Finite Group Databasebeta

← #410877

Subgroups — #410877

Order 512.

1442 conjugacy classes of subgroups.

The subgroup lattice has 534 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#13881C211yesmaximal, characteristic
5#11339C214yesmaximal, 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×#384—21no—
25C2×#382—21no—
26C2×#382C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1854—22no—
30#1854—22no—
31#1853—21no—
32#1853—21no—
33#1852—21no—
34#1852—21no—
35C2×#343C2214yes—
36#1736C2212yes—
37#1736C2212yes—
38C2×#339C2214yes—
39C2×#339C2214yes—
40#1442—22no—
41C4⋊#95C2212yes—
42C22⋊#95—22no—
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—44no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2312yes—
79#428—28no—
80#428—44no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#106—22no—
84C2×#106—22no—
85#407—22no—
86#407—22no—
87#407Q814yes—
88#407Q814yes—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93#389—24no—
94#389—24no—
95#389—24no—
96#389—24no—
97#385—28no—
98#385—28no—
99#384—24no—
100#384—24no—
101#382—24no—
102#382—24no—
103#382—44no—
104#380—21no—
105#380—21no—
106#380—21no—
107#380—21no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4Q814yes—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—24no—
114C2×#95—21no—
115C2×#95—24no—
116#343—44no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121#339—28no—
122#339—44no—
123C25C2×D411yescharacteristic
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4—82no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D414yes—
144C23×C4C2×Q812yes—
145C23×C4C2×Q812yes—
146C23×C4C2×Q812yes—
147C23×C4#4112yes—
148C23×C4C2411yescharacteristic
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—24no—
152C2×M16—24no—
153C2×M16—24no—
154C2×M16—24no—
155C2×M16C2×Q814yes—
156C2×M16C2×Q814yes—
157C22×C8—22no—
158C22×C8—22no—
159#117—28no—
160#117—28no—
161#117—24no—
162#117—24no—
163#117—24no—
164#117—24no—
165#117—24no—
166#117—42no—
167#117—42no—
168#117—44no—
169#117—44no—
170#117#4118yes—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—84no—
199C2×C4⋊C4—84no—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C42—28no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—44no—
225C2×C42C2×Q814yes—
226#107—44no—
227#107—44no—
228#106—44no—
229#106—44no—
230C22⋊C8—24no—
231C22⋊C8—24no—
232C22⋊C8—24no—
233C22⋊C8—24no—
234#95—28no—
235#95—28no—
236#95—24no—
237#95—48no—
238#95—41no—
239#95—41no—
240#95—84no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—41no—
246C24—41no—
247C24#12011yescharacteristic
248C24#12011yescharacteristic
249C24C22⋊Q812yes—
250C24C4⋊Q812yes—
251C24C22×D411yescharacteristic
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
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—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—816no—
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4C22⋊Q812yes—
312C22×C4C22⋊Q812yes—
313C22×C4C22⋊Q812yes—
314C22×C4C22⋊Q812yes—
315C22×C4C22⋊Q812yes—
316C22×C4C4⋊Q812yes—
317C22×C4C4⋊Q812yes—
318C22×C4C4⋊Q812yes—
319C22×C4C4⋊Q812yes—
320C22×C4C4⋊Q812yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×Q811yescharacteristic
324C22×C4C22×Q811yescharacteristic
325C22×C4C22×Q811yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4114yes—
328C22×C4H(5,𝔽2)14yes—
329M16—28no—
330M16—28no—
331M16—44no—
332M16—44no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—88no—
358C4⋊C4—88no—
359C4⋊C4#12314yes—
360C4⋊C4#12314yes—
361C22⋊C4—28no—
362C22⋊C4—28no—
363C22⋊C4—48no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C42—28no—
369C42—28no—
370C42—28no—
371C42—44no—
372C42—44no—
373C42—44no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—24no—
383C23—24no—
384C23—24no—
385C23—24no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—42no—
393C23—42no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23#39512yes—
399C23#39512yes—
400C23#45412yes—
401C23#45512yes—
402C23C2×#12011yescharacteristic
403C23C2×C22⋊Q811yescharacteristic
404C23#52512yes—
405C23C2×C4⋊Q811yescharacteristic
406C23#53412yes—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—48no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4—88no—
449C2×C4—88no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4#39512yes—
460C2×C4#39512yes—
461C2×C4C2×C22⋊Q811yescharacteristic
462C2×C4C2×C22⋊Q811yescharacteristic
463C2×C4C2×#12312yes—
464C2×C4#52512yes—
465C2×C4C2×C4⋊Q811yescharacteristic
466C2×C4C2×C4⋊Q811yescharacteristic
467C2×C4#53412yes—
468C2×C4#53512yes—
469C2×C4#55314yes—
470C2×C4#55414yes—
471C2×C4#55414yes—
472C2×C4#55414yes—
473C8—44no—
474C8—44no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—48no—
493C22—48no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22#206114yes—
499C22C2×#39511yescharacteristic
500C22#257912yes—
501C22#258312yes—
502C22#261311yescharacteristic
503C22#284911yescharacteristic
504C22C2×#45511yescharacteristic
505C22C2×#45411yescharacteristic
506C22#301911yescharacteristic
507C22#302011yescharacteristic
508C4—22no—
509C4—22no—
510C4—44no—
511C4—44no—
512C4—84no—
513C4—82no—
514C4—84no—
515C4—82no—
516C4—161no—
517C4—161no—
518C4#248712yes—
519C4#250212yes—
520C4#257912yes—
521C4#258312yes—
522C2—22no—
523C2—24no—
524C2—22no—
525C2—42no—
526C2#1509211yescharacteristic
527C2C2×#206111yescharacteristic
528C2#2356111yescharacteristic
529C2#2961111yescharacteristic
530C2#2961211yescharacteristic
531C2#3299711yescharacteristic
532C2#3299911yescharacteristic
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. #11339C2semidirect
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
#138812
#113392
#112172
#107332
#107272