Finite Group Databasebeta

← #360334

Subgroups — #360334

Order 512.

1520 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1823C211yesmaximal, characteristic
2C2×#1807C211yesmaximal, characteristic
3C22×#340C211yesmaximal, characteristic
4#13904C211yesmaximal, characteristic
5C22⋊C4×C4⋊C4C211yesmaximal, characteristic
6#11316C212yesmaximal, complemented
7#11257C212yesmaximal, complemented
8#10908C212yesmaximal, complemented
9#10807C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#106C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C2×#381—21no—
18C2×#381—21no—
19C2×#380C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×C4×C4⋊C4C2212yes—
22C2×C4×C22⋊C4—21no—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1823—24no—
25#1807—24no—
26C2×#340C418yescomplemented
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1445—22no—
39#1436—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C22⋊#95—21no—
44C24×C4—21no—
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—24no—
50C23×C8—42no—
51C23×C8C2×C414yes—
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4C2×C414yes—
66C22×C4⋊C4C2×C414yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—22no—
78C22×C42C2312yes—
79C2×#106—24no—
80C2×#106—24no—
81C2×#106—42no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85#381—44no—
86#381—44no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380D414yes—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—42no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—44no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95D412yes—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—42no—
111#340C2×C414yes—
112#340C2×C414yes—
113#340C2×C414yes—
114#340C2×C414yes—
115#336—24no—
116#336—42no—
117C25C2×D411yescharacteristic
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—24no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—44no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—82no—
140C23×C4C22×C412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4#4112yes—
146C23×C4#4112yes—
147C23×C4#4112yes—
148C23×C4#4112yes—
149C23×C4C2411yescharacteristic
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—28no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—28no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—44no—
167C22×C8—48no—
168C22×C8—48no—
169C22×C8C22×C412yes—
170C22×C8C22×C412yes—
171C22×C8#4114yes—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4#4114yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4SD1614yes—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—44no—
235C2×C42—44no—
236C2×C42—82no—
237#106—48no—
238C22⋊C8—44no—
239#95—24no—
240#95—24no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—42no—
246#95—44no—
247#95—44no—
248#95—82no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24C4×D412yes—
254C24#12112yes—
255C24#12312yes—
256C24C2×SD1612yes—
257C24#13512yes—
258C24AGL(1,ℤ/8)12yes—
259C24AGL(1,ℤ/8)12yes—
260C24C22×D411yescharacteristic
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—48no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—48no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—84no—
334C22×C4—88no—
335C22×C4—88no—
336C22×C4—84no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4#11714yes—
341C22×C4C4×D414yes—
342C22×C4C4×D412yes—
343C22×C4#12112yes—
344C22×C4#12312yes—
345C22×C4#12414yes—
346C22×C4#12614yes—
347C22×C4C2×SD1614yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4112yes—
353C22×C4C2×#4112yes—
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4H(5,𝔽2)12yes—
357C22×C42−1+412yes—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—416no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8#11714yes—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—48no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—48no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4#11714yes—
400C22⋊C4—28no—
401C22⋊C4—48no—
402C22⋊C4—44no—
403C22⋊C4—84no—
404C42—42no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—88no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—22no—
415C23—22no—
416C23—24no—
417C23—24no—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23#38814yes—
426C23C4×SD1614yes—
427C23#44214yes—
428C23#46014yes—
429C23#46314yes—
430C23#48114yes—
431C23#48214yes—
432C23C2×C4×D411yescharacteristic
433C23C2×#12111yescharacteristic
434C23C2×#12311yescharacteristic
435C23#54612yes—
436C23#54712yes—
437C23C22×SD1611yescharacteristic
438C23C2×#13511yescharacteristic
439C23C2×AGL(1,ℤ/8)11yescharacteristic
440C23C2×AGL(1,ℤ/8)11yescharacteristic
441C23#57712yes—
442C23#57812yes—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—816no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—816no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4#38814yes—
508C2×C4C2×#11711yescharacteristic
509C2×C4C2×#11711yescharacteristic
510C2×C4C2×#11711yescharacteristic
511C2×C4C2×C4×D411yescharacteristic
512C2×C4#51811yescharacteristic
513C2×C4#51911yescharacteristic
514C2×C4C2×#12411yescharacteristic
515C2×C4C2×#12611yescharacteristic
516C2×C4#52912yes—
517C2×C4#52912yes—
518C2×C4#53912yes—
519C2×C4#53912yes—
520C2×C4#54612yes—
521C2×C4#54712yes—
522C2×C4#55112yes—
523C2×C4#55112yes—
524C2×C4#55212yes—
525C2×C4#55212yes—
526C8—24no—
527C8—42no—
528C8—82no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—28no—
536C22—28no—
537C22—28no—
538C22—28no—
539C22#192618yes—
540C22C2×#38811yescharacteristic
541C22#234211yescharacteristic
542C22#234411yescharacteristic
543C22#241912yes—
544C22#242512yes—
545C22C2×C4×SD1611yescharacteristic
546C22C2×#44211yescharacteristic
547C22#294911yescharacteristic
548C22#295011yescharacteristic
549C22C2×#46311yescharacteristic
550C22C2×#46011yescharacteristic
551C22#304612yes—
552C22C2×#48211yescharacteristic
553C22C2×#48111yescharacteristic
554C22#309712yes—
555C22#329812yes—
556C22#330012yes—
557C22#330612yes—
558C22#331012yes—
559C22#333512yes—
560C22#333712yes—
561C22#334312yes—
562C22#334312yes—
563C4—24no—
564C4—44no—
565C4—44no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—84no—
571C4—162no—
572C4#228311yescharacteristic
573C4#231311yescharacteristic
574C4#234211yescharacteristic
575C4#234411yescharacteristic
576C4#248812yes—
577C4#249312yes—
578C2—28no—
579C2#1387411yescharacteristic
580C2C2×#192611yescharacteristic
581C2#2106611yescharacteristic
582C2#2256211yescharacteristic
583C2#2257111yescharacteristic
584C2#2260911yescharacteristic
585C2#2261711yescharacteristic
586C2#2493712yes—
587C2#2494512yes—
588C2#2520412yes—
589C2#2521212yes—
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. #11316C2semidirect
7. #11257C2semidirect
8. #10908C2semidirect
9. #10807C2semidirect
26. C2×#340C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18232
C2×#18072
C22×#3402
#139042
C22⋊C4×C4⋊C42
#113162
#112572
#109082
#108072
#107302
#107142