Finite Group Databasebeta

← #770457

Subgroups — #770457

Order 512.

2434 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#400C211yesmaximal, characteristic
2C2×#2329C211yesmaximal, characteristic
3#15626C214yesmaximal, complemented
4#15624C214yesmaximal, complemented
5#15609C214yesmaximal, complemented
6#15533C214yesmaximal, complemented
7#15014C211yesmaximal, characteristic
8#14914C211yesmaximal, characteristic
9#14821C211yesmaximal, characteristic
10#14720C211yesmaximal, characteristic
11#14149C212yesmaximal
12#14130C212yesmaximal
13#14044C212yesmaximal
14#14044C212yesmaximal
15#13778C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#400—28no—
20C2×#400C2214yes—
21C2×#400C2214yes—
22C2×#398C2212yes—
23C2×#395C2212yes—
24#2329—28no—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#389C2214yes—
28C2×#387C2212yes—
29C2×#387C2212yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44#1446C2212yes—
45#1445C2212yes—
46#1444C2212yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2212yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2212yes—
60#1441C2212yes—
61#1441C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2214yes—
69C4⋊#95C2214yes—
70C4⋊#95C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—28no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2314yes—
112#400—216no—
113#400—216no—
114#398—44no—
115#395—44no—
116#391—28no—
117#389—28no—
118#389—48no—
119#387—28no—
120#387—28no—
121#385—28no—
122#385—28no—
123#385—28no—
124#385—44no—
125#385—44no—
126#380—24no—
127#380Q8116yescomplemented
128C4×C22⋊C4—28no—
129C4×C22⋊C4—44no—
130C2×#95—24no—
131C2×#95—24no—
132C2×#95—24no—
133C2×#95—24no—
134C2×#95D418yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C25C2411yescharacteristic
166C22×Q8—22no—
167C22×Q8—24no—
168C22×Q8—42no—
169C22×Q8—42no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4C2×D414yes—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4C2×Q818yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C22⋊Q8—232no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—416no—
240C2×C4⋊C4—416no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4C2×Q8116yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4C2×Q818yes—
265C2×C22⋊C4C2×Q818yes—
266C2×C22⋊C4#4118yes—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—44no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42—44no—
274#95—28no—
275#95—24no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—42no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—42no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—42no—
299#95—42no—
300#95—44no—
301#95C2×D418yes—
302#95C2×D418yes—
303#95C2×Q8116yes—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24C22×D412yes—
309C24C22×Q814yes—
310C24C2×#4114yes—
311C24H(5,𝔽2)12yes—
312C242−1+412yes—
313C24C2511yescharacteristic
314C2×Q8—28no—
315C2×Q8—82no—
316C2×Q8—82no—
317C2×Q8—88no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—28no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—416no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—88no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4—88no—
408C22×C4—88no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4C4⋊Q8116yes—
412C22×C4C22×D414yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D414yes—
416C22×C4C22×Q818yes—
417C22×C4C22×Q818yes—
418C22×C4C22×Q814yes—
419C22×C4C22×Q814yes—
420C22×C4C2×#4114yes—
421C22×C4C2×#4114yes—
422C22×C4H(5,𝔽2)14yes—
423C4⋊C4—28no—
424C4⋊C4—216no—
425C4⋊C4—416no—
426C4⋊C4—416no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—82no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4—82no—
433C4⋊C4—88no—
434C4⋊C4—88no—
435C4⋊C4—88no—
436C4⋊C4—88no—
437C22⋊C4—216no—
438C22⋊C4—216no—
439C22⋊C4—216no—
440C22⋊C4—48no—
441C42—84no—
442C42—84no—
443C42—88no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—28no—
449C23—28no—
450C23—28no—
451C23—28no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23#52514yes—
457C23C2×C4⋊Q814yes—
458C23#53714yes—
459C23#55418yes—
460C23#55718yes—
461C23C23×D411yescharacteristic
462C23C23×Q812yes—
463C23C22×#4112yes—
464C23C2×H(5,𝔽2)11yescharacteristic
465C23C2×H(5,𝔽2)12yes—
466C23C2×2−1+411yescharacteristic
467C23#58512yes—
468C23#58512yes—
469C23#58512yes—
470Q8—88no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—416no—
498C2×C4—416no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
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—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—88no—
547C2×C4—88no—
548C2×C4—88no—
549C2×C4—88no—
550C2×C4—88no—
551C2×C4C2×C4⋊Q818yes—
552C22—216no—
553C22—216no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—24no—
559C22—24no—
560C22#258418yes—
561C22C2×#52511yescharacteristic
562C22C22×C4⋊Q811yescharacteristic
563C22C2×#53711yescharacteristic
564C22C2×#55412yes—
565C22C2×#55712yes—
566C22#348411yescharacteristic
567C22#348511yescharacteristic
568C22#348511yescharacteristic
569C22#348711yescharacteristic
570C22#350114yes—
571C22#350414yes—
572C22#351414yes—
573C22#352114yes—
574C22#353412yes—
575C22#353512yes—
576C22#356012yes—
577C22#356412yes—
578C4—28no—
579C4—48no—
580C4—48no—
581C4—44no—
582C4—44no—
583C4—42no—
584C4—42no—
585C4—44no—
586C4—44no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4—82no—
592C2—28no—
593C2C2×#258411yescharacteristic
594C2#3539111yescharacteristic
595C2#3545111yescharacteristic
596C2#3545411yescharacteristic
597C2#3564011yescharacteristic
598C2#3672211yescharacteristic
599C2#3696011yescharacteristic
600C2#4355414yes—
601C2#4355714yes—
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
3. #15626C2semidirect
4. #15624C2semidirect
5. #15609C2semidirect
6. #15533C2semidirect
127. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4002
C2×#23292
#156262
#156242
#156092
#155332
#150142
#149142
#148212
#147202
#141492
#141302
#140442
#137782