Finite Group Databasebeta

← #210959

Subgroups — #210959

Order 512.

709 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#23701C211yesmaximal, characteristic
2#22133C211yesmaximal, characteristic
3#22098C211yesmaximal, characteristic, complemented
4C2×C8×SD16C211yesmaximal, characteristic, complemented
5C2×#1454C211yesmaximal, characteristic, complemented
6#9666C211yesmaximal, characteristic, complemented
7#9599C211yesmaximal, characteristic, complemented
8#9558C211yesmaximal, characteristic, complemented
9#9557C211yesmaximal, characteristic, complemented
10#9076C211yesmaximal, characteristic
11#9075C211yesmaximal, characteristic, complemented
12#8600C211yesmaximal, characteristic, complemented
13C8×#107C211yesmaximal, characteristic, complemented
14#8201C211yesmaximal, characteristic, complemented
15#8200C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C4×C22⋊Q8C2211yescharacteristic
23C4×#121C2211yescharacteristic
24#2088C411yescharacteristic
25#2077C411yescharacteristic
26#2068C411yescharacteristic
27#2067C411yescharacteristic
28#2055C411yescharacteristic
29#2036C411yescharacteristic
30#2033C411yescharacteristic
31#2000C411yescharacteristic
32#1984C2211yescharacteristic
33#1954—21no—
34#1954C2211yescharacteristic
35#1939—21no—
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1923—21no—
39#1923—21no—
40#1923—21no—
41#1916—21no—
42#1869—21no—
43#1850—21no—
44#1797C2211yescharacteristic
45#1797C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C4×#103C2211yescharacteristic
50C4×#102C2211yescharacteristic
51C8×SD16—24no—
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58#1454—24no—
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C23×C8—21no—
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×#121C2×C411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C8×Q8—22no—
78C8×Q8—22no—
79C4×SD16—22no—
80C4×SD16—22no—
81C8×D4—22no—
82C8×D4—22no—
83#431—22no—
84#431—22no—
85#431—22no—
86#431—22no—
87#431—22no—
88#431—22no—
89C2×#107C2×C411yescharacteristic
90C2×#107C2×C411yescharacteristic
91C2×#106C2×C411yescharacteristic
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2×C411yescharacteristic
97C2×C4⋊C8C2×C411yescharacteristic
98C2×C4⋊C8D411yescharacteristic
99C2×C4⋊C8D411yescharacteristic
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⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8C2×C411yescharacteristic
117C2×C22⋊C8C2×C411yescharacteristic
118C2×C22⋊C8D411yescharacteristic
119C2×C22⋊C8D411yescharacteristic
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129#391C2×C411yescharacteristic
130#390C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#388—21no—
134#387—21no—
135#386—21no—
136#385—21no—
137#380C2×C411yescharacteristic
138#380C2×C411yescharacteristic
139C4×C4⋊C4—21no—
140C4×C4⋊C4C2311yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#335—22no—
159#334—22no—
160Q8⋊C8—22no—
161Q8⋊C8—22no—
162D4⋊C8—22no—
163D4⋊C8—22no—
164#322—22no—
165#322—22no—
166#322—22no—
167C82D414yescomplemented
168C22×Q8C22×C411yescharacteristic
169C22×D4—21no—
170C22×D4C22×C411yescharacteristic
171C23×C4—21no—
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4#4111yescharacteristic
175C2×SD16—22no—
176C2×SD16—22no—
177C2×SD16—22no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—42no—
185C22×C8—81no—
186C22×C8—81no—
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197#121—22no—
198#121—22no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—81no—
208C4×D4—81no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C22⋊C4—21no—
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×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42C2411yescharacteristic
237#107—22no—
238#107—22no—
239#106—22no—
240C4⋊C8—24no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—42no—
254C4⋊C8—81no—
255C4⋊C8—81no—
256#103—22no—
257#103—22no—
258#103—22no—
259#102—22no—
260#102—22no—
261#102—22no—
262C22⋊C8—24no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—42no—
268C22⋊C8—81no—
269C22⋊C8—81no—
270C22⋊C8—81no—
271C22⋊C8—81no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—42no—
279C4×C8—81no—
280C4×C8—81no—
281C4×C8C2×D412yes—
282C4×C8C2×D412yes—
283C4×C8C2×D412yes—
284C4×C8#4112yes—
285C4×C8#4112yes—
286C4×C8#4112yes—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295C24—21no—
296C24C4×D411yescharacteristic
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8#13112yes—
303C2×D4—24no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—81no—
310C2×D4—81no—
311C2×D4#13112yes—
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—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4C4×D411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4C4×D411yescharacteristic
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C2×#4111yescharacteristic
347SD16—28no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—22no—
361C2×C8—22no—
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—41no—
371C2×C8—41no—
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—84no—
387C2×C8C4×D412yes—
388C2×C8C4×D412yes—
389C2×C8#12112yes—
390C2×C8#12112yes—
391C4⋊C4—24no—
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—41no—
403C4⋊C4—41no—
404C4⋊C4—81no—
405C4⋊C4—81no—
406C4⋊C4#13112yes—
407C4⋊C4#13112yes—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—81no—
413C22⋊C4—81no—
414C22⋊C4—81no—
415C22⋊C4—81no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42C22×D411yescharacteristic
429C42C2×#4111yescharacteristic
430C42C2×#4111yescharacteristic
431C42C2×#4111yescharacteristic
432C23—21no—
433C23—22no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23C2×C4×D411yescharacteristic
447Q8—22no—
448Q8—22no—
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—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—42no—
471C2×C4—42no—
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—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4C2×C4×D411yescharacteristic
503C2×C4C4×#4111yescharacteristic
504C2×C4C2×#12111yescharacteristic
505C2×C4#52511yescharacteristic
506C2×C4#52911yescharacteristic
507C2×C4#53211yescharacteristic
508C2×C4C2×#13111yescharacteristic
509C2×C4C2×#13111yescharacteristic
510C2×C4C2×#13111yescharacteristic
511C2×C4C2×#13111yescharacteristic
512C2×C4#57611yescharacteristic
513C2×C4#57711yescharacteristic
514C2×C4#57711yescharacteristic
515C2×C4#57811yescharacteristic
516C8—22no—
517C8—22no—
518C8—22no—
519C8—22no—
520C8—42no—
521C8—42no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—81no—
527C8—81no—
528C8—81no—
529C8—81no—
530C22—41no—
531C22—41no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—42no—
537C22—41no—
538C22—42no—
539C22—41no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22C4×#12111yescharacteristic
545C22#292911yescharacteristic
546C22#292911yescharacteristic
547C22#294611yescharacteristic
548C22#294711yescharacteristic
549C22#295711yescharacteristic
550C22#295711yescharacteristic
551C4—21no—
552C4—21no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—42no—
558C4—42no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4#292911yescharacteristic
564C4#293111yescharacteristic
565C4#296611yescharacteristic
566C4#296611yescharacteristic
567C4#305711yescharacteristic
568C4#306111yescharacteristic
569C4#317811yescharacteristic
570C4#317911yescharacteristic
571C2—81no—
572C2—82no—
573C2#1791811yescharacteristic
574C2#1791911yescharacteristic
575C2#1792011yescharacteristic
576C2#1792111yescharacteristic
577C2#2149111yescharacteristic
578C2#2153511yescharacteristic
579C2#2154511yescharacteristic
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 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22098C2semidirect
4. C2×C8×SD16C2semidirect
5. C2×#1454C2semidirect
6. #9666C2semidirect
7. #9599C2semidirect
8. #9558C2semidirect
9. #9557C2semidirect
11. #9075C2semidirect
12. #8600C2semidirect
13. C8×#107C2semidirect
14. #8201C2semidirect
15. #8200C2semidirect
167. C82D4semidirect

Maximal subgroups

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

subgroupindex
#237012
#221332
#220982
C2×C8×SD162
C2×#14542
#96662
#95992
#95582
#95572
#90762
#90752
#86002
C8×#1072
#82012
#82002