Finite Group Databasebeta

← #370170

Subgroups — #370170

Order 512.

1538 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1852C211yesmaximal, characteristic
2C2×#1850C211yesmaximal, characteristic
3C22×#340C211yesmaximal, characteristic
4#13881C211yesmaximal, characteristic
5C22⋊C4×C4⋊C4C211yesmaximal, characteristic
6#11268C212yesmaximal, complemented
7#11232C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10850C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C2×#382C2212yes—
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C22⋊C4—21no—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1852—24no—
26#1850C418yescomplemented
27C2×#340—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37#1442—22no—
38#1436—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C22⋊#95—21no—
42C22⋊#95—21no—
43C22⋊#95—21no—
44C24×C4—21no—
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—42no—
50C23×C8—42no—
51C23×C8C2×C414yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2312yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—42no—
63C22×C4⋊C4—42no—
64C22×C4⋊C4D414yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4D411yescharacteristic
74C22×C22⋊C4D411yescharacteristic
75C22×C42—22no—
76C22×C42C2312yes—
77C2×#107—22no—
78C2×#107—22no—
79C2×#107—24no—
80C2×#107—42no—
81C2×C4⋊C8—42no—
82C2×C4⋊C8—42no—
83C2×C4⋊C8C2×C418yescomplemented
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8C2×C414yes—
87#382—24no—
88#382—42no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380Q814yes—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—42no—
95C4×C22⋊C4—44no—
96C4×C22⋊C4C2×C414yes—
97C4×C22⋊C4Q814yes—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95D412yes—
108#340—22no—
109#340—22no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—42no—
114#340—42no—
115#340—42no—
116#336C2×C418yescomplemented
117C25C2×D411yescharacteristic
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—24no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—44no—
130C23×C4—44no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—82no—
137C23×C4—82no—
138C23×C4C22×C412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×Q812yes—
144C23×C4C2×Q812yes—
145C23×C4#4112yes—
146C23×C4#4112yes—
147C23×C4C2411yescharacteristic
148C22×C8—28no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—48no—
164C22×C8—44no—
165C22×C8—44no—
166C22×C8—48no—
167C22×C8C22×C414yes—
168C22×C8C22×C412yes—
169C22×C8C22×C412yes—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4—48no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×Q814yes—
206C2×C4⋊C4#4114yes—
207C2×C4⋊C4#4114yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4C4⋊C414yes—
219C2×C22⋊C4C4⋊C414yes—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—44no—
225C2×C42—44no—
226C2×C42—82no—
227C2×C42C22×C414yes—
228C2×C42C2×Q814yes—
229#107—44no—
230#107—44no—
231C4⋊C8—48no—
232C22⋊C8—28no—
233#95—24no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—42no—
239#95—82no—
240#95—82no—
241#95—82no—
242#95C4⋊C418yes—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24C2×C4⋊C412yes—
248C24#12112yes—
249C24C22⋊Q812yes—
250C24#13512yes—
251C24#13512yes—
252C24AGL(1,ℤ/8)12yes—
253C24#13712yes—
254C24C22×D411yescharacteristic
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—48no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—48no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—48no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—84no—
321C22×C4—88no—
322C22×C4—88no—
323C22×C4—88no—
324C22×C4—88no—
325C22×C4—84no—
326C22×C4—84no—
327C22×C4C2×C4⋊C414yes—
328C22×C4C2×C4⋊C414yes—
329C22×C4C2×C4⋊C412yes—
330C22×C4C4×D414yes—
331C22×C4C4×Q814yes—
332C22×C4#12112yes—
333C22×C4C22⋊Q812yes—
334C22×C4#12514yes—
335C22×C4C4⋊Q814yes—
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×Q811yescharacteristic
339C22×C4C22×Q811yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4112yes—
343C22×C4C2×#4112yes—
344C22×C4H(5,𝔽2)12yes—
345C22×C4H(5,𝔽2)12yes—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—416no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—48no—
373C4⋊C4—48no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4#12314yes—
386C22⋊C4—28no—
387C22⋊C4—28no—
388C22⋊C4—48no—
389C22⋊C4—84no—
390C42—24no—
391C42—24no—
392C42—24no—
393C42—24no—
394C42—88no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23—22no—
399C23—22no—
400C23—24no—
401C23—22no—
402C23—24no—
403C23—24no—
404C23—22no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23#38914yes—
412C23#42714yes—
413C23#42814yes—
414C23#46314yes—
415C23#46414yes—
416C23#47814yes—
417C23#47914yes—
418C23C22×C4⋊C411yescharacteristic
419C23C2×#12111yescharacteristic
420C23C2×C22⋊Q811yescharacteristic
421C23#54612yes—
422C23#54612yes—
423C23C2×#13511yescharacteristic
424C23C2×#13511yescharacteristic
425C23C2×AGL(1,ℤ/8)11yescharacteristic
426C23C2×#13711yescharacteristic
427C23#57612yes—
428C23#57812yes—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—28no—
436C2×C4—24no—
437C2×C4—28no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—816no—
468C2×C4—816no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4#38914yes—
490C2×C4C2×C4×D411yescharacteristic
491C2×C4C2×C4×Q811yescharacteristic
492C2×C4C4×#4112yes—
493C2×C4#51812yes—
494C2×C4C2×#12311yescharacteristic
495C2×C4#52511yescharacteristic
496C2×C4C2×#12511yescharacteristic
497C2×C4#52912yes—
498C2×C4C2×C4⋊Q811yescharacteristic
499C2×C4#53411yescharacteristic
500C2×C4#53511yescharacteristic
501C2×C4#53812yes—
502C2×C4#55112yes—
503C2×C4#55112yes—
504C2×C4#55412yes—
505C2×C4#55412yes—
506C2×C4#55412yes—
507C2×C4#55412yes—
508C8—42no—
509C8—44no—
510C8—42no—
511C22—28no—
512C22—28no—
513C22—24no—
514C22—24no—
515C22—28no—
516C22—24no—
517C22—28no—
518C22—24no—
519C22—24no—
520C22—24no—
521C22#194918yes—
522C22C2×#38911yescharacteristic
523C22#234912yes—
524C22#241111yescharacteristic
525C22#241211yescharacteristic
526C22#245012yes—
527C22C2×#42711yescharacteristic
528C22C2×#42811yescharacteristic
529C22#291312yes—
530C22C2×#46311yescharacteristic
531C22C2×#46411yescharacteristic
532C22#304111yescharacteristic
533C22#304411yescharacteristic
534C22C2×#47811yescharacteristic
535C22C2×#47911yescharacteristic
536C22#308312yes—
537C22#329712yes—
538C22#329712yes—
539C22#330012yes—
540C22#330012yes—
541C22#330812yes—
542C22#331212yes—
543C22#331412yes—
544C22#331812yes—
545C4—24no—
546C4—44no—
547C4—44no—
548C4—44no—
549C4—84no—
550C4—84no—
551C4—82no—
552C4—161no—
553C4—161no—
554C4C4×#12311yescharacteristic
555C4#232111yescharacteristic
556C4#236812yes—
557C4#250211yescharacteristic
558C4#250512yes—
559C4#258311yescharacteristic
560C2—28no—
561C2#1404411yescharacteristic
562C2C2×#194911yescharacteristic
563C2#2117311yescharacteristic
564C2#2270412yes—
565C2#2270912yes—
566C2#2447911yescharacteristic
567C2#2448111yescharacteristic
568C2#2452911yescharacteristic
569C2#2453111yescharacteristic
570C2#2641812yes—
571C2#2642012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11268C2semidirect
7. #11232C2semidirect
8. #10907C2semidirect
9. #10850C2semidirect
26. #1850C4semidirect
83. C2×C4⋊C8C2×C4semidirect
116. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18522
C2×#18502
C22×#3402
#138812
C22⋊C4×C4⋊C42
#112682
#112322
#109072
#108502
#107302
#107212