Finite Group Databasebeta

← #360730

Subgroups — #360730

Order 512.

1520 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34184C211yesmaximal, characteristic
2C2×#1860C212yesmaximal
3C2×#1859C212yesmaximal
4#20800C211yesmaximal, characteristic, complemented
5#20648C211yesmaximal, characteristic
6C2×#1759C211yesmaximal, characteristic
7#14354C211yesmaximal, characteristic, complemented
8#11340C212yesmaximal, complemented
9#11339C212yesmaximal, complemented
10#10858C211yesmaximal, characteristic, complemented
11#10840C211yesmaximal, characteristic, complemented
12C2×#520C2211yescharacteristic
13C2×#520C2211yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C2×#428C2212yes—
16C2×#428C2212yes—
17C2×#423C2211yescharacteristic
18C22×#104C2212yes—
19C2×#417C2212yes—
20C2×C4×M16C2211yescharacteristic
21#2351—22no—
22#2349—24no—
23#2349—22no—
24#2317—21no—
25#2317—21no—
26#2317C2211yescharacteristic
27#2317C2211yescharacteristic
28#2285—21no—
29#2285C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#384C2212yes—
32C2×#382—21no—
33C2×#382C2212yes—
34C2×#382C2211yescharacteristic
35C2×#376C2211yescharacteristic
36#1860—28no—
37#1859—28no—
38#1847—21no—
39#1847—21no—
40#1845—21no—
41#1844—21no—
42#1803—22no—
43#1802—22no—
44#1759C418yescomplemented
45C2×#343C2212yes—
46C2×#343C2212yes—
47#1736C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50C2×#339C2212yes—
51#1444—24no—
52C22×#41C2311yescharacteristic
53C22×M16C2312yes—
54C22×M16C2312yes—
55#520—24no—
56#520—24no—
57#520—42no—
58#520—42no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41C2×C414yes—
62C4×#41Q814yes—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8D412yes—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4D412yes—
72C2×#117—21no—
73C2×#117—21no—
74C2×#117—21no—
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C42—21no—
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88#428—24no—
89#428—24no—
90#428—42no—
91#428—42no—
92C2×#107—22no—
93C2×#107—22no—
94C2×#106—22no—
95C2×#106—22no—
96#423—22no—
97#423—22no—
98C2×C4⋊C8—22no—
99C2×#104C2×C418yes—
100C2×#104D418yes—
101C2×#104Q818yes—
102#417—24no—
103#417—24no—
104C2×#103—22no—
105C2×#102—22no—
106C4×M16—22no—
107C4×M16C2×C414yescomplemented
108C2×#97—21no—
109C2×C4×C8—21no—
110#389—24no—
111#389—24no—
112#389—22no—
113#389—22no—
114#385—24no—
115#385—24no—
116#385—22no—
117#385—22no—
118#384—21no—
119#384—21no—
120#384—22no—
121#384—21no—
122#384—21no—
123#384—24no—
124#384—42no—
125#384D411yescharacteristic
126#384D411yescharacteristic
127#382—24no—
128#382—21no—
129#382—21no—
130#382—21no—
131#382—22no—
132#382—21no—
133#382—24no—
134#382—42no—
135#382D411yescharacteristic
136#382D411yescharacteristic
137#380—24no—
138#380—22no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—22no—
141#376—22no—
142#376C2×C414yescomplemented
143C2×#95—24no—
144C2×#95—22no—
145C2×#95—24no—
146C2×#95D414yes—
147#343—24no—
148#343—24no—
149#340—24no—
150#339—24no—
151#339—24no—
152#339C2×C418yescomplemented
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41C22×C412yes—
158C2×#41C2×D414yes—
159C2×#41C2×Q812yes—
160C2×#41C2×Q814yes—
161C22×Q8C2×D411yescharacteristic
162C22×D4—21no—
163C22×D4C2×D411yescharacteristic
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2411yescharacteristic
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16C22×C414yes—
184C2×M16C2×D414yes—
185C2×M16C2×Q814yes—
186C22×C8—22no—
187C22×C8—22no—
188C4×Q8—24no—
189C4×Q8—42no—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×Q8C4⋊C418yescomplemented
194C4×D4—28no—
195C4×D4—28no—
196C4×D4—24no—
197C4×D4—48no—
198C4×D4—48no—
199C4×D4—42no—
200C4×D4—42no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4C4⋊C418yescomplemented
204#117—21no—
205#117—21no—
206#117—24no—
207#117—24no—
208#117—22no—
209#117—22no—
210#117—22no—
211#117—22no—
212#117—44no—
213#117—44no—
214#117C22×C412yes—
215#117C2×Q812yes—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—84no—
243C2×C4⋊C4—88no—
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D412yes—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42C22×C414yes—
267C2×C42C22×C412yes—
268C2×C42C2×D414yes—
269C2×C42C2×D412yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×Q814yes—
273C2×C42C2×Q812yes—
274C2×C42#4112yes—
275C2×C42#4112yes—
276#107—44no—
277#107—44no—
278#106—44no—
279#106—44no—
280C4⋊C8—44no—
281#104C4⋊C4132yescomplemented
282#103—44no—
283#102—44no—
284#97—24no—
285C4×C8—24no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—22no—
290#95—24no—
291#95—24no—
292#95—22no—
293#95—48no—
294C24—21no—
295C24C22×D411yescharacteristic
296#41—24no—
297#41—24no—
298#41—24no—
299#41—24no—
300#41—24no—
301#41—24no—
302#41C2×C4⋊C418yes—
303#41C4⋊Q818yes—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8C2×C4⋊C412yes—
308C2×Q8#12712yes—
309C2×Q8C4⋊Q812yes—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4C2×C4⋊C412yes—
318C2×D4#12712yes—
319C2×D4C4⋊Q812yes—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—48no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4C4×D414yes—
355C22×C4#12012yes—
356C22×C4#12012yes—
357C22×C4C22⋊Q814yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D412yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×Q811yescharacteristic
363C22×C4C22×Q812yes—
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4H(5,𝔽2)12yes—
367C22×C42−1+412yes—
368M16—24no—
369M16—24no—
370M16—24no—
371M16—42no—
372M16—42no—
373M16—42no—
374M16—42no—
375M16C2×C4⋊C418yes—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C4⋊C4—22no—
383C4⋊C4—28no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C4⋊C4—84no—
405C4⋊C4—84no—
406C4⋊C4—816no—
407C4⋊C4C2×C4⋊C414yes—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—28no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C42—22no—
416C42—24no—
417C42—22no—
418C42—24no—
419C42—24no—
420C42—28no—
421C42—44no—
422C42—44no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42C2×C4⋊C418yes—
428C42C2×C4⋊C414yes—
429C42C4×D412yes—
430C42C4×D412yes—
431C42C22⋊Q812yes—
432C42C22⋊Q812yes—
433C42#12314yes—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—42no—
440C23—42no—
441C23C2×C4×D411yescharacteristic
442C23C2×#12011yescharacteristic
443C23C2×C22⋊Q811yescharacteristic
444C23D4212yes—
445C23D4×Q812yes—
446Q8—22no—
447Q8—22no—
448Q8—22no—
449Q8#38414yes—
450D4—28no—
451D4—28no—
452D4—22no—
453D4—22no—
454D4—22no—
455D4#38414yes—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—24no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—48no—
471C2×C4—44no—
472C2×C4—48no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—88no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—82no—
493C2×C4—84no—
494C2×C4—164no—
495C2×C4—164no—
496C2×C4#38514yes—
497C2×C4#38514yes—
498C2×C4C22×C4⋊C411yescharacteristic
499C2×C4C22×C4⋊C412yes—
500C2×C4C2×C4×D411yescharacteristic
501C2×C4#52012yes—
502C2×C4C2×C22⋊Q811yescharacteristic
503C2×C4C2×#12311yescharacteristic
504C2×C4C2×#12711yescharacteristic
505C2×C4C2×C4⋊Q812yes—
506C2×C4C2×C4⋊Q811yescharacteristic
507C2×C4#53511yescharacteristic
508C2×C4#53611yescharacteristic
509C2×C4#54412yes—
510C2×C4D4212yes—
511C2×C4#54712yes—
512C2×C4#54712yes—
513C2×C4D4×Q812yes—
514C2×C4#55412yes—
515C2×C4#55412yes—
516C8—28no—
517C8—82no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—24no—
523C22—24no—
524C22—44no—
525C22—42no—
526C22—42no—
527C22—44no—
528C22C2×#38511yescharacteristic
529C22D4×C4⋊C412yes—
530C22#243311yescharacteristic
531C22#243411yescharacteristic
532C22#244012yes—
533C22#302112yes—
534C22#302212yes—
535C4—24no—
536C4—24no—
537C4—24no—
538C4—48no—
539C4—44no—
540C4—81no—
541C4—82no—
542C4—81no—
543C4—81no—
544C4—82no—
545C4—81no—
546C4—162no—
547C4C2×#38411yescharacteristic
548C4#231711yescharacteristic
549C4D4×C4⋊C412yes—
550C4#244012yes—
551C4#250111yescharacteristic
552C4#250211yescharacteristic
553C2—22no—
554C2—22no—
555C2—44no—
556C2#1388011yescharacteristic
557C2#2086512yes—
558C2#2575111yescharacteristic
559C2#2575211yescharacteristic
560C2#2583312yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20800C2semidirect
7. #14354C2semidirect
8. #11340C2semidirect
9. #11339C2semidirect
10. #10858C2semidirect
11. #10840C2semidirect
44. #1759C4semidirect
107. C4×M16C2×C4semidirect
142. #376C2×C4semidirect
152. #339C2×C4semidirect
193. C4×Q8C4⋊C4semidirect
203. C4×D4C4⋊C4semidirect
281. #104C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#341842
C2×#18602
C2×#18592
#208002
#206482
C2×#17592
#143542
#113402
#113392
#108582
#108402