Finite Group Databasebeta

← #231793

Subgroups — #231793

Order 512.

697 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23797C211yesmaximal, characteristic
2#23209C211yesmaximal, characteristic
3#23149C211yesmaximal, characteristic, complemented
4C2×#1584C211yesmaximal, characteristic, complemented
5C2×SD16⋊C8C211yesmaximal, characteristic
6#9677C211yesmaximal, characteristic, complemented
7#9675C211yesmaximal, characteristic
8#9661C211yesmaximal, characteristic
9#9658C211yesmaximal, characteristic, complemented
10#8950C211yesmaximal, characteristic, complemented
11#8948C211yesmaximal, characteristic
12#8946C211yesmaximal, characteristic, complemented
13#8945C211yesmaximal, characteristic
14#8279C211yesmaximal, characteristic, complemented
15#8265C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22C2×C4×SD16C2211yescharacteristic
23C2×C8×D4C2211yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×#124C2211yescharacteristic
26#2097C411yescharacteristic
27#2097C411yescharacteristic
28#2096C411yescharacteristic
29#2093C411yescharacteristic
30#2016C411yescharacteristic
31#2015C411yescharacteristic
32#2015C411yescharacteristic
33#2013C411yescharacteristic
34#1984C2211yescharacteristic
35#1984C2211yescharacteristic
36#1984C2211yescharacteristic
37#1954C2211yescharacteristic
38#1939—21no—
39#1924—21no—
40#1924—21no—
41#1916—21no—
42#1916—21no—
43#1850—21no—
44C8×C4⋊C4—21no—
45#1767C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×C4⋊C8C2211yescharacteristic
49#1752—21no—
50#1744C2211yescharacteristic
51#1584—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×#322C2211yescharacteristic
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#446—22no—
81#446—41no—
82C8×Q8—22no—
83C8×Q8—22no—
84C8×Q8—41no—
85C8×Q8—41no—
86C4×Q16—22no—
87C4×Q16—22no—
88C4×Q16—41no—
89C4×Q16—41no—
90C4×SD16—22no—
91C4×SD16—22no—
92C4×SD16—41no—
93C4×SD16—41no—
94C8×D4—22no—
95C8×D4—41no—
96C2×#107C2×C411yescharacteristic
97C2×#107C2×C411yescharacteristic
98C2×#106C2×C411yescharacteristic
99C2×#106C2×C411yescharacteristic
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8D411yescharacteristic
105C2×C4⋊C8D411yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×#97—21no—
122C2×#97—21no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8—21no—
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132#391C2×C411yescharacteristic
133#391C2×C411yescharacteristic
134#391C2×C411yescharacteristic
135#390C2×C411yescharacteristic
136#389—21no—
137#389—21no—
138#388—21no—
139#388—21no—
140#384C2×C411yescharacteristic
141#382C2×C411yescharacteristic
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C43C2311yescharacteristic
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336C2×C411yescharacteristic
157#336C2×C411yescharacteristic
158#336C2×C411yescharacteristic
159#336C2×C411yescharacteristic
160#336D411yescharacteristic
161#336D411yescharacteristic
162#335D414yes—
163#334—22no—
164Q8⋊C8—22no—
165Q8⋊C8—22no—
166Q8⋊C8—22no—
167Q8⋊C8—41no—
168Q8⋊C8—41no—
169Q8⋊C8—41no—
170D4⋊C8—22no—
171D4⋊C8—41no—
172#322—22no—
173#322—22no—
174C22×Q8C22×C411yescharacteristic
175C22×Q8C22×C411yescharacteristic
176C22×Q8C22×C411yescharacteristic
177C22×D4C22×C411yescharacteristic
178C23×C4—21no—
179C2×Q16—22no—
180C2×Q16—22no—
181C2×Q16—22no—
182C2×Q16—41no—
183C2×Q16—41no—
184C2×Q16—41no—
185C2×SD16—22no—
186C2×SD16—22no—
187C2×SD16—22no—
188C2×SD16—22no—
189C2×SD16—22no—
190C2×SD16—41no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8—81no—
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C22×C8C22×C411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210#124—22no—
211#124—41no—
212C4×Q8—21no—
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×D4—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C2×C4⋊C4—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⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42#4111yescharacteristic
252C2×C42#4111yescharacteristic
253C2×C42C2411yescharacteristic
254#107—22no—
255#107—41no—
256#106—22no—
257#106—41no—
258C4⋊C8—22no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—41no—
266C4⋊C8—42no—
267C4⋊C8—42no—
268C4⋊C8—42no—
269C4⋊C8—42no—
270C4⋊C8C2×D412yes—
271C4⋊C8C2×D412yes—
272#103—22no—
273#103—22no—
274#103—22no—
275#103—41no—
276#103—41no—
277#103—41no—
278#102—22no—
279#102—41no—
280C22⋊C8—42no—
281C22⋊C8—42no—
282C22⋊C8—81no—
283#97—42no—
284#97—81no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—42no—
296C4×C8—42no—
297C4×C8—42no—
298C4×C8C2×D412yes—
299C4×C8#4112yes—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308C24—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—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#13112yes—
326C2×Q8#13112yes—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4#13112yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4C4×D411yescharacteristic
347C22×C4C4×D411yescharacteristic
348C22×C4C4×D411yescharacteristic
349C22×C4C4×D411yescharacteristic
350C22×C4C23×C411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C2×#4111yescharacteristic
353Q16—44no—
354Q16—44no—
355SD16—28no—
356SD16—44no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
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—22no—
369C2×C8—22no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
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—84no—
397C2×C8C4×D412yes—
398C2×C8C4×D412yes—
399C2×C8#13712yes—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—81no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—42no—
423C22⋊C4—81no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—22no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42C22×D411yescharacteristic
441C42C2×#4111yescharacteristic
442C42H(5,𝔽2)11yescharacteristic
443C42H(5,𝔽2)11yescharacteristic
444C23—22no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23C2×C4×D411yescharacteristic
452Q8—22no—
453Q8—22no—
454Q8—22no—
455Q8—22no—
456Q8—41no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460Q8—41no—
461Q8—41no—
462Q8—41no—
463Q8—41no—
464D4—22no—
465D4—22no—
466D4—41no—
467D4—41no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
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—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4#44412yes—
507C2×C4C2×C4×D411yescharacteristic
508C2×C4#51811yescharacteristic
509C2×C4D4211yescharacteristic
510C2×C4#55011yescharacteristic
511C2×C4#55011yescharacteristic
512C2×C4#55911yescharacteristic
513C2×C4C2×#13111yescharacteristic
514C2×C4C2×#13111yescharacteristic
515C2×C4C2×#13111yescharacteristic
516C2×C4#56811yescharacteristic
517C2×C4C2×#13711yescharacteristic
518C2×C4#57511yescharacteristic
519C2×C4#57711yescharacteristic
520C2×C4#57811yescharacteristic
521C8—22no—
522C8—22no—
523C8—22no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—42no—
529C8—42no—
530C8—41no—
531C8—81no—
532C8—81no—
533C8—81no—
534C8—81no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22#236611yescharacteristic
543C22#292911yescharacteristic
544C22#293311yescharacteristic
545C22C4×#13711yescharacteristic
546C22#294711yescharacteristic
547C22C2×#44411yescharacteristic
548C22#295711yescharacteristic
549C4—21no—
550C4—21no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—42no—
555C4—42no—
556C4—41no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4#292911yescharacteristic
563C4#293311yescharacteristic
564C4#297611yescharacteristic
565C4#297711yescharacteristic
566C4#326711yescharacteristic
567C4#327011yescharacteristic
568C4#340011yescharacteristic
569C4#340211yescharacteristic
570C2—82no—
571C2#1791511yescharacteristic
572C2#1791911yescharacteristic
573C2#1800911yescharacteristic
574C2#1802111yescharacteristic
575C2#2317711yescharacteristic
576C2#2318011yescharacteristic
577C2#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
3. #23149C2semidirect
4. C2×#1584C2semidirect
6. #9677C2semidirect
9. #9658C2semidirect
10. #8950C2semidirect
12. #8946C2semidirect
14. #8279C2semidirect
15. #8265C2semidirect

Maximal subgroups

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

subgroupindex
#237972
#232092
#231492
C2×#15842
C2×SD16⋊C82
#96772
#96752
#96612
#96582
#89502
#89482
#89462
#89452
#82792
#82652