Finite Group Databasebeta

← #202850

Subgroups — #202850

Order 512.

696 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21963C211yesmaximal, characteristic
2#21864C211yesmaximal, characteristic
3#21837C211yesmaximal, characteristic
4#20545C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6#9576C211yesmaximal, characteristic, complemented
7#9575C211yesmaximal, characteristic, complemented
8#9570C211yesmaximal, characteristic, complemented
9#9569C211yesmaximal, characteristic, complemented
10#9568C211yesmaximal, characteristic, complemented
11#9567C211yesmaximal, characteristic, complemented
12#9562C211yesmaximal, characteristic, complemented
13#9561C211yesmaximal, characteristic, complemented
14#8039C211yesmaximal, characteristic
15#7880C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2285C2211yescharacteristic
22#2283C2211yescharacteristic
23#1991—21no—
24#1991—21no—
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1988C2211yescharacteristic
28#1988C2211yescharacteristic
29#1987C2211yescharacteristic
30#1987C2211yescharacteristic
31#1956C2211yescharacteristic
32#1956C2211yescharacteristic
33#1949—21no—
34#1949—21no—
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2211yescharacteristic
39#1873—21no—
40#1873—21no—
41#1825—21no—
42#1825—21no—
43#1809—21no—
44#1809—21no—
45#1805—21no—
46#1805—21no—
47#1770C2211yescharacteristic
48#1770C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1768—21no—
52#1768—21no—
53#1752—21no—
54C8×C22⋊C4—21no—
55#1550C418yescomplemented
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—
69C23×C8—21no—
70#533—22no—
71#533—22no—
72#533—22no—
73#533C2×C414yes—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×#125—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×#117—21no—
82C2×#117—21no—
83C22×C42C2311yescharacteristic
84#431—22no—
85#431—22no—
86#431—22no—
87#431—22no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#106—21no—
91C2×#106—21no—
92#423—22no—
93#423C2×C414yes—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8C2×C414yes—
98C2×C4⋊C8D411yescharacteristic
99C2×C4⋊C8D411yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8D411yescharacteristic
112C2×C22⋊C8D411yescharacteristic
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
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—81no—
168C2×M16—81no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—24no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—81no—
177C22×C8—81no—
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8C22×C412yes—
189C4⋊Q8C22×C412yes—
190#125—24no—
191#125—42no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C22⋊Q8—42no—
195C22⋊Q8—42no—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—42no—
199C4×Q8—42no—
200#117—24no—
201#117—42no—
202#117—81no—
203#117—81no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42C22×C412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42#4112yes—
233C2×C42#4112yes—
234C2×C42#4112yes—
235C2×C42C2411yescharacteristic
236#107—81no—
237#107—81no—
238#106—81no—
239#106—81no—
240C4⋊C8—24no—
241C4⋊C8—24no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8C22×C412yes—
253C4⋊C8C22×C412yes—
254C4⋊C8C22×C412yes—
255C4⋊C8C22×C412yes—
256#103—81no—
257#103—81no—
258#103—81no—
259#103—81no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—42no—
263#97—81no—
264#97—81no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—42no—
270C4×C8—81no—
271C4×C8—81no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—41no—
281#95—41no—
282C24#12111yescharacteristic
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—81no—
292C2×Q8—81no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
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—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4#11712yes—
312C22×C4C4×D412yes—
313C22×C4C4×D412yes—
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12312yes—
318C22×C4#12412yes—
319C22×C4#12612yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C2×#4111yescharacteristic
323M16—84no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
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—84no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—81no—
382C4⋊C4—81no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—44no—
386C22⋊C4—82no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—44no—
392C42—82no—
393C42#11712yes—
394C42C4×D412yes—
395C42C4×D412yes—
396C42#12312yes—
397C42#12412yes—
398C42#12612yes—
399C42C23×C411yescharacteristic
400C42C2×#4111yescharacteristic
401C42C2×#4111yescharacteristic
402C42C2×#4111yescharacteristic
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23#38812yes—
410C23C2×#12111yescharacteristic
411Q8—42no—
412Q8—42no—
413Q8—82no—
414Q8—82no—
415Q8—81no—
416Q8—81no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4#38812yes—
460C2×C4#38812yes—
461C2×C4#38812yes—
462C2×C4C2×#11711yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×C4×D411yescharacteristic
465C2×C4C2×#12311yescharacteristic
466C2×C4C2×#12411yescharacteristic
467C2×C4C2×#12611yescharacteristic
468C2×C4#57611yescharacteristic
469C2×C4#57711yescharacteristic
470C2×C4#57711yescharacteristic
471C2×C4#57711yescharacteristic
472C2×C4#57711yescharacteristic
473C2×C4#57811yescharacteristic
474C2×C4#57811yescharacteristic
475C2×C4#57811yescharacteristic
476C8—41no—
477C8—41no—
478C8—41no—
479C8—41no—
480C8—42no—
481C8—42no—
482C8—42no—
483C8—42no—
484C8—81no—
485C8—81no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22C2×#38811yescharacteristic
494C22#304511yescharacteristic
495C22#304611yescharacteristic
496C22#306311yescharacteristic
497C22#306311yescharacteristic
498C22#306711yescharacteristic
499C22#306711yescharacteristic
500C4—22no—
501C4—21no—
502C4—21no—
503C4—22no—
504C4—22no—
505C4—82no—
506C4—82no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—81no—
511C4—161no—
512C4—161no—
513C4#294611yescharacteristic
514C4#294711yescharacteristic
515C4#294911yescharacteristic
516C4#295011yescharacteristic
517C4#309811yescharacteristic
518C4#309911yescharacteristic
519C4#313711yescharacteristic
520C4#313711yescharacteristic
521C2—42no—
522C2#1957711yescharacteristic
523C2#1957911yescharacteristic
524C2#1958511yescharacteristic
525C2#1958711yescharacteristic
526C2#2106211yescharacteristic
527C2#2107511yescharacteristic
528C2#2108811yescharacteristic
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. #9570C2semidirect
9. #9569C2semidirect
10. #9568C2semidirect
11. #9567C2semidirect
12. #9562C2semidirect
13. #9561C2semidirect
55. #1550C4semidirect

Maximal subgroups

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

subgroupindex
#219632
#218642
#218372
#205452
C2×#15502
#95762
#95752
#95702
#95692
#95682
#95672
#95622
#95612
#80392
#78802