Finite Group Databasebeta

← #121510

Subgroups — #121510

Order 512.

1056 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21938C211yesmaximal, characteristic
2C2×#1853C211yesmaximal, characteristic
3#13880C211yesmaximal, characteristic
4#10854C211yesmaximal, characteristic
5#10730C211yesmaximal, characteristic
6#8023C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C23×C4⋊C4C2211yescharacteristic
9C22×#106—21no—
10C22×C22⋊C8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C2×#385—22no—
13C2×#384—21no—
14C2×#380C2211yescharacteristic
15C2×#380C2211yescharacteristic
16C2×#380C2211yescharacteristic
17C2×C4×C22⋊C4C2211yescharacteristic
18#1853—22no—
19#1853—22no—
20#1853—21no—
21#1853—21no—
22#1853—21no—
23#1853—21no—
24#1807—22no—
25C2×#342—22no—
26C2×#340—22no—
27C2×#340—21no—
28#1444—22no—
29C4⋊#95—21no—
30#1294—22no—
31#1282C414yescomplemented
32C24×C4—21no—
33C24×C4C2311yescharacteristic
34C23×C8—21no—
35C23×C8D412yes—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—24no—
43C22×C4⋊C4—44no—
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4D412yes—
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4D411yescharacteristic
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4D412yes—
53C22×C42—21no—
54C2×#106—24no—
55C2×#106—22no—
56C2×#106—22no—
57C2×#106—44no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8C2×C412yes—
66C2×C22⋊C8C2×C412yes—
67C2×C22⋊C8D412yes—
68C2×C22⋊C8Q812yes—
69C2×C22⋊C8Q812yes—
70#385—44no—
71#385—44no—
72#384—44no—
73#380—21no—
74#380—21no—
75#380—21no—
76#380—21no—
77#380—21no—
78#380—21no—
79#380—21no—
80#380—21no—
81#380—21no—
82#380—21no—
83#380—21no—
84#380Q812yes—
85C4×C22⋊C4—21no—
86C4×C22⋊C4—21no—
87C4×C22⋊C4C2×C412yes—
88C4×C22⋊C4Q812yes—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—24no—
92C2×#95—21no—
93C2×#95—21no—
94#342—44no—
95#340—22no—
96#340—22no—
97#340—24no—
98#340—44no—
99C25C2×D411yescharacteristic
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—44no—
109C23×C4—41no—
110C23×C4—41no—
111C23×C4—41no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—82no—
115C23×C4C22×C411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×Q811yescharacteristic
122C23×C4C2×Q811yescharacteristic
123C23×C4C2×Q811yescharacteristic
124C23×C4C2×Q811yescharacteristic
125C23×C4#4111yescharacteristic
126C23×C4#4111yescharacteristic
127C23×C4#4111yescharacteristic
128C23×C4#4111yescharacteristic
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—42no—
135C22×C8—42no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8C4⋊C414yescomplemented
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—48no—
168C2×C4⋊C4—48no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—84no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4C4⋊C414yes—
187C2×C22⋊C4SD1614yes—
188C2×C22⋊C4SD1614yes—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—42no—
192C2×C42—42no—
193#106—48no—
194#106—82no—
195#106—82no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8C4⋊C414yescomplemented
204#95—22no—
205#95—22no—
206#95—22no—
207#95—48no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213C24—41no—
214C24—41no—
215C24—41no—
216C24—41no—
217C24C2×C4⋊C411yescharacteristic
218C24#12011yescharacteristic
219C24#12111yescharacteristic
220C24C22⋊Q811yescharacteristic
221C24#12311yescharacteristic
222C24#12711yescharacteristic
223C24C4⋊Q811yescharacteristic
224C24C2×SD1612yes—
225C24C2×SD1612yes—
226C24AGL(1,ℤ/8)12yes—
227C24#13712yes—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—88no—
276C22×C4—88no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4—81no—
292C22×C4C2×C4⋊C411yescharacteristic
293C22×C4C2×C4⋊C411yescharacteristic
294C22×C4C4×D411yescharacteristic
295C22×C4C4×D411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×Q811yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4C22⋊Q811yescharacteristic
303C22×C4C22⋊Q811yescharacteristic
304C22×C4C22⋊Q811yescharacteristic
305C22×C4C22⋊Q811yescharacteristic
306C22×C4C22⋊Q811yescharacteristic
307C22×C4C22⋊Q811yescharacteristic
308C22×C4C22⋊Q811yescharacteristic
309C22×C4C22⋊Q811yescharacteristic
310C22×C4#12311yescharacteristic
311C22×C4#12311yescharacteristic
312C22×C4#12511yescharacteristic
313C22×C4#12511yescharacteristic
314C22×C4#12511yescharacteristic
315C22×C4C4⋊Q811yescharacteristic
316C22×C4C4⋊Q811yescharacteristic
317C22×C4C4⋊Q811yescharacteristic
318C22×C4C4⋊Q811yescharacteristic
319C22×C4C4⋊Q811yescharacteristic
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—44no—
325C2×C8—44no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—44no—
329C2×C8—82no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C4⋊C4—82no—
346C4⋊C4—816no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—44no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—81no—
355C22⋊C4—81no—
356C22⋊C4—81no—
357C22⋊C4—81no—
358C22⋊C4#10618yescomplemented
359C42—44no—
360C42—44no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23#38411yescharacteristic
390C23#38511yescharacteristic
391C23#38911yescharacteristic
392C23#39211yescharacteristic
393C23#39511yescharacteristic
394C23#39811yescharacteristic
395C23#39911yescharacteristic
396C23C2×#10612yes—
397C23#42812yes—
398C23#44812yes—
399C23#45012yes—
400C23#46012yes—
401C23#46112yes—
402C23#47512yes—
403C23#47612yes—
404C23#48112yes—
405C23#48312yes—
406C23#49211yescharacteristic
407C23#49211yescharacteristic
408C23#49611yescharacteristic
409C23#49711yescharacteristic
410C23#49812yes—
411C23#50112yes—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—28no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—48no—
450C2×C4—44no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—88no—
454C2×C4—84no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—161no—
475C2×C4—161no—
476C2×C4—161no—
477C2×C4—161no—
478C2×C4—162no—
479C2×C4—162no—
480C2×C4#38511yescharacteristic
481C2×C4#38511yescharacteristic
482C2×C4#38911yescharacteristic
483C2×C4#38911yescharacteristic
484C2×C4#39511yescharacteristic
485C2×C4#39511yescharacteristic
486C2×C4#39811yescharacteristic
487C2×C4#39811yescharacteristic
488C2×C4B(4,𝔽2)12yes—
489C2×C4#45812yes—
490C8—44no—
491C8—44no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—22no—
505C22—22no—
506C22—21no—
507C22—21no—
508C22—22no—
509C22—22no—
510C22—21no—
511C22—21no—
512C22—22no—
513C22—21no—
514C22—22no—
515C22—21no—
516C22—22no—
517C22—21no—
518C22—22no—
519C22—21no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22#144411yescharacteristic
529C22#184411yescharacteristic
530C22#184711yescharacteristic
531C22#185312yes—
532C22#194812yes—
533C22#200811yescharacteristic
534C22#200911yescharacteristic
535C22#201311yescharacteristic
536C22#201411yescharacteristic
537C22#202111yescharacteristic
538C22#202211yescharacteristic
539C22#205812yes—
540C22#208412yes—
541C22#209411yescharacteristic
542C22#209611yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—48no—
550C4—84no—
551C4—162no—
552C4—161no—
553C4—161no—
554C4—161no—
555C4—161no—
556C4#186712yes—
557C4#206212yes—
558C2—22no—
559C2—21no—
560C2—22no—
561C2—22no—
562C2—21no—
563C2—21no—
564C2—21no—
565C2—22no—
566C2#994911yescharacteristic
567C2#995111yescharacteristic
568C2#995411yescharacteristic
569C2#996011yescharacteristic
570C2#1132311yescharacteristic
571C2#1133311yescharacteristic
572C2#1134411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
31. #1282C4semidirect
138. C22×C8C4⋊C4semidirect
203. C22⋊C8C4⋊C4semidirect
358. C22⋊C4#106semidirect

Maximal subgroups

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

subgroupindex
#219382
C2×#18532
#138802
#108542
#107302
#80232
C2×#12822