Finite Group Databasebeta

← #365550

Subgroups — #365550

Order 512.

1546 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#34208C211yesmaximal, characteristic
2C2×#1963C211yesmaximal, characteristic
3#21200C211yesmaximal, characteristic
4#21171C211yesmaximal, characteristic, complemented
5C2×#1860C212yesmaximal, complemented
6C2×#1859C212yesmaximal, complemented
7#14354C211yesmaximal, characteristic, complemented
8#11079C212yesmaximal, complemented
9#11078C212yesmaximal, complemented
10#10858C211yesmaximal, characteristic, complemented
11#10701C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#520C2211yescharacteristic
14C2×#520C2211yescharacteristic
15C2×#428C2212yes—
16C2×#423C2211yescharacteristic
17C22×#104C2212yes—
18C22×#104C2212yes—
19C2×#417C2212yes—
20C2×#417C2212yes—
21C2×C4×M16C2211yescharacteristic
22#2364—24no—
23#2352—22no—
24#2352—22no—
25#2328—21no—
26#2327—21no—
27#2311C2211yescharacteristic
28#2311C2211yescharacteristic
29C4×#124—21no—
30C4×#124—21no—
31C2×#384C2211yescharacteristic
32C2×#384C2212yes—
33C2×#382C2212yes—
34C2×#382C2211yescharacteristic
35C4×#117C2211yescharacteristic
36C2×#376C2211yescharacteristic
37#1963C418yescomplemented
38#1959—21no—
39#1959—21no—
40#1957—21no—
41#1956—21no—
42#1948—22no—
43#1947—22no—
44#1860—28no—
45#1859—28no—
46C2×#343C2212yes—
47C2×#343C2212yes—
48#1736C2212yes—
49C2×#339C2212yes—
50C2×#339C2212yes—
51#1444—24no—
52C22×#41C2311yescharacteristic
53C22×#41C2311yescharacteristic
54C22×M16C2312yes—
55C22×M16C2312yes—
56#532—22no—
57#532—22no—
58#532C2×C414yes—
59#532Q814yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×#127D411yescharacteristic
62C2×#124D411yescharacteristic
63C2×#124D411yescharacteristic
64C2×#121—22no—
65#520—24no—
66#520—24no—
67#520—42no—
68#520—42no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×#117—21no—
77C2×#117—21no—
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89#428—24no—
90#428—24no—
91C2×#107—22no—
92C2×#106—22no—
93#423—22no—
94#423C2×C414yescomplemented
95C2×C4⋊C8—22no—
96C2×#104—24no—
97C2×#104—24no—
98C2×#104—24no—
99C2×#104—24no—
100C2×#104C2×C418yescomplemented
101C2×#104C2×C418yescomplemented
102#417—24no—
103#417—24no—
104#417—42no—
105#417—42no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#102—22no—
109C2×#102—22no—
110C4×M16—22no—
111C4×M16—22no—
112C2×#97—21no—
113C2×C4×C8—21no—
114#391—21no—
115#391—21no—
116#390—21no—
117#390—21no—
118#388—24no—
119#388—24no—
120#387—24no—
121#387—24no—
122#386—24no—
123#386—24no—
124#386—22no—
125#386—22no—
126#384—24no—
127#384—22no—
128#384—21no—
129#384—21no—
130#384—42no—
131#382—21no—
132#382—22no—
133#382—21no—
134#382—24no—
135#382—42no—
136#380—22no—
137#380—22no—
138C4×C4⋊C4D412yes—
139C4×C4⋊C4D412yes—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—24no—
142C4×C22⋊C4—22no—
143#376—21no—
144#376—22no—
145#376C2×C414yescomplemented
146C2×#95—22no—
147C2×#95—24no—
148C2×#95D414yes—
149C43—21no—
150#343—24no—
151#343—24no—
152#340—24no—
153#339—24no—
154#339D418yescomplemented
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—22no—
160C2×#41—22no—
161C2×#41—22no—
162C2×#41—22no—
163C2×#41—22no—
164C2×#41—22no—
165C2×#41—22no—
166C2×#41—22no—
167C2×#41C22×C412yes—
168C2×#41C22×C412yes—
169C2×#41C2×Q812yes—
170C2×#41C2×Q812yes—
171C22×Q8C2×D411yescharacteristic
172C22×Q8C2×D411yescharacteristic
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—22no—
176C22×D4C2×D411yescharacteristic
177C22×D4C2×D411yescharacteristic
178C23×C4—22no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2411yescharacteristic
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16C22×C414yes—
199C2×M16C2×D414yes—
200C22×C8—22no—
201C22×C8—22no—
202C4⋊Q8—22no—
203C4⋊Q8C4⋊C414yescomplemented
204#127—22no—
205#127C4⋊C414yescomplemented
206#124—22no—
207#124—22no—
208#124C4⋊C414yescomplemented
209#124C4⋊C414yescomplemented
210#121—28no—
211#121—28no—
212C4×Q8—44no—
213C4×Q8—44no—
214C4×D4—28no—
215C4×D4—28no—
216C4×D4—44no—
217C4×D4—44no—
218C4×D4—48no—
219C4×D4—48no—
220#117—22no—
221#117—22no—
222#117—22no—
223#117—42no—
224#117—42no—
225#117—42no—
226#117—42no—
227#117C2×D414yes—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—84no—
253C2×C4⋊C4—88no—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42C22×C414yes—
275C2×C42C22×C412yes—
276C2×C42C2×D414yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D412yes—
280C2×C42C2×Q812yes—
281C2×C42#4114yes—
282C2×C42#4112yes—
283C2×C42#4112yes—
284#107—44no—
285#106—44no—
286C4⋊C8—28no—
287#104—216no—
288#104—216no—
289#103—44no—
290#103—44no—
291#102—44no—
292#102—44no—
293#97—42no—
294C4×C8—42no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—24no—
299#95—24no—
300#95—22no—
301#95—22no—
302#95—48no—
303C24—21no—
304C24—21no—
305C24C22×D411yescharacteristic
306#41—24no—
307#41—24no—
308#41—24no—
309#41—24no—
310#41—24no—
311#41—24no—
312#41—24no—
313#41—24no—
314#41—42no—
315#41—42no—
316#41—42no—
317#41—42no—
318#41—42no—
319#41—42no—
320#41—42no—
321#41—42no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8C2×C4⋊C412yes—
329C2×Q8C2×C4⋊C412yes—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—28no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—82no—
346C2×D4—82no—
347C2×D4C2×C4⋊C412yes—
348C2×D4C2×C4⋊C412yes—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—48no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4C4×D414yes—
388C22×C4#12012yes—
389C22×C4#12012yes—
390C22×C4C22⋊Q814yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C22×Q811yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4112yes—
398C22×C4C2×#4111yescharacteristic
399C22×C4H(5,𝔽2)12yes—
400C22×C42−1+412yes—
401M16—24no—
402M16—24no—
403M16—24no—
404M16—24no—
405M16—42no—
406M16—42no—
407M16—42no—
408M16—42no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—28no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—816no—
435C4⋊C4#12712yes—
436C4⋊C4#12712yes—
437C22⋊C4—24no—
438C22⋊C4—28no—
439C22⋊C4—28no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—42no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—24no—
449C42—44no—
450C42—44no—
451C42—42no—
452C42—82no—
453C42—82no—
454C42C2×C4⋊C412yes—
455C42C2×C4⋊C412yes—
456C42C4×D418yes—
457C42C4×D414yes—
458C42C22⋊Q814yes—
459C42#12312yes—
460C42#12312yes—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23C2×C4×D411yescharacteristic
474C23C2×#12011yescharacteristic
475C23C2×C22⋊Q811yescharacteristic
476C23D4212yes—
477C23#54812yes—
478Q8—24no—
479Q8—24no—
480Q8—42no—
481Q8—42no—
482D4—24no—
483D4—24no—
484D4—44no—
485D4—44no—
486D4—44no—
487D4—44no—
488D4—42no—
489D4—42no—
490D4—88no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—28no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4—82no—
532C2×C4—84no—
533C2×C4—164no—
534C2×C4—164no—
535C2×C4#38514yes—
536C2×C4#38514yes—
537C2×C4C22×C4⋊C411yescharacteristic
538C2×C4C2×C4×D411yescharacteristic
539C2×C4C2×C4×D412yes—
540C2×C4#52012yes—
541C2×C4C2×C22⋊Q811yescharacteristic
542C2×C4C2×#12311yescharacteristic
543C2×C4C2×#12711yescharacteristic
544C2×C4#53211yescharacteristic
545C2×C4#53212yes—
546C2×C4#53511yescharacteristic
547C2×C4#53611yescharacteristic
548C2×C4#54212yes—
549C2×C4D4212yes—
550C2×C4#54712yes—
551C2×C4#54712yes—
552C2×C4D4×Q812yes—
553C2×C4#55312yes—
554C2×C4#55412yes—
555C8—44no—
556C8—44no—
557C22—24no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—44no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22—82no—
567C22—82no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22C2×#38511yescharacteristic
573C22#236112yes—
574C22#243211yescharacteristic
575C22#243311yescharacteristic
576C22#245212yes—
577C22#302112yes—
578C22#302212yes—
579C4—24no—
580C4—24no—
581C4—48no—
582C4—42no—
583C4—42no—
584C4—42no—
585C4—42no—
586C4—82no—
587C4—82no—
588C4—82no—
589C4—162no—
590C4C4×#12711yescharacteristic
591C4#232611yescharacteristic
592C4D4×C4⋊C412yes—
593C4#242212yes—
594C4#254211yescharacteristic
595C4#256611yescharacteristic
596C2—22no—
597C2—22no—
598C2—82no—
599C2—82no—
600C2#1397711yescharacteristic
601C2#2086512yes—
602C2#2567012yes—
603C2#2575111yescharacteristic
604C2#2575211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
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. #21171C2semidirect
5. C2×#1860C2semidirect
6. C2×#1859C2semidirect
7. #14354C2semidirect
8. #11079C2semidirect
9. #11078C2semidirect
10. #10858C2semidirect
11. #10701C2semidirect
37. #1963C4semidirect
94. #423C2×C4semidirect
100. C2×#104C2×C4semidirect
101. C2×#104C2×C4semidirect
145. #376C2×C4semidirect
154. #339D4semidirect
203. C4⋊Q8C4⋊C4semidirect
205. #127C4⋊C4semidirect
208. #124C4⋊C4semidirect
209. #124C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342082
C2×#19632
#212002
#211712
C2×#18602
C2×#18592
#143542
#110792
#110782
#108582
#107012