Finite Group Databasebeta

← #231439

Subgroups — #231439

Order 512.

707 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23209C211yesmaximal, characteristic
2#23150C211yesmaximal, characteristic, complemented
3C4×#497C211yesmaximal, characteristic
4C2×#1591C211yesmaximal, characteristic, complemented
5C2×SD16⋊C8C211yesmaximal, characteristic
6#9677C211yesmaximal, characteristic, complemented
7#9676C211yesmaximal, characteristic
8#9658C211yesmaximal, characteristic, complemented
9#9655C211yesmaximal, characteristic
10#8952C211yesmaximal, characteristic
11#8950C211yesmaximal, characteristic, complemented
12#8946C211yesmaximal, characteristic, complemented
13#8945C211yesmaximal, characteristic
14#8276C211yesmaximal, characteristic, complemented
15#8262C211yesmaximal, characteristic, complemented
16C2×#497C411yescharacteristic
17C2×#446C2211yescharacteristic
18C2×#446C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×SD16C2211yescharacteristic
23C2×C4×SD16C2211yescharacteristic
24C2×C8×D4C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×#124C2211yescharacteristic
27#1984C2211yescharacteristic
28#1984C2211yescharacteristic
29#1984C2211yescharacteristic
30#1973C411yescharacteristic
31#1973C411yescharacteristic
32#1971C411yescharacteristic
33#1971C411yescharacteristic
34#1960C411yescharacteristic
35#1959C411yescharacteristic
36#1954C2211yescharacteristic
37#1940—21no—
38#1924—21no—
39#1924—21no—
40#1917—21no—
41#1916—21no—
42#1850—21no—
43#1847C411yescharacteristic
44#1771—21no—
45#1767C2211yescharacteristic
46#1767C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×C4⋊C8C2211yescharacteristic
49#1752—21no—
50C4×#97C2211yescharacteristic
51#1591—24no—
52SD16⋊C8—24no—
53C2×#335C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×C82C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×Q16C2×C411yescharacteristic
70C22×Q16C2×C411yescharacteristic
71C22×SD16C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C23×C8—21no—
74C2×C4⋊Q8C2×C411yescharacteristic
75C2×#124C2×C411yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C2×C4×D4C2311yescharacteristic
80#497—24no—
81#446—22no—
82#446—22no—
83#446—41no—
84#446—41no—
85C8×Q8—22no—
86C8×Q8—41no—
87C4×Q16—22no—
88C4×Q16—22no—
89C4×Q16—41no—
90C4×Q16—41no—
91C4×SD16—22no—
92C4×SD16—22no—
93C4×SD16—41no—
94C4×SD16—41no—
95C8×D4—22no—
96C8×D4—41no—
97C2×#107C2×C411yescharacteristic
98C2×#107C2×C411yescharacteristic
99C2×#106C2×C411yescharacteristic
100C2×#106C2×C411yescharacteristic
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8D411yescharacteristic
106C2×C4⋊C8D411yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×#97—21no—
123C2×#97—21no—
124C2×#97—21no—
125C2×#97—21no—
126C2×#97C2×C411yescharacteristic
127C2×#97C2×C411yescharacteristic
128C2×#97C2×C411yescharacteristic
129C2×#97C2×C411yescharacteristic
130C2×C4×C8—21no—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137#391C2×C411yescharacteristic
138#391C2×C411yescharacteristic
139#391C2×C411yescharacteristic
140#390C2×C411yescharacteristic
141#389—21no—
142#389—21no—
143#388—21no—
144#388—21no—
145#384C2×C411yescharacteristic
146#382C2×C411yescharacteristic
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C43C2311yescharacteristic
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#336D411yescharacteristic
162#336D411yescharacteristic
163#335D414yes—
164#334—22no—
165Q8⋊C8—22no—
166Q8⋊C8—22no—
167Q8⋊C8—22no—
168Q8⋊C8—41no—
169Q8⋊C8—41no—
170Q8⋊C8—41no—
171D4⋊C8—22no—
172D4⋊C8—41no—
173#322—22no—
174C82—22no—
175C22×Q8C22×C411yescharacteristic
176C22×Q8C22×C411yescharacteristic
177C22×Q8C22×C411yescharacteristic
178C22×D4C22×C411yescharacteristic
179C23×C4—21no—
180C2×Q16—22no—
181C2×Q16—22no—
182C2×Q16—22no—
183C2×Q16—22no—
184C2×Q16—22no—
185C2×Q16—41no—
186C2×SD16—22no—
187C2×SD16—22no—
188C2×SD16—22no—
189C2×SD16—22no—
190C2×SD16—41no—
191C2×SD16M1614yes—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—42no—
199C22×C8—42no—
200C22×C8—81no—
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C22×C8C22×C411yescharacteristic
204C22×C8C22×C411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C4⋊Q8—22no—
210C4⋊Q8—41no—
211#124—22no—
212#124—41no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42#4111yescharacteristic
253C2×C42#4111yescharacteristic
254C2×C42C2411yescharacteristic
255#107—22no—
256#107—41no—
257#106—22no—
258#106—41no—
259C4⋊C8—22no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—41no—
267C4⋊C8—42no—
268C4⋊C8—42no—
269C4⋊C8—42no—
270C4⋊C8—42no—
271C4⋊C8C2×D412yes—
272C4⋊C8C2×D412yes—
273#103—22no—
274#103—22no—
275#103—22no—
276#103—41no—
277#103—41no—
278#103—41no—
279#102—22no—
280#102—41no—
281C22⋊C8—42no—
282C22⋊C8—42no—
283C22⋊C8—81no—
284#97—22no—
285#97—22no—
286#97—22no—
287#97—41no—
288#97—42no—
289#97—42no—
290#97—42no—
291#97—81no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—42no—
303C4×C8C2×D412yes—
304C4×C8#4112yes—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313C24—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8C2×M1612yes—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4C2×M1612yes—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4C4×D411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4C4×D411yescharacteristic
354C22×C4C4×D411yescharacteristic
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4111yescharacteristic
358Q16—44no—
359Q16—44no—
360SD16—28no—
361SD16—44no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—22no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
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—84no—
399C2×C8C4×D412yes—
400C2×C8C4×D412yes—
401C2×C8C2×M1612yes—
402C2×C8#13112yes—
403C2×C8#13112yes—
404C2×C8#13712yes—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—81no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—81no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42C22×D411yescharacteristic
446C42C2×#4111yescharacteristic
447C42H(5,𝔽2)11yescharacteristic
448C42H(5,𝔽2)11yescharacteristic
449C23—22no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23C2×C4×D411yescharacteristic
457Q8—22no—
458Q8—22no—
459Q8—41no—
460Q8—41no—
461Q8—41no—
462Q8—41no—
463Q8—41no—
464Q8—41no—
465Q8—41no—
466Q8—41no—
467Q8—41no—
468Q8—41no—
469D4—22no—
470D4—22no—
471D4—41no—
472D4—41no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
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—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4C2×C4×D411yescharacteristic
513C2×C4#51811yescharacteristic
514C2×C4D4211yescharacteristic
515C2×C4#55011yescharacteristic
516C2×C4#55011yescharacteristic
517C2×C4#55911yescharacteristic
518C2×C4C22×M1611yescharacteristic
519C2×C4C2×#13111yescharacteristic
520C2×C4C2×#13111yescharacteristic
521C2×C4#56811yescharacteristic
522C2×C4C2×#13711yescharacteristic
523C2×C4#57511yescharacteristic
524C2×C4#57711yescharacteristic
525C2×C4#57811yescharacteristic
526C8—22no—
527C8—22no—
528C8—22no—
529C8—42no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—42no—
534C8—42no—
535C8—41no—
536C8—81no—
537C8—81no—
538C8—81no—
539C8#43514yes—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22#236611yescharacteristic
548C22#293111yescharacteristic
549C22D4×M1611yescharacteristic
550C22C4×#13711yescharacteristic
551C22#294711yescharacteristic
552C22#295511yescharacteristic
553C22#295611yescharacteristic
554C4—21no—
555C4—21no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—42no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4C2×#43511yescharacteristic
568C4#293511yescharacteristic
569C4#297411yescharacteristic
570C4#298111yescharacteristic
571C4#326711yescharacteristic
572C4#327011yescharacteristic
573C4#340011yescharacteristic
574C4#340211yescharacteristic
575C2—82no—
576C2#1796711yescharacteristic
577C2#1796911yescharacteristic
578C2#1804711yescharacteristic
579C2#1805111yescharacteristic
580C2#2317411yescharacteristic
581C2#2318111yescharacteristic
582C2#2319411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23150C2semidirect
4. C2×#1591C2semidirect
6. #9677C2semidirect
8. #9658C2semidirect
11. #8950C2semidirect
12. #8946C2semidirect
14. #8276C2semidirect
15. #8262C2semidirect

Maximal subgroups

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

subgroupindex
#232092
#231502
C4×#4972
C2×#15912
C2×SD16⋊C82
#96772
#96762
#96582
#96552
#89522
#89502
#89462
#89452
#82762
#82622