Finite Group Databasebeta

← #1045265

Subgroups — #1045265

Order 512.

1906 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17209C212yesmaximal, complemented
2#16824C212yesmaximal
3#16712C214yesmaximal, complemented
4#16236C212yesmaximal, complemented
5#15594C214yesmaximal
6#15234C212yesmaximal, complemented
7#15156C212yesmaximal, complemented
8#15135C211yesmaximal, characteristic
9#14592C212yesmaximal
10#14130C211yesmaximal, characteristic
11#14006C212yesmaximal, complemented
12#13973C212yesmaximal
13#13909C212yesmaximal, complemented
14#13881C212yesmaximal
15C4×#399C211yesmaximal, characteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#400C2214yes—
18C2×#399C2212yes—
19C2×#399C2211yescharacteristic
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#385C2212yes—
24C2×#384C2211yescharacteristic
25C2×#382C2212yes—
26C2×#382C2212yes—
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#1445C2212yes—
35#1445C2212yes—
36#1445C2211yescharacteristic
37#1444C2212yes—
38#1444C2211yescharacteristic
39#1444C2212yes—
40#1443C2214yes—
41#1443C2214yes—
42#1442C2212yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1439C2211yescharacteristic
56#1439C2212yes—
57#1439C2212yes—
58#1439C2214yes—
59#1437C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70#1434C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C4×#95C2211yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C22×C4⋊C4—42no—
78C22×C4⋊C4—44no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2311yescharacteristic
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2314yes—
104#400—48no—
105#399—24no—
106#399—28no—
107#389—28no—
108#389—44no—
109#389—44no—
110#385—44no—
111#384—42no—
112#382—44no—
113#382—44no—
114#380—44no—
115#380Q818yes—
116C4×C4⋊C4—28no—
117C4×C4⋊C4—42no—
118C4×C4⋊C4—44no—
119C4×C22⋊C4—28no—
120C4×C22⋊C4—42no—
121C2×#95C2311yescharacteristic
122C2×#95C2312yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2311yescharacteristic
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C25C2411yescharacteristic
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—42no—
163C23×C4C2×Q814yes—
164C23×C4C2414yes—
165C23×C4C2411yescharacteristic
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2412yes—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2411yescharacteristic
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—416no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4C2×Q818yes—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4C2×Q818yes—
235C2×C22⋊C4#4114yes—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42#4118yes—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—28no—
254#95—22no—
255#95—24no—
256#95—28no—
257#95—24no—
258#95—42no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—41no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—41no—
267#95—41no—
268#95—44no—
269#95—42no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95C2×Q818yes—
281C24—22no—
282C24—21no—
283C24—21no—
284C24—22no—
285C24—21no—
286C24—22no—
287C24—21no—
288C24—22no—
289C24C22×Q812yes—
290C24C2×#4112yes—
291C242−1+412yes—
292C24C2511yescharacteristic
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—28no—
328C22×C4—216no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
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—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—48no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—48no—
373C22×C4—84no—
374C22×C4—88no—
375C22×C4—88no—
376C22×C4—84no—
377C22×C4C22×Q812yes—
378C22×C4C22×Q818yes—
379C22×C4C22×Q812yes—
380C22×C4C2×#4114yes—
381C22×C4C2×#4112yes—
382C22×C4C2×#4114yes—
383C22×C4C2×#4112yes—
384C22×C4C2×#4114yes—
385C22×C4H(5,𝔽2)12yes—
386C22×C4H(5,𝔽2)14yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C42−1+412yes—
389C4⋊C4—216no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—88no—
399C4⋊C4—84no—
400C4⋊C4—88no—
401C4⋊C4—84no—
402C4⋊C4—88no—
403C22⋊C4—48no—
404C22⋊C4—44no—
405C22⋊C4—48no—
406C22⋊C4—48no—
407C22⋊C4—44no—
408C22⋊C4—88no—
409C42—216no—
410C42—44no—
411C42—48no—
412C42—44no—
413C42—88no—
414C23—22no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—22no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—24no—
423C23—22no—
424C23—24no—
425C23—22no—
426C23—42no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23#55714yes—
437C23C23×Q811yescharacteristic
438C23C22×#4112yes—
439C23C22×#4111yescharacteristic
440C23C2×H(5,𝔽2)12yes—
441C23C2×H(5,𝔽2)11yescharacteristic
442C23C2×H(5,𝔽2)11yescharacteristic
443C23C2×2−1+411yescharacteristic
444C23C2×2−1+411yescharacteristic
445C23#58512yes—
446C23#58512yes—
447C23#58511yescharacteristic
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—28no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—416no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—41no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—816no—
531C2×C4#53918yes—
532C2×C4#54414yes—
533C2×C4#55418yes—
534C2×C4#55914yes—
535C2×C4#56014yes—
536C22—24no—
537C22—28no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—44no—
543C22—44no—
544C22—42no—
545C22—42no—
546C22—44no—
547C22C2×#53912yes—
548C22C2×#54411yescharacteristic
549C22C2×#55412yes—
550C22#347512yes—
551C22C2×#55711yescharacteristic
552C22C2×#55911yescharacteristic
553C22#349412yes—
554C22#349512yes—
555C22#350111yescharacteristic
556C22#350412yes—
557C22#350511yescharacteristic
558C22#351111yescharacteristic
559C22#351212yes—
560C22#351412yes—
561C22C2×#56011yescharacteristic
562C22#353611yescharacteristic
563C22#354011yescharacteristic
564C22#354612yes—
565C22#355111yescharacteristic
566C22#356112yes—
567C22#356212yes—
568C22#356311yescharacteristic
569C22#356512yes—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—48no—
574C4—44no—
575C4—42no—
576C4—44no—
577C4—81no—
578C4—84no—
579C4—84no—
580C4—82no—
581C4—82no—
582C4—84no—
583C4#280718yes—
584C2—44no—
585C2#3975111yescharacteristic
586C2#3991212yes—
587C2#4294812yes—
588C2#4298111yescharacteristic
589C2#4494212yes—
590C2#4543311yescharacteristic
591C2#4568311yescharacteristic
592C2C2×#280711yescharacteristic
593C2#5301811yescharacteristic
594C2#5932512yes—
595C2#5986511yescharacteristic
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. #17209C2semidirect
3. #16712C2semidirect
4. #16236C2semidirect
6. #15234C2semidirect
7. #15156C2semidirect
11. #14006C2semidirect
13. #13909C2semidirect

Maximal subgroups

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

subgroupindex
#172092
#168242
#167122
#162362
#155942
#152342
#151562
#151352
#145922
#141302
#140062
#139732
#139092
#138812
C4×#3992