Finite Group Databasebeta

← #186632

Subgroups — #186632

Order 512.

3518 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#547C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic, complemented
3#20732C211yesmaximal, characteristic, complemented
4#20646C211yesmaximal, characteristic, complemented
5C2×#1549C211yesmaximal, characteristic, complemented
6C2×#1486C211yesmaximal, characteristic, complemented
7C2×#1478C211yesmaximal, characteristic, complemented
8#8446C218yesmaximal, complemented
9C23×#41—21no—
10C2×#547—24no—
11C2×#547—22no—
12C2×#547—24no—
13C2×#547—22no—
14C2×#547—24no—
15C2×#547—22no—
16C2×#547C418yescomplemented
17C2×#547C2214yes—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#123—22no—
20C22×C22⋊Q8—21no—
21C22×#121—21no—
22C22×C4×D4C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C23×C4⋊C4—21no—
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#418—21no—
29#2285C2211yescharacteristic
30#1990C2214yescomplemented
31#1987—22no—
32#1824—22no—
33#1823—21no—
34#1804—22no—
35#1802C2214yescomplemented
36#1802C2214yescomplemented
37#1786—21no—
38C4×#102—22no—
39#1736—21no—
40#1549—24no—
41C2×#332C2214yescomplemented
42#1486—24no—
43#1478—24no—
44C2×D4⋊C8C2214yescomplemented
45C2×D4⋊C8C2214yescomplemented
46C22×#41—24no—
47C22×#41—28no—
48C22×#41—28no—
49C22×#41—24no—
50C22×#41—416no—
51C23×Q8—21no—
52C23×D4—22no—
53C23×D4D411yescharacteristic
54C23×D4D411yescharacteristic
55C24×C4—22no—
56C24×C4—21no—
57C24×C4—21no—
58C22×M16—21no—
59#547—216no—
60#547—216no—
61#547—216no—
62#547—216no—
63#547—28no—
64#547—216no—
65#547—216no—
66#547—28no—
67#547—28no—
68#547C2×C4116yes—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8C2×C414yes—
74C2×C4⋊Q8D414yes—
75C2×C4⋊Q8D412yes—
76C2×C4⋊Q8C2312yes—
77C2×#123—28no—
78C2×#123—216no—
79C2×#123—216no—
80C2×#123—28no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—24no—
85C2×#121—24no—
86C2×#121—28no—
87C2×#121—28no—
88C2×#121—24no—
89C2×C4×D4—21no—
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93C2×C4×D4—22no—
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×C4×D4—22no—
97C2×C4×D4C2×C414yes—
98C2×C4×D4C2×C414yes—
99C2×C4×D4D414yes—
100C2×C4×D4D414yes—
101C2×C4×D4D412yes—
102C2×C4×D4D412yes—
103C2×C4×D4C2312yes—
104C2×C4×D4C2312yes—
105C2×#117—21no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—28no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—28no—
111C22×C4⋊C4—24no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D412yes—
114C22×C4⋊C4D411yescharacteristic
115C22×C22⋊C4—22no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C42C2311yescharacteristic
119#431—22no—
120#431—22no—
121#431—22no—
122#431—22no—
123#423—22no—
124#423—22no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129#418—44no—
130C2×#102—24no—
131C2×#102—24no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C4×C8—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#340—28no—
145#332—28no—
146D4⋊C8—28no—
147D4⋊C8—28no—
148C25—21no—
149C25—21no—
150C2×#41—216no—
151C2×#41—232no—
152C2×#41—216no—
153C2×#41—432no—
154C2×#41—464no—
155C2×#41—464no—
156C2×#41—416no—
157C2×#41—416no—
158C22×Q8—24no—
159C22×Q8—22no—
160C22×Q8—24no—
161C22×Q8—22no—
162C22×Q8—48no—
163C22×D4—28no—
164C22×D4—28no—
165C22×D4—216no—
166C22×D4—22no—
167C22×D4—21no—
168C22×D4—22no—
169C22×D4—21no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—416no—
175C22×D4C22⋊C414yes—
176C22×D4C22⋊C414yes—
177C22×D4C2×D412yes—
178C22×D4C2×D412yes—
179C23×C4—28no—
180C23×C4—216no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—48no—
193C23×C4—416no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C2×M16—42no—
202C2×M16—42no—
203C2×M16—42no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C4⋊Q8—24no—
208C4⋊Q8—24no—
209C4⋊Q8—24no—
210C4⋊Q8—24no—
211C4⋊Q8—24no—
212C4⋊Q8—22no—
213C4⋊Q8—22no—
214C4⋊Q8C22⋊C418yes—
215C4⋊Q8C22×C414yes—
216C4⋊Q8C2×D414yes—
217#123—232no—
218#123—264no—
219#123—232no—
220C22⋊Q8—216no—
221C22⋊Q8—232no—
222C22⋊Q8—216no—
223#121—216no—
224#121—232no—
225#121—216no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—24no—
231C4×D4—24no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4C22⋊C418yes—
241C4×D4C22⋊C418yes—
242C4×D4C22×C414yes—
243C4×D4C22×C414yes—
244C4×D4C2×D414yes—
245C4×D4C2×D414yes—
246#117—42no—
247#117—42no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—216no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—232no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4C22⋊C414yes—
267C2×C4⋊C4C22⋊C414yes—
268C2×C4⋊C4C22⋊C418yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D414yes—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—41no—
283C2×C42—41no—
284C2×C42C22×C412yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2411yescharacteristic
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—42no—
297C4⋊C8—42no—
298#102—48no—
299#102—48no—
300C22⋊C8—42no—
301C22⋊C8—42no—
302C4×C8—22no—
303C4×C8—22no—
304C4×C8—42no—
305#95—41no—
306#95—41no—
307C24—28no—
308C24—22no—
309C24—24no—
310C24—22no—
311C24—24no—
312C24—44no—
313C24—44no—
314C24—44no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24#12011yescharacteristic
320#41—4128no—
321#41—432no—
322#41—464no—
323#41—432no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—28no—
327C2×Q8—48no—
328C2×Q8—416no—
329C2×Q8—416no—
330C2×Q8—48no—
331C2×D4—28no—
332C2×D4—216no—
333C2×D4—216no—
334C2×D4—232no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—28no—
345C2×D4—28no—
346C2×D4—464no—
347C2×D4—432no—
348C2×D4—44no—
349C2×D4—42no—
350C2×D4—44no—
351C2×D4—42no—
352C2×D4C2×C22⋊C414yes—
353C2×D4C2×C22⋊C414yes—
354C2×D4AGL(1,ℤ/8)18yes—
355C22×C4—24no—
356C22×C4—232no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—42no—
370C22×C4—48no—
371C22×C4—416no—
372C22×C4—416no—
373C22×C4—44no—
374C22×C4—48no—
375C22×C4—464no—
376C22×C4—432no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—48no—
383C22×C4—42no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—44no—
387C22×C4—48no—
388C22×C4—44no—
389C22×C4—48no—
390C22×C4—42no—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4#12012yes—
395C22×C4#12011yescharacteristic
396C22×C4#12011yescharacteristic
397C22×C4#12012yes—
398C22×C4#12012yes—
399C22×C4#12011yescharacteristic
400C22×C4#13512yes—
401C22×C4#13512yes—
402C22×C4AGL(1,ℤ/8)12yes—
403C22×C4#13712yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407M16—84no—
408C2×C8—44no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—44no—
415C4⋊C4—232no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—28no—
423C4⋊C4—24no—
424C4⋊C4—28no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4C2×C22⋊C414yes—
430C4⋊C4C2×C22⋊C414yes—
431C4⋊C4C2×C22⋊C418yes—
432C22⋊C4—232no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—28no—
438C22⋊C4—28no—
439C22⋊C4—82no—
440C42—82no—
441C42C2×C22⋊C412yes—
442C42C2×C22⋊C412yes—
443C42C2×C22⋊C412yes—
444C42#12012yes—
445C42#12012yes—
446C42#12012yes—
447C42C23×C411yescharacteristic
448C42C22×D411yescharacteristic
449C42C22×D411yescharacteristic
450C42C22×D411yescharacteristic
451C23—22no—
452C23—22no—
453C23—216no—
454C23—28no—
455C23—24no—
456C23—24no—
457C23—416no—
458C23—416no—
459C23—48no—
460C23—48no—
461C23—48no—
462C23—42no—
463C23—42no—
464C23—44no—
465C23—48no—
466C23—42no—
467C23—42no—
468C23—44no—
469C23—44no—
470C23—48no—
471C23—88no—
472C23—88no—
473C23#37912yes—
474C23#44912yes—
475C23#45212yes—
476C23C2×#12011yescharacteristic
477Q8—416no—
478Q8—48no—
479Q8—48no—
480D4—216no—
481D4—24no—
482D4—24no—
483D4—24no—
484D4—28no—
485D4—464no—
486D4—42no—
487D4—42no—
488D4—48no—
489D4#41818yescomplemented
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—416no—
501C2×C4—48no—
502C2×C4—416no—
503C2×C4—464no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—44no—
510C2×C4—48no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—416no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4#37912yes—
520C2×C4#37912yes—
521C2×C4#37912yes—
522C2×C4#41614yes—
523C2×C4#41714yes—
524C2×C4#44914yes—
525C2×C4#44912yes—
526C2×C4#45214yes—
527C2×C4#45212yes—
528C2×C4C22×C22⋊C411yescharacteristic
529C2×C4C22×C22⋊C411yescharacteristic
530C2×C4C22×C22⋊C411yescharacteristic
531C2×C4C2×#12011yescharacteristic
532C2×C4C2×#12011yescharacteristic
533C2×C4C2×#12011yescharacteristic
534C2×C4C2×#13511yescharacteristic
535C2×C4C2×#13511yescharacteristic
536C2×C4C2×AGL(1,ℤ/8)14yes—
537C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
538C2×C4C2×#13711yescharacteristic
539C8—82no—
540C8—82no—
541C8—82no—
542C22—24no—
543C22—24no—
544C22—22no—
545C22—22no—
546C22—28no—
547C22—432no—
548C22—48no—
549C22—48no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—816no—
556C22—84no—
557C22—84no—
558C22#178814yes—
559C22C2×#37911yescharacteristic
560C22C2×#44911yescharacteristic
561C22C2×#45211yescharacteristic
562C22#300312yes—
563C22#302012yes—
564C4—22no—
565C4—21no—
566C4—21no—
567C4—22no—
568C4—44no—
569C4—44no—
570C4—48no—
571C4—162no—
572C4#178814yes—
573C4C2×#41611yescharacteristic
574C4C2×#41812yes—
575C4C2×#41711yescharacteristic
576C4C2×#44911yescharacteristic
577C4C2×#45211yescharacteristic
578C4#300312yes—
579C2—24no—
580C2—48no—
581C2—84no—
582C2#1878212yes—
583C2#1878312yes—
584C2C2×#178811yescharacteristic
585C2#2059811yescharacteristic
586C2#2061711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#547C2
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. #21834C2semidirect
3. #20732C2semidirect
4. #20646C2semidirect
5. C2×#1549C2semidirect
6. C2×#1486C2semidirect
7. C2×#1478C2semidirect
8. #8446C2semidirect
16. C2×#547C4semidirect
30. #1990C22semidirect
35. #1802C22semidirect
36. #1802C22semidirect
41. C2×#332C22semidirect
44. C2×D4⋊C8C22semidirect
45. C2×D4⋊C8C22semidirect
489. D4#418semidirect

Maximal subgroups

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

subgroupindex
C22×#5472
#218342
#207322
#206462
C2×#15492
C2×#14862
C2×#14782
#84462