Finite Group Databasebeta

← #536523

Subgroups — #536523

Order 512.

2330 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic
2C2×#2356C211yesmaximal, characteristic
3C2×#2331C211yesmaximal, characteristic
4#14582C214yesmaximal
5#14380C214yesmaximal
6#14235C214yesmaximal
7#14161C211yesmaximal, characteristic
8#14142C212yesmaximal
9#14090C214yesmaximal
10#14030C212yesmaximal
11#14021C212yesmaximal
12#13848C211yesmaximal, characteristic
13#13804C212yesmaximal
14#13804C211yesmaximal, characteristic
15#13786C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18#2568C4116yescomplemented
19C2×#400C2212yes—
20C2×#400C2212yes—
21#2356—28no—
22#2331—28no—
23C2×#391C2214yes—
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2214yes—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#387C2214yes—
31C2×#387C2214yes—
32C2×#383C2212yes—
33C2×#382C2212yes—
34C2×#382C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×#376C2212yes—
43C2×#376C2211yescharacteristic
44#1446C2212yes—
45#1446C2212yes—
46#1444C2212yes—
47#1443C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1442C2212yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2212yes—
55#1441C2212yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2211yescharacteristic
68C4⋊#95C2211yescharacteristic
69#1434C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72#1434C2214yes—
73#1434C2214yes—
74#1434C2212yes—
75C4×#95C2214yes—
76C4×#95C2212yes—
77C4×#95C2212yes—
78C23×Q8C2311yescharacteristic
79C23×Q8C2312yes—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8C2×C418yes—
83C2×C4×Q8—24no—
84C2×C4×Q8—24no—
85C2×C4×Q8C2×C418yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C42C2311yescharacteristic
99C22×C42C2312yes—
100C22×C42C2314yes—
101C22×C42C2312yes—
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2314yes—
105C22×C42C2312yes—
106C22×C42C2314yes—
107C22×C42C2311yescharacteristic
108#400C2×C4116yes—
109#400C2×C4116yes—
110#391—28no—
111#391—24no—
112#391—24no—
113#391—216no—
114#391C2×C4132yes—
115#391C2×C4116yes—
116#389—28no—
117#387—216no—
118#387—216no—
119#383—28no—
120#382C2×C4116yes—
121#382D418yes—
122C4×C4⋊C4—28no—
123C4×C4⋊C4—28no—
124C4×C4⋊C4—28no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—28no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4C2×C418yes—
129#376—24no—
130#376—28no—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C22×Q8—22no—
167C22×Q8—24no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—44no—
175C22×Q8—44no—
176C22×Q8C22×C414yes—
177C22×Q8C22×C418yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C4⋊Q8—232no—
194C4×Q8—232no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—28no—
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—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4C22×C414yes—
225C2×C4⋊C4C22×C418yes—
226C2×C4⋊C4C22×C418yes—
227C2×C4⋊C4C22×C418yes—
228C2×C4⋊C4C22×C418yes—
229C2×C4⋊C4C2×D418yes—
230C2×C4⋊C4#4114yes—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—28no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—28no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—28no—
246C2×C42—28no—
247C2×C42—24no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42—44no—
252C2×C42—44no—
253C2×C42—44no—
254C2×C42C22×C418yes—
255C2×C42C22×C418yes—
256C2×C42C22×C414yes—
257C2×C42C2×D414yes—
258C2×C42#4114yes—
259C2×C42#4114yes—
260#95—22no—
261#95—24no—
262#95—28no—
263#95—24no—
264#95—22no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—28no—
274#95—28no—
275#95—24no—
276#95—44no—
277#95—42no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—42no—
285#95—42no—
286#95—44no—
287#95C22×C414yes—
288#95C22×C414yes—
289#95C22×C4116yes—
290#95C22×C4116yes—
291#95C22×C4116yes—
292#95C22×C4116yes—
293#95C2×D418yes—
294#95C2×D418yes—
295C24C2511yescharacteristic
296C2×Q8—28no—
297C2×Q8—216no—
298C2×Q8—28no—
299C2×Q8—28no—
300C2×Q8—416no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—88no—
306C2×Q8—84no—
307C2×Q8—84no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
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—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4C23×C414yes—
382C22×C4C23×C414yes—
383C22×C4C23×C414yes—
384C22×C4C23×C414yes—
385C22×C4C23×C418yes—
386C22×C4C23×C412yes—
387C22×C4C23×C412yes—
388C22×C4C23×C412yes—
389C22×C4C22×D418yes—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C2×#4114yes—
394C22×C4C2×#4114yes—
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4C2×#4112yes—
398C22×C4C2×#4112yes—
399C22×C4C2×#4112yes—
400C22×C42−1+414yes—
401C22×C42−1+412yes—
402C22×C42−1+412yes—
403C4⋊C4—28no—
404C4⋊C4—216no—
405C4⋊C4—216no—
406C4⋊C4—216no—
407C4⋊C4—216no—
408C4⋊C4—28no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—44no—
414C4⋊C4—88no—
415C42—216no—
416C42—216no—
417C42—28no—
418C42—416no—
419C42—44no—
420C42—48no—
421C42—84no—
422C42—88no—
423C42—88no—
424C42C4×D4116yes—
425C23C24×C411yescharacteristic
426C23C23×D411yescharacteristic
427C23C22×#4111yescharacteristic
428C23C22×#4111yescharacteristic
429C23C22×#4111yescharacteristic
430C23C2×2−1+411yescharacteristic
431C23C2×2−1+411yescharacteristic
432C23C2×2−1+412yes—
433C23#58512yes—
434C23#58512yes—
435C23#58512yes—
436Q8—416no—
437Q8—816no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—28no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4C2×C4×D414yes—
525C2×C4C2×C4×D414yes—
526C2×C4C2×C4×D414yes—
527C2×C4C4×#4114yes—
528C2×C4#51914yes—
529C2×C4#54714yes—
530C2×C4#54814yes—
531C2×C4#55214yes—
532C2×C4#55614yes—
533C22C22×C4×D411yescharacteristic
534C22C2×C4×#4111yescharacteristic
535C22C2×#51911yescharacteristic
536C22#342812yes—
537C22C4×2−1+412yes—
538C22C2×#54711yescharacteristic
539C22C2×#54811yescharacteristic
540C22C2×#55211yescharacteristic
541C22C2×#55611yescharacteristic
542C22#348912yes—
543C22#349312yes—
544C22#349512yes—
545C22#350012yes—
546C22#350911yescharacteristic
547C22#351011yescharacteristic
548C22#351011yescharacteristic
549C22#351011yescharacteristic
550C22#351812yes—
551C22#352112yes—
552C22#355512yes—
553C22#355712yes—
554C22#356312yes—
555C22#356512yes—
556C4—24no—
557C4—24no—
558C4—44no—
559C4—44no—
560C4—44no—
561C4—44no—
562C4—44no—
563C4—48no—
564C4—48no—
565C4—44no—
566C4—44no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4#236218yes—
571C2C2×#236211yescharacteristic
572C2#3421111yescharacteristic
573C2#3426811yescharacteristic
574C2#3430712yes—
575C2#3449012yes—
576C2#4515712yes—
577C2#4516311yescharacteristic
578C2#4522712yes—
579C2#4523111yescharacteristic
580C2#4530611yescharacteristic
581C2#4543111yescharacteristic
subgroupG/H
center, Frattini, socle, prime radical, cosocleC24C25
commutatorC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #2568C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#23562
C2×#23312
#145822
#143802
#142352
#141612
#141422
#140902
#140302
#140212
#138482
#138042
#137862