Finite Group Databasebeta

← #495579

Subgroups — #495579

Order 512.

1060 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34971C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3C4×#413C211yesmaximal, characteristic
4#13293C212yesmaximal, complemented
5#13217C212yesmaximal, complemented
6#13081C214yesmaximal
7C4×#352C214yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10C2×#413C2212yes—
11C2×C4×M16—21no—
12#2599—21no—
13#2599—21no—
14#2581—21no—
15#2536—21no—
16#2486C411yescharacteristic, complemented
17#2486C2211yescharacteristic
18#2484C412yescomplemented
19#2409C411yescharacteristic, complemented
20#2409C2211yescharacteristic
21#2406C2212yes—
22#2399—21no—
23#2398—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26C4×#123C412yescomplemented
27C4×#123C2212yes—
28#2283—21no—
29#2283C411yescharacteristic, complemented
30C2×#352C2218yes—
31#1820—21no—
32#1799—21no—
33#1799C2212yes—
34#1782C2212yes—
35C4×#101—21no—
36C4×#101—21no—
37C4×#100C2212yes—
38C4×#100C2212yes—
39C4×#99—21no—
40C4×#99C2212yes—
41#1742—21no—
42#1390C2218yes—
43C22×#41C2311yescharacteristic
44C23×Q8C2×C411yescharacteristic
45C22×M16—22no—
46C2×#123C2×C412yes—
47C2×#123C2312yes—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C4×#41—41no—
51C4×#41—41no—
52C4×#41—41no—
53C4×#41—41no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—41no—
59C2×C4×Q8—41no—
60C2×C4×Q8—41no—
61C2×C4×Q8—41no—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117D411yescharacteristic
72C2×#117D411yescharacteristic
73C22×C4⋊C4C2×C412yes—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C412yes—
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79#413—28no—
80C2×#101—22no—
81C2×#100C2314yes—
82C2×#99—22no—
83C2×#99C2314yes—
84C4×M16—22no—
85C4×M16—22no—
86C4×M16—41no—
87C4×M16—41no—
88C2×#97—41no—
89C2×C4×C8—41no—
90#400—22no—
91#398—21no—
92#398—22no—
93#398—21no—
94#397—21no—
95#397—21no—
96#395—22no—
97#395—21no—
98#395—21no—
99#394—22no—
100#393—21no—
101#393—21no—
102#391—21no—
103#391—21no—
104#389—21no—
105#389—21no—
106#388—22no—
107#388—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#386C2×C412yes—
113#386C2312yes—
114#385—22no—
115#385—22no—
116#385C2×C412yes—
117#385C2×C412yes—
118#383—21no—
119#383—41no—
120#382—21no—
121#382—41no—
122#381—22no—
123#381C2×C412yes—
124#381C2×C412yes—
125#381C2×C412yes—
126#380—21no—
127#380—21no—
128#380—22no—
129#380C2×C411yescharacteristic
130#380C2×C411yescharacteristic
131C4×C4⋊C4—22no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—41no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4C2×C412yes—
139#376—21no—
140#376—41no—
141#352—216no—
142#343—22no—
143#343—22no—
144#342—22no—
145#342—22no—
146C2×#41—24no—
147C2×#41—41no—
148C2×#41—41no—
149C2×#41—42no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8—44no—
155C22×Q8C2×D411yescharacteristic
156C22×D4C22×C411yescharacteristic
157C22×D4C2×D411yescharacteristic
158C22×D4C2411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C2×M16—28no—
171C2×M16—24no—
172C2×M16—44no—
173C2×M16—42no—
174C22×C8—42no—
175#123—28no—
176#123—44no—
177C22⋊Q8—44no—
178C22⋊Q8—44no—
179C4×Q8—41no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—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—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205#117—41no—
206#117—41no—
207#117—41no—
208#117—41no—
209#117—41no—
210#117—41no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
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—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4C22⋊C411yescharacteristic
238C2×C22⋊C4C22⋊C411yescharacteristic
239C2×C22⋊C4C22×C412yes—
240C2×C22⋊C4C22×C412yes—
241C2×C22⋊C4C22×C411yescharacteristic
242C2×C22⋊C4C22×C411yescharacteristic
243C2×C22⋊C4C2×D411yescharacteristic
244C2×C22⋊C4C2×D411yescharacteristic
245C2×C42—21no—
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—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273#101—28no—
274#100—24no—
275#100#4118yes—
276#99—28no—
277#99—44no—
278#97—42no—
279#97—42no—
280C4×C8—42no—
281C4×C8—42no—
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—22no—
294#95—22no—
295#95—42no—
296#95—42no—
297#95—42no—
298#95—42no—
299C24C2×C22⋊C411yescharacteristic
300C24C23×C411yescharacteristic
301C24C22×D411yescharacteristic
302#41—48no—
303#41—48no—
304C2×Q8—24no—
305C2×Q8—22no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—48no—
313C2×Q8—44no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4C2×#4114yes—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
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—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—44no—
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×C4C22×D411yescharacteristic
377M16—28no—
378M16—48no—
379M16—44no—
380C2×C8—44no—
381C2×C8—48no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—84no—
401C22⋊C4—24no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—42no—
413C22⋊C4—44no—
414C22⋊C4—44no—
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—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—44no—
434C42—44no—
435C42—42no—
436C42—42no—
437C42—84no—
438C42#9911yescharacteristic
439C42#9911yescharacteristic
440C42#9911yescharacteristic
441C42#9911yescharacteristic
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23#41012yes—
455C23C22×C22⋊C411yescharacteristic
456C23C4×#4112yes—
457C23#53212yes—
458Q8—44no—
459Q8—44no—
460Q8—48no—
461Q8—44no—
462D4—44no—
463D4—44no—
464D4—44no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—21no—
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—21no—
485C2×C4—21no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4C2×#9911yescharacteristic
526C2×C4C2×#9911yescharacteristic
527C2×C4C2×#9911yescharacteristic
528C2×C4C2×#9911yescharacteristic
529C2×C4C2×#9911yescharacteristic
530C2×C4C2×#9911yescharacteristic
531C2×C4#52512yes—
532C8—48no—
533C22—21no—
534C22—21no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—81no—
549C22—81no—
550C22#229711yescharacteristic
551C22C22×#9911yescharacteristic
552C22C2×#41011yescharacteristic
553C4—21no—
554C4—21no—
555C4—22no—
556C4—21no—
557C4—21no—
558C4—44no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—164no—
571C4#212012yes—
572C4#212312yes—
573C2—21no—
574C2—21no—
575C2—42no—
576C2—42no—
577C2—42no—
578C2#2142011yescharacteristic
579C2C2×#212011yescharacteristic
580C2C2×#212311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
socleC22C22×#99
prime radicalC22×#41C23
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
4. #13293C2semidirect
5. #13217C2semidirect
7. C4×#352C2semidirect
9. C22×C4×Q8C4semidirect
16. #2486C4semidirect
18. #2484C4semidirect
19. #2409C4semidirect
26. C4×#123C4semidirect
29. #2283C4semidirect

Maximal subgroups

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

subgroupindex
#349712
#207232
C4×#4132
#132932
#132172
#130812
C4×#3522