Finite Group Databasebeta

← #877419

Subgroups — #877419

Order 512.

2426 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17080C211yesmaximal, characteristic, complemented
2#17076C211yesmaximal, characteristic, complemented
3#17012C212yesmaximal, complemented
4#16120C214yesmaximal, complemented
5#15691C214yesmaximal, complemented
6#14934C214yesmaximal
7#14822C212yesmaximal
8#14702C212yesmaximal
9#14604C212yesmaximal, complemented
10#14471C212yesmaximal, complemented
11#13878C214yesmaximal
12#13870C212yesmaximal
13C2×C4⋊#95C211yesmaximal, characteristic
14C22×C22⋊Q8C2212yes—
15C22×#117C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C2×#400C2212yes—
20C2×#400C2212yes—
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#381C2214yes—
24C2×#381C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C22×#95C2214yes—
28#1446C2214yes—
29#1446C2214yes—
30#1444C2214yes—
31#1444C2214yes—
32#1442C2214yes—
33#1442C2214yes—
34#1442C2214yes—
35#1442C2214yes—
36#1442C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1441C2212yes—
43#1441C2212yes—
44#1441C2212yes—
45#1441C2212yes—
46#1441C2212yes—
47#1441C2212yes—
48#1436C2214yes—
49#1436C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2214yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2211yescharacteristic
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C23×Q8C2312yes—
68C24×C4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2312yes—
71C24×C4C2312yes—
72C2×C22⋊Q8—28no—
73C2×C22⋊Q8—44no—
74C2×C22⋊Q8—44no—
75C2×#117—44no—
76C2×#117—48no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4—42no—
82C22×C4⋊C4Q818yescomplemented
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C42—24no—
100C22×C42—42no—
101C22×C42—42no—
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2314yes—
105C22×C42C2311yescharacteristic
106#400—28no—
107#400—44no—
108#391—28no—
109#391—44no—
110#381—48no—
111#381—44no—
112#380—44no—
113#380—44no—
114C2×#95—216no—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2312yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C25C2411yescharacteristic
144C22×Q8—24no—
145C22×Q8—24no—
146C22×Q8—24no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8—42no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—28no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—28no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—28no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—44no—
172C23×C4C2×Q814yes—
173C23×C4C2×Q818yes—
174C23×C4#4118yes—
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186C23×C4C2412yes—
187C22⋊Q8—432no—
188#117—432no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—416no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—416no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C2×Q814yes—
227C2×C4⋊C4C2×Q814yes—
228C2×C4⋊C4C2×Q818yes—
229C2×C4⋊C4#4114yes—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—44no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—41no—
243C2×C42—41no—
244C2×C42—42no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42—416no—
248C2×C42#4114yes—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—24no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—42no—
272#95—44no—
273#95—42no—
274#95—44no—
275#95—42no—
276#95—44no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24C22×Q812yes—
284C24C2×#4114yes—
285C24H(5,𝔽2)12yes—
286C24H(5,𝔽2)14yes—
287C24H(5,𝔽2)12yes—
288C24C2511yescharacteristic
289C2×Q8—48no—
290C2×Q8—48no—
291C2×Q8—48no—
292C2×Q8—84no—
293C2×Q8—84no—
294C2×Q8—84no—
295C2×Q8—84no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—28no—
299C22×C4—28no—
300C22×C4—28no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—216no—
316C22×C4—216no—
317C22×C4—232no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—216no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—416no—
377C22×C4—416no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—88no—
382C22×C4—84no—
383C22×C4C22×Q814yes—
384C22×C4C22×Q814yes—
385C22×C4C22×Q812yes—
386C22×C4C22×Q812yes—
387C22×C4C2×#4114yes—
388C22×C4C2×#4114yes—
389C22×C4C2×#4112yes—
390C22×C4C2×#4112yes—
391C22×C4C2×#4114yes—
392C22×C4C2×#4114yes—
393C22×C4H(5,𝔽2)12yes—
394C22×C42−1+412yes—
395C22×C42−1+414yes—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—48no—
399C4⋊C4—44no—
400C4⋊C4—48no—
401C4⋊C4—44no—
402C4⋊C4—48no—
403C4⋊C4—416no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—84no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C22⋊C4—416no—
411C22⋊C4—416no—
412C42—44no—
413C42—48no—
414C42—44no—
415C42—416no—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23#53818yes—
435C23#54314yes—
436C23#55418yes—
437C23#55914yes—
438C23H(3,𝔽4)14yes—
439C23C23×Q811yescharacteristic
440C23C22×#4111yescharacteristic
441C23C22×#4111yescharacteristic
442C23C22×#4112yes—
443C23C2×H(5,𝔽2)11yescharacteristic
444C23C2×H(5,𝔽2)12yes—
445C23C2×H(5,𝔽2)11yescharacteristic
446C23C2×H(5,𝔽2)11yescharacteristic
447C23C2×2−1+411yescharacteristic
448C23C2×2−1+412yes—
449C23#58512yes—
450Q8—816no—
451C2×C4—232no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—432no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—432no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—816no—
528C2×C4#53314yes—
529C2×C4#54414yes—
530C2×C4#55218yes—
531C22—24no—
532C22—28no—
533C22—28no—
534C22—24no—
535C22—28no—
536C22—24no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—28no—
541C22#280418yes—
542C22C2×#53311yescharacteristic
543C22C2×#53812yes—
544C22C2×#54311yescharacteristic
545C22C2×#54411yescharacteristic
546C22C2×#55212yes—
547C22C2×#55412yes—
548C22#347512yes—
549C22C2×#55911yescharacteristic
550C22#349011yescharacteristic
551C22#349011yescharacteristic
552C22#349211yescharacteristic
553C22#349211yescharacteristic
554C22#349412yes—
555C22#350212yes—
556C22#350512yes—
557C22#350912yes—
558C22#352212yes—
559C22C2×H(3,𝔽4)11yescharacteristic
560C22#353512yes—
561C22#353612yes—
562C22#354411yescharacteristic
563C22#354511yescharacteristic
564C22#355912yes—
565C4—24no—
566C4—24no—
567C4—28no—
568C4—44no—
569C4—44no—
570C4—44no—
571C4—44no—
572C4—84no—
573C4—84no—
574C4—84no—
575C4—82no—
576C4—84no—
577C2—28no—
578C2#3505211yescharacteristic
579C2#3619312yes—
580C2#3719411yescharacteristic
581C2#3788112yes—
582C2#3846712yes—
583C2#3949612yes—
584C2#4599111yescharacteristic
585C2#4609911yescharacteristic
586C2C2×#280411yescharacteristic
587C2#5932911yescharacteristic
588C2#5986111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17080C2semidirect
2. #17076C2semidirect
3. #17012C2semidirect
4. #16120C2semidirect
5. #15691C2semidirect
9. #14604C2semidirect
10. #14471C2semidirect
82. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#170802
#170762
#170122
#161202
#156912
#149342
#148222
#147022
#146042
#144712
#138782
#138702
C2×C4⋊#952