Finite Group Databasebeta

← #481865

Subgroups — #481865

Order 512.

1025 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2396C211yesmaximal, characteristic
2#20694C211yesmaximal, characteristic
3#17357C211yesmaximal, characteristic
4#11816C214yesmaximal, complemented
5#11815C212yesmaximal, complemented
6#11814C212yesmaximal, complemented
7#11721C214yesmaximal, complemented
8C22×C4×Q8C411yescharacteristic
9C22×#117C2211yescharacteristic
10#2920—21no—
11C2×#407—21no—
12#2396—21no—
13#2396—21no—
14#2396—21no—
15#2396—21no—
16#2396C412yes—
17#2396C412yes—
18#2396C2212yes—
19#2396C2212yes—
20C2×#400C2211yescharacteristic
21C2×#399C411yescharacteristic
22C2×#398—21no—
23C2×#396—21no—
24C2×#395C2211yescharacteristic
25C2×#393C411yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C411yescharacteristic
28C2×C4×C22⋊C4—21no—
29#1822—21no—
30#1822C2212yes—
31#1821—21no—
32#1821C2212yes—
33#1742—21no—
34#1742C2212yes—
35#1464—21no—
36#1464C2212yes—
37#1463C2212yes—
38#1462—21no—
39#1458—21no—
40#1458C2212yes—
41#1354C2218yescomplemented
42#1353C2218yescomplemented
43C23×Q8C2×C411yescharacteristic
44C24×C4D411yescharacteristic
45C22×M16—22no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—41no—
49C2×C4×Q8—41no—
50C2×C4×Q8—41no—
51C2×C4×Q8—41no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C412yes—
54C2×#117—41no—
55C2×#117—41no—
56C2×#117—41no—
57C2×#117—41no—
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C411yescharacteristic
60C2×#117D411yescharacteristic
61C2×#117D411yescharacteristic
62C2×#117D411yescharacteristic
63C2×#117D411yescharacteristic
64C2×#117C2311yescharacteristic
65C2×#117C2311yescharacteristic
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2×C411yescharacteristic
80C22×C42D411yescharacteristic
81#432—22no—
82#432—41no—
83#431—22no—
84#431—41no—
85#423—41no—
86#407—22no—
87#407—22no—
88#407—22no—
89#407—41no—
90#407—41no—
91C2×C22⋊C8—41no—
92C2×C22⋊C8—41no—
93C4×M16—41no—
94#400—21no—
95#400—21no—
96#400C2×C412yes—
97#400C2312yes—
98#399—21no—
99#399—21no—
100#399C2×C412yes—
101#399C2×C412yes—
102#398—22no—
103#398—22no—
104#398—22no—
105#398—22no—
106#396—22no—
107#396—22no—
108#396—22no—
109#396—22no—
110#395—21no—
111#395—21no—
112#395C2×C412yes—
113#395C2312yes—
114#393—21no—
115#393—21no—
116#393C2×C412yes—
117#393C2×C412yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C412yes—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95C2×C411yescharacteristic
135C2×#95C2×C411yescharacteristic
136#343—22no—
137#343—22no—
138#343—22no—
139#343C2314yes—
140#324—22no—
141#324C2314yes—
142#323—22no—
143#323—22no—
144C25C22⋊C411yescharacteristic
145C22×Q8—21no—
146C22×Q8—44no—
147C22×Q8—41no—
148C22×Q8—41no—
149C22×Q8C22×C412yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—24no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—42no—
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C2×M16—24no—
171C2×M16—42no—
172C2×M16—42no—
173C2×M16—44no—
174C2×M16—44no—
175C22×C8—42no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—42no—
184#117—41no—
185#117—41no—
186#117—41no—
187#117—41no—
188#117—41no—
189#117—41no—
190#117—41no—
191#117—41no—
192#117—41no—
193#117—41no—
194#117—41no—
195#117—41no—
196#117—41no—
197#117—41no—
198#117—41no—
199#117—41no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2411yescharacteristic
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4C22⋊C411yescharacteristic
240C2×C22⋊C4C22⋊C411yescharacteristic
241C2×C22⋊C4C22⋊C411yescharacteristic
242C2×C22⋊C4C22⋊C411yescharacteristic
243C2×C22⋊C4C22×C411yescharacteristic
244C2×C22⋊C4C22×C411yescharacteristic
245C2×C22⋊C4C2×D411yescharacteristic
246C2×C22⋊C4C2×D411yescharacteristic
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—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—44no—
260C2×C42—41no—
261C2×C42—41no—
262C2×C42—41no—
263C2×C42—41no—
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C4⋊C8—42no—
273C4⋊C8—42no—
274C22⋊C8—24no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278C22⋊C8—42no—
279C22⋊C8—42no—
280#97—42no—
281C4×C8—42no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—21no—
288#95—21no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95C22×C412yes—
293#95C22×C412yes—
294C24—22no—
295C24—21no—
296C24—21no—
297C24—41no—
298C24—42no—
299C24—42no—
300C24C2×C22⋊C411yescharacteristic
301C24C2×C22⋊C411yescharacteristic
302C24C2×C22⋊C411yescharacteristic
303C2×Q8—44no—
304C2×Q8—42no—
305C2×Q8—44no—
306C2×Q8—42no—
307C2×Q8—48no—
308C2×Q8—44no—
309C2×Q8—44no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
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—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4#9911yescharacteristic
366C22×C4#9911yescharacteristic
367C22×C4#9911yescharacteristic
368C22×C4#9911yescharacteristic
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×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4#13714yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382M16—44no—
383M16—44no—
384M16—84no—
385M16—84no—
386C2×C8—44no—
387C2×C8—44no—
388C2×C8—84no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
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—21no—
434C42—21no—
435C42—44no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—44no—
441C23—22no—
442C23—21no—
443C23—22no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—42no—
451C23—42no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—48no—
457C23—42no—
458C23—42no—
459C23—81no—
460C23—81no—
461C23—84no—
462C23C2×#9911yescharacteristic
463C23C2×#9911yescharacteristic
464C23#41912yes—
465C23C22×C22⋊C411yescharacteristic
466Q8—44no—
467Q8—44no—
468Q8—48no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—84no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4C2×#9911yescharacteristic
533C2×C4C2×#9911yescharacteristic
534C2×C4C2×#9911yescharacteristic
535C2×C4C2×#9911yescharacteristic
536C2×C4#41912yes—
537C2×C4#42112yes—
538C2×C4C2×#13712yes—
539C8—84no—
540C8—84no—
541C22—24no—
542C22—24no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—22no—
550C22—42no—
551C22—42no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—81no—
558C22—84no—
559C22—84no—
560C22—81no—
561C22#212012yes—
562C22#213212yes—
563C22C22×#9911yescharacteristic
564C22C2×#41911yescharacteristic
565C22C2×#42111yescharacteristic
566C4—22no—
567C4—21no—
568C4—21no—
569C4—44no—
570C4—41no—
571C4—44no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—82no—
581C4—82no—
582C4#151112yes—
583C4#151112yes—
584C2—21no—
585C2—21no—
586C2—22no—
587C2—22no—
588C2—42no—
589C2—81no—
590C2C2×#151111yescharacteristic
591C2C2×#212011yescharacteristic
592C2C2×#213211yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
socleC22C22×#99
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11816C2semidirect
5. #11815C2semidirect
6. #11814C2semidirect
7. #11721C2semidirect
41. #1354C22semidirect
42. #1353C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23962
#206942
#173572
#118162
#118152
#118142
#117212