Finite Group Databasebeta

← #410876

Subgroups — #410876

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2#20846C211yesmaximal, characteristic
3#20846C211yesmaximal, characteristic
4#13880C211yesmaximal, characteristic
5#11340C214yesmaximal, complemented
6#11217C214yesmaximal, complemented
7#10733C211yesmaximal, characteristic
8#10733C211yesmaximal, characteristic
9#10727C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C2×#428C2212yes—
14C2×#428C2212yes—
15C2×#407C2211yescharacteristic
16C2×#407C2211yescharacteristic
17#2317—22no—
18#2317—22no—
19#2317—22no—
20#2317C2214yes—
21C2×#389—21no—
22C2×#389—21no—
23C2×#385—22no—
24C2×#385—22no—
25C2×#384—21no—
26C2×#384C2212yes—
27C2×#382—21no—
28C2×#380C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1854—22no—
31#1854—22no—
32#1853—21no—
33#1853—21no—
34#1852—21no—
35#1852—21no—
36C2×#343C2214yes—
37#1736C2212yes—
38#1736C2212yes—
39C2×#339C2214yes—
40C2×#339C2214yes—
41#1444—22no—
42C4⋊#95C2212yes—
43C24×C4D412yes—
44C24×C4C2311yescharacteristic
45C22×M16C2312yes—
46C22×M16C2312yes—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117—28no—
50C2×#117—21no—
51C2×#117—21no—
52C2×#117—44no—
53C2×#117D412yes—
54C2×#117D412yes—
55C2×#117C2312yes—
56C2×#117C2312yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—44no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D412yes—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2312yes—
80#428—28no—
81#428—44no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#106—22no—
85C2×#106—22no—
86#407—22no—
87#407—22no—
88#407Q814yes—
89#407Q814yes—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94#389—24no—
95#389—24no—
96#389—24no—
97#389—24no—
98#385—28no—
99#385—28no—
100#385—48no—
101#384—24no—
102#384—24no—
103#384—44no—
104#382—24no—
105#382—24no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380—21no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4Q814yes—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—24no—
116C2×#95—21no—
117#343—44no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#339—28no—
123#339—44no—
124C25C2×D411yescharacteristic
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—44no—
137C23×C4—44no—
138C23×C4—82no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D414yes—
145C23×C4C2×Q812yes—
146C23×C4C2×Q812yes—
147C23×C4C2×Q812yes—
148C23×C4#4112yes—
149C23×C4C2411yescharacteristic
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—24no—
153C2×M16—24no—
154C2×M16—24no—
155C2×M16—24no—
156C2×M16C2×Q814yes—
157C2×M16C2×Q814yes—
158C22×C8—22no—
159C22×C8—22no—
160#117—28no—
161#117—28no—
162#117—24no—
163#117—24no—
164#117—24no—
165#117—24no—
166#117—24no—
167#117—42no—
168#117—42no—
169#117—44no—
170#117—44no—
171#117#4118yes—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4—84no—
202C2×C4⋊C4—84no—
203C2×C4⋊C4—84no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C42—28no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—44no—
229C2×C42C2×Q814yes—
230#107—44no—
231#107—44no—
232#106—44no—
233#106—44no—
234C22⋊C8—24no—
235C22⋊C8—24no—
236C22⋊C8—24no—
237C22⋊C8—24no—
238#95—28no—
239#95—28no—
240#95—24no—
241#95—48no—
242#95—41no—
243#95—41no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—41no—
249C24—41no—
250C24#12011yescharacteristic
251C24#12011yescharacteristic
252C24C22⋊Q812yes—
253C24C4⋊Q812yes—
254C24C22×D411yescharacteristic
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—28no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—816no—
312C22×C4#12011yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4C22⋊Q812yes—
315C22×C4C22⋊Q812yes—
316C22×C4C22⋊Q812yes—
317C22×C4C22⋊Q812yes—
318C22×C4C22⋊Q812yes—
319C22×C4C4⋊Q812yes—
320C22×C4C4⋊Q812yes—
321C22×C4C4⋊Q812yes—
322C22×C4C4⋊Q812yes—
323C22×C4C4⋊Q812yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×Q811yescharacteristic
327C22×C4C22×Q811yescharacteristic
328C22×C4C22×Q811yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4114yes—
331C22×C4H(5,𝔽2)14yes—
332M16—28no—
333M16—28no—
334M16—44no—
335M16—44no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—88no—
361C4⋊C4—88no—
362C4⋊C4—816no—
363C4⋊C4#12314yes—
364C4⋊C4#12314yes—
365C22⋊C4—216no—
366C22⋊C4—28no—
367C22⋊C4—28no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—82no—
371C22⋊C4—82no—
372C42—28no—
373C42—28no—
374C42—28no—
375C42—44no—
376C42—44no—
377C42—44no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—24no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—42no—
397C23—42no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23#39512yes—
403C23#39512yes—
404C23#45312yes—
405C23#45612yes—
406C23C2×#12011yescharacteristic
407C23C2×C22⋊Q811yescharacteristic
408C23#52512yes—
409C23C2×C4⋊Q811yescharacteristic
410C23#53412yes—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—28no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—48no—
434C2×C4—48no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—88no—
453C2×C4—88no—
454C2×C4—84no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4#39512yes—
464C2×C4#39512yes—
465C2×C4C2×C22⋊Q811yescharacteristic
466C2×C4C2×C22⋊Q811yescharacteristic
467C2×C4C2×#12312yes—
468C2×C4#52512yes—
469C2×C4C2×C4⋊Q811yescharacteristic
470C2×C4C2×C4⋊Q811yescharacteristic
471C2×C4#53412yes—
472C2×C4#53512yes—
473C2×C4#55314yes—
474C2×C4#55414yes—
475C2×C4#55414yes—
476C2×C4#55414yes—
477C8—44no—
478C8—44no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—48no—
497C22—48no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22#206014yes—
503C22C2×#39511yescharacteristic
504C22#257912yes—
505C22#258312yes—
506C22#261311yescharacteristic
507C22#284911yescharacteristic
508C22C2×#45311yescharacteristic
509C22C2×#45611yescharacteristic
510C22#301911yescharacteristic
511C22#302011yescharacteristic
512C4—22no—
513C4—22no—
514C4—48no—
515C4—44no—
516C4—44no—
517C4—84no—
518C4—82no—
519C4—82no—
520C4—161no—
521C4—161no—
522C4#248712yes—
523C4#250212yes—
524C4#257912yes—
525C4#258312yes—
526C2—22no—
527C2—24no—
528C2—22no—
529C2—42no—
530C2#1509211yescharacteristic
531C2C2×#206011yescharacteristic
532C2#2356111yescharacteristic
533C2#2961011yescharacteristic
534C2#2961311yescharacteristic
535C2#3299611yescharacteristic
536C2#3300011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
5. #11340C2semidirect
6. #11217C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
#208462
#138802
#113402
#112172
#107332
#107272