Finite Group Databasebeta

← #360789

Subgroups — #360789

Order 512.

1510 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1836C211yesmaximal, characteristic, complemented
3C2×#1834C211yesmaximal, characteristic, complemented
4C2×#1750C211yesmaximal, characteristic
5#14340C211yesmaximal, characteristic
6#11268C212yesmaximal, complemented
7#11267C212yesmaximal, complemented
8#10843C211yesmaximal, characteristic, complemented
9#10842C211yesmaximal, characteristic, complemented
10#10731C212yesmaximal
11#10730C212yesmaximal
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2317—24no—
19#2317—24no—
20C2×#389—21no—
21C2×#389—21no—
22C2×#385—21no—
23C2×#385—21no—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31#1836—24no—
32#1834—24no—
33#1750C418yescomplemented
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43#1441—21no—
44#1441—21no—
45#1439—21no—
46#1439—21no—
47C4⋊#95C2212yes—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—24no—
59C2×#117—42no—
60C2×#117C2×C414yes—
61C2×#117Q814yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8C2×C418yes—
91C2×#97—22no—
92C2×#97—22no—
93C2×#97C2×C414yescomplemented
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96C2×C4×C8C2×C414yescomplemented
97#389—28no—
98#389—28no—
99#385—28no—
100#385—28no—
101#384—22no—
102#384—22no—
103#384—22no—
104#384—22no—
105#384—22no—
106#384D414yes—
107#382—22no—
108#382—22no—
109#382—22no—
110#382—22no—
111#382—22no—
112#382D414yes—
113#380—24no—
114#380—24no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—42no—
129#340—42no—
130#340—42no—
131#340—42no—
132#336C2×C418yescomplemented
133C25C2×D411yescharacteristic
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—82no—
150C23×C4—82no—
151C23×C4C22×C412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×Q812yes—
158C23×C4#4114yes—
159C23×C4#4114yes—
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×C8C22×C414yes—
175#117—28no—
176#117—28no—
177#117—28no—
178#117—28no—
179#117—48no—
180#117—48no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
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—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4C4⋊C414yes—
227C2×C4⋊C4C4⋊C414yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4C4⋊C414yes—
243C2×C22⋊C4C4⋊C414yes—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—24no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42C22×C412yes—
252C2×C42C22×C412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×Q812yes—
256C2×C42C2×Q812yes—
257C2×C42#4112yes—
258C2×C42#4112yes—
259C22⋊C8—216no—
260#97—28no—
261C4×C8—28no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—42no—
267#95—42no—
268#95—44no—
269#95—82no—
270#95—82no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24C2×C4⋊C412yes—
280C24#12412yes—
281C24#12412yes—
282C24C22×D411yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
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—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—48no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
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—84no—
349C22×C4—84no—
350C22×C4—88no—
351C22×C4—88no—
352C22×C4C2×C4⋊C412yes—
353C22×C4C2×C4⋊C412yes—
354C22×C4C2×C4⋊C412yes—
355C22×C4C2×C4⋊C412yes—
356C22×C4C2×C4⋊C412yes—
357C22×C4#11718yes—
358C22×C4#12012yes—
359C22×C4#12012yes—
360C22×C4#12312yes—
361C22×C4#12312yes—
362C22×C4#12412yes—
363C22×C4#12412yes—
364C22×C4#12518yes—
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×Q811yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4C2×#4112yes—
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4H(5,𝔽2)12yes—
374C22×C42−1+412yes—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C4⋊C4—28no—
390C4⋊C4—28no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—48no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C4⋊C4—84no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C22⋊C4—28no—
415C22⋊C4—28no—
416C22⋊C4—48no—
417C22⋊C4—88no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—48no—
423C42C4×D414yes—
424C42C4×D414yes—
425C42C22⋊Q814yes—
426C42C22⋊Q814yes—
427C23—22no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23#38214yes—
450C23C22×C4⋊C411yescharacteristic
451C23C2×#12011yescharacteristic
452C23C2×#12311yescharacteristic
453C23C2×#12411yescharacteristic
454C23C2×#12411yescharacteristic
455C23#53511yescharacteristic
456C23#53611yescharacteristic
457C23#57612yes—
458C23#57712yes—
459C23#57712yes—
460C23#57812yes—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—816no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4#38214yes—
525C2×C4#38514yes—
526C2×C4#38514yes—
527C2×C4C2×#11712yes—
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×C4×D411yescharacteristic
530C2×C4#52012yes—
531C2×C4C2×C22⋊Q811yescharacteristic
532C2×C4C2×C22⋊Q811yescharacteristic
533C2×C4C2×#12512yes—
534C2×C4#54412yes—
535C2×C4#54612yes—
536C2×C4#54612yes—
537C2×C4#54812yes—
538C2×C4#54812yes—
539C2×C4#55212yes—
540C2×C4#55212yes—
541C2×C4#55312yes—
542C2×C4#55312yes—
543C8—44no—
544C8—44no—
545C22—28no—
546C22—24no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—44no—
552C22—44no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22C2×#38211yescharacteristic
557C22C2×#38511yescharacteristic
558C22#231711yescharacteristic
559C22#243511yescharacteristic
560C22#243611yescharacteristic
561C22#250311yescharacteristic
562C22#250411yescharacteristic
563C22#291212yes—
564C22#291312yes—
565C22#300811yescharacteristic
566C22#300911yescharacteristic
567C22#301011yescharacteristic
568C22#301111yescharacteristic
569C22#309812yes—
570C22#309912yes—
571C22#313612yes—
572C22#313612yes—
573C22#313712yes—
574C22#313712yes—
575C22#322412yes—
576C22#322512yes—
577C22#324112yes—
578C22#324212yes—
579C4—24no—
580C4—24no—
581C4—24no—
582C4—42no—
583C4—42no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—162no—
589C4—162no—
590C4#235212yes—
591C4#235212yes—
592C4#244212yes—
593C4#244212yes—
594C2—44no—
595C2#1388111yescharacteristic
596C2#2076312yes—
597C2#2084711yescharacteristic
598C2#2084811yescharacteristic
599C2#2199212yes—
600C2#2586511yescharacteristic
601C2#2586611yescharacteristic
602C2#2591411yescharacteristic
603C2#2591511yescharacteristic
604C2#2797012yes—
605C2#2798412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1836C2semidirect
3. C2×#1834C2semidirect
6. #11268C2semidirect
7. #11267C2semidirect
8. #10843C2semidirect
9. #10842C2semidirect
33. #1750C4semidirect
93. C2×#97C2×C4semidirect
96. C2×C4×C8C2×C4semidirect
132. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#18362
C2×#18342
C2×#17502
#143402
#112682
#112672
#108432
#108422
#107312
#107302