Finite Group Databasebeta

← #393284

Subgroups — #393284

Order 512.

977 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1937C211yesmaximal, characteristic
2#21086C211yesmaximal, characteristic
3#21046C211yesmaximal, characteristic
4#21020C211yesmaximal, characteristic, complemented
5C2×#1726C211yesmaximal, characteristic, complemented
6#10930C211yesmaximal, characteristic
7#10930C211yesmaximal, characteristic
8#10929C211yesmaximal, characteristic
9#10924C211yesmaximal, characteristic
10#10883C211yesmaximal, characteristic, complemented
11#10883C211yesmaximal, characteristic, complemented
12#10882C211yesmaximal, characteristic, complemented
13#10879C211yesmaximal, characteristic, complemented
14#10823C211yesmaximal, characteristic, complemented
15#10818C211yesmaximal, characteristic, complemented
16C2×#424C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C22×#104C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C2×#416C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#408C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30#2297C2211yescharacteristic
31#2296C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1937—24no—
34#1936C414yescomplemented
35#1935C414yescomplemented
36#1923—24no—
37#1916—24no—
38C2×#341C2211yescharacteristic
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41C2×#341C2211yescharacteristic
42#1735C2211yescharacteristic
43#1735C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50#1729C2211yescharacteristic
51#1729C2211yescharacteristic
52#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1728C2211yescharacteristic
55#1728C2211yescharacteristic
56#1726—24no—
57C22×#41C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C2×#117C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#424—22no—
73#424—22no—
74#424—22no—
75#424—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×#104—22no—
79C2×#104—22no—
80C2×#104—22no—
81C2×#104—22no—
82C2×#104—22no—
83C2×#104—22no—
84#416—22no—
85#416—22no—
86#416—22no—
87#416—22no—
88C2×#103C2×C412yes—
89C2×#103C2×C412yes—
90C2×#102C2×C412yes—
91C2×#102C2×C412yes—
92#408—22no—
93#408—22no—
94#408—22no—
95#408—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×#97—22no—
99C2×#97—22no—
100C2×#97—22no—
101C2×#97—22no—
102C2×#97—22no—
103C2×#97—22no—
104C2×#97—22no—
105C2×#97—22no—
106C2×#97C2×C414yes—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8D414yes—
115C2×C4×C8D414yes—
116#391C2×C412yes—
117#390C2×C412yes—
118#389C2×C414yes—
119#386—22no—
120#385—22no—
121C4×C4⋊C4—22no—
122C4×C22⋊C4—22no—
123#376—22no—
124#376—22no—
125C2×#95—22no—
126#341—22no—
127#341—22no—
128#341—22no—
129#341—22no—
130#340C2×C414yes—
131#340C2×C414yes—
132#339—22no—
133#339—22no—
134#339—22no—
135#339—22no—
136#339—22no—
137#339—22no—
138#336—22no—
139#336—22no—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144C2×#41—21no—
145C2×#41—21no—
146C2×#41—21no—
147C2×#41—21no—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C22×Q8C22×C411yescharacteristic
157C22×D4—21no—
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4#4112yes—
163C23×C4C2411yescharacteristic
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8C22×C412yes—
209C22×C8C22×C412yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C22×C8C2×D412yes—
214C22×C8#4112yes—
215C22×C8#4112yes—
216C22×C8#4112yes—
217C22×C8#4112yes—
218C22×C8#4112yes—
219C22×C8#4112yes—
220#117—21no—
221#117—21no—
222#117—21no—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—21no—
227#117—21no—
228#117—21no—
229#117—21no—
230#117—21no—
231#117—21no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—24no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42C22×C412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42#4112yes—
257C2×C42#4112yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260C2×C42#4112yes—
261C4⋊C8—44no—
262C4⋊C8—44no—
263#104—44no—
264#104—44no—
265#103—24no—
266#103—24no—
267#102—24no—
268#102—24no—
269C22⋊C8—44no—
270C22⋊C8—44no—
271#97—28no—
272#97—28no—
273#97—28no—
274C4×C8—28no—
275C4×C8—28no—
276C4×C8—28no—
277#95—28no—
278#95—24no—
279#95—24no—
280C24—21no—
281C24C2×#4111yescharacteristic
282#41—41no—
283#41—41no—
284#41—41no—
285#41—41no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4C4×D414yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352M16—41no—
353M16—41no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8#11714yes—
405C2×C8C4×D414yes—
406C2×C8#12114yes—
407C2×C8#12414yes—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42#12314yes—
439C42#12614yes—
440C23—24no—
441C23—44no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23C2×C4×D411yescharacteristic
446C23#52511yescharacteristic
447C23#52911yescharacteristic
448C23#52911yescharacteristic
449C23#52911yescharacteristic
450C23#52911yescharacteristic
451C23#52911yescharacteristic
452Q8—41no—
453Q8—41no—
454Q8—41no—
455Q8—41no—
456D4—41no—
457D4—41no—
458D4—41no—
459D4—41no—
460D4—81no—
461D4—81no—
462D4—81no—
463D4—81no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4C2×#11711yescharacteristic
491C2×C4C2×C4×D411yescharacteristic
492C2×C4C4×#4111yescharacteristic
493C2×C4C4×#4111yescharacteristic
494C2×C4C4×#4111yescharacteristic
495C2×C4C4×#4111yescharacteristic
496C2×C4C2×#12111yescharacteristic
497C2×C4C2×#12311yescharacteristic
498C2×C4#52511yescharacteristic
499C2×C4#52511yescharacteristic
500C2×C4#52511yescharacteristic
501C2×C4C2×#12411yescharacteristic
502C2×C4C2×#12611yescharacteristic
503C2×C4#52911yescharacteristic
504C2×C4#52911yescharacteristic
505C2×C4#52911yescharacteristic
506C2×C4#52911yescharacteristic
507C2×C4#52911yescharacteristic
508C2×C4#52911yescharacteristic
509C2×C4#52911yescharacteristic
510C2×C4#52911yescharacteristic
511C2×C4#52911yescharacteristic
512C2×C4#52911yescharacteristic
513C2×C4#52911yescharacteristic
514C2×C4#53211yescharacteristic
515C2×C4#53211yescharacteristic
516C2×C4#53211yescharacteristic
517C2×C4#53211yescharacteristic
518C8—24no—
519C8—24no—
520C8—24no—
521C8—24no—
522C8—81no—
523C8—81no—
524C8—81no—
525C8—81no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—82no—
533C22—82no—
534C22—81no—
535C22—82no—
536C22#229311yescharacteristic
537C22C4×#12111yescharacteristic
538C22C4×#12411yescharacteristic
539C22#285211yescharacteristic
540C22#285311yescharacteristic
541C22#286511yescharacteristic
542C22#286611yescharacteristic
543C22#295514yes—
544C4—44no—
545C4—44no—
546C4—44no—
547C4—82no—
548C4—82no—
549C4—81no—
550C4—82no—
551C4#229011yescharacteristic
552C4#229811yescharacteristic
553C4C4×#12311yescharacteristic
554C4C4×#12611yescharacteristic
555C4#285011yescharacteristic
556C4#285411yescharacteristic
557C4#286011yescharacteristic
558C4#286111yescharacteristic
559C2—21no—
560C2—21no—
561C2—21no—
562C2—21no—
563C2—161no—
564C2#1470111yescharacteristic
565C2#2129612yes—
566C2#2154812yes—
567C2#2166312yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21020C2semidirect
5. C2×#1726C2semidirect
10. #10883C2semidirect
11. #10883C2semidirect
12. #10882C2semidirect
13. #10879C2semidirect
14. #10823C2semidirect
15. #10818C2semidirect
34. #1936C4semidirect
35. #1935C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19372
#210862
#210462
#210202
C2×#17262
#109302
#109292
#109242
#108832
#108822
#108792
#108232
#108182