Finite Group Databasebeta

← #575051

Subgroups — #575051

Order 512.

3014 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
1C2×#2379C211yesmaximal, characteristic
2C2×#2372C211yesmaximal, characteristic
3C2×#2326C211yesmaximal, characteristic
4#14467C212yesmaximal, complemented
5#14405C214yesmaximal, complemented
6#14403C214yesmaximal, complemented
7#14401C212yesmaximal
8#14354C212yesmaximal, complemented
9#14015C212yesmaximal
10#13976C212yesmaximal, complemented
11#13963C212yesmaximal, complemented
12#13915C214yesmaximal, complemented
13#13880C214yesmaximal, complemented
14C22×#124C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18#2379—28no—
19#2372C4116yescomplemented
20#2326—28no—
21C2×#391C2211yescharacteristic
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#390C2211yescharacteristic
25C2×#389C2212yes—
26C2×#388C2214yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#385C2214yes—
34C2×#385C2214yes—
35C2×#385C2212yes—
36C2×#384C2211yescharacteristic
37C2×#384C2214yes—
38C2×#384C2212yes—
39C2×#382C2212yes—
40C2×#382C2214yes—
41C2×#380C2214yes—
42C2×C4×C22⋊C4C2214yes—
43C2×#376C2211yescharacteristic
44C2×#376C2212yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2212yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2212yes—
53#1442C2212yes—
54#1442C2212yes—
55#1441C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66#1434C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C4×#95C2212yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2312yes—
74C24×C4C2312yes—
75C2×#124C2×C418yes—
76C2×#124D418yes—
77C2×#124Q818yes—
78C2×C4×Q8—24no—
79C2×C4×Q8—42no—
80C2×C4×Q8—42no—
81C2×C4×D4—24no—
82C2×C4×D4—42no—
83C2×C4×D4—42no—
84C22×C4⋊C4—44no—
85C22×C4⋊C4—48no—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2314yes—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#391—24no—
115#391—28no—
116#391C2×C4116yescomplemented
117#390—24no—
118#389—44no—
119#388—216no—
120#388—216no—
121#388C2×C4132yescomplemented
122#386—28no—
123#386C2×C4116yes—
124#385—216no—
125#385—216no—
126#385—44no—
127#385—48no—
128#385—48no—
129#384—24no—
130#384—48no—
131#384—44no—
132#382—28no—
133#382—48no—
134#380—48no—
135C4×C22⋊C4C2×C4132yescomplemented
136#376—24no—
137#376C2×C4116yescomplemented
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C25C2412yes—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—41no—
162C22×Q8—41no—
163C22×Q8C22×C414yes—
164C22×Q8C2×D414yes—
165C22×Q8C2×Q814yes—
166C22×D4—41no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4—41no—
170C22×D4C22×C414yes—
171C22×D4C2×D414yes—
172C22×D4C2×Q814yes—
173C23×C4—28no—
174C23×C4—28no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—42no—
187C23×C4C22×C418yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200#124C4⋊C4164yescomplemented
201C4×Q8—416no—
202C4×D4—416no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—432no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C22×C418yes—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4C22×C4116yes—
261C2×C22⋊C4C22×C418yes—
262C2×C22⋊C4C22×C4116yes—
263C2×C22⋊C4C2×D4116yes—
264C2×C22⋊C4C2×Q8116yes—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—28no—
269C2×C42—28no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—41no—
278C2×C42—41no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42C22×C4116yes—
282C2×C42C22×C418yes—
283C2×C42C22×C414yes—
284C2×C42C2×D414yes—
285C2×C42C2×Q814yes—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95C22×C418yes—
303#95C22×C4116yes—
304#95C22×C418yes—
305#95C22×C4116yes—
306C24—24no—
307C24—22no—
308C24—24no—
309C24—22no—
310C24—24no—
311C24—24no—
312C24—22no—
313C24—22no—
314C24C23×C414yes—
315C24C22×D414yes—
316C24C22×Q814yes—
317C24C2511yescharacteristic
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8C2×C4⋊C4116yescomplemented
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4C2×C4⋊C4116yes—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—232no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—416no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—84no—
396C22×C4—88no—
397C22×C4—84no—
398C22×C4—88no—
399C22×C4—88no—
400C22×C4—84no—
401C22×C4—88no—
402C22×C4C23×C418yes—
403C22×C4C23×C412yes—
404C22×C4C23×C414yes—
405C22×C4C23×C414yes—
406C22×C4C23×C418yes—
407C22×C4C22×D412yes—
408C22×C4C22×D414yes—
409C22×C4C22×D414yes—
410C22×C4C22×Q814yes—
411C22×C4C22×Q812yes—
412C22×C4C22×Q814yes—
413C22×C4H(5,𝔽2)12yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C4H(5,𝔽2)18yes—
416C22×C42−1+418yes—
417C4⋊C4—48no—
418C4⋊C4—416no—
419C4⋊C4—48no—
420C4⋊C4—416no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—88no—
426C4⋊C4—84no—
427C4⋊C4—88no—
428C4⋊C4—88no—
429C4⋊C4—88no—
430C4⋊C4—88no—
431C22⋊C4—232no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—84no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4C2×C4⋊C4164yescomplemented
438C42—28no—
439C42—232no—
440C42—48no—
441C42—48no—
442C42—44no—
443C42—84no—
444C42C2×C4⋊C4116yescomplemented
445C23—28no—
446C23—28no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—28no—
452C23—28no—
453C23—24no—
454C23—24no—
455C23—28no—
456C23—28no—
457C23—42no—
458C23—42no—
459C23—44no—
460C23—44no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23C22×C4⋊C418yes—
468C23C24×C411yescharacteristic
469C23C23×D411yescharacteristic
470C23C23×Q811yescharacteristic
471C23C2×H(5,𝔽2)14yes—
472C23C2×H(5,𝔽2)11yescharacteristic
473C23C2×H(5,𝔽2)11yescharacteristic
474C23C2×2−1+414yes—
475C23#58511yescharacteristic
476C23#58511yescharacteristic
477Q8—416no—
478D4—416no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—432no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—816no—
547C2×C4—816no—
548C2×C4C22×C4⋊C418yes—
549C2×C4C22×C4⋊C414yes—
550C2×C4C22×C4⋊C418yes—
551C2×C4#520116yes—
552C22—216no—
553C22—28no—
554C22—44no—
555C22—48no—
556C22—48no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—48no—
562C22—44no—
563C22C23×C4⋊C411yescharacteristic
564C22C2×#52014yes—
565C22C4×H(5,𝔽2)11yescharacteristic
566C22C4×H(5,𝔽2)11yescharacteristic
567C22#348411yescharacteristic
568C22#348511yescharacteristic
569C22#349812yes—
570C22#349912yes—
571C22#350111yescharacteristic
572C22#350111yescharacteristic
573C22#350314yes—
574C22#353212yes—
575C22#353514yes—
576C22#353812yes—
577C22#354412yes—
578C22#354714yes—
579C22#354912yes—
580C4—28no—
581C4—216no—
582C4—24no—
583C4—28no—
584C4—42no—
585C4—48no—
586C4—42no—
587C4—82no—
588C4—84no—
589C4—82no—
590C4—82no—
591C4—84no—
592C2—48no—
593C2#3418312yes—
594C2#3424811yescharacteristic
595C2#3445712yes—
596C2#3448512yes—
597C2#5154211yescharacteristic
598C2#5154612yes—
599C2#5160112yes—
600C2#5160512yes—
601C2#5190211yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #14467C2semidirect
5. #14405C2semidirect
6. #14403C2semidirect
8. #14354C2semidirect
10. #13976C2semidirect
11. #13963C2semidirect
12. #13915C2semidirect
13. #13880C2semidirect
19. #2372C4semidirect
116. #391C2×C4semidirect
121. #388C2×C4semidirect
135. C4×C22⋊C4C2×C4semidirect
137. #376C2×C4semidirect
200. #124C4⋊C4semidirect
324. C2×Q8C2×C4⋊C4semidirect
437. C22⋊C4C2×C4⋊C4semidirect
444. C42C2×C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23792
C2×#23722
C2×#23262
#144672
#144052
#144032
#144012
#143542
#140152
#139762
#139632
#139152
#138802