Finite Group Databasebeta

← #412234

Subgroups — #412234

Order 512.

1182 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2C2×#2061C212yesmaximal
3#21170C211yesmaximal, characteristic
4C4×#428C211yesmaximal, characteristic
5#13911C211yesmaximal, characteristic
6#11392C212yesmaximal
7#11339C212yesmaximal
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—
28#2285C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#382C2212yes—
31C4×#117C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#2061—28no—
35#1949—22no—
36#1948—21no—
37#1947—21no—
38#1776—21no—
39#1776—21no—
40C4×#107—21no—
41C4×#106—21no—
42C2×#343C2212yes—
43C2×#343C2212yes—
44#1736C2211yescharacteristic
45#1736C2211yescharacteristic
46#1736C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50C2×#339C2212yes—
51#1442—22no—
52C4⋊#95—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⋊C4D412yes—
73C22×C4⋊C4D412yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C42—21no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80#428—22no—
81#428—22no—
82#428—24no—
83#428—42no—
84#428D414yes—
85#428Q814yescomplemented
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—22no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—22no—
92#423—22no—
93#423Q814yescomplemented
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C4×M16—22no—
97C4×M16Q814yescomplemented
98C2×#97—21no—
99C2×C4×C8—21no—
100#398—24no—
101#398—24no—
102#395—24no—
103#395—24no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#385—24no—
111#384—22no—
112#384—22no—
113#384—22no—
114#384—21no—
115#384—21no—
116#384—21no—
117#384—21no—
118#384D411yescharacteristic
119#384D411yescharacteristic
120#382—22no—
121#382—24no—
122#382—42no—
123#382D412yes—
124#380—21no—
125#380—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4Q814yescomplemented
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140#376—21no—
141C2×#95—21no—
142C2×#95—24no—
143C2×#95—21no—
144C2×#95—24no—
145C43—21no—
146#343—24no—
147#343—42no—
148#340—22no—
149#340—22no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#339—24no—
155#339—24no—
156#339—24no—
157#339—42no—
158C22×Q8—24no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D414yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—22no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—22no—
179C2×M16C2×D412yes—
180C2×M16C2×Q812yes—
181C2×M16C2×Q814yes—
182C2×M16C2×Q812yes—
183C2×M16C2×Q812yes—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C4⋊Q8—44no—
188C4⋊Q8—44no—
189C4⋊Q8—44no—
190C4⋊Q8—44no—
191C4⋊Q8—84no—
192C4⋊Q8—84no—
193#125—44no—
194#125—44no—
195#125—44no—
196#125—44no—
197C22⋊Q8—48no—
198C22⋊Q8—48no—
199#117—21no—
200#117—21no—
201#117—21no—
202#117—21no—
203#117—22no—
204#117—22no—
205#117—22no—
206#117—22no—
207#117C2×D412yes—
208#117C2×Q812yes—
209#117#4112yes—
210#117#4112yes—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—82no—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×Q812yes—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—44no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×Q814yes—
276C2×C42C2×Q812yes—
277C2×C42C2×Q812yes—
278C2×C42C2×Q812yes—
279C2×C42#4112yes—
280#107—24no—
281#107—24no—
282#107—44no—
283#106—24no—
284#106—24no—
285#106—44no—
286C4⋊C8—24no—
287C4⋊C8—24no—
288#97—24no—
289C4×C8—24no—
290#95—24no—
291#95—24no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—48no—
300#95—41no—
301#95—41no—
302#95—84no—
303C24C22×D411yescharacteristic
304C2×Q8—48no—
305C2×Q8—84no—
306C2×Q8—84no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4#12011yescharacteristic
340C22×C4#12011yescharacteristic
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4C22⋊Q812yes—
344C22×C4C22⋊Q812yes—
345C22×C4C4⋊Q812yes—
346C22×C4C4⋊Q812yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×Q811yescharacteristic
351C22×C4C22×Q811yescharacteristic
352C22×C4C22×Q811yescharacteristic
353C22×C4C22×Q811yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C42−1+411yescharacteristic
360C22×C42−1+411yescharacteristic
361M16—22no—
362M16—22no—
363M16—22no—
364M16—22no—
365M16—22no—
366M16—24no—
367M16—24no—
368M16—42no—
369M16—41no—
370M16—41no—
371M16—42no—
372M16C4⋊Q814yescomplemented
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—24no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—42no—
381C2×C8—42no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—42no—
395C4⋊C4—48no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4—42no—
412C4⋊C4—82no—
413C4⋊C4—82no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C4⋊C4#12112yes—
417C4⋊C4#12112yes—
418C4⋊C4C22⋊Q812yes—
419C4⋊C4C22⋊Q812yes—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—24no—
431C42—24no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—82no—
441C42—82no—
442C42C22⋊Q812yes—
443C42C22⋊Q812yes—
444C42C22⋊Q812yes—
445C42C22⋊Q812yes—
446C42C4⋊Q812yes—
447C42C4⋊Q812yes—
448C42C4⋊Q812yes—
449C42C4⋊Q812yes—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23C2×#12011yescharacteristic
455C23C2×C22⋊Q811yescharacteristic
456C23C2×C4⋊Q811yescharacteristic
457C23D4211yescharacteristic
458C23#54611yescharacteristic
459C23#54811yescharacteristic
460C23D4×Q811yescharacteristic
461Q8—168no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—84no—
511C2×C4—84no—
512C2×C4#39512yes—
513C2×C4#39512yes—
514C2×C4#39512yes—
515C2×C4#39512yes—
516C2×C4#45412yes—
517C2×C4#45512yes—
518C2×C4C2×#12111yescharacteristic
519C2×C4C2×C22⋊Q811yescharacteristic
520C2×C4C2×C22⋊Q811yescharacteristic
521C2×C4C2×C22⋊Q811yescharacteristic
522C2×C4C2×C4⋊Q811yescharacteristic
523C2×C4C2×C4⋊Q811yescharacteristic
524C2×C4C2×C4⋊Q811yescharacteristic
525C2×C4#53211yescharacteristic
526C2×C4#53311yescharacteristic
527C2×C4#53311yescharacteristic
528C2×C4#53511yescharacteristic
529C2×C4#53611yescharacteristic
530C2×C4#54211yescharacteristic
531C2×C4#54211yescharacteristic
532C2×C4#54411yescharacteristic
533C2×C4#54411yescharacteristic
534C2×C4D4211yescharacteristic
535C2×C4#54611yescharacteristic
536C2×C4#54811yescharacteristic
537C2×C4D4×Q811yescharacteristic
538C2×C4D4×Q811yescharacteristic
539C2×C4D4×Q811yescharacteristic
540C2×C4#55411yescharacteristic
541C2×C4#55411yescharacteristic
542C2×C4#55711yescharacteristic
543C2×C4#55711yescharacteristic
544C2×C4Q8211yescharacteristic
545C2×C4Q8211yescharacteristic
546C8—24no—
547C8—42no—
548C8—44no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22C2×#39511yescharacteristic
562C22#240811yescharacteristic
563C22#240911yescharacteristic
564C22#245111yescharacteristic
565C22#245211yescharacteristic
566C22#255011yescharacteristic
567C22#255111yescharacteristic
568C22C2×#45511yescharacteristic
569C22C2×#45411yescharacteristic
570C22#301911yescharacteristic
571C22#302011yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—84no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—84no—
584C4—82no—
585C4#206114yescomplemented
586C4#245111yescharacteristic
587C4#245211yescharacteristic
588C4#254611yescharacteristic
589C4#255011yescharacteristic
590C4#255111yescharacteristic
591C4#256711yescharacteristic
592C4#262811yescharacteristic
593C4#285711yescharacteristic
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2#1512811yescharacteristic
599C2C2×#206111yescharacteristic
600C2#2356111yescharacteristic
601C2#2423111yescharacteristic
602C2#2423211yescharacteristic
603C2#2432111yescharacteristic
604C2#2432211yescharacteristic
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
85. #428Q8semidirect
93. #423Q8semidirect
97. C4×M16Q8semidirect
137. C4×C4⋊C4Q8semidirect
372. M16C4⋊Q8semidirect
585. C4#2061semidirect

Maximal subgroups

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

subgroupindex
#352752
C2×#20612
#211702
C4×#4282
#139112
#113922
#113392
#112172
#108572
#108522
#108512