Finite Group Databasebeta

← #231190

Subgroups — #231190

Order 512.

672 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23700C211yesmaximal, characteristic
2#22986C211yesmaximal, characteristic
3#22956C211yesmaximal, characteristic, complemented
4#9557C211yesmaximal, characteristic, complemented
5#9554C211yesmaximal, characteristic, complemented
6#9539C211yesmaximal, characteristic, complemented
7#9538C211yesmaximal, characteristic, complemented
8#8952C211yesmaximal, characteristic, complemented
9#8945C211yesmaximal, characteristic, complemented
10#8942C211yesmaximal, characteristic, complemented
11#8939C211yesmaximal, characteristic
12#8207C211yesmaximal, characteristic, complemented
13#8202C211yesmaximal, characteristic, complemented
14#8148C211yesmaximal, characteristic, complemented
15#7877C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×#435—21no—
19C2×C8×D4C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24C4×#121C2211yescharacteristic
25#2088C411yescharacteristic
26#2077C411yescharacteristic
27#2066C411yescharacteristic
28#2065C411yescharacteristic
29#2055C411yescharacteristic
30#2035C411yescharacteristic
31#2034C411yescharacteristic
32#2000C411yescharacteristic
33#1984C2211yescharacteristic
34#1954C2211yescharacteristic
35#1940—21no—
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1924—21no—
39#1917—21no—
40#1870—21no—
41#1850—21no—
42#1850—21no—
43C4×#106C2211yescharacteristic
44#1770C2211yescharacteristic
45#1770C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1752—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×M16—21no—
69C23×C8—21no—
70C23×C8—21no—
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×#121C2×C411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#446—22no—
78#446—41no—
79C4×SD16—41no—
80C4×SD16—41no—
81#435—44no—
82C8×D4—22no—
83C8×D4—41no—
84#431—41no—
85#431—41no—
86C2×#106C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88C2×#106C2×C411yescharacteristic
89#423—22no—
90#423—41no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—41no—
98C2×C4⋊C8C2×C411yescharacteristic
99C2×C4⋊C8C2×C411yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×#103C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8C2×C411yescharacteristic
115C2×C22⋊C8C2×C411yescharacteristic
116C2×C22⋊C8D411yescharacteristic
117C2×C22⋊C8D411yescharacteristic
118C2×#97—21no—
119C2×#97—21no—
120C2×#97—21no—
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#391C2×C411yescharacteristic
128#390C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#388—21no—
132#387—21no—
133#386—21no—
134#385—21no—
135#380C2×C411yescharacteristic
136#380C2×C411yescharacteristic
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336D411yescharacteristic
156#336D411yescharacteristic
157Q8⋊C8—22no—
158Q8⋊C8—41no—
159D4⋊C8—22no—
160D4⋊C8—41no—
161#322—41no—
162C82—41no—
163C22×Q8C22×C411yescharacteristic
164C22×D4—21no—
165C22×D4C22×C411yescharacteristic
166C23×C4—21no—
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4#4111yescharacteristic
170C2×SD16—22no—
171C2×SD16—41no—
172C2×SD16—41no—
173C2×M16—42no—
174C2×M16—42no—
175C2×M16—81no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—42no—
183C22×C8—42no—
184C22×C8—42no—
185C22×C8—42no—
186C22×C8—81no—
187C22×C8—81no—
188C22×C8C22×C411yescharacteristic
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C22×C411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198#121—22no—
199#121—41no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—42no—
209C4×D4—42no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42C2411yescharacteristic
238#106—41no—
239#106—41no—
240#106—41no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—41no—
248C4⋊C8—41no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252C4⋊C8—42no—
253C4⋊C8—81no—
254C4⋊C8#4112yes—
255C4⋊C8#4112yes—
256#103—41no—
257#103—41no—
258#103—41no—
259#102—22no—
260#102—41no—
261#102—41no—
262C22⋊C8—22no—
263C22⋊C8—41no—
264C22⋊C8—41no—
265C22⋊C8—42no—
266C22⋊C8—42no—
267C22⋊C8—42no—
268C22⋊C8—81no—
269C22⋊C8SD1614yescomplemented
270#97—42no—
271#97—42no—
272#97—81no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293C24—21no—
294C24C4×D411yescharacteristic
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—81no—
309C2×D4#13112yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4C4×D411yescharacteristic
340C22×C4C2×SD1612yes—
341C22×C4#13512yes—
342C22×C4C23×C411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C2×#4111yescharacteristic
345SD16—44no—
346M16—84no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—84no—
388C2×C8—84no—
389C2×C8C2×SD1612yes—
390C2×C8C2×SD1612yes—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—42no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—81no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—42no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42C2×#4111yescharacteristic
429C42C2×#4111yescharacteristic
430C42H(5,𝔽2)11yescharacteristic
431C42H(5,𝔽2)11yescharacteristic
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—22no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23C4×SD1612yes—
446C23C2×C4×D411yescharacteristic
447Q8—41no—
448Q8—41no—
449Q8—41no—
450Q8—41no—
451D4—22no—
452D4—22no—
453D4—41no—
454D4—41no—
455D4—84no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4C4×SD1612yes—
502C2×C4C4×#4111yescharacteristic
503C2×C4#51811yescharacteristic
504C2×C4#54611yescharacteristic
505C2×C4#54611yescharacteristic
506C2×C4#55311yescharacteristic
507C2×C4#55311yescharacteristic
508C2×C4C2×#13111yescharacteristic
509C2×C4#56811yescharacteristic
510C2×C4#56811yescharacteristic
511C2×C4#56811yescharacteristic
512C2×C4C22×SD1611yescharacteristic
513C2×C4C2×#13511yescharacteristic
514C2×C4#57611yescharacteristic
515C2×C4#57811yescharacteristic
516C8—22no—
517C8—22no—
518C8—42no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—42no—
524C8—41no—
525C8—42no—
526C8—41no—
527C8—81no—
528C8—81no—
529C8—81no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22#236411yescharacteristic
545C22#293011yescharacteristic
546C22#293311yescharacteristic
547C22C2×C4×SD1611yescharacteristic
548C22#295111yescharacteristic
549C22#295511yescharacteristic
550C22#295611yescharacteristic
551C4—22no—
552C4—21no—
553C4—21no—
554C4—41no—
555C4—41no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4#296711yescharacteristic
564C4#296811yescharacteristic
565C4#297611yescharacteristic
566C4#298011yescharacteristic
567C4#329511yescharacteristic
568C4#329711yescharacteristic
569C4#329811yescharacteristic
570C4#330011yescharacteristic
571C2—42no—
572C2—82no—
573C2#1792911yescharacteristic
574C2#1795411yescharacteristic
575C2#1799511yescharacteristic
576C2#1802711yescharacteristic
577C2#2312411yescharacteristic
578C2#2312611yescharacteristic
579C2#2314811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22956C2semidirect
4. #9557C2semidirect
5. #9554C2semidirect
6. #9539C2semidirect
7. #9538C2semidirect
8. #8952C2semidirect
9. #8945C2semidirect
10. #8942C2semidirect
12. #8207C2semidirect
13. #8202C2semidirect
14. #8148C2semidirect
15. #7877C2semidirect
269. C22⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#237002
#229862
#229562
#95572
#95542
#95392
#95382
#89522
#89452
#89422
#89392
#82072
#82022
#81482
#78772