Finite Group Databasebeta

← #194184

Subgroups — #194184

Order 512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3076C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#21885C211yesmaximal, characteristic, complemented
4#21860C211yesmaximal, characteristic, complemented
5C2×#1477C211yesmaximal, characteristic
6#8984C212yesmaximal, complemented
7#8982C212yesmaximal, complemented
8#8976C212yesmaximal, complemented
9#8974C212yesmaximal, complemented
10C2×#1296C211yesmaximal, characteristic, complemented
11#7893C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14#3076—24no—
15#3076C418yescomplemented
16C2×#479—21no—
17C2×#477C2211yescharacteristic
18C2×#475C2211yescharacteristic
19C2×#478—21no—
20C2×#476C2211yescharacteristic
21C2×#474C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#427—21no—
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C2×#416—21no—
27#2355—21no—
28#2355C2212yes—
29#2354—21no—
30#2354C2212yes—
31#2352—22no—
32#2352—22no—
33#2314—22no—
34#2311—21no—
35#2285—21no—
36#2285—21no—
37#2285C2211yescharacteristic
38C2×#384C2211yescharacteristic
39#2274—21no—
40#1990C2212yes—
41#1987C2212yes—
42#1981C2212yes—
43#1956C2212yes—
44#1892—22no—
45#1872—22no—
46#1803—22no—
47#1803—21no—
48#1803—21no—
49#1477—24no—
50C2×Q8⋊C8C2212yes—
51C2×D4⋊C8C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C22×#41D411yescharacteristic
57C23×C8—21no—
58C23×C8—21no—
59#533—22no—
60#533—22no—
61#533—22no—
62#533C2×C414yes—
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C4×#41—22no—
69C4×#41—22no—
70C4×#41C2×C414yes—
71C4×#41D414yes—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4D411yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×#117—21no—
81C2×#117—22no—
82C2×#117—22no—
83C2×#117—21no—
84C2×#117—21no—
85C2×#117D411yescharacteristic
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4D412yes—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#479—28no—
92#478—28no—
93#477—22no—
94#477C2×C414yescomplemented
95#476—22no—
96#476C2×C414yescomplemented
97#475—22no—
98#475C2×C414yescomplemented
99#474—22no—
100#474C2×C414yescomplemented
101#431—22no—
102#431—22no—
103#427—28no—
104C2×#107—21no—
105C2×#107—21no—
106C2×#106—21no—
107C2×#106—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—22no—
110C2×C4⋊C8—22no—
111C2×C4⋊C8—22no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8C2×C414yes—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118#416—28no—
119C2×#103—21no—
120C2×#103—21no—
121C2×#103—24no—
122C2×#103—22no—
123C2×#103—22no—
124C2×#102—21no—
125C2×#102—21no—
126C2×C22⋊C8—21no—
127C2×C4×C8C2312yes—
128#391—22no—
129#390—22no—
130#388—24no—
131#388—24no—
132#387—24no—
133#387—24no—
134#386—24no—
135#384—21no—
136#384—21no—
137#384D412yes—
138#384D411yescharacteristic
139#384D411yescharacteristic
140#384C2312yes—
141#382—22no—
142#382—22no—
143#382—21no—
144#380—22no—
145#380—22no—
146#380—22no—
147#380—21no—
148#380—22no—
149#380D412yes—
150C4×C4⋊C4—22no—
151C4×C4⋊C4—22no—
152C4×C4⋊C4—22no—
153C4×C4⋊C4—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4C2312yes—
158C4×C22⋊C4—24no—
159#340—24no—
160#340—22no—
161#340—22no—
162Q8⋊C8—24no—
163D4⋊C8—24no—
164C2×#41—22no—
165C2×#41—22no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—22no—
169C2×#41C22⋊C414yes—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8C2×D411yescharacteristic
173C22×D4—21no—
174C22×D4C2×D411yescharacteristic
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C22×C8—24no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—81no—
191C22×C8—81no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8C22×C412yes—
199C4⋊Q8C22×C412yes—
200#125—24no—
201#125—42no—
202C22⋊Q8—24no—
203C22⋊Q8—24no—
204C22⋊Q8—42no—
205C22⋊Q8—42no—
206C4×Q8—22no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—42no—
212C4×Q8—42no—
213C4×Q8C22⋊C414yes—
214C4×Q8C22×C412yes—
215C4×Q8C2×D412yes—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—22no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4C22⋊C414yes—
224C4×D4C22×C412yes—
225C4×D4C2×D412yes—
226#117—22no—
227#117—24no—
228#117—24no—
229#117—24no—
230#117—42no—
231#117—42no—
232#117—42no—
233#117—82no—
234#117—82no—
235#117—84no—
236#117C22⋊C414yes—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—82no—
257C2×C4⋊C4—81no—
258C2×C4⋊C4—81no—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D414yes—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—41no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—41no—
280C2×C42C22×C412yes—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42#4112yes—
286C2×C42#4112yes—
287C2×C42C2411yescharacteristic
288#107—24no—
289#107—24no—
290#106—24no—
291#106—24no—
292C4⋊C8—24no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8—42no—
298C4⋊C8—42no—
299C4⋊C8C22×C412yes—
300C4⋊C8C22×C412yes—
301#103—24no—
302#103—24no—
303#103—82no—
304#103—82no—
305#103—84no—
306#102—24no—
307#102—24no—
308C22⋊C8—42no—
309C4×C8—22no—
310C4×C8—22no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—22no—
316#95—22no—
317C24—21no—
318C24#12011yescharacteristic
319#41—28no—
320#41—28no—
321#41—44no—
322#41—44no—
323C2×Q8—21no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8C2×C22⋊C412yes—
339C2×D4—24no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4C2×C22⋊C412yes—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—44no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4C2×C22⋊C412yes—
378C22×C4C4×D412yes—
379C22×C4C4×D412yes—
380C22×C4#12011yescharacteristic
381C22×C4#12011yescharacteristic
382C22×C4#12011yescharacteristic
383C22×C4#12112yes—
384C22×C4#12112yes—
385C22×C4#12312yes—
386C22×C4#13512yes—
387C22×C4#13512yes—
388C22×C4#13712yes—
389C22×C4#13712yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—48no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—84no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—82no—
438C4⋊C4C2×C22⋊C412yes—
439C4⋊C4C2×C22⋊C412yes—
440C22⋊C4—24no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—24no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C42—22no—
449C42—22no—
450C42—24no—
451C42—24no—
452C42—42no—
453C42—42no—
454C42—44no—
455C42—84no—
456C42—84no—
457C42C2×C22⋊C412yes—
458C42C4×D412yes—
459C42C4×D412yes—
460C42#12112yes—
461C42#12112yes—
462C42#12312yes—
463C42C23×C411yescharacteristic
464C42C22×D411yescharacteristic
465C42C2×#4111yescharacteristic
466C42C2×#4111yescharacteristic
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23#38612yes—
479C23#45212yes—
480C23#45212yes—
481C23C2×#12011yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—44no—
485Q8—41no—
486Q8—41no—
487Q8—42no—
488Q8—42no—
489Q8—82no—
490Q8—81no—
491Q8—81no—
492D4—22no—
493D4—22no—
494D4—48no—
495D4—41no—
496D4—41no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—48no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—164no—
540C2×C4#38612yes—
541C2×C4#38612yes—
542C2×C4#38612yes—
543C2×C4#41614yes—
544C2×C4#41914yes—
545C2×C4#45212yes—
546C2×C4#45212yes—
547C2×C4#48514yes—
548C2×C4#48514yes—
549C2×C4C22×C22⋊C411yescharacteristic
550C2×C4C2×C4×D411yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C2×#12111yescharacteristic
553C2×C4C2×#12111yescharacteristic
554C2×C4C2×#12311yescharacteristic
555C2×C4C2×#13511yescharacteristic
556C2×C4C2×#13511yescharacteristic
557C2×C4C2×#13711yescharacteristic
558C2×C4C2×#13711yescharacteristic
559C2×C4#57611yescharacteristic
560C2×C4#57711yescharacteristic
561C2×C4#57711yescharacteristic
562C2×C4#57811yescharacteristic
563C8—42no—
564C8—42no—
565C8—82no—
566C8—82no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22#187314yes—
581C22C2×#38611yescharacteristic
582C22C2×#45211yescharacteristic
583C22C2×#45211yescharacteristic
584C22#301211yescharacteristic
585C22#301311yescharacteristic
586C22#301911yescharacteristic
587C22#302011yescharacteristic
588C4—22no—
589C4—21no—
590C4—22no—
591C4—21no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—84no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4#187314yes—
601C4C2×#41611yescharacteristic
602C4C2×#41911yescharacteristic
603C4#295111yescharacteristic
604C4#295211yescharacteristic
605C4#304511yescharacteristic
606C4#304611yescharacteristic
607C4C2×#48511yescharacteristic
608C4C2×#48511yescharacteristic
609C2—24no—
610C2—82no—
611C2#1887511yescharacteristic
612C2#1887711yescharacteristic
613C2#1888711yescharacteristic
614C2#1888911yescharacteristic
615C2C2×#187311yescharacteristic
616C2#2093711yescharacteristic
617C2#2094311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21885C2semidirect
4. #21860C2semidirect
6. #8984C2semidirect
7. #8982C2semidirect
8. #8976C2semidirect
9. #8974C2semidirect
10. C2×#1296C2semidirect
11. #7893C2semidirect
15. #3076C4semidirect
94. #477C2×C4semidirect
96. #476C2×C4semidirect
98. #475C2×C4semidirect
100. #474C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30762
#341882
#218852
#218602
C2×#14772
#89842
#89822
#89762
#89742
C2×#12962
#78932