Finite Group Databasebeta

← #493694

Subgroups — #493694

Order 512.

1410 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
1C4×#525C211yesmaximal, characteristic
2C2×#2120C212yesmaximal
3#20637C211yesmaximal, characteristic, complemented
4#20632C211yesmaximal, characteristic, complemented
5#12794C214yesmaximal, complemented
6#12793C214yesmaximal, complemented
7#12573C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10C23×C42C411yescharacteristic, complemented
11C2×#423—21no—
12C2×#413C2212yes—
13C2×#410C2212yes—
14C4×C22⋊Q8—22no—
15C4×#123C412yescomplemented
16C4×#123C2212yes—
17C4×#121—22no—
18C4×#120C412yescomplemented
19C4×#120C2212yes—
20#2297C2211yescharacteristic
21#2296C411yescharacteristic, complemented
22#2290—21no—
23#2290—21no—
24#2285—21no—
25#2280C411yescharacteristic, complemented
26C42×D4—21no—
27C42×D4—21no—
28C4×#117C411yescharacteristic, complemented
29#2120—28no—
30C2×#352C2214yes—
31C2×AΣL(1,𝔽16)C2214yes—
32#1801—22no—
33#1800C2214yes—
34#1799—22no—
35#1799C2214yes—
36#1742—21no—
37#1740—21no—
38#1390C2218yes—
39C22×#41C2311yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C412yes—
42C22×M16—22no—
43#525—22no—
44#525—22no—
45#525D414yes—
46#525Q814yes—
47C2×#123C2×C411yescharacteristic
48C2×#123C2311yescharacteristic
49C2×C22⋊Q8—21no—
50C2×#121—21no—
51C2×#120C2×C411yescharacteristic
52C2×#120C2311yescharacteristic
53C4×#41—42no—
54C4×#41—41no—
55C4×#41—41no—
56C2×C4×Q8D412yes—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4C2×C412yes—
62C2×C4×D4D412yes—
63C2×C4×D4C2312yes—
64C2×#117—22no—
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117D411yescharacteristic
69C2×#117D411yescharacteristic
70C22×C42—21no—
71C22×C42—22no—
72C22×C42—21no—
73C22×C42—41no—
74C22×C42—41no—
75C22×C42—42no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C412yes—
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81#423—24no—
82#423—41no—
83#423—41no—
84C2×C4⋊C8—42no—
85#413—24no—
86#413—24no—
87C2×#101—22no—
88C2×#100C2314yes—
89#410—24no—
90#410—24no—
91C2×#99—22no—
92C2×#99C2314yes—
93#391—21no—
94#390—21no—
95#389—22no—
96#388—22no—
97#388—22no—
98#387—22no—
99#387—22no—
100#386—22no—
101#386—22no—
102#386C2×C412yes—
103#386C2312yes—
104#385—22no—
105#385—22no—
106#385C2×C412yes—
107#385C2×C412yes—
108#384—41no—
109#382—41no—
110#381C2×C412yes—
111#380—22no—
112#380—24no—
113#379C2×C412yes—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—42no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4C2×C412yes—
129C4×C22⋊C4C2×C412yes—
130#376—21no—
131C43—21no—
132#352—28no—
133AΣL(1,𝔽16)—28no—
134#343—24no—
135#342—24no—
136C25C22×C411yescharacteristic
137C2×#41—41no—
138C2×#41—41no—
139C2×#41—42no—
140C2×#41C2×D412yes—
141C2×#41C2×Q812yes—
142C2×#41#4112yes—
143C2×#41#4112yes—
144C22×Q8C2×D411yescharacteristic
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C22×C411yescharacteristic
148C22×D4C2×D411yescharacteristic
149C22×D4C2411yescharacteristic
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—42no—
156C23×C4—48no—
157C23×C4—41no—
158C23×C4—41no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4C22⋊C412yes—
162C23×C4C4⋊C414yescomplemented
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C2×M16—24no—
171C2×M16—42no—
172C2×M16—44no—
173C2×M16—44no—
174C22×C8—42no—
175#123—22no—
176#123—21no—
177#123—21no—
178#123C4⋊C414yescomplemented
179#123C2×D412yes—
180#123C2×Q812yes—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183#121—24no—
184#121—24no—
185#120—22no—
186#120—21no—
187#120—21no—
188#120C4⋊C414yescomplemented
189#120C2×D412yes—
190#120C2×Q812yes—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—42no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—48no—
204C4×D4—48no—
205#117—22no—
206#117—41no—
207#117—41no—
208#117—41no—
209#117—41no—
210#117—41no—
211#117—41no—
212#117—42no—
213#117—44no—
214#117—44no—
215#117C4⋊C414yescomplemented
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4C22⋊C412yes—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4C22⋊C412yes—
237C2×C22⋊C4C22×C411yescharacteristic
238C2×C22⋊C4C22×C411yescharacteristic
239C2×C22⋊C4C22×C412yes—
240C2×C22⋊C4C22×C412yes—
241C2×C22⋊C4C2×D412yes—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42—42no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42—42no—
263C2×C42—41no—
264C2×C42—41no—
265C2×C42—42no—
266C2×C42—41no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—41no—
270C2×C42—42no—
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C412yes—
275C2×C42C22⋊C412yes—
276C2×C42C22⋊C411yescharacteristic
277C2×C42C2×D412yes—
278C4⋊C8—44no—
279C4⋊C8—44no—
280#101—44no—
281#100—24no—
282#100—24no—
283#99—24no—
284#99—24no—
285#99—44no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—44no—
295C24—21no—
296C24—21no—
297C24—41no—
298C24—44no—
299C24—41no—
300C24C2×C22⋊C411yescharacteristic
301C24C2×C4⋊C412yes—
302C24C23×C411yescharacteristic
303C24C22×D411yescharacteristic
304#41—42no—
305#41—42no—
306#41—42no—
307#41—42no—
308#41—42no—
309#41—42no—
310#41—42no—
311#41—42no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8#12112yes—
315C2×Q8C22⋊Q812yes—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—24no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4C2×C4⋊C412yes—
328C2×D4#11712yes—
329C2×D4#12112yes—
330C2×D4C22⋊Q812yes—
331C2×D4C22×D411yescharacteristic
332C2×D4C22×Q811yescharacteristic
333C2×D4C2×#4111yescharacteristic
334C2×D4C2×#4111yescharacteristic
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—28no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—48no—
385C22×C4—44no—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C412yes—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C4⋊C412yes—
394C22×C4C2×C4⋊C412yes—
395C22×C4#11712yes—
396C22×C4#12112yes—
397C22×C4#12112yes—
398C22×C4C22⋊Q812yes—
399C22×C4C22⋊Q812yes—
400C22×C4C22×D411yescharacteristic
401M16—44no—
402M16—44no—
403M16—84no—
404C2×C8—44no—
405C2×C8—84no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—84no—
417C4⋊C4—88no—
418C22⋊C4—21no—
419C22⋊C4—21no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—24no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—24no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—42no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4C2×C4⋊C412yes—
437C22⋊C4C2×C4⋊C412yes—
438C42—24no—
439C42—22no—
440C42—22no—
441C42—21no—
442C42—22no—
443C42—22no—
444C42—21no—
445C42—21no—
446C42—22no—
447C42—22no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—42no—
452C42—44no—
453C42—44no—
454C42—44no—
455C42—42no—
456C42—41no—
457C42—42no—
458C42—41no—
459C42—42no—
460C42—84no—
461C42#9911yescharacteristic
462C42#9911yescharacteristic
463C42#9912yes—
464C23—24no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—21no—
471C23—21no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—48no—
483C23—41no—
484C23—41no—
485C23#38012yes—
486C23C22×C22⋊C411yescharacteristic
487C23C22×C4⋊C411yescharacteristic
488C23C2×#11711yescharacteristic
489C23C2×#12111yescharacteristic
490C23C2×C22⋊Q811yescharacteristic
491Q8—42no—
492Q8—42no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—41no—
501D4—44no—
502D4—44no—
503D4—48no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—48no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—416no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—84no—
559C2×C4—88no—
560C2×C4#38012yes—
561C2×C4#38012yes—
562C2×C4#38012yes—
563C2×C4C2×#9912yes—
564C2×C4C2×#9911yescharacteristic
565C2×C4C2×#9912yes—
566C2×C4C2×#9911yescharacteristic
567C2×C4#41012yes—
568C2×C4C2×#12111yescharacteristic
569C2×C4C2×C22⋊Q811yescharacteristic
570C8—84no—
571C22—22no—
572C22—24no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—22no—
578C22—41no—
579C22—41no—
580C22—42no—
581C22—41no—
582C22—41no—
583C22—42no—
584C22—44no—
585C22—42no—
586C22—44no—
587C22—44no—
588C22—48no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—81no—
595C22—81no—
596C22C2×#38011yescharacteristic
597C22C22×#9911yescharacteristic
598C22C2×#41011yescharacteristic
599C4—22no—
600C4—21no—
601C4—22no—
602C4—21no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—44no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—82no—
612C4—164no—
613C4#179912yes—
614C4#179912yes—
615C4#212014yescomplemented
616C2—21no—
617C2—21no—
618C2—41no—
619C2—44no—
620C2—41no—
621C2—41no—
622C2—41no—
623C2—42no—
624C2C2×#179911yescharacteristic
625C2C2×#212012yes—
subgroupG/H
centerC2×C4C2×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
socleC22C22×#99
prime radicalC2×#525C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20637C2semidirect
4. #20632C2semidirect
5. #12794C2semidirect
6. #12793C2semidirect
7. #12573C2semidirect
10. C23×C42C4semidirect
15. C4×#123C4semidirect
18. C4×#120C4semidirect
21. #2296C4semidirect
25. #2280C4semidirect
28. C4×#117C4semidirect
162. C23×C4C4⋊C4semidirect
178. #123C4⋊C4semidirect
188. #120C4⋊C4semidirect
215. #117C4⋊C4semidirect
615. C4#2120semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
C2×#21202
#206372
#206322
#127942
#127932
#125732