Finite Group Databasebeta

← #379908

Subgroups — #379908

Order 512.

1736 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic
2C2×#2296C211yesmaximal, characteristic
3C2×#1873C211yesmaximal, characteristic
4C2×#1825C211yesmaximal, characteristic
5#14702C211yesmaximal, characteristic
6#11220C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#11067C212yesmaximal, complemented
9#11067C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10712C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C2×#485—24no—
17C2×#485—24no—
18C2×#485C418yescomplemented
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26#2296—24no—
27#2296—24no—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#385—22no—
32C2×#380C2211yescharacteristic
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4—21no—
35#1873—24no—
36#1825—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1436—21no—
45#1436—21no—
46C4⋊#95C2212yes—
47C22⋊#95—22no—
48C23×Q8C2311yescharacteristic
49C24×C4D412yes—
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8C2×C414yes—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—42no—
65C2×#117—42no—
66C2×#117C2×C414yes—
67C2×#117D414yes—
68C2×#117D414yes—
69C2×#117D414yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C42—21no—
83C22×C42—22no—
84C22×C42—22no—
85C22×C42—21no—
86C22×C42—42no—
87C22×C42—42no—
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42C2312yes—
91#485—232no—
92C2×#107—22no—
93C2×#107—22no—
94C2×#107C2×C414yescomplemented
95C2×#106—22no—
96C2×#106—22no—
97C2×#106C2×C414yescomplemented
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—24no—
103C2×#103—42no—
104C2×#103—42no—
105C2×#103C2×C414yescomplemented
106C2×#103C2×C414yescomplemented
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8C2×C414yes—
113#391—42no—
114#391—42no—
115#389—24no—
116#389—24no—
117#389—24no—
118#389—24no—
119#385—216no—
120#380—22no—
121#380—22no—
122C4×C4⋊C4—28no—
123C4×C4⋊C4—28no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—21no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—42no—
135#340—42no—
136#336—42no—
137#336—42no—
138C25C2×D411yescharacteristic
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4C22×C412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4#4112yes—
174C23×C4#4112yes—
175C23×C4C2411yescharacteristic
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8C22×C412yes—
192C22×C8C22×C412yes—
193C22⋊Q8—28no—
194C22⋊Q8—28no—
195C22⋊Q8—44no—
196C22⋊Q8—44no—
197#117—28no—
198#117—28no—
199#117—28no—
200#117—48no—
201#117—44no—
202#117—44no—
203#117C22⋊C4116yes—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—44no—
268C2×C42—48no—
269C2×C42—44no—
270C2×C42—44no—
271#107—28no—
272#106—28no—
273#103—28no—
274#103—28no—
275#103—48no—
276C22⋊C8—28no—
277C22⋊C8—44no—
278#95—28no—
279#95—28no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—42no—
284#95—44no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24C4×D412yes—
292C24#12012yes—
293C24#12112yes—
294C24C22×D411yescharacteristic
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4C2×C22⋊C414yes—
383C22×C4C4×D414yes—
384C22×C4C4×D412yes—
385C22×C4#12012yes—
386C22×C4#12114yes—
387C22×C4#12112yes—
388C22×C4#12314yes—
389C22×C4#13514yes—
390C22×C4#13714yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4H(5,𝔽2)12yes—
398C22×C4H(5,𝔽2)12yes—
399C22×C42−1+412yes—
400C22×C42−1+412yes—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—416no—
429C4⋊C4—416no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—48no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4C2×C22⋊C414yes—
440C4⋊C4C2×C22⋊C414yes—
441C22⋊C4—216no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—84no—
445C42—216no—
446C42—44no—
447C42—44no—
448C42—48no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—42no—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—22no—
458C23—22no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23#38614yes—
474C23C2×C4×D411yescharacteristic
475C23C2×#12011yescharacteristic
476C23C2×#12111yescharacteristic
477C23#53412yes—
478C23#53612yes—
479C23C2×#13512yes—
480C23C2×#13712yes—
481C23#57512yes—
482C23#57512yes—
483Q8—42no—
484Q8—42no—
485Q8—44no—
486Q8—82no—
487Q8—81no—
488Q8—81no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—432no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4#38614yes—
549C2×C4C22×C22⋊C411yescharacteristic
550C2×C4C2×C4×D411yescharacteristic
551C2×C4#52012yes—
552C2×C4#52012yes—
553C2×C4C2×#12111yescharacteristic
554C2×C4C2×#12311yescharacteristic
555C2×C4#53412yes—
556C2×C4#53511yescharacteristic
557C2×C4#53511yescharacteristic
558C2×C4#53611yescharacteristic
559C2×C4#53611yescharacteristic
560C2×C4#53612yes—
561C2×C4#53712yes—
562C2×C4#53712yes—
563C2×C4#53912yes—
564C2×C4#53912yes—
565C2×C4#53912yes—
566C2×C4#53912yes—
567C8—42no—
568C8—42no—
569C8—82no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22C2×#38611yescharacteristic
580C22#232212yes—
581C22#261311yescharacteristic
582C22#261411yescharacteristic
583C22#261812yes—
584C22C4×#13511yescharacteristic
585C22#294311yescharacteristic
586C22C4×#13712yes—
587C22#300212yes—
588C22#300712yes—
589C22#303512yes—
590C22#303911yescharacteristic
591C22#304011yescharacteristic
592C22#319012yes—
593C22#319112yes—
594C22#319612yes—
595C22#319712yes—
596C22#323112yes—
597C22#323312yes—
598C22#323712yes—
599C22#324012yes—
600C4—24no—
601C4—44no—
602C4—44no—
603C4—44no—
604C4—44no—
605C4—84no—
606C4—82no—
607C4—82no—
608C4—162no—
609C4#231111yescharacteristic
610C4#231111yescharacteristic
611C4#232212yes—
612C4#261812yes—
613C4#262311yescharacteristic
614C4#262311yescharacteristic
615C2—44no—
616C2#1425611yescharacteristic
617C2#2092411yescharacteristic
618C2#2092711yescharacteristic
619C2#2207712yes—
620C2#2208312yes—
621C2#2960112yes—
622C2#2964512yes—
623C2#2975912yes—
624C2#2976212yes—
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
6. #11220C2semidirect
7. #11219C2semidirect
8. #11067C2semidirect
9. #11067C2semidirect
18. C2×#485C4semidirect
94. C2×#107C2×C4semidirect
97. C2×#106C2×C4semidirect
105. C2×#103C2×C4semidirect
106. C2×#103C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4852
C2×#22962
C2×#18732
C2×#18252
#147022
#112202
#112192
#110672
#107312
#107122