Finite Group Databasebeta

← #393468

Subgroups — #393468

Order 512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2274C211yesmaximal, characteristic
2#20846C211yesmaximal, characteristic
3C2×#1826C211yesmaximal, characteristic
4#20702C211yesmaximal, characteristic
5#13881C211yesmaximal, characteristic
6#10899C214yesmaximal, complemented
7#10892C214yesmaximal, complemented
8#10733C211yesmaximal, characteristic
9#10727C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C2×#428C2212yes—
14C2×#407C2211yescharacteristic
15C2×#407C2211yescharacteristic
16C2×#382C2212yes—
17C2×#380C2211yescharacteristic
18#2274—22no—
19#2274—22no—
20#2274—22no—
21#2274C2214yes—
22C2×C4×C4⋊C4—21no—
23C2×C4×C4⋊C4—21no—
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C22⋊C4—22no—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1854—22no—
29#1853—21no—
30#1852—21no—
31#1826C418yescomplemented
32#1825—22no—
33#1824—21no—
34#1823—21no—
35C2×#343C2214yes—
36#1736C2212yes—
37#1736C2212yes—
38#1736C2212yes—
39C2×#339C2214yes—
40C2×#339C2214yes—
41#1442—22no—
42C4⋊#95C2212yes—
43C22⋊#95—22no—
44C24×C4D412yes—
45C24×C4C2311yescharacteristic
46C22×M16C2312yes—
47C22×M16C2312yes—
48C2×#117—21no—
49C2×#117—21no—
50C2×#117—21no—
51C2×#117—21no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—28no—
55C2×#117C2×C414yes—
56C2×#117D412yes—
57C2×#117D412yes—
58C2×#117Q814yes—
59C2×#117C2312yes—
60C2×#117C2312yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—28no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42—22no—
81C22×C42—22no—
82C22×C42C2312yes—
83#428—28no—
84C2×#107—22no—
85C2×#106—22no—
86#407—22no—
87#407—22no—
88#407—22no—
89#407C2×C414yes—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94#382—28no—
95#380—21no—
96#380—21no—
97#380—21no—
98#380—21no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—24no—
107C4×C22⋊C4—28no—
108C4×C22⋊C4—28no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4C2×C414yes—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—24no—
115C2×#95—21no—
116C2×#95—24no—
117#343—28no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#339—28no—
125#339C2×C4116yescomplemented
126C25C2×D411yescharacteristic
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—28no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—44no—
140C23×C4—82no—
141C23×C4C22×C412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D414yes—
148C23×C4C2×Q812yes—
149C23×C4#4112yes—
150C23×C4#4112yes—
151C23×C4C2411yescharacteristic
152C2×M16—22no—
153C2×M16—24no—
154C2×M16—24no—
155C2×M16—24no—
156C2×M16—24no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16C22×C414yes—
160C22×C8—22no—
161C22×C8—22no—
162#117—28no—
163#117—28no—
164#117—24no—
165#117—24no—
166#117—24no—
167#117—24no—
168#117—24no—
169#117—22no—
170#117—22no—
171#117—24no—
172#117—24no—
173#117—24no—
174#117C4⋊C418yes—
175#117C22×C414yes—
176#117C2×Q814yes—
177#117#4118yes—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4C4⋊C414yes—
203C2×C4⋊C4C4⋊C414yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4#4118yes—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C42—28no—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—28no—
233C2×C42—28no—
234C2×C42—84no—
235C2×C42—84no—
236C2×C42C22×C414yes—
237#107—44no—
238#106—44no—
239C22⋊C8—24no—
240C22⋊C8—24no—
241C22⋊C8—42no—
242C22⋊C8—42no—
243#95—28no—
244#95—28no—
245#95—24no—
246#95—41no—
247#95—41no—
248#95—48no—
249#95—84no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—41no—
255C24—41no—
256C24C4×D412yes—
257C24#12011yescharacteristic
258C24#12011yescharacteristic
259C24C22⋊Q812yes—
260C24C22×D411yescharacteristic
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—28no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—28no—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—28no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—28no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—816no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4C2×C4⋊C412yes—
318C22×C4C2×C4⋊C412yes—
319C22×C4C4×D412yes—
320C22×C4C4×D412yes—
321C22×C4C4×D412yes—
322C22×C4#12011yescharacteristic
323C22×C4#12011yescharacteristic
324C22×C4C22⋊Q812yes—
325C22×C4C22⋊Q812yes—
326C22×C4C22⋊Q812yes—
327C22×C4#12312yes—
328C22×C4#12312yes—
329C22×C4#12318yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×Q811yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4114yes—
336C22×C4C2×#4114yes—
337C22×C4C2×#4111yescharacteristic
338M16—28no—
339M16—44no—
340M16—44no—
341M16—44no—
342C2×C8—24no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C4⋊C4—28no—
349C4⋊C4—28no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4C2×C4⋊C414yes—
365C4⋊C4C2×C4⋊C414yes—
366C4⋊C4#12314yes—
367C4⋊C4#12314yes—
368C22⋊C4—28no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—28no—
372C22⋊C4—28no—
373C22⋊C4—48no—
374C22⋊C4—82no—
375C22⋊C4—82no—
376C42—28no—
377C42—24no—
378C42—24no—
379C42—28no—
380C42—28no—
381C42—28no—
382C42—44no—
383C42—88no—
384C42—88no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—24no—
403C23—42no—
404C23—42no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23#38512yes—
410C23#38512yes—
411C23#45412yes—
412C23#45512yes—
413C23C2×C4×D411yescharacteristic
414C23C2×#12011yescharacteristic
415C23C2×C22⋊Q811yescharacteristic
416C23C2×#12312yes—
417C23#52512yes—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—48no—
438C2×C4—48no—
439C2×C4—48no—
440C2×C4—416no—
441C2×C4—48no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—48no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—84no—
457C2×C4—88no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4#38512yes—
471C2×C4#38512yes—
472C2×C4C22×C4⋊C411yescharacteristic
473C2×C4C2×C4×D411yescharacteristic
474C2×C4C4×#4114yes—
475C2×C4C2×C22⋊Q811yescharacteristic
476C2×C4C2×#12311yescharacteristic
477C2×C4C2×#12312yes—
478C2×C4#52512yes—
479C2×C4#52512yes—
480C2×C4#52512yes—
481C2×C4#52914yes—
482C2×C4#52914yes—
483C2×C4#53314yes—
484C8—44no—
485C8—82no—
486C22—22no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—22no—
491C22—21no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—48no—
504C22—48no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22#186014yes—
510C22C2×#38511yescharacteristic
511C22C4×#12312yes—
512C22#284911yescharacteristic
513C22#284911yescharacteristic
514C22#285512yes—
515C22C2×#45511yescharacteristic
516C22C2×#45411yescharacteristic
517C22#301911yescharacteristic
518C22#302011yescharacteristic
519C4—22no—
520C4—22no—
521C4—44no—
522C4—44no—
523C4—48no—
524C4—44no—
525C4—44no—
526C4—84no—
527C4—161no—
528C4—161no—
529C4#229612yes—
530C4C4×#12312yes—
531C4#285312yes—
532C4#285512yes—
533C2—22no—
534C2—22no—
535C2—24no—
536C2—42no—
537C2#1470511yescharacteristic
538C2C2×#186011yescharacteristic
539C2#2086411yescharacteristic
540C2#3299111yescharacteristic
541C2#3299311yescharacteristic
542C2#3299711yescharacteristic
543C2#3299911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #10899C2semidirect
7. #10892C2semidirect
31. #1826C4semidirect
125. #339C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22742
#208462
C2×#18262
#207022
#138812
#108992
#108922
#107332
#107272