Finite Group Databasebeta

← #202831

Subgroups — #202831

Order 512.

720 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21963C211yesmaximal, characteristic
2#21892C211yesmaximal, characteristic
3#21318C211yesmaximal, characteristic
4#20476C211yesmaximal, characteristic
5C2×#1559C211yesmaximal, characteristic
6#9576C211yesmaximal, characteristic, complemented
7#9575C211yesmaximal, characteristic, complemented
8#9574C211yesmaximal, characteristic, complemented
9#9573C211yesmaximal, characteristic, complemented
10#9568C211yesmaximal, characteristic, complemented
11#9567C211yesmaximal, characteristic, complemented
12#9566C211yesmaximal, characteristic, complemented
13#9565C211yesmaximal, characteristic, complemented
14#8042C211yesmaximal, characteristic
15#7874C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2285C2211yescharacteristic
22#2283C2211yescharacteristic
23#1991—21no—
24#1991—21no—
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1987C2211yescharacteristic
30#1987C2211yescharacteristic
31#1981C2211yescharacteristic
32#1981C2211yescharacteristic
33#1949—21no—
34#1949—21no—
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2211yescharacteristic
39#1894—21no—
40#1894—21no—
41#1809—21no—
42#1809—21no—
43#1805—21no—
44#1805—21no—
45#1770C2211yescharacteristic
46#1770C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1752—21no—
50C8×C22⋊C4—21no—
51#1746—21no—
52#1735—21no—
53#1735—21no—
54#1726—21no—
55#1559C418yescomplemented
56C2×#333C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×M16—21no—
69C22×M16—21no—
70C23×C8D411yescharacteristic
71C23×C8D411yescharacteristic
72#533—22no—
73#533—22no—
74#533—22no—
75#533C2×C414yes—
76C2×C4⋊Q8C2311yescharacteristic
77C2×C4⋊Q8C2311yescharacteristic
78C2×#125—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×#117—21no—
84C2×#117—21no—
85C22×C42C2311yescharacteristic
86#432—22no—
87#432C2×C414yes—
88C2×#107—21no—
89C2×#107—21no—
90C2×#106—21no—
91C2×#106—21no—
92#423—22no—
93#423—22no—
94#423—22no—
95#423C2×C414yes—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×C22⋊C8—21no—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8D411yescharacteristic
117C2×C4×C8D411yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#383—21no—
123#382—21no—
124#382D411yescharacteristic
125#382D411yescharacteristic
126#381—21no—
127#380—21no—
128#380—21no—
129#380D411yescharacteristic
130#380D411yescharacteristic
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136#376—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#333C2×C414yes—
158#333C2×C414yes—
159#332C2×C414yes—
160#332C2×C414yes—
161C22×Q8—21no—
162C22×Q8—21no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4#4111yescharacteristic
166C2×M16—24no—
167C2×M16—42no—
168C2×M16—81no—
169C2×M16—81no—
170C2×M16—81no—
171C2×M16—81no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—41no—
183C22×C8—41no—
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8C22×C412yes—
195C4⋊Q8C22×C412yes—
196#125—24no—
197#125—42no—
198C22⋊Q8—24no—
199C22⋊Q8—24no—
200C22⋊Q8—42no—
201C22⋊Q8—42no—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×Q8—42no—
205C4×Q8—42no—
206#117—24no—
207#117—42no—
208#117—42no—
209#117—42no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42C22×C412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42C2411yescharacteristic
242#107—81no—
243#107—81no—
244#106—81no—
245#106—81no—
246C4⋊C8—24no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—42no—
256C4⋊C8C22×C412yes—
257C4⋊C8C22×C412yes—
258C4⋊C8C22×C412yes—
259C4⋊C8C22×C412yes—
260#103—81no—
261#103—81no—
262#103—81no—
263#103—81no—
264C22⋊C8—24no—
265#97—24no—
266#97—42no—
267#97—42no—
268C4×C8—22no—
269C4×C8—22no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—42no—
275C4×C8—42no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—41no—
285#95—41no—
286C24#12111yescharacteristic
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—81no—
296C2×Q8—81no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4#11712yes—
316C22×C4C4×D412yes—
317C22×C4C4×D412yes—
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12312yes—
322C22×C4#12412yes—
323C22×C4#12612yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C2×#4111yescharacteristic
327M16—84no—
328M16—84no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—44no—
344C2×C8—44no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8#13712yes—
360C2×C8#13712yes—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—81no—
388C4⋊C4—81no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—44no—
392C22⋊C4—82no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—44no—
398C42—82no—
399C42#11712yes—
400C42C4×D412yes—
401C42C4×D412yes—
402C42#12312yes—
403C42#12412yes—
404C42#12612yes—
405C42C23×C411yescharacteristic
406C42C2×#4111yescharacteristic
407C42C2×#4111yescharacteristic
408C42C2×#4111yescharacteristic
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23#38812yes—
416C23C2×#12111yescharacteristic
417Q8—42no—
418Q8—42no—
419Q8—82no—
420Q8—82no—
421Q8—81no—
422Q8—81no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4#38812yes—
466C2×C4#38812yes—
467C2×C4#38812yes—
468C2×C4C2×#11711yescharacteristic
469C2×C4C2×C4×D411yescharacteristic
470C2×C4C2×C4×D411yescharacteristic
471C2×C4C2×#12311yescharacteristic
472C2×C4C2×#12411yescharacteristic
473C2×C4C2×#12611yescharacteristic
474C2×C4C2×#13711yescharacteristic
475C2×C4C2×#13711yescharacteristic
476C2×C4#57511yescharacteristic
477C2×C4#57511yescharacteristic
478C2×C4#57611yescharacteristic
479C2×C4#57711yescharacteristic
480C2×C4#57711yescharacteristic
481C2×C4#57811yescharacteristic
482C8—22no—
483C8—22no—
484C8—41no—
485C8—41no—
486C8—42no—
487C8—42no—
488C8—42no—
489C8—42no—
490C8—81no—
491C8—81no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22C2×#38811yescharacteristic
500C22#303911yescharacteristic
501C22#304011yescharacteristic
502C22#306011yescharacteristic
503C22#306111yescharacteristic
504C22#306711yescharacteristic
505C22#306711yescharacteristic
506C4—22no—
507C4—21no—
508C4—21no—
509C4—22no—
510C4—22no—
511C4—82no—
512C4—82no—
513C4—81no—
514C4—81no—
515C4—81no—
516C4—81no—
517C4—161no—
518C4—161no—
519C4C4×#13711yescharacteristic
520C4C4×#13711yescharacteristic
521C4#294611yescharacteristic
522C4#294711yescharacteristic
523C4#309511yescharacteristic
524C4#309511yescharacteristic
525C4#313611yescharacteristic
526C4#313711yescharacteristic
527C2—42no—
528C2#1936011yescharacteristic
529C2#1936111yescharacteristic
530C2#1936611yescharacteristic
531C2#1936711yescharacteristic
532C2#2106111yescharacteristic
533C2#2107511yescharacteristic
534C2#2109211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
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
6. #9576C2semidirect
7. #9575C2semidirect
8. #9574C2semidirect
9. #9573C2semidirect
10. #9568C2semidirect
11. #9567C2semidirect
12. #9566C2semidirect
13. #9565C2semidirect
55. #1559C4semidirect

Maximal subgroups

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

subgroupindex
#219632
#218922
#213182
#204762
C2×#15592
#95762
#95752
#95742
#95732
#95682
#95672
#95662
#95652
#80422
#78742