Finite Group Databasebeta

← #188328

Subgroups — #188328

Order 512.

1684 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2342C211yesmaximal, characteristic
2#21932C212yesmaximal, complemented
3C2×#1462C211yesmaximal, characteristic
4C22×#323C211yesmaximal, characteristic
5#8561C214yesmaximal, complemented
6#8559C212yesmaximal, complemented
7#8557C212yesmaximal, complemented
8#7915C212yesmaximal, complemented
9C22×#117C411yescharacteristic
10C23×C22⋊C4C411yescharacteristic
11C23×C22⋊C4C2211yescharacteristic
12C2×#407C2212yes—
13C22×C22⋊C8C2212yes—
14#2342—22no—
15#2342—22no—
16#2342—24no—
17C2×#388C412yes—
18C2×#388C2212yes—
19C2×#385C412yes—
20C2×#381C411yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#379C411yescharacteristic
23C2×C4×C22⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1851—24no—
26#1850—22no—
27#1850—22no—
28#1797C2214yescomplemented
29#1797C2214yescomplemented
30#1462—24no—
31C2×#323—24no—
32C2×#323C2211yescharacteristic
33C2×#323C2212yes—
34C2×#323C2211yescharacteristic
35C2×#323C2212yes—
36C2×#323C2212yes—
37#1280C2214yescomplemented
38#1280C2214yescomplemented
39C26C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C412yes—
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C23×C8—22no—
45C2×#117—21no—
46C2×#117—22no—
47C2×#117—21no—
48C2×#117—44no—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C412yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—24no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—44no—
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4D414yes—
68C22×C22⋊C4Q814yes—
69C22×C22⋊C4C2311yescharacteristic
70C22×C22⋊C4C2311yescharacteristic
71C22×C42C2×C412yes—
72C22×C42C2312yes—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—24no—
75#407—24no—
76#407—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8C2314yes—
82C2×C22⋊C8C2312yes—
83C2×C22⋊C8C2312yes—
84#388—24no—
85#388—24no—
86#388—24no—
87#388—24no—
88#385—24no—
89#385—24no—
90#381—22no—
91#381—22no—
92#380—22no—
93#380—22no—
94#379—22no—
95#379—22no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4D412yes—
99C4×C22⋊C4D412yes—
100C4×C22⋊C4D414yes—
101C2×#95C2×C412yes—
102C2×#95C2×C412yes—
103C2×#95C2×C412yes—
104C2×#95C2312yes—
105#336—24no—
106#336—24no—
107#323—24no—
108#323—24no—
109#323—22no—
110#323—22no—
111#323—24no—
112C25—21no—
113C25—21no—
114C25—22no—
115C25—24no—
116C25—42no—
117C25—42no—
118C25—42no—
119C25—82no—
120C25C4⋊C414yes—
121C25C22×C411yescharacteristic
122C25C22×C411yescharacteristic
123C25C22×C411yescharacteristic
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—42no—
131C23×C4—44no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—82no—
135C23×C4—82no—
136C23×C4C4⋊C414yes—
137C23×C4C22×C412yes—
138C23×C4C22×C412yes—
139C23×C4C22×C412yes—
140C23×C4C22×C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×Q812yes—
148C23×C4#4114yes—
149C23×C4C2411yescharacteristic
150C2×M16—44no—
151C2×M16—84no—
152C22×C8—24no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—44no—
156C22×C8—84no—
157#117—44no—
158#117—48no—
159#117—44no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—82no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—48no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4C22⋊C411yescharacteristic
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22⋊C411yescharacteristic
201C2×C22⋊C4C4⋊C414yes—
202C2×C22⋊C4C4⋊C414yes—
203C2×C22⋊C4C2×D412yes—
204C2×C22⋊C4C2×D412yes—
205C2×C22⋊C4C2×Q812yes—
206C2×C22⋊C4C2×Q812yes—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—44no—
210C2×C42—82no—
211C2×C42C2×D414yes—
212C2×C42C2×D414yes—
213C4⋊C8—84no—
214C4⋊C8—84no—
215C22⋊C8—24no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—24no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—44no—
222C22⋊C8—44no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95C22⋊C414yes—
227#95C22⋊C414yes—
228C24—21no—
229C24—21no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—21no—
236C24—21no—
237C24—24no—
238C24—24no—
239C24—22no—
240C24—22no—
241C24—28no—
242C24—42no—
243C24—44no—
244C24—44no—
245C24—44no—
246C24—48no—
247C24—41no—
248C24—42no—
249C24—41no—
250C24—42no—
251C24—48no—
252C24—42no—
253C24—41no—
254C24—42no—
255C24—48no—
256C24—41no—
257C24—42no—
258C24—42no—
259C24—82no—
260C24—82no—
261C24—84no—
262C24—84no—
263C24—88no—
264C24—88no—
265C24—88no—
266C24—88no—
267C24C2×C22⋊C411yescharacteristic
268C24C2×C22⋊C411yescharacteristic
269C24C2×C4⋊C412yes—
270C24C2×C4⋊C412yes—
271C24C2×C4⋊C412yes—
272C24#11712yes—
273C24#13114yes—
274C24C23×C411yescharacteristic
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—48no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—48no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—84no—
322C22×C4—88no—
323C22×C4—88no—
324C22×C4C2×C22⋊C414yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C4⋊C412yes—
329C22×C4C2×C4⋊C412yes—
330C22×C4C2×C4⋊C412yes—
331C22×C4#11712yes—
332C22×C4#12114yes—
333C22×C4#12114yes—
334C22×C4C22⋊Q814yes—
335C22×C4C22⋊Q814yes—
336C22×C4C22×D412yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×Q811yescharacteristic
339C22×C4C2×#4112yes—
340M16—88no—
341C2×C8—44no—
342C2×C8—44no—
343C2×C8—44no—
344C2×C8—44no—
345C2×C8—44no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—88no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—84no—
352C22⋊C4—22no—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—28no—
359C22⋊C4—28no—
360C22⋊C4—24no—
361C22⋊C4—42no—
362C22⋊C4—48no—
363C22⋊C4—44no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—44no—
369C22⋊C4—84no—
370C22⋊C4#10014yes—
371C42—44no—
372C42—44no—
373C42—44no—
374C42—44no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—21no—
379C23—21no—
380C23—24no—
381C23—24no—
382C23—24no—
383C23—22no—
384C23—22no—
385C23—21no—
386C23—21no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—24no—
391C23—24no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—42no—
397C23—42no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—48no—
403C23—48no—
404C23—44no—
405C23—42no—
406C23—42no—
407C23—44no—
408C23—44no—
409C23—42no—
410C23—44no—
411C23—42no—
412C23—42no—
413C23—48no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—41no—
419C23—41no—
420C23—48no—
421C23—42no—
422C23—42no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23—84no—
427C23—84no—
428C23—88no—
429C23—88no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—816no—
435C23—816no—
436C23—832no—
437C23—88no—
438C23—88no—
439C23#38012yes—
440C23#38012yes—
441C23C2×#10012yes—
442C23#41312yes—
443C23#42414yes—
444C23C22×C22⋊C411yescharacteristic
445C23C22×C4⋊C411yescharacteristic
446C23C2×#11711yescharacteristic
447C23C2×#13112yes—
448C23#56812yes—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—88no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—88no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4#38014yes—
497C2×C4C2×#10014yes—
498C2×C4C2×#12112yes—
499C2×C4C2×C22⋊Q812yes—
500C8—84no—
501C8—82no—
502C8—82no—
503C22—22no—
504C22—22no—
505C22—24no—
506C22—22no—
507C22—22no—
508C22—24no—
509C22—22no—
510C22—21no—
511C22—21no—
512C22—22no—
513C22—21no—
514C22—21no—
515C22—22no—
516C22—24no—
517C22—22no—
518C22—22no—
519C22—44no—
520C22—44no—
521C22—48no—
522C22—44no—
523C22—44no—
524C22—48no—
525C22—44no—
526C22—42no—
527C22—42no—
528C22—44no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—816no—
535C22—82no—
536C22—82no—
537C22—84no—
538C22—816no—
539C22—82no—
540C22—82no—
541C22#180014yes—
542C22C2×#38011yescharacteristic
543C22#288012yes—
544C22#288312yes—
545C22C22×#10011yescharacteristic
546C22C2×#41311yescharacteristic
547C22C2×#42411yescharacteristic
548C22#290511yescharacteristic
549C22#292412yes—
550C4—24no—
551C4—22no—
552C4—22no—
553C4—24no—
554C4—42no—
555C4—44no—
556C4—42no—
557C4—42no—
558C4—82no—
559C4—82no—
560C2—21no—
561C2—21no—
562C2—22no—
563C2—24no—
564C2—42no—
565C2—44no—
566C2—82no—
567C2#1742112yes—
568C2#1742212yes—
569C2#2062811yescharacteristic
570C2#2063311yescharacteristic
571C2C2×#180011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21932C2semidirect
5. #8561C2semidirect
6. #8559C2semidirect
7. #8557C2semidirect
8. #7915C2semidirect
28. #1797C22semidirect
29. #1797C22semidirect
37. #1280C22semidirect
38. #1280C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23422
#219322
C2×#14622
C22×#3232
#85612
#85592
#85572
#79152