Finite Group Databasebeta

← #545390

Subgroups — #545390

Order 512.

2254 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2581C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3C2×#2320C211yesmaximal, characteristic
4#14626C212yesmaximal
5#14411C214yesmaximal
6#14395C214yesmaximal
7#14327C212yesmaximal
8#14307C212yesmaximal
9#14277C211yesmaximal, characteristic
10#14250C212yesmaximal
11#14142C212yesmaximal
12#14132C212yesmaximal
13#14006C212yesmaximal
14#14003C212yesmaximal
15#13926C211yesmaximal, characteristic
16#13850C211yesmaximal, characteristic
17#13804C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20#2581C4116yescomplemented
21C2×#400C2212yes—
22C2×#395C2212yes—
23#2355—28no—
24#2320—28no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#387C2212yes—
35C2×#387C2212yes—
36C2×#387C2214yes—
37C2×#387C2214yes—
38C2×#384C2211yescharacteristic
39C2×#383C2212yes—
40C2×#382C2212yes—
41C2×#382C2212yes—
42C2×#382C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×#376C2211yescharacteristic
49#1446C2211yescharacteristic
50#1446C2211yescharacteristic
51#1445C2211yescharacteristic
52#1445C2211yescharacteristic
53#1445C2214yes—
54#1444C2212yes—
55#1444C2212yes—
56#1443C2214yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2212yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2212yes—
63#1441C2212yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2212yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2214yes—
77C4⋊#95C2214yes—
78#1434C2212yes—
79#1434C2214yes—
80#1434C2214yes—
81#1434C2214yes—
82#1434C2214yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×Q8C2311yescharacteristic
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8C2×C418yes—
93C2×C4×Q8C2×C418yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2314yes—
115C22×C42C2314yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118#400C2×C4116yes—
119#395C2×C4116yes—
120#391—24no—
121#391—24no—
122#391—28no—
123#391—44no—
124#391C2×C4116yes—
125#389—24no—
126#389—24no—
127#389—44no—
128#389C2×C4116yes—
129#387—28no—
130#387—28no—
131#387—216no—
132#387C2×C4132yes—
133#384—24no—
134#383—28no—
135#382—28no—
136#382—28no—
137#382C2×C418yes—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—28no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—44no—
142C4×C4⋊C4D418yes—
143#376—24no—
144C2×#95C2311yescharacteristic
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2311yescharacteristic
158C2×#95C2314yes—
159C2×#95C2311yescharacteristic
160C2×#95C2311yescharacteristic
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8C22×C414yes—
194C22×Q8C22×C414yes—
195C23×C4C2411yescharacteristic
196C23×C4C2414yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C4×Q8—232no—
213C4×Q8—232no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C22×C418yes—
248C2×C4⋊C4C22×C414yes—
249C2×C4⋊C4C22×C418yes—
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4C22×C418yes—
252C2×C4⋊C4C22×C4116yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D414yes—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—28no—
269C2×C42—28no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42C22×C418yes—
276C2×C42C22×C418yes—
277C2×C42C22×C414yes—
278C2×C42C2×D418yes—
279C2×C42C2×D414yes—
280C2×C42#4114yes—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—28no—
286#95—28no—
287#95—24no—
288#95—24no—
289#95—28no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95—44no—
299#95—42no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—42no—
304#95—42no—
305#95—42no—
306#95—42no—
307#95—42no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95C22×C418yes—
315#95C22×C418yes—
316#95C22×C4116yes—
317#95C22×C4116yes—
318#95C22×C4116yes—
319#95C2×D418yes—
320C24C2511yescharacteristic
321C2×Q8—28no—
322C2×Q8—28no—
323C2×Q8—28no—
324C2×Q8—28no—
325C2×Q8—48no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—48no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—44no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4C23×C418yes—
414C22×C4C23×C414yes—
415C22×C4C23×C414yes—
416C22×C4C23×C414yes—
417C22×C4C23×C414yes—
418C22×C4C23×C412yes—
419C22×C4C23×C412yes—
420C22×C4C23×C412yes—
421C22×C4C22×D414yes—
422C22×C4C22×D414yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4H(5,𝔽2)12yes—
430C22×C42−1+414yes—
431C22×C42−1+412yes—
432C22×C42−1+412yes—
433C22×C42−1+412yes—
434C4⋊C4—28no—
435C4⋊C4—28no—
436C4⋊C4—216no—
437C4⋊C4—216no—
438C4⋊C4—216no—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—416no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C4⋊C4—88no—
449C42—216no—
450C42—216no—
451C42—48no—
452C42—44no—
453C42—44no—
454C42—48no—
455C42—88no—
456C42—88no—
457C42C4×D4116yes—
458C23C24×C411yescharacteristic
459C23C23×D411yescharacteristic
460C23C22×#4111yescharacteristic
461C23C2×H(5,𝔽2)11yescharacteristic
462C23C2×2−1+411yescharacteristic
463C23C2×2−1+411yescharacteristic
464C23C2×2−1+411yescharacteristic
465C23C2×2−1+412yes—
466C23#58512yes—
467C23#58512yes—
468C23#58512yes—
469Q8—416no—
470Q8—416no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
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—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4C2×C4×D414yes—
564C2×C4C2×C4×D414yes—
565C2×C4C2×C4×D414yes—
566C2×C4#51914yes—
567C2×C4#52014yes—
568C2×C4#53614yes—
569C2×C4#53714yes—
570C2×C4#54114yes—
571C2×C4#54214yes—
572C22C22×C4×D411yescharacteristic
573C22C2×#51911yescharacteristic
574C22C2×#52011yescharacteristic
575C22#342812yes—
576C22C4×2−1+412yes—
577C22C2×#53611yescharacteristic
578C22C2×#53711yescharacteristic
579C22C2×#54111yescharacteristic
580C22C2×#54211yescharacteristic
581C22#348511yescharacteristic
582C22#348711yescharacteristic
583C22#349312yes—
584C22#349512yes—
585C22#349911yescharacteristic
586C22#350011yescharacteristic
587C22#355512yes—
588C22#355512yes—
589C22#355712yes—
590C22#355712yes—
591C22#356112yes—
592C22#356212yes—
593C22#356312yes—
594C22#356512yes—
595C4—24no—
596C4—24no—
597C4—44no—
598C4—44no—
599C4—44no—
600C4—44no—
601C4—44no—
602C4—48no—
603C4—44no—
604C4—44no—
605C4—81no—
606C4—84no—
607C4—81no—
608C4—81no—
609C4—81no—
610C4—82no—
611C4#232318yes—
612C2C2×#232311yescharacteristic
613C2#3420111yescharacteristic
614C2#3427611yescharacteristic
615C2#3449012yes—
616C2#3449912yes—
617C2#4264711yescharacteristic
618C2#4264911yescharacteristic
619C2#4268912yes—
620C2#4269112yes—
621C2#4287711yescharacteristic
622C2#4287911yescharacteristic
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
20. #2581C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25812
C2×#23552
C2×#23202
#146262
#144112
#143952
#143272
#143072
#142772
#142502
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