Finite Group Databasebeta

← #364263

Subgroups — #364263

Order 512.

1604 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2347C211yesmaximal, characteristic
2C2×#1851C211yesmaximal, characteristic
3C2×#1825C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5#13880C211yesmaximal, characteristic
6#11227C212yesmaximal, complemented
7#11207C212yesmaximal, complemented
8#10849C212yesmaximal, complemented
9#10807C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10722C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17#2347—24no—
18#2347C418yescomplemented
19C2×#389C2212yes—
20C2×#388—22no—
21C2×#385—22no—
22C2×#384C2212yes—
23C2×#383—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1851—24no—
30#1825—24no—
31#1736—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43#1444—21no—
44#1444—21no—
45C4⋊#95C2212yes—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—24no—
52C22×M16—42no—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117C2×C414yes—
60C2×#117D414yes—
61C2×#117Q814yes—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—42no—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4D412yes—
80C22×C42—21no—
81C22×C42—22no—
82C22×C42C2312yes—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—24no—
85C2×C4⋊C8—42no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89#389—24no—
90#389C2×C418yescomplemented
91#388—216no—
92#385—48no—
93#384—24no—
94#384—42no—
95#383—28no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380C2×C414yes—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—28no—
102C4×C4⋊C4C2×C418yescomplemented
103C4×C22⋊C4C2×C414yes—
104C4×C22⋊C4D414yes—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95D411yescharacteristic
112C2×#95D411yescharacteristic
113#340—24no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#340—42no—
123#336—24no—
124C25C2×D411yescharacteristic
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—44no—
136C23×C4—42no—
137C23×C4—44no—
138C23×C4—81no—
139C23×C4—81no—
140C23×C4C22×C412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×Q812yes—
147C23×C4#4112yes—
148C23×C4#4112yes—
149C23×C4C2411yescharacteristic
150C2×M16—28no—
151C2×M16—48no—
152C2×M16—48no—
153C2×M16—44no—
154C2×M16—44no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8#4114yes—
172#117—28no—
173#117—28no—
174#117—28no—
175#117C4⋊C4116yescomplemented
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—28no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4C22×C414yes—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×Q812yes—
220C2×C4⋊C4C2×Q812yes—
221C2×C4⋊C4#4114yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—81no—
231C2×C22⋊C4—81no—
232C2×C22⋊C4C22⋊C418yes—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—28no—
241C2×C42—82no—
242C2×C42—82no—
243C2×C42C22×C414yes—
244C2×C42C2×D414yes—
245C4⋊C8—48no—
246C22⋊C8—44no—
247#95—24no—
248#95—28no—
249#95—28no—
250#95—44no—
251#95—42no—
252#95—44no—
253#95C22⋊C414yes—
254#95C22⋊C414yes—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24C2×C22⋊C412yes—
264C24#12112yes—
265C24C22⋊Q812yes—
266C24C22×D411yescharacteristic
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—48no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—88no—
331C22×C4—88no—
332C22×C4—84no—
333C22×C4—84no—
334C22×C4C2×C22⋊C414yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C414yes—
337C22×C4C2×C4⋊C414yes—
338C22×C4#11714yes—
339C22×C4#12114yes—
340C22×C4#12112yes—
341C22×C4C22⋊Q814yes—
342C22×C4C22⋊Q812yes—
343C22×C4AGL(1,ℤ/8)14yes—
344C22×C4#13714yes—
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×Q811yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4112yes—
352C22×C4C2×#4112yes—
353C22×C4H(5,𝔽2)12yes—
354C22×C42−1+412yes—
355M16—416no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—48no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—416no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—88no—
396C4⋊C4—88no—
397C4⋊C4C2×C4⋊C414yes—
398C4⋊C4C2×C4⋊C414yes—
399C22⋊C4—216no—
400C22⋊C4—216no—
401C22⋊C4—84no—
402C22⋊C4—84no—
403C42—24no—
404C42—24no—
405C42—24no—
406C42—24no—
407C42—216no—
408C42—88no—
409C23—24no—
410C23—22no—
411C23—22no—
412C23—24no—
413C23—24no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23#38014yes—
432C23C22×C22⋊C411yescharacteristic
433C23C2×#12111yescharacteristic
434C23C2×C22⋊Q811yescharacteristic
435C23#54612yes—
436C23#54712yes—
437C23C2×AGL(1,ℤ/8)12yes—
438C23C2×#13712yes—
439C23#57611yescharacteristic
440C23#57711yescharacteristic
441C23#57711yescharacteristic
442C23#57811yescharacteristic
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—28no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—816no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4#38014yes—
504C2×C4#41718yes—
505C2×C4C22×C4⋊C411yescharacteristic
506C2×C4C2×#11711yescharacteristic
507C2×C4C4×#4112yes—
508C2×C4#52012yes—
509C2×C4C2×#12111yescharacteristic
510C2×C4C2×C22⋊Q811yescharacteristic
511C2×C4#52511yescharacteristic
512C2×C4#52511yescharacteristic
513C2×C4#53312yes—
514C2×C4#53511yescharacteristic
515C2×C4#53611yescharacteristic
516C2×C4#54412yes—
517C2×C4#54612yes—
518C2×C4#54712yes—
519C2×C4#55112yes—
520C2×C4#55112yes—
521C2×C4#55212yes—
522C2×C4#55212yes—
523C8—44no—
524C8—81no—
525C8—81no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—42no—
530C22—42no—
531C22—44no—
532C22—44no—
533C22—42no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22C2×#38011yescharacteristic
538C22#233912yes—
539C22#241111yescharacteristic
540C22#241311yescharacteristic
541C22#243812yes—
542C22C2×#41712yes—
543C22#289711yescharacteristic
544C22#289811yescharacteristic
545C22#303811yescharacteristic
546C22#303911yescharacteristic
547C22#304511yescharacteristic
548C22#304611yescharacteristic
549C22#307912yes—
550C22#308311yescharacteristic
551C22#308411yescharacteristic
552C22#331812yes—
553C22#332012yes—
554C22#332412yes—
555C22#332812yes—
556C22#334512yes—
557C22#334612yes—
558C22#335312yes—
559C22#335612yes—
560C4—24no—
561C4—28no—
562C4—44no—
563C4—44no—
564C4—44no—
565C4—84no—
566C4—84no—
567C4—161no—
568C4—161no—
569C4#229611yescharacteristic
570C4#231711yescharacteristic
571C4#234312yes—
572C4#241111yescharacteristic
573C4#241311yescharacteristic
574C4#243812yes—
575C2—44no—
576C2#1396211yescharacteristic
577C2#2064811yescharacteristic
578C2#2065211yescharacteristic
579C2#2249312yes—
580C2#2250412yes—
581C2#2449211yescharacteristic
582C2#2449611yescharacteristic
583C2#2459211yescharacteristic
584C2#2459611yescharacteristic
585C2#2600512yes—
586C2#2600712yes—
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. #11227C2semidirect
7. #11207C2semidirect
8. #10849C2semidirect
9. #10807C2semidirect
18. #2347C4semidirect
90. #389C2×C4semidirect
102. C4×C4⋊C4C2×C4semidirect
175. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23472
C2×#18512
C2×#18252
C2×#17362
#138802
#112272
#112072
#108492
#108072
#107302
#107222