Finite Group Databasebeta

← #186695

Subgroups — #186695

Order 512.

2602 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2#21202C211yesmaximal, characteristic, complemented
3#20740C211yesmaximal, characteristic, complemented
4#20738C211yesmaximal, characteristic
5C2×#1558C211yesmaximal, characteristic, complemented
6C2×#1491C211yesmaximal, characteristic, complemented
7C2×#1490C211yesmaximal, characteristic
8#8450C218yesmaximal, complemented
9C2×#554—24no—
10C2×#554—22no—
11C2×#554—24no—
12C2×#554—22no—
13C2×#554—24no—
14C2×#554—22no—
15C2×#554C418yescomplemented
16C2×#554C2214yes—
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125—21no—
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8—22no—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4—21no—
23C22×C4×D4C2211yescharacteristic
24C23×C4⋊C4—21no—
25C2×#432C2211yescharacteristic
26C2×#432C2211yescharacteristic
27C2×#419—22no—
28C2×C4×M16C2211yescharacteristic
29#2285C2211yescharacteristic
30#1960—22no—
31#1956C2214yescomplemented
32#1824—21no—
33#1823—21no—
34#1803C2214yescomplemented
35#1802C2214yes—
36#1787—21no—
37#1786—21no—
38#1763—22no—
39#1762—22no—
40#1558—24no—
41C2×#333C2214yescomplemented
42#1491—24no—
43#1490—24no—
44C2×Q8⋊C8C2214yescomplemented
45C2×D4⋊C8C2214yes—
46C23×Q8D412yes—
47C23×Q8D411yescharacteristic
48C23×D4D411yescharacteristic
49C24×C4—22no—
50C24×C4—21no—
51C22×M16—21no—
52#554—216no—
53#554—216no—
54#554—216no—
55#554—216no—
56#554—28no—
57#554—216no—
58#554—28no—
59#554—216no—
60#554—28no—
61#554C2×C4116yes—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8C2×C414yes—
67C2×C4⋊Q8D412yes—
68C2×C4⋊Q8D414yes—
69C2×C4⋊Q8C2312yes—
70C2×#125—28no—
71C2×#125—24no—
72C2×#125—28no—
73C2×#125—24no—
74C2×C22⋊Q8—24no—
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—216no—
78C2×C22⋊Q8—28no—
79C2×C22⋊Q8—216no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—28no—
82C2×C4×Q8—22no—
83C2×C4×Q8—21no—
84C2×C4×Q8—22no—
85C2×C4×Q8—21no—
86C2×C4×Q8C2×C414yes—
87C2×C4×Q8D414yes—
88C2×C4×Q8D412yes—
89C2×C4×Q8C2312yes—
90C2×C4×D4—28no—
91C2×C4×D4—24no—
92C2×C4×D4—28no—
93C2×C4×D4—24no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4C2×C414yes—
99C2×C4×D4D414yes—
100C2×C4×D4D412yes—
101C2×C4×D4C2312yes—
102C2×#117—21no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—28no—
111C22×C4⋊C4—24no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C22⋊C4—22no—
115C22×C22⋊C4—21no—
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118#432—22no—
119#432—22no—
120#432—22no—
121#432—22no—
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2312yes—
124#419—48no—
125C2×#103—24no—
126C2×#103—28no—
127C2×#102—24no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C4×M16—22no—
131C4×M16—22no—
132C2×#97—21no—
133C2×#97—21no—
134C2×#97—21no—
135C2×C4×C8C2312yes—
136#384C2312yes—
137#382—21no—
138#380—21no—
139#380—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142#340—24no—
143#340—24no—
144#333—28no—
145Q8⋊C8—28no—
146D4⋊C8—28no—
147C25—21no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—24no—
152C22×Q8—44no—
153C22×Q8—44no—
154C22×Q8—44no—
155C22×Q8C22⋊C414yes—
156C22×Q8C22⋊C418yes—
157C22×Q8C2×D412yes—
158C22×Q8C2×D414yes—
159C22×D4—28no—
160C22×D4—22no—
161C22×D4—21no—
162C22×D4C22⋊C414yes—
163C22×D4C2×D412yes—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—28no—
169C23×C4—216no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—48no—
175C23×C4—84no—
176C23×C4—84no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C2×M16—24no—
181C2×M16—42no—
182C2×M16—42no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C4⋊Q8—24no—
187C4⋊Q8—24no—
188C4⋊Q8—24no—
189C4⋊Q8—24no—
190C4⋊Q8—24no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8C22⋊C418yes—
194C4⋊Q8C22×C414yes—
195C4⋊Q8C2×D414yes—
196#125—216no—
197#125—232no—
198#125—216no—
199C22⋊Q8—216no—
200C22⋊Q8—232no—
201C22⋊Q8—232no—
202C22⋊Q8—264no—
203C22⋊Q8—216no—
204C22⋊Q8—232no—
205C4×Q8—24no—
206C4×Q8—24no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8C22⋊C418yes—
213C4×Q8C22×C414yes—
214C4×Q8C2×D414yes—
215C4×D4—216no—
216C4×D4—232no—
217C4×D4—216no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4C22⋊C418yes—
226C4×D4C22×C414yes—
227C4×D4C2×D414yes—
228#117—44no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—216no—
231C2×C4⋊C4—216no—
232C2×C4⋊C4—232no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—216no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—216no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4C22⋊C414yes—
256C2×C4⋊C4C22⋊C414yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—216no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—41no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—41no—
271C2×C42C22×C412yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2411yescharacteristic
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283#103—416no—
284#103—48no—
285#102—48no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288#97—42no—
289#97—42no—
290#97—42no—
291C4×C8—22no—
292C4×C8—22no—
293#95—41no—
294#95—41no—
295C24—28no—
296C24—22no—
297C24—24no—
298C24—44no—
299C24—42no—
300C24—42no—
301C24#12011yescharacteristic
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—28no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—48no—
309C2×Q8—44no—
310C2×Q8—48no—
311C2×Q8—44no—
312C2×Q8—416no—
313C2×Q8—44no—
314C2×Q8—88no—
315C2×Q8—82no—
316C2×Q8—82no—
317C2×Q8C2×C22⋊C414yes—
318C2×Q8C2×C22⋊C418yes—
319C2×D4—28no—
320C2×D4—216no—
321C2×D4—216no—
322C2×D4—22no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4C2×C22⋊C414yes—
326C2×D4#13718yes—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—232no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—28no—
343C22×C4—48no—
344C22×C4—416no—
345C22×C4—416no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—48no—
353C22×C4—42no—
354C22×C4—832no—
355C22×C4—88no—
356C22×C4—88no—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C412yes—
360C22×C4#12012yes—
361C22×C4#12012yes—
362C22×C4#12012yes—
363C22×C4#12011yescharacteristic
364C22×C4#12011yescharacteristic
365C22×C4#12011yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369M16—48no—
370C2×C8—24no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—44no—
376C2×C8—44no—
377C4⋊C4—232no—
378C4⋊C4—232no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—28no—
389C4⋊C4—28no—
390C4⋊C4—28no—
391C4⋊C4—24no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4C2×C22⋊C414yes—
396C4⋊C4C2×C22⋊C414yes—
397C22⋊C4—232no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—28no—
401C22⋊C4—82no—
402C42—24no—
403C42—24no—
404C42—28no—
405C42—82no—
406C42C2×C22⋊C412yes—
407C42C2×C22⋊C412yes—
408C42C2×C22⋊C412yes—
409C42#12012yes—
410C42#12012yes—
411C42#12012yes—
412C42C23×C411yescharacteristic
413C42C22×D411yescharacteristic
414C42C22×D411yescharacteristic
415C42C22×D411yescharacteristic
416C23—22no—
417C23—22no—
418C23—216no—
419C23—28no—
420C23—24no—
421C23—48no—
422C23—416no—
423C23—416no—
424C23—48no—
425C23—44no—
426C23—42no—
427C23—42no—
428C23#37912yes—
429C23#45312yes—
430C23#45512yes—
431C23C2×#12011yescharacteristic
432Q8—24no—
433Q8—24no—
434Q8—48no—
435Q8—48no—
436Q8—44no—
437Q8—84no—
438Q8—88no—
439D4—216no—
440D4—24no—
441D4—28no—
442D4#41918yescomplemented
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—28no—
459C2×C4—416no—
460C2×C4—48no—
461C2×C4—48no—
462C2×C4—48no—
463C2×C4—416no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—48no—
471C2×C4—48no—
472C2×C4—832no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4#37912yes—
477C2×C4#37912yes—
478C2×C4#37912yes—
479C2×C4C22×C22⋊C411yescharacteristic
480C2×C4C22×C22⋊C411yescharacteristic
481C2×C4C22×C22⋊C411yescharacteristic
482C2×C4C2×#12011yescharacteristic
483C2×C4C2×#12011yescharacteristic
484C2×C4C2×#12011yescharacteristic
485C2×C4C2×#13714yes—
486C2×C4#57514yes—
487C8—44no—
488C8—82no—
489C8—82no—
490C22—24no—
491C22—24no—
492C22—22no—
493C22—22no—
494C22—28no—
495C22—432no—
496C22—48no—
497C22—48no—
498C22—44no—
499C22—44no—
500C22#179214yes—
501C22C2×#37911yescharacteristic
502C22#300412yes—
503C22#300512yes—
504C22C2×#45311yescharacteristic
505C22C2×#45511yescharacteristic
506C4—22no—
507C4—22no—
508C4—21no—
509C4—21no—
510C4—22no—
511C4—44no—
512C4—44no—
513C4—44no—
514C4—48no—
515C4—162no—
516C4C2×#41912yes—
517C4#289612yes—
518C4#300412yes—
519C4#300512yes—
520C2—24no—
521C2—48no—
522C2#1852512yes—
523C2#1853012yes—
524C2#2059911yescharacteristic
525C2#2060011yescharacteristic
526C2C2×#179211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21202C2semidirect
3. #20740C2semidirect
5. C2×#1558C2semidirect
6. C2×#1491C2semidirect
8. #8450C2semidirect
15. C2×#554C4semidirect
31. #1956C22semidirect
34. #1803C22semidirect
41. C2×#333C22semidirect
44. C2×Q8⋊C8C22semidirect
442. D4#419semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
#212022
#207402
#207382
C2×#15582
C2×#14912
C2×#14902
#84502