Finite Group Databasebeta

← #368441

Subgroups — #368441

Order 512.

1652 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C2×#1928C212yesmaximal, complemented
3C2×#1894C211yesmaximal, characteristic
4C2×#1873C211yesmaximal, characteristic
5#14139C211yesmaximal, characteristic
6#13801C211yesmaximal, characteristic
7#11257C212yesmaximal, complemented
8#11122C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#428—28no—
17C2×#428—24no—
18C22×#107C2212yes—
19C22×#106C2212yes—
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C2×#391C2212yes—
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1928—28no—
29#1894C418yescomplemented
30#1873—24no—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1441—21no—
39#1441—21no—
40#1437—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43#1434—21no—
44C22⋊#95—22no—
45C4×#95—21no—
46C23×Q8C2311yescharacteristic
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—24no—
52C22×M16—24no—
53C22×M16C2×C414yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8C2×C414yes—
63C2×#117—22no—
64C2×#117—24no—
65C2×#117—42no—
66C2×#117D414yes—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4D412yes—
82C22×C42—21no—
83C22×C42—22no—
84C22×C42C2312yes—
85#428—232no—
86C2×#107—24no—
87C2×#107—24no—
88C2×#107D418yes—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103C2×C418yescomplemented
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102#391—24no—
103#391—42no—
104#380—22no—
105#380C2×C414yes—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—42no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95D411yescharacteristic
119C2×#95D411yescharacteristic
120#340—22no—
121#340—22no—
122#340—24no—
123#340—42no—
124#340C2×C418yescomplemented
125#336—24no—
126#336—24no—
127C25C2×D411yescharacteristic
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8C22×C412yes—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—44no—
149C23×C4—42no—
150C23×C4—44no—
151C23×C4—81no—
152C23×C4—81no—
153C23×C4C22×C412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4#4112yes—
162C23×C4C2411yescharacteristic
163C2×M16—216no—
164C2×M16—28no—
165C2×M16—28no—
166C2×M16—28no—
167C2×M16—216no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8C22×C414yes—
183C22×C8C2×D414yes—
184C22×C8#4114yes—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189#117—28no—
190#117—44no—
191#117—48no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C414yes—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D412yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—81no—
236C2×C22⋊C4—81no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42—82no—
248#107—216no—
249#106—216no—
250#103—216no—
251#103—44no—
252#103—44no—
253C22⋊C8—44no—
254#95—24no—
255#95—22no—
256#95—22no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—82no—
262#95—82no—
263#95—82no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24C4×D412yes—
273C24#12112yes—
274C24#12312yes—
275C24C22×D411yescharacteristic
276C2×Q8—22no—
277C2×Q8—24no—
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—48no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4#11714yes—
365C22×C4C4×D414yes—
366C22×C4C4×D412yes—
367C22×C4#12112yes—
368C22×C4#12312yes—
369C22×C4#12414yes—
370C22×C4#12614yes—
371C22×C4#13714yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D412yes—
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4112yes—
379C22×C4C2×#4111yescharacteristic
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+412yes—
382M16—232no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8C4×D418yes—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—48no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—48no—
419C4⋊C4#12014yes—
420C22⋊C4—48no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—84no—
424C42—48no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—88no—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—24no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23#38814yes—
453C23C2×C4×D411yescharacteristic
454C23C2×#12111yescharacteristic
455C23C2×#12311yescharacteristic
456C23D4212yes—
457C23#54712yes—
458C23C2×#13712yes—
459C23#57512yes—
460C23#57611yescharacteristic
461C23#57711yescharacteristic
462C23#57711yescharacteristic
463C23#57811yescharacteristic
464Q8—42no—
465Q8—44no—
466Q8—42no—
467Q8—81no—
468Q8—81no—
469Q8—82no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—816no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4#38814yes—
532C2×C4C2×#11711yescharacteristic
533C2×C4C2×C4×D411yescharacteristic
534C2×C4C2×C4×D412yes—
535C2×C4#52012yes—
536C2×C4C2×#12011yescharacteristic
537C2×C4#52511yescharacteristic
538C2×C4#52512yes—
539C2×C4C2×#12411yescharacteristic
540C2×C4C2×#12611yescharacteristic
541C2×C4#53411yescharacteristic
542C2×C4#53611yescharacteristic
543C2×C4#53912yes—
544C2×C4#54612yes—
545C2×C4#54812yes—
546C2×C4#54812yes—
547C2×C4#55212yes—
548C2×C4#55312yes—
549C2×C4#55312yes—
550C8—28no—
551C8—81no—
552C8—81no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—44no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22C2×#38811yescharacteristic
565C22#236112yes—
566C22#241011yescharacteristic
567C22#241311yescharacteristic
568C22#244312yes—
569C22C4×#13712yes—
570C22#294611yescharacteristic
571C22#294711yescharacteristic
572C22#303911yescharacteristic
573C22#304011yescharacteristic
574C22#304511yescharacteristic
575C22#304611yescharacteristic
576C22#309512yes—
577C22#309811yescharacteristic
578C22#309911yescharacteristic
579C22#326712yes—
580C22#327012yes—
581C22#327312yes—
582C22#327812yes—
583C22#334612yes—
584C22#334812yes—
585C22#335212yes—
586C22#335612yes—
587C4—24no—
588C4—44no—
589C4—44no—
590C4—84no—
591C4—82no—
592C4—84no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4C4×#12011yescharacteristic
597C4#232211yescharacteristic
598C4#235212yes—
599C4#243611yescharacteristic
600C4#248112yes—
601C4#259111yescharacteristic
602C2—44no—
603C2#1401911yescharacteristic
604C2#2106111yescharacteristic
605C2#2106211yescharacteristic
606C2#2303612yes—
607C2#2304112yes—
608C2#2437311yescharacteristic
609C2#2437411yescharacteristic
610C2#2459511yescharacteristic
611C2#2459611yescharacteristic
612C2#2614512yes—
613C2#2615512yes—
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
2. C2×#1928C2semidirect
7. #11257C2semidirect
8. #11122C2semidirect
9. #10907C2semidirect
29. #1894C4semidirect
98. C2×#103C2×C4semidirect
124. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#19282
C2×#18942
C2×#18732
#141392
#138012
#112572
#111222
#109072
#107302
#107152