Finite Group Databasebeta

← #495416

Subgroups — #495416

Order 512.

1077 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34947C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3C4×#410C211yesmaximal, characteristic
4#13300C214yesmaximal, complemented
5#13299C212yesmaximal, complemented
6#13298C212yesmaximal, complemented
7#12581C214yesmaximal
8C2×#533C411yescharacteristic, complemented
9C2×C4×#41C2211yescharacteristic
10C2×#410C2212yes—
11C2×#410C2212yes—
12#2568C411yescharacteristic, complemented
13#2567C411yescharacteristic, complemented
14#2566C2212yes—
15#2565C412yescomplemented
16#2564—21no—
17#2564—21no—
18#2563C2211yescharacteristic
19#2542C2211yescharacteristic
20#2417—21no—
21#2417—21no—
22#2396—21no—
23#2396—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26#2297C2212yes—
27#2285—21no—
28#2285C412yescomplemented
29C4×#117—21no—
30C2×C4×C4⋊C4C411yescharacteristic, complemented
31#1799—21no—
32#1799—21no—
33#1799C2212yes—
34#1799C2212yes—
35C4×#99—21no—
36C4×#99—21no—
37C4×#99C2212yes—
38C4×#99C2212yes—
39#1740—21no—
40#1740—21no—
41#1393C2218yes—
42#1393C2218yes—
43C22×#41C2311yescharacteristic
44#533—41no—
45#533—41no—
46#533—41no—
47#533—41no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#125C2×C412yes—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C4×#41—41no—
56C4×#41—41no—
57C4×#41—41no—
58C4×#41—41no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—22no—
70C2×#117—22no—
71C2×#117C2×C412yes—
72C2×#117C2×C411yescharacteristic
73C2×#117D411yescharacteristic
74C2×#117D411yescharacteristic
75C22×C4⋊C4—21no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C412yes—
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80#410—24no—
81#410—28no—
82#410Q818yescomplemented
83C2×#99—22no—
84C2×#99—22no—
85C2×#99C2314yes—
86C2×#99C2314yes—
87#400—21no—
88#400—21no—
89#399—21no—
90#399—21no—
91#398—22no—
92#398—22no—
93#396—22no—
94#396—22no—
95#395—21no—
96#395—21no—
97#393—21no—
98#393—21no—
99#391—21no—
100#391—21no—
101#391C2×C412yes—
102#391C2×C412yes—
103#390C2312yes—
104#390C2312yes—
105#389—21no—
106#389—21no—
107#387—22no—
108#387—22no—
109#386—22no—
110#386—22no—
111#385—22no—
112#385—22no—
113#384—41no—
114#384C2×C412yes—
115#384C2×C412yes—
116#382—41no—
117#382C2×C412yes—
118#382C2×C412yes—
119#380—21no—
120#380—21no—
121#380—22no—
122#380—22no—
123#380—22no—
124#380—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—41no—
130C4×C4⋊C4—41no—
131C4×C4⋊C4—41no—
132C4×C4⋊C4—41no—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C412yes—
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140#376—41no—
141C2×#95—21no—
142C43—41no—
143#342—22no—
144#342—22no—
145#342—22no—
146#342—22no—
147C2×#41—41no—
148C2×#41—41no—
149C2×#41—42no—
150C2×#41C2×Q814yes—
151C2×#41#4114yes—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8C22×C411yescharacteristic
155C22×D4C2×D411yescharacteristic
156C22×D4C2×D411yescharacteristic
157C22×D4C2411yescharacteristic
158C23×C4—22no—
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172#125—42no—
173#125—42no—
174C22⋊Q8—44no—
175C22⋊Q8—44no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202#117—28no—
203#117—41no—
204#117—41no—
205#117—41no—
206#117—41no—
207#117—41no—
208#117—41no—
209#117—44no—
210#117—42no—
211#117—42no—
212#117—44no—
213#117—44no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4C22⋊C411yescharacteristic
246C2×C22⋊C4C22⋊C411yescharacteristic
247C2×C22⋊C4C2×D411yescharacteristic
248C2×C22⋊C4C2×D411yescharacteristic
249C2×C22⋊C4C2×D411yescharacteristic
250C2×C22⋊C4C2×D411yescharacteristic
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—42no—
266C2×C42—44no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C412yes—
281C2×C42C22×C412yes—
282#99—28no—
283#99—24no—
284#99—28no—
285#99—44no—
286#99C2×Q818yes—
287#95—22no—
288#95—21no—
289#95—21no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—42no—
299#95—42no—
300#95—42no—
301#95—42no—
302C24C2×C22⋊C411yescharacteristic
303C24C22×D411yescharacteristic
304C24C22×D411yescharacteristic
305#41—44no—
306#41—44no—
307#41—44no—
308#41—44no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—82no—
317C2×Q8C4×Q814yes—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4C22⋊Q814yes—
325C2×D4#12412yes—
326C2×D4C4⋊Q812yes—
327C2×D4C22×Q812yes—
328C2×D4C2×#4112yes—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—82no—
364C22×C4—81no—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C4×Q814yes—
377C22×C4C22⋊Q814yes—
378C22×C4#12412yes—
379C22×C4C4⋊Q812yes—
380C22×C4C23×C411yescharacteristic
381C4⋊C4—24no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—48no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—84no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—84no—
405C4⋊C4—84no—
406C22⋊C4—28no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—82no—
420C22⋊C4—82no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—42no—
440C42—42no—
441C42—48no—
442C42—42no—
443C42—42no—
444C42—44no—
445C42—84no—
446C42—84no—
447C42—84no—
448C42#9911yescharacteristic
449C42#9911yescharacteristic
450C42#9911yescharacteristic
451C42#9911yescharacteristic
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23#39112yes—
465C23C22×C22⋊C411yescharacteristic
466C23C2×C22⋊Q812yes—
467C23C2×#12411yescharacteristic
468C23C2×C4⋊Q811yescharacteristic
469Q8—44no—
470Q8—82no—
471Q8—84no—
472Q8—82no—
473D4—44no—
474D4—42no—
475D4—42no—
476D4—42no—
477D4—42no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—41no—
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—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—88no—
532C2×C4—84no—
533C2×C4—84no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4#39112yes—
538C2×C4#39112yes—
539C2×C4#39112yes—
540C2×C4C2×#9911yescharacteristic
541C2×C4C2×#9911yescharacteristic
542C2×C4C2×#9911yescharacteristic
543C2×C4C2×#9911yescharacteristic
544C2×C4C2×#9911yescharacteristic
545C2×C4C2×#9911yescharacteristic
546C2×C4#41012yes—
547C2×C4C2×C4×Q812yes—
548C22—21no—
549C22—21no—
550C22—22no—
551C22—21no—
552C22—21no—
553C22—22no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—42no—
558C22—42no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—81no—
566C22—81no—
567C22C2×#39111yescharacteristic
568C22C22×#9911yescharacteristic
569C22C2×#41011yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—44no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—82no—
584C4—88no—
585C4—82no—
586C4—82no—
587C4—164no—
588C4#199612yes—
589C4#199612yes—
590C4#212612yes—
591C2—21no—
592C2—21no—
593C2—42no—
594C2—41no—
595C2—42no—
596C2—41no—
597C2C2×#199611yescharacteristic
598C2C2×#212611yescharacteristic
599C2#3335811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
socleC22C22×#99
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ 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
4. #13300C2semidirect
5. #13299C2semidirect
6. #13298C2semidirect
8. C2×#533C4semidirect
12. #2568C4semidirect
13. #2567C4semidirect
15. #2565C4semidirect
28. #2285C4semidirect
30. C2×C4×C4⋊C4C4semidirect
82. #410Q8semidirect

Maximal subgroups

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

subgroupindex
#349472
#206322
C4×#4102
#133002
#132992
#132982
#125812