Finite Group Databasebeta

← #4414816

Subgroups — #4414816

Order 512.

2318 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#17076C211yesmaximal, characteristic, complemented
2#17076C212yesmaximal, complemented
3#17072C211yesmaximal, characteristic, complemented
4#16572C214yesmaximal, complemented
5#16524C214yesmaximal, complemented
6#14822C212yesmaximal
7#14767C214yesmaximal
8#14758C214yesmaximal
9#14720C212yesmaximal
10#14682C212yesmaximal
11#14639C212yesmaximal, complemented
12#14638C212yesmaximal, complemented
13C2×C4⋊#95C211yesmaximal, characteristic
14C22×C22⋊Q8C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C42C2211yescharacteristic
18C2×#400C2212yes—
19C2×#400C2214yes—
20C2×#400C2212yes—
21C2×#395C2212yes—
22C2×#395C2212yes—
23C2×#395C2214yes—
24C2×#385C2214yes—
25C2×#382C2212yes—
26C2×#382C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C22×#95C2214yes—
31#1446C2214yes—
32#1445C2214yes—
33#1444C2214yes—
34#1444C2214yes—
35#1444C2214yes—
36#1444C2214yes—
37#1444C2214yes—
38#1443C2214yes—
39#1443C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1441C2212yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2212yes—
49#1441C2212yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2214yes—
53C4⋊#95C2211yescharacteristic
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2212yes—
56C4⋊#95C2212yes—
57C4⋊#95C2214yes—
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60#1434C2214yes—
61#1434C2214yes—
62C22⋊#95C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C23×Q8C2312yes—
68C24×C4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2312yes—
71C24×C4C2312yes—
72C2×C22⋊Q8—44no—
73C2×C22⋊Q8—48no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—42no—
77C22×C4⋊C4—42no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C42—42no—
99C22×C42—42no—
100C22×C42Q818yescomplemented
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104#400—28no—
105#400—48no—
106#400—44no—
107#395—28no—
108#395—48no—
109#395—44no—
110#385—48no—
111#382—44no—
112#382—44no—
113#380—44no—
114#380—44no—
115#380—44no—
116C2×#95—48no—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2312yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2312yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C25C2411yescharacteristic
145C22×Q8—24no—
146C22×Q8—44no—
147C22×Q8—42no—
148C22×Q8—44no—
149C22×Q8—42no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—28no—
163C23×C4—28no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4C2×Q814yes—
173C23×C4C2×Q818yes—
174C23×C4C2414yes—
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C23×C4C2412yes—
186C22⋊Q8—432no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—416no—
215C2×C4⋊C4—416no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
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⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4#4114yes—
234C2×C4⋊C4#4114yes—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—416no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42—41no—
248C2×C42C2×Q818yes—
249C2×C42C2×Q814yes—
250C2×C42C2×Q814yes—
251#95—24no—
252#95—28no—
253#95—24no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
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—44no—
272#95—42no—
273#95—42no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—22no—
283C24—22no—
284C24C22×Q812yes—
285C24H(5,𝔽2)14yes—
286C24H(5,𝔽2)12yes—
287C242−1+412yes—
288C242−1+414yes—
289C24C2511yescharacteristic
290C2×Q8—48no—
291C2×Q8—84no—
292C2×Q8—88no—
293C2×Q8—88no—
294C2×Q8—84no—
295C22×C4—28no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—28no—
299C22×C4—28no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—28no—
313C22×C4—28no—
314C22×C4—216no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—216no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
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—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—416no—
376C22×C4—416no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—88no—
380C22×C4—88no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4C22×Q814yes—
384C22×C4C22×Q812yes—
385C22×C4C22×Q812yes—
386C22×C4C22×Q814yes—
387C22×C4C2×#4114yes—
388C22×C4C2×#4114yes—
389C22×C4C2×#4112yes—
390C22×C4C2×#4112yes—
391C22×C4H(5,𝔽2)12yes—
392C22×C4H(5,𝔽2)12yes—
393C22×C4H(5,𝔽2)14yes—
394C22×C42−1+414yes—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—48no—
402C4⋊C4—416no—
403C4⋊C4—84no—
404C4⋊C4—84no—
405C4⋊C4—84no—
406C4⋊C4—84no—
407C4⋊C4—84no—
408C4⋊C4—88no—
409C4⋊C4—84no—
410C4⋊C4—88no—
411C22⋊C4—416no—
412C22⋊C4—416no—
413C42—216no—
414C42—44no—
415C42—44no—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23#54418yes—
435C23#54414yes—
436C23H(3,𝔽4)14yes—
437C23#56318yes—
438C23#56314yes—
439C23C23×Q811yescharacteristic
440C23C22×#4111yescharacteristic
441C23C22×#4111yescharacteristic
442C23C2×H(5,𝔽2)11yescharacteristic
443C23C2×H(5,𝔽2)11yescharacteristic
444C23C2×H(5,𝔽2)11yescharacteristic
445C23C2×H(5,𝔽2)12yes—
446C23C2×H(5,𝔽2)12yes—
447C23C2×2−1+411yescharacteristic
448C23C2×2−1+412yes—
449C23C2×2−1+412yes—
450Q8—816no—
451C2×C4—232no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
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—24no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—432no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
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—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—816no—
527C2×C4—816no—
528C2×C4#53314yes—
529C2×C4#54314yes—
530C2×C4#54418yes—
531C22—24no—
532C22—28no—
533C22—28no—
534C22—24no—
535C22—28no—
536C22—24no—
537C22—24no—
538C22—28no—
539C22—24no—
540C22—24no—
541C22#284418yes—
542C22C2×#53311yescharacteristic
543C22C2×#54311yescharacteristic
544C22C2×#54412yes—
545C22C2×#54411yescharacteristic
546C22C2×#54412yes—
547C22#349011yescharacteristic
548C22#349011yescharacteristic
549C22#349211yescharacteristic
550C22#349211yescharacteristic
551C22#351912yes—
552C22#352012yes—
553C22#352212yes—
554C22#352312yes—
555C22C2×H(3,𝔽4)11yescharacteristic
556C22C2×#56312yes—
557C22C2×#56311yescharacteristic
558C22#353612yes—
559C22#353612yes—
560C22#353712yes—
561C22#353812yes—
562C22#354311yescharacteristic
563C22#354411yescharacteristic
564C22#355212yes—
565C4—28no—
566C4—24no—
567C4—24no—
568C4—44no—
569C4—44no—
570C4—44no—
571C4—82no—
572C4—84no—
573C4—84no—
574C4—84no—
575C4—84no—
576C4—82no—
577C2—28no—
578C2#3505211yescharacteristic
579C2#3523912yes—
580C2C2×#284411yescharacteristic
581C2#3747712yes—
582C2#3762311yescharacteristic
583C2#4608911yescharacteristic
584C2#4609112yes—
585C2#4609712yes—
586C2#4609911yescharacteristic
587C2#4619911yescharacteristic
588C2#4620511yescharacteristic
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. #17076C2semidirect
2. #17076C2semidirect
3. #17072C2semidirect
4. #16572C2semidirect
5. #16524C2semidirect
11. #14639C2semidirect
12. #14638C2semidirect
100. C22×C42Q8semidirect

Maximal subgroups

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

subgroupindex
#170762
#170722
#165722
#165242
#148222
#147672
#147582
#147202
#146822
#146392
#146382
C2×C4⋊#952