Finite Group Databasebeta

← #368455

Subgroups — #368455

Order 512.

1722 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2C2×#1935C212yesmaximal, complemented
3C2×#1873C211yesmaximal, characteristic
4C2×#1873C211yesmaximal, characteristic
5C2×#1749C211yesmaximal, characteristic
6#14702C211yesmaximal, characteristic
7#11293C212yesmaximal, complemented
8#11067C212yesmaximal, complemented
9#10881C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10731C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×#97C2212yes—
21#2296—28no—
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#385—22no—
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4—22no—
29#1935—28no—
30#1873—24no—
31#1873C418yescomplemented
32#1749—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1436—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C22⋊#95—22no—
44C23×Q8C2311yescharacteristic
45C24×C4D412yes—
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8C2×C414yes—
55C2×#117—22no—
56C2×#117—24no—
57C2×#117—24no—
58C2×#117—24no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—42no—
62C2×#117C2×C414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C42—21no—
76C22×C42—22no—
77C22×C42—21no—
78C22×C42—22no—
79C22×C42—24no—
80C22×C42—42no—
81C22×C42D412yes—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103C2×C418yescomplemented
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8C2×C414yes—
95C2×C22⋊C8D414yes—
96C2×#97—24no—
97C2×#97—24no—
98C2×#97—42no—
99#391—24no—
100#391—42no—
101#389—24no—
102#389—24no—
103#389—24no—
104#389D418yes—
105#385—216no—
106#380—22no—
107#380—22no—
108C4×C4⋊C4—216no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—42no—
121#340C2×C418yescomplemented
122#336—24no—
123C25C2×D411yescharacteristic
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8C22×C412yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4C22×C412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4C2411yescharacteristic
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8C22×C414yes—
174C22×C8C2×D414yes—
175C22⋊Q8—28no—
176C22⋊Q8—28no—
177C22⋊Q8—44no—
178C22⋊Q8—44no—
179#117—216no—
180#117—216no—
181#117—48no—
182#117—48no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
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—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C414yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4#4114yes—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—42no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—48no—
243C2×C42—48no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42C2×D414yes—
248#103—216no—
249#103—48no—
250C22⋊C8—28no—
251C22⋊C8—28no—
252#97—48no—
253#95—28no—
254#95—28no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—42no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24C4×D412yes—
269C24#12112yes—
270C24#12412yes—
271C24C22×D411yescharacteristic
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—24no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—48no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4#11714yes—
360C22×C4C4×D414yes—
361C22×C4C4×D412yes—
362C22×C4#12014yes—
363C22×C4#12112yes—
364C22×C4#12314yes—
365C22×C4#12412yes—
366C22×C4#12614yes—
367C22×C4#13714yes—
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4112yes—
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4H(5,𝔽2)12yes—
377C22×C42−1+412yes—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—416no—
400C4⋊C4—416no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—48no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4C4×D418yes—
411C22⋊C4—216no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—84no—
415C42—216no—
416C42—24no—
417C42—24no—
418C42—48no—
419C42—48no—
420C42—42no—
421C42—42no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—22no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23#38814yes—
445C23C2×C4×D411yescharacteristic
446C23C2×#12011yescharacteristic
447C23C2×#12111yescharacteristic
448C23#52511yescharacteristic
449C23C2×#12411yescharacteristic
450C23#53411yescharacteristic
451C23#53611yescharacteristic
452C23C2×#13712yes—
453C23#57512yes—
454C23#57512yes—
455C23#57512yes—
456Q8—44no—
457Q8—44no—
458Q8—82no—
459Q8—82no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—432no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—81no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4#38814yes—
522C2×C4C2×#11711yescharacteristic
523C2×C4C2×C4×D412yes—
524C2×C4C2×C4×D411yescharacteristic
525C2×C4#52012yes—
526C2×C4C2×#12311yescharacteristic
527C2×C4#52512yes—
528C2×C4C2×#12611yescharacteristic
529C2×C4#53912yes—
530C2×C4D4212yes—
531C2×C4#54612yes—
532C2×C4#54712yes—
533C2×C4#54812yes—
534C2×C4#54812yes—
535C2×C4#55212yes—
536C2×C4#55312yes—
537C2×C4#55312yes—
538C8—44no—
539C8—44no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—44no—
544C22—42no—
545C22—42no—
546C22—44no—
547C22—44no—
548C22—42no—
549C22—42no—
550C22—44no—
551C22C2×#38811yescharacteristic
552C22C4×#12011yescharacteristic
553C22#232211yescharacteristic
554C22#241011yescharacteristic
555C22#241311yescharacteristic
556C22#243611yescharacteristic
557C22#259111yescharacteristic
558C22#294312yes—
559C22C4×#13712yes—
560C22#300512yes—
561C22#300712yes—
562C22#305412yes—
563C22#305612yes—
564C22#311112yes—
565C22#311612yes—
566C22#313511yescharacteristic
567C22#313511yescharacteristic
568C22#313911yescharacteristic
569C22#314011yescharacteristic
570C22#319612yes—
571C22#319712yes—
572C22#323712yes—
573C22#324012yes—
574C4—24no—
575C4—44no—
576C4—44no—
577C4—48no—
578C4—84no—
579C4—82no—
580C4—82no—
581C4—161no—
582C4—161no—
583C4#235212yes—
584C4#236112yes—
585C4#244312yes—
586C4#248112yes—
587C2—44no—
588C2#1401911yescharacteristic
589C2#2108611yescharacteristic
590C2#2109411yescharacteristic
591C2#2145712yes—
592C2#2208312yes—
593C2#2438912yes—
594C2#2461112yes—
595C2#2590911yescharacteristic
596C2#2591011yescharacteristic
597C2#2933811yescharacteristic
598C2#2933911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1935C2semidirect
7. #11293C2semidirect
8. #11067C2semidirect
9. #10881C2semidirect
31. #1873C4semidirect
89. C2×#103C2×C4semidirect
121. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
C2×#19352
C2×#18732
C2×#17492
#147022
#112932
#110672
#108812
#107312