Finite Group Databasebeta

← #222569

Subgroups — #222569

Order 512.

896 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3385C211yesmaximal, characteristic
2#22767C211yesmaximal, characteristic
3Q8×#103C211yesmaximal, characteristic, complemented
4#22502C211yesmaximal, characteristic
5C2×#1588C211yesmaximal, characteristic
6#9831C211yesmaximal, characteristic, complemented
7#9730C211yesmaximal, characteristic, complemented
8#9223C211yesmaximal, characteristic
9#9216C211yesmaximal, characteristic, complemented
10#8976C211yesmaximal, characteristic
11#8975C211yesmaximal, characteristic, complemented
12#8655C211yesmaximal, characteristic, complemented
13#8451C211yesmaximal, characteristic, complemented
14#8450C211yesmaximal, characteristic
15#8213C211yesmaximal, characteristic, complemented
16C2×Q82C2211yescharacteristic
17C2×#554C2211yescharacteristic
18#3385—24no—
19#3385C418yescomplemented
20C2×#501—21no—
21C2×#500C2211yescharacteristic
22C2×#479—21no—
23C2×#477C2211yescharacteristic
24C2×#476C2211yescharacteristic
25C2×#474—21no—
26C2×#446C2211yescharacteristic
27C2×C8×Q8C2211yescharacteristic
28C2×#440—21no—
29C2×C4×SD16C2211yescharacteristic
30C2×#436C2211yescharacteristic
31#2354C2211yescharacteristic
32Q8×C4⋊C4C2211yescharacteristic
33#1990—21no—
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1987—21no—
37#1987C2211yescharacteristic
38#1987C2211yescharacteristic
39#1956C2211yescharacteristic
40#1956C2211yescharacteristic
41#1934—21no—
42#1920—21no—
43#1894—21no—
44#1880—21no—
45#1873—21no—
46#1834C2211yescharacteristic
47#1823—21no—
48#1803—21no—
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1802C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×#103—21no—
55C4×#103—21no—
56C4×#103C2211yescharacteristic
57#1736—21no—
58#1588—24no—
59C2×#335C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×Q8⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×#322C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1279C2211yescharacteristic
69#1277C2211yescharacteristic
70C22×SD16D411yescharacteristic
71C22×M16—21no—
72Q82—22no—
73Q82—22no—
74Q82—41no—
75Q82C2×C414yes—
76#554—22no—
77#554—22no—
78#554—41no—
79#554C2×C414yes—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—41no—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×#125—21no—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—41no—
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×C4×Q8C2311yescharacteristic
92C2×C4×Q8C2311yescharacteristic
93C2×C4×D4—21no—
94C2×C4×D4C2311yescharacteristic
95C2×#117—21no—
96C22×C4⋊C4—21no—
97#501—28no—
98#500—22no—
99#500C2×C414yes—
100#479—28no—
101#477—22no—
102#477C2×C414yes—
103#476—22no—
104#476C2×C414yes—
105#474—28no—
106#446—22no—
107#446C2×C414yes—
108C8×Q8—22no—
109C8×Q8—41no—
110#440—28no—
111C4×SD16C2×C414yes—
112C4×SD16D414yes—
113#436—22no—
114#436—41no—
115C2×#107—21no—
116C2×#107—21no—
117C2×#107—21no—
118C2×#106—21no—
119C2×#106D411yescharacteristic
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×#103—21no—
127C2×#103—21no—
128C2×#103—21no—
129C2×#103—21no—
130C2×#103—41no—
131C2×#103—41no—
132C2×#103D411yescharacteristic
133C2×#103D411yescharacteristic
134C2×#103D411yescharacteristic
135C2×#102—21no—
136C2×#102—21no—
137C2×#102D411yescharacteristic
138C2×C22⋊C8—21no—
139C2×#97—21no—
140C2×C4×C8—21no—
141C2×C4×C8—21no—
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145#390—21no—
146#389—21no—
147#389—41no—
148#388—41no—
149#384—21no—
150#384C2311yescharacteristic
151#384C2311yescharacteristic
152#380—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4C2311yescharacteristic
156#340—21no—
157#340—21no—
158#340—21no—
159#340—21no—
160#340—21no—
161#340—21no—
162#340—21no—
163#340—21no—
164#340—21no—
165#340—21no—
166#340—21no—
167#340—21no—
168#340—21no—
169#340—21no—
170#340D411yescharacteristic
171#340D411yescharacteristic
172#336—21no—
173#335—22no—
174#333—41no—
175#333—41no—
176Q8⋊C8—22no—
177Q8⋊C8—22no—
178D4⋊C8—22no—
179#322—22no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×Q8—41no—
183C22×Q8C2×D411yescharacteristic
184C22×Q8C2×D411yescharacteristic
185C22×D4C2×D411yescharacteristic
186C23×C4—21no—
187C23×C4—41no—
188C2×SD16—22no—
189C2×SD16—22no—
190C2×SD16C22⋊C414yes—
191C2×M16—42no—
192C2×M16—42no—
193C2×M16—81no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C4⋊Q8—24no—
204C4⋊Q8—24no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—42no—
212C4⋊Q8—42no—
213C4⋊Q8—44no—
214C4⋊Q8—44no—
215C4⋊Q8C22×C412yes—
216C4⋊Q8C22×C412yes—
217#125—24no—
218#125—42no—
219C22⋊Q8—24no—
220C22⋊Q8—44no—
221C22⋊Q8—44no—
222C22⋊Q8—42no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—42no—
230C4×Q8—42no—
231C4×Q8—42no—
232C4×Q8—81no—
233C4×Q8C22×C412yes—
234C4×Q8C22×C412yes—
235C4×Q8C2×D412yes—
236C4×Q8#4112yes—
237C4×D4—24no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—42no—
241C4×D4C22×C412yes—
242C4×D4C2×D412yes—
243#117—42no—
244#117—81no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2×D411yescharacteristic
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—41no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42C2411yescharacteristic
281#107—24no—
282#107—24no—
283#107—24no—
284#106—24no—
285#106C22⋊C414yes—
286C4⋊C8—24no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—42no—
295C4⋊C8C22×C412yes—
296#103—24no—
297#103—22no—
298#103—41no—
299#103—42no—
300#103—42no—
301#103—81no—
302#103—82no—
303#103—82no—
304#103C22⋊C414yes—
305#102—24no—
306#102—81no—
307#102C22⋊C414yes—
308C22⋊C8—42no—
309#97—24no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—42no—
315C4×C8—81no—
316C4×C8C22×C412yes—
317C4×C8C2×D412yes—
318#95—21no—
319#95—41no—
320#95—41no—
321C24—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—44no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—81no—
336C2×Q8—81no—
337C2×Q8—81no—
338C2×Q8—81no—
339C2×Q8—161no—
340C2×Q8—161no—
341C2×Q8C2×C22⋊C412yes—
342C2×Q8C4×D412yes—
343C2×Q8#13712yes—
344C2×Q8#13712yes—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4C2×C22⋊C412yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—44no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—161no—
370C22×C4—161no—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4H(5,𝔽2)11yescharacteristic
374SD16—28no—
375M16—84no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8C2×C22⋊C412yes—
402C2×C8C2×C22⋊C412yes—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—82no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C4⋊C4—161no—
437C4⋊C4C2×C22⋊C412yes—
438C4⋊C4C2×C22⋊C412yes—
439C4⋊C4C4×D412yes—
440C4⋊C4#13712yes—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—44no—
444C22⋊C4—82no—
445C22⋊C4—161no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—44no—
451C42—41no—
452C42—41no—
453C42—82no—
454C42—82no—
455C42C23×C411yescharacteristic
456C42C22×D411yescharacteristic
457C42C2×#4111yescharacteristic
458C42H(5,𝔽2)11yescharacteristic
459C23—22no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23D4211yescharacteristic
467Q8—22no—
468Q8—22no—
469Q8—22no—
470Q8—22no—
471Q8—22no—
472Q8—41no—
473Q8—41no—
474Q8—82no—
475Q8—82no—
476Q8—161no—
477Q8—161no—
478Q8—164no—
479Q8#41914yes—
480D4—22no—
481D4—22no—
482D4—41no—
483D4—41no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—164no—
524C2×C4C22×C22⋊C411yescharacteristic
525C2×C4C2×C4×D411yescharacteristic
526C2×C4#51811yescharacteristic
527C2×C4D4211yescharacteristic
528C2×C4#54611yescharacteristic
529C2×C4#54611yescharacteristic
530C2×C4C2×#13711yescharacteristic
531C2×C4C2×#13711yescharacteristic
532C2×C4C2×#13711yescharacteristic
533C2×C4#57511yescharacteristic
534C2×C4#57711yescharacteristic
535C2×C4#57711yescharacteristic
536C2×C4#57811yescharacteristic
537C2×C4#57811yescharacteristic
538C8—22no—
539C8—22no—
540C8—42no—
541C8—42no—
542C8—41no—
543C8—41no—
544C8—81no—
545C8—81no—
546C8—81no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22D4×C22⋊C411yescharacteristic
555C22#326611yescharacteristic
556C22#326611yescharacteristic
557C22#326911yescharacteristic
558C22#327011yescharacteristic
559C22#329011yescharacteristic
560C22#329211yescharacteristic
561C4—24no—
562C4—21no—
563C4—21no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—81no—
570C4—82no—
571C4—82no—
572C4—161no—
573C4—161no—
574C4C2×#41911yescharacteristic
575C4#289811yescharacteristic
576C4C4×#13711yescharacteristic
577C4#294711yescharacteristic
578C4#327511yescharacteristic
579C4#327611yescharacteristic
580C4#331211yescharacteristic
581C4#332011yescharacteristic
582C2—82no—
583C2#2004611yescharacteristic
584C2#2005011yescharacteristic
585C2#2009611yescharacteristic
586C2#2010011yescharacteristic
587C2#2242511yescharacteristic
588C2#2244411yescharacteristic
589C2#2246611yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. Q8×#103C2semidirect
6. #9831C2semidirect
7. #9730C2semidirect
9. #9216C2semidirect
11. #8975C2semidirect
12. #8655C2semidirect
13. #8451C2semidirect
15. #8213C2semidirect
19. #3385C4semidirect

Maximal subgroups

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

subgroupindex
C2×#33852
#227672
Q8×#1032
#225022
C2×#15882
#98312
#97302
#92232
#92162
#89762
#89752
#86552
#84512
#84502
#82132