Finite Group Databasebeta

← #412233

Subgroups — #412233

Order 512.

1254 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2C2×#2060C212yesmaximal
3#21170C211yesmaximal, characteristic
4C4×#428C211yesmaximal, characteristic
5#13910C211yesmaximal, characteristic
6#11392C212yesmaximal
7#11340C212yesmaximal
8#11217C212yesmaximal
9#10857C211yesmaximal, characteristic
10#10852C211yesmaximal, characteristic
11#10851C211yesmaximal, characteristic
12C2×#544C2212yes—
13C2×#428C2212yes—
14C2×#428C2211yescharacteristic
15C2×#428C2211yescharacteristic
16C2×#423C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18#2868—21no—
19#2859—21no—
20#2660—22no—
21#2552—21no—
22#2552—21no—
23#2551—21no—
24#2551—21no—
25#2440—22no—
26#2440—22no—
27#2317C2212yes—
28C2×#389—22no—
29#2285C2211yescharacteristic
30C2×#384C2212yes—
31C2×#384C2211yescharacteristic
32C4×#117C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#2060—28no—
36#1949—22no—
37#1948—21no—
38#1947—21no—
39#1776—21no—
40#1776—21no—
41C4×#107—21no—
42C4×#106—21no—
43C2×#343C2212yes—
44C2×#343C2212yes—
45#1736C2211yescharacteristic
46#1736C2211yescharacteristic
47#1736C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50C2×#339C2212yes—
51C2×#339C2212yes—
52#1444—22no—
53C4⋊#95C2212yes—
54C22×M16C2311yescharacteristic
55C22×M16C2312yes—
56C22×M16C2311yescharacteristic
57#544—24no—
58#544—24no—
59#544—42no—
60#544—42no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×#125—22no—
66C2×#125—22no—
67C2×C22⋊Q8—24no—
68C2×#117C2312yes—
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4D412yes—
74C22×C4⋊C4D412yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81#428—22no—
82#428—22no—
83#428—24no—
84#428—42no—
85#428D414yes—
86#428Q814yescomplemented
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—22no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—22no—
93#423—22no—
94#423Q814yescomplemented
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C4×M16—22no—
98C4×M16Q814yescomplemented
99C2×#97—21no—
100C2×C4×C8—21no—
101#398—24no—
102#398—24no—
103#395—24no—
104#395—24no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—22no—
111#389—48no—
112#385—24no—
113#384—22no—
114#384—22no—
115#384—22no—
116#384—24no—
117#384—21no—
118#384—21no—
119#384—21no—
120#384—21no—
121#384—42no—
122#384D411yescharacteristic
123#384D411yescharacteristic
124#382—22no—
125#382D412yes—
126#380—21no—
127#380—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4Q814yescomplemented
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142#376—21no—
143C2×#95—21no—
144C2×#95—24no—
145C2×#95—21no—
146C43—21no—
147#343—24no—
148#343—42no—
149#340—22no—
150#340—22no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—21no—
155#339—24no—
156#339—24no—
157#339—24no—
158#339—42no—
159C22×Q8—24no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D414yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—22no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—22no—
180C2×M16C2×D412yes—
181C2×M16C2×Q812yes—
182C2×M16C2×Q814yes—
183C2×M16C2×Q812yes—
184C2×M16C2×Q812yes—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C4⋊Q8—44no—
189C4⋊Q8—44no—
190C4⋊Q8—44no—
191C4⋊Q8—44no—
192C4⋊Q8—84no—
193C4⋊Q8—84no—
194#125—44no—
195#125—44no—
196#125—44no—
197#125—44no—
198C22⋊Q8—48no—
199C22⋊Q8—48no—
200#117—21no—
201#117—21no—
202#117—21no—
203#117—21no—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—22no—
208#117C2×D412yes—
209#117C2×Q812yes—
210#117#4112yes—
211#117#4112yes—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—82no—
248C2×C4⋊C4—84no—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C4⋊C4C2×Q812yes—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—44no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×Q814yes—
280C2×C42C2×Q812yes—
281C2×C42C2×Q812yes—
282C2×C42C2×Q812yes—
283C2×C42#4112yes—
284#107—24no—
285#107—24no—
286#107—44no—
287#106—24no—
288#106—24no—
289#106—44no—
290C4⋊C8—24no—
291C4⋊C8—24no—
292#97—24no—
293C4×C8—24no—
294#95—24no—
295#95—24no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—22no—
301#95—22no—
302#95—24no—
303#95—48no—
304#95—41no—
305#95—41no—
306C24C22×D411yescharacteristic
307C2×Q8—48no—
308C2×Q8—84no—
309C2×Q8—84no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4#12011yescharacteristic
345C22×C4#12011yescharacteristic
346C22×C4C22⋊Q812yes—
347C22×C4C22⋊Q812yes—
348C22×C4C4⋊Q812yes—
349C22×C4C4⋊Q812yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×Q811yescharacteristic
354C22×C4C22×Q811yescharacteristic
355C22×C4C22×Q811yescharacteristic
356C22×C4C22×Q811yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C42−1+411yescharacteristic
363C22×C42−1+411yescharacteristic
364M16—22no—
365M16—22no—
366M16—22no—
367M16—22no—
368M16—22no—
369M16—24no—
370M16—24no—
371M16—42no—
372M16—41no—
373M16—41no—
374M16—42no—
375M16C4⋊Q814yescomplemented
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—42no—
385C4⋊C4—216no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—41no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C4⋊C4—816no—
420C4⋊C4#12112yes—
421C4⋊C4#12112yes—
422C4⋊C4C22⋊Q812yes—
423C4⋊C4C22⋊Q812yes—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—24no—
435C42—24no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—82no—
445C42—82no—
446C42C22⋊Q812yes—
447C42C22⋊Q812yes—
448C42C22⋊Q812yes—
449C42C22⋊Q812yes—
450C42C4⋊Q812yes—
451C42C4⋊Q812yes—
452C42C4⋊Q812yes—
453C42C4⋊Q812yes—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23C2×#12011yescharacteristic
459C23C2×C22⋊Q811yescharacteristic
460C23C2×C4⋊Q811yescharacteristic
461C23D4211yescharacteristic
462C23#54611yescharacteristic
463C23#54811yescharacteristic
464C23D4×Q811yescharacteristic
465Q8—168no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—28no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
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—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4#39512yes—
517C2×C4#39512yes—
518C2×C4#39512yes—
519C2×C4#39512yes—
520C2×C4#45312yes—
521C2×C4#45612yes—
522C2×C4C2×#12111yescharacteristic
523C2×C4C2×C22⋊Q811yescharacteristic
524C2×C4C2×C22⋊Q811yescharacteristic
525C2×C4C2×C22⋊Q811yescharacteristic
526C2×C4C2×C4⋊Q811yescharacteristic
527C2×C4C2×C4⋊Q811yescharacteristic
528C2×C4C2×C4⋊Q811yescharacteristic
529C2×C4#53211yescharacteristic
530C2×C4#53311yescharacteristic
531C2×C4#53311yescharacteristic
532C2×C4#53511yescharacteristic
533C2×C4#53611yescharacteristic
534C2×C4#54211yescharacteristic
535C2×C4#54211yescharacteristic
536C2×C4#54411yescharacteristic
537C2×C4#54411yescharacteristic
538C2×C4D4211yescharacteristic
539C2×C4#54611yescharacteristic
540C2×C4#54811yescharacteristic
541C2×C4D4×Q811yescharacteristic
542C2×C4D4×Q811yescharacteristic
543C2×C4D4×Q811yescharacteristic
544C2×C4#55411yescharacteristic
545C2×C4#55411yescharacteristic
546C2×C4#55711yescharacteristic
547C2×C4#55711yescharacteristic
548C2×C4Q8211yescharacteristic
549C2×C4Q8211yescharacteristic
550C8—24no—
551C8—42no—
552C8—44no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22C2×#39511yescharacteristic
566C22#240811yescharacteristic
567C22#240911yescharacteristic
568C22#245111yescharacteristic
569C22#245211yescharacteristic
570C22#255011yescharacteristic
571C22#255111yescharacteristic
572C22C2×#45311yescharacteristic
573C22C2×#45611yescharacteristic
574C22#301911yescharacteristic
575C22#302011yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—48no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—82no—
585C4—82no—
586C4—84no—
587C4—82no—
588C4—82no—
589C4#206014yescomplemented
590C4#245111yescharacteristic
591C4#245211yescharacteristic
592C4#254611yescharacteristic
593C4#255011yescharacteristic
594C4#255111yescharacteristic
595C4#256711yescharacteristic
596C4#262811yescharacteristic
597C4#285711yescharacteristic
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2#1512811yescharacteristic
603C2C2×#206011yescharacteristic
604C2#2356111yescharacteristic
605C2#2423011yescharacteristic
606C2#2423311yescharacteristic
607C2#2432011yescharacteristic
608C2#2432311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×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
86. #428Q8semidirect
94. #423Q8semidirect
98. C4×M16Q8semidirect
139. C4×C4⋊C4Q8semidirect
375. M16C4⋊Q8semidirect
589. C4#2060semidirect

Maximal subgroups

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

subgroupindex
#352752
C2×#20602
#211702
C4×#4282
#139102
#113922
#113402
#112172
#108572
#108522
#108512