76C2×#117—44no—
77C2×#117C2×C411yescharacteristic
78C2×#117C2×C411yescharacteristic
79C2×#117D411yescharacteristic
80C2×#117D411yescharacteristic
81C2×#117D411yescharacteristic
82C2×#117D411yescharacteristic
83C2×#117C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C22×C4⋊C4—41no—
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C4⋊C4D411yescharacteristic
88C22×C22⋊C4—22no—
89C22×C22⋊C4D411yescharacteristic
90C22×C22⋊C4C2311yescharacteristic
91C22×C42—41no—
92C22×C42C2×C411yescharacteristic
93C22×C42D411yescharacteristic
94#431—22no—
95#431—22no—
96#431—41no—
97#431—41no—
98C2×C4⋊C8—41no—
99C2×C22⋊C8—22no—
100C2×C4×C8—41no—
101#398—21no—
102#398—21no—
103#398C2×C412yescomplemented
104#398C2×C412yescomplemented
105#397—21no—
106#397—21no—
107#397C2×C412yes—
108#397C2312yes—
109#396—22no—
110#396—22no—
111#396—22no—
112#396—22no—
113#394—21no—
114#394—21no—
115#394—22no—
116#394—22no—
117#394—22no—
118#394—22no—
119#394C2×C412yescomplemented
120#394C2×C412yescomplemented
121#392—21no—
122#392—21no—
123#392C2×C412yes—
124#392C2312yes—
125#391—21no—
126#391—21no—
127#390—21no—
128#390—21no—
129#386—22no—
130#386—22no—
131#384—21no—
132#382C2×C411yescharacteristic, complemented
133#382C2×C411yescharacteristic, complemented
134#380—21no—
135#380C2×C412yes—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2×C412yescomplemented
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140#342—22no—
141#342—22no—
142#342—22no—
143#342C2314yes—
144#324—22no—
145#324—22no—
146#323—22no—
147#323C2314yes—
148C25C2×D411yescharacteristic
149C2×#41—41no—
150C2×#41—41no—
151C2×#41—41no—
152C2×#41—41no—
153C2×#41—44no—
154C2×#41—44no—
155C22×Q8—21no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4C22×C412yes—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—42no—
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4D814yes—
176C23×C4SD1614yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C22×C8—42no—
182C4⋊Q8—41no—
183C4⋊Q8—41no—
184#127—41no—
185#127—41no—
186#124—42no—
187#124—42no—
188#123—48no—
189C22⋊Q8—44no—
190C22⋊Q8—42no—
191C22⋊Q8—42no—
192#121—42no—
193#121—42no—
194#121—42no—
195#121—42no—
196#121—44no—
197#121—42no—
198#121—42no—
199#120—44no—
200#120—44no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—42no—
204C4×D4—42no—
205C4×D4—48no—
206C4×D4—44no—
207C4×D4—44no—
208#117—28no—
209#117—28no—
210#117—41no—
211#117—41no—
212#117—41no—
213#117—41no—
214#117—41no—
215#117—41no—
216#117—41no—
217#117—41no—
218#117—41no—
219#117—41no—
220#117—41no—
221#117—41no—
222#117—41no—
223#117—41no—
224#117—41no—
225#117—41no—
226#117—48no—
227#117—48no—
228#117—44no—
229#117—44no—
230#117—44no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22×C411yescharacteristic
276C2×C22⋊C4C2×D411yescharacteristic
277C2×C22⋊C4C2×D411yescharacteristic
278C2×C22⋊C4C2×D411yescharacteristic
279C2×C22⋊C4C2×D411yescharacteristic
280C2×C22⋊C4C2411yescharacteristic
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—22no—
290C2×C42—48no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42C22⋊C411yescharacteristic
294C2×C42C22⋊C411yescharacteristic
295C2×C42C22⋊C411yescharacteristic
296C2×C42C22⋊C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C4⋊C8—42no—
302C4⋊C8—42no—
303C22⋊C8—24no—
304C22⋊C8—42no—
305C4×C8—42no—
306C4×C8—42no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—21no—
311#95—21no—
312#95—22no—
313#95C22×C412yes—
314#95C22×C412yes—
315C24—21no—
316C24—21no—
317C24—22no—
318C24—22no—
319C24—81no—
320C24—81no—
321C24C2×C22⋊C411yescharacteristic
322C24C2×C22⋊C411yescharacteristic
323C24C2×D812yes—
324C24C2×SD1612yes—
325C24C22×D411yescharacteristic
326#41—84no—
327#41—84no—
328#41—84no—
329#41—84no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—82no—
348C2×D4—82no—
349C2×D4—82no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—82no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
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—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
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—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4#9911yescharacteristic
400C22×C4#9911yescharacteristic
401C22×C4#9911yescharacteristic
402C22×C4#9911yescharacteristic
403C22×C4#10214yes—
404C22×C4#10214yes—
405C22×C4#10214yes—
406C22×C4#10214yes—
407C22×C4#10414yes—
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×D812yes—
419C22×C4C2×D812yes—
420C22×C4C2×SD1612yes—
421C22×C4C2×SD1612yes—
422C22×C4C23×C411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C2×C8—44no—
425C2×C8—84no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C22⋊C4—28no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—48no—
456C22⋊C4—44no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—48no—
484C42—48no—
485C42—44no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—22no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—81no—
510C23—81no—
511C23—82no—
512C23—82no—
513C23—84no—
514C23C2×#9911yescharacteristic
515C23C2×#9911yescharacteristic
516C23C2×#10212yes—
517C23C2×#10212yes—
518C23C2×#10412yes—
519C23C22×C22⋊C411yescharacteristic
520C23C22×D811yescharacteristic
521C23C22×SD1611yescharacteristic
522Q8—82no—
523Q8—82no—
524D4—84no—
525D4—84no—
526D4—84no—
527D4—84no—
528D4—88no—
529D4—84no—
530D4—84no—
531D4—82no—
532D4—82no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—816no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4#32714yes—
591C2×C4#32714yes—
592C2×C4#32714yes—
593C2×C4#32714yes—
594C2×C4C2×#9911yescharacteristic
595C2×C4C2×#9911yescharacteristic
596C2×C4C2×#9911yescharacteristic
597C2×C4C2×#9911yescharacteristic
598C2×C4C2×#10212yes—
599C2×C4C2×#10212yes—
600C2×C4C2×#10212yes—
601C2×C4C2×#10212yes—
602C2×C4C2×#10412yes—
603C2×C4C2×#10412yes—
604C8—84no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—22no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—44no—
620C22—44no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—81no—
626C22—81no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22C2×#32712yes—
637C22C2×#32712yes—
638C22C22×#9911yescharacteristic
639C22C22×#10211yescharacteristic
640C22C22×#10411yescharacteristic
641C4—22no—
642C4—22no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—82no—
652C4—82no—
653C4—88no—
654C4—82no—
655C4C2×#32712yes—
656C4C2×#32712yes—
657C4C2×#32712yes—
658C4C2×#32712yes—
659C2—21no—
660C2—22no—
661C2—21no—
662C2—22no—
663C2—41no—
664C2—41no—
665C2—42no—
666C2—162no—
667C2C22×#32711yescharacteristic
668C2#3334711yescharacteristic
669C2#3335811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radical#34633C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1740C2semidirect
3. C2×#1458C2semidirect
4. #11694C2semidirect
5. #11693C2semidirect
6. #11692C2semidirect
7. #11691C2semidirect
8. C2×#532C4semidirect
13. #2621C4semidirect
17. #2619C4semidirect
20. #2618C4semidirect
25. #2392C4semidirect
28. #2285C4semidirect
49. C2×C4⋊Q8C2×C4semidirect
50. C2×#127C2×C4semidirect
65. C2×C4×D4C2×C4semidirect
103. #398C2×C4semidirect
104. #398C2×C4semidirect
119. #394C2×C4semidirect
120. #394C2×C4semidirect
132. #382C2×C4semidirect
133. #382C2×C4semidirect
137. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#346332
C2×#17402
C2×#14582
#116942
#116932
#116922
#116912