Finite Group Databasebeta

← #495604

Subgroups — #495604

Order 512.

1045 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34979C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3C4×#410C211yesmaximal, characteristic
4#13298C212yesmaximal, complemented
5#13224C212yesmaximal, complemented
6#13091C214yesmaximal
7C4×#354C214yesmaximal, complemented
8C2×#533C411yescharacteristic, complemented
9C2×C4×#41C2211yescharacteristic
10C2×#410C2212yes—
11#2611—21no—
12#2611—21no—
13#2568C411yescharacteristic, complemented
14#2547C2211yescharacteristic
15#2546C411yescharacteristic, complemented
16#2542C2211yescharacteristic
17#2499C412yescomplemented
18#2432C2212yes—
19#2415—21no—
20#2415—21no—
21#2395—21no—
22#2395—21no—
23C4×C4⋊Q8C412yescomplemented
24C4×#124C2212yes—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—21no—
27#2283—21no—
28C4×#117—21no—
29C4×#117C411yescharacteristic, complemented
30C2×#354C2218yes—
31#1820—21no—
32#1799—21no—
33#1799C2212yes—
34#1782C2212yes—
35C4×#99—21no—
36C4×#99—21no—
37C4×#99—21no—
38C4×#99C2212yes—
39C4×#99C2212yes—
40C4×#99C2212yes—
41#1740—21no—
42#1393C2218yes—
43C22×#41C2311yescharacteristic
44#533—41no—
45#533—41no—
46#533—41no—
47#533—41no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#125—21no—
52C2×#124C2312yes—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2×C412yes—
57C4×#41—41no—
58C4×#41—41no—
59C4×#41—41no—
60C4×#41—41no—
61C2×C4×Q8—21no—
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C2×#117—21no—
73C2×#117—22no—
74C2×#117C2×C411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117D411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80#410—28no—
81C2×#99—22no—
82C2×#99—22no—
83C2×#99C2314yes—
84C2×#99C2314yes—
85#400—21no—
86#400—21no—
87#399—21no—
88#399—21no—
89#398—21no—
90#398—21no—
91#397—22no—
92#397—22no—
93#397—21no—
94#397—21no—
95#394—22no—
96#394—22no—
97#391—21no—
98#391—21no—
99#391C2×C412yes—
100#391C2×C412yes—
101#391C2×C412yes—
102#390C2312yes—
103#389—22no—
104#389—22no—
105#389—21no—
106#389—21no—
107#388—22no—
108#388—22no—
109#387—22no—
110#387—22no—
111#385—22no—
112#385—22no—
113#384C2×C412yes—
114#383—41no—
115#382—41no—
116#382C2×C412yes—
117#381—22no—
118#380—21no—
119#380—21no—
120#380—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—41no—
125C4×C4⋊C4—41no—
126C4×C4⋊C4—41no—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4C2×C412yes—
136#376—21no—
137#376—41no—
138#376—41no—
139C43—41no—
140C43C2×C412yes—
141#354—216no—
142#342—22no—
143#342—22no—
144#342—22no—
145#342—22no—
146C2×#41—24no—
147C2×#41—41no—
148C2×#41—41no—
149C2×#41—42no—
150C22×Q8—21no—
151C22×Q8C22×C412yes—
152C22×Q8C22×C411yescharacteristic
153C22×D4C2×D411yescharacteristic
154C22×D4C2×D411yescharacteristic
155C22×D4C2411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—44no—
170#125—42no—
171#125—42no—
172#124—28no—
173C22⋊Q8—44no—
174C22⋊Q8—44no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C4×Q8—41no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×D4—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—
201#117—28no—
202#117—41no—
203#117—41no—
204#117—41no—
205#117—41no—
206#117—41no—
207#117—41no—
208#117—44no—
209#117—44no—
210#117—42no—
211#117—42no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4C22⋊C411yescharacteristic
241C2×C22⋊C4C22⋊C411yescharacteristic
242C2×C22⋊C4C2×D411yescharacteristic
243C2×C22⋊C4C2×D411yescharacteristic
244C2×C22⋊C4C2×D411yescharacteristic
245C2×C22⋊C4C2×D411yescharacteristic
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22×C412yes—
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C412yes—
277#99—28no—
278#99—24no—
279#99—28no—
280#99—44no—
281#99#4118yes—
282#95—21no—
283#95—21no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—42no—
294#95—42no—
295#95—42no—
296#95—42no—
297#95—42no—
298#95—42no—
299C24C2×C22⋊C411yescharacteristic
300C24C22×D411yescharacteristic
301C24C22×D411yescharacteristic
302#41—48no—
303#41—48no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—41no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4C2×#4114yes—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C23×C411yescharacteristic
371C4⋊C4—24no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—84no—
393C22⋊C4—28no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—24no—
428C42—42no—
429C42—44no—
430C42—44no—
431C42—48no—
432C42—42no—
433C42—42no—
434C42—84no—
435C42—84no—
436C42—84no—
437C42#9911yescharacteristic
438C42#9911yescharacteristic
439C42#9911yescharacteristic
440C42#9911yescharacteristic
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23C22×C22⋊C411yescharacteristic
455C23#52512yes—
456C23#53212yes—
457Q8—44no—
458Q8—84no—
459Q8—82no—
460Q8—82no—
461D4—44no—
462D4—44no—
463D4—44no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4C2×#9911yescharacteristic
524C2×C4C2×#9911yescharacteristic
525C2×C4C2×#9911yescharacteristic
526C2×C4C2×#9911yescharacteristic
527C2×C4C2×#9911yescharacteristic
528C2×C4C2×#9911yescharacteristic
529C2×C4#41012yes—
530C2×C4C4×#4112yes—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—22no—
535C22—42no—
536C22—42no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22#229711yescharacteristic
549C22C22×#9911yescharacteristic
550C22C2×#41011yescharacteristic
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—22no—
556C4—22no—
557C4—44no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—82no—
566C4—88no—
567C4—82no—
568C4—82no—
569C4—164no—
570C4#212612yes—
571C4#212912yes—
572C2—21no—
573C2—21no—
574C2—42no—
575C2—42no—
576C2—81no—
577C2#2142011yescharacteristic
578C2C2×#212611yescharacteristic
579C2C2×#212911yescharacteristic
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. #13298C2semidirect
5. #13224C2semidirect
7. C4×#354C2semidirect
8. C2×#533C4semidirect
13. #2568C4semidirect
15. #2546C4semidirect
17. #2499C4semidirect
23. C4×C4⋊Q8C4semidirect
29. C4×#117C4semidirect

Maximal subgroups

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

subgroupindex
#349792
#207232
C4×#4102
#132982
#132242
#130912
C4×#3542