Finite Group Databasebeta

← #356268

Subgroups — #356268

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2353C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×C4×#99C211yesmaximal, characteristic
4#13802C212yesmaximal, complemented
5#10960C214yesmaximal
6#10854C212yesmaximal, complemented
7#10801C212yesmaximal, complemented
8C4×#342C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C22×#99C2212yes—
11C22×#99C2212yes—
12#2353—22no—
13#2353—22no—
14#2353—24no—
15C2×#388C412yescomplemented
16C2×#387C412yescomplemented
17C2×#386C2212yes—
18C2×#383C411yescharacteristic, complemented
19C2×#381C411yescharacteristic, complemented
20C2×#380C2212yes—
21C2×#380C2211yescharacteristic
22C2×C4×C22⋊C4C411yescharacteristic, complemented
23C2×C4×C22⋊C4C2214yes—
24C2×C4×C22⋊C4C2212yes—
25C2×C4×C22⋊C4C2212yes—
26C2×#376C411yescharacteristic, complemented
27C22×#95C2211yescharacteristic
28#1799C418yescomplemented
29C4×#99C418yescomplemented
30C2×#342C2214yes—
31C2×#342C2214yes—
32C2×#342C2211yescharacteristic
33C2×#342C2212yes—
34C2×#342C2211yescharacteristic
35#1436—24no—
36#1434—22no—
37C22⋊#95C2214yes—
38C4×#95—22no—
39C23×D4C2311yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2312yes—
42C24×C4C2311yescharacteristic
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4C2×C414yes—
47C2×C4×D4C2×C414yes—
48C22×C4⋊C4—22no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C412yes—
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C412yes—
53C22×C22⋊C4C2×C412yes—
54C22×C22⋊C4C2312yes—
55C22×C22⋊C4C2312yes—
56C22×C22⋊C4C2314yes—
57C22×C22⋊C4C2311yescharacteristic
58C22×C22⋊C4C2312yes—
59C22×C42—22no—
60C22×C42—24no—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C412yes—
63C22×C42C2×C412yes—
64C2×#99—24no—
65C2×#99—24no—
66C2×#99—42no—
67C2×#99—42no—
68C2×#99C2×C418yes—
69C2×#99C2×C418yes—
70#388—24no—
71#388—24no—
72#387—24no—
73#387—24no—
74#386—24no—
75#386—24no—
76#383—22no—
77#383—22no—
78#381—22no—
79#381—22no—
80#380—22no—
81#380—22no—
82#380—24no—
83#380C2×C418yes—
84C4×C22⋊C4—24no—
85C4×C22⋊C4—24no—
86C4×C22⋊C4—24no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—28no—
91C4×C22⋊C4—44no—
92C4×C22⋊C4C2×C418yes—
93#376—21no—
94#376—22no—
95#376—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—24no—
99C2×#95—24no—
100C2×#95—24no—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C412yes—
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C412yes—
105C2×#95C2×C412yes—
106C2×#95C2×C412yes—
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C412yes—
110C2×#95D412yes—
111C2×#95D412yes—
112C2×#95D414yes—
113#342—28no—
114#342—44no—
115#342C2×C418yes—
116#342C2×C414yes—
117#342C2×C414yes—
118C25C22×C411yescharacteristic
119C25C2411yescharacteristic
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4C22×C412yes—
129C22×D4C22×C412yes—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—44no—
140C23×C4—82no—
141C23×C4—82no—
142C23×C4—82no—
143C23×C4C4214yescomplemented
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C412yes—
146C23×C4C22×C412yes—
147C23×C4C22×C412yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C412yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C412yes—
154C23×C4C22×C412yes—
155C23×C4C2×D414yes—
156C23×C4C2×D414yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4#4114yes—
160C4×D4—28no—
161C4×D4—28no—
162C4×D4—28no—
163C4×D4—28no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—84no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C4214yescomplemented
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4C4214yescomplemented
198C2×C22⋊C4C22×C414yes—
199C2×C22⋊C4C22×C418yes—
200C2×C22⋊C4C22×C412yes—
201C2×C22⋊C4C22×C412yes—
202C2×C22⋊C4C22×C414yes—
203C2×C22⋊C4#4114yes—
204C2×C22⋊C4#4114yes—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—28no—
213C2×C42—44no—
214C2×C42—48no—
215C2×C42—48no—
216C2×C42—84no—
217C2×C42—84no—
218C2×C42—82no—
219C2×C42C4214yescomplemented
220#99—216no—
221#99—48no—
222#95—21no—
223#95—21no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—28no—
232#95—28no—
233#95—48no—
234#95—48no—
235#95—84no—
236#95C22⋊C411yescharacteristic
237#95C22⋊C412yes—
238#95C22⋊C411yescharacteristic
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241#95C22⋊C414yes—
242#95C22⋊C414yes—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24C2×C4212yes—
254C24C23×C411yescharacteristic
255C24C23×C411yescharacteristic
256C24C23×C411yescharacteristic
257C24C22×D412yes—
258C24C2×#4112yes—
259C24C2×#4112yes—
260C24C2×#4112yes—
261C24H(5,𝔽2)11yescharacteristic
262C24H(5,𝔽2)11yescharacteristic
263C242−1+411yescharacteristic
264C242−1+411yescharacteristic
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—28no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—28no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4—88no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4C2×C4212yes—
353C22×C4C2×C4212yes—
354C22×C4C2×C4212yes—
355C22×C4C2×C4212yes—
356C22×C4C2×C4212yes—
357C22×C4C2×C22⋊C414yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4#11712yes—
364C22×C4#11712yes—
365C22×C4C4×D414yes—
366C22×C4C4×D414yes—
367C22×C4C4×D414yes—
368C22×C4C4×D414yes—
369C4⋊C4—28no—
370C4⋊C4—84no—
371C4⋊C4—88no—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—28no—
378C22⋊C4—24no—
379C22⋊C4—28no—
380C22⋊C4—28no—
381C22⋊C4—24no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—42no—
387C22⋊C4—44no—
388C22⋊C4—42no—
389C22⋊C4—44no—
390C22⋊C4—84no—
391C22⋊C4#11718yes—
392C42—24no—
393C42—24no—
394C42—44no—
395C42—44no—
396C42—82no—
397C42—82no—
398C42—88no—
399C42—816no—
400C23—24no—
401C23—22no—
402C23—24no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—44no—
411C23—42no—
412C23—42no—
413C23—41no—
414C23—41no—
415C23—44no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23C22×C4211yescharacteristic
434C23C22×C22⋊C411yescharacteristic
435C23C2×#11711yescharacteristic
436C23C2×#11712yes—
437C23C2×C4×D412yes—
438C23C2×C4×D412yes—
439C23C4×#4111yescharacteristic
440C23C4×#4111yescharacteristic
441C23C4×#4112yes—
442C23#51811yescharacteristic
443C23#51911yescharacteristic
444C23#52011yescharacteristic
445C23#52011yescharacteristic
446C23#52011yescharacteristic
447C23#52011yescharacteristic
448C23#54612yes—
449C23#54712yes—
450C23#54812yes—
451C23#55012yes—
452C23#55212yes—
453C23#55312yes—
454C23#55512yes—
455C23#55612yes—
456D4—44no—
457D4—44no—
458D4—44no—
459D4—44no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—48no—
486C2×C4—48no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—48no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4—84no—
526C2×C4—84no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4C4×C22⋊C412yes—
532C2×C4C4×C22⋊C412yes—
533C2×C4C4×C22⋊C414yes—
534C2×C4#41014yes—
535C22—24no—
536C22—24no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—22no—
545C22—21no—
546C22—21no—
547C22—24no—
548C22—24no—
549C22—22no—
550C22—22no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22C2×C4×C22⋊C411yescharacteristic
571C22C4×#11711yescharacteristic
572C22#227411yescharacteristic
573C22#233911yescharacteristic
574C22#234111yescharacteristic
575C22#234611yescharacteristic
576C22#234711yescharacteristic
577C22#235212yes—
578C22#235412yes—
579C22#236212yes—
580C22#236312yes—
581C22C2×#41012yes—
582C22#288312yes—
583C4—22no—
584C4—22no—
585C4—24no—
586C4—24no—
587C4—48no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—44no—
592C4—42no—
593C4—88no—
594C4—84no—
595C4—82no—
596C4—82no—
597C2—22no—
598C2—22no—
599C2—22no—
600C2—21no—
601C2—21no—
602C2—21no—
603C2—21no—
604C2—22no—
605C2—82no—
606C2#1380111yescharacteristic
607C2C4×#41011yescharacteristic
608C2#2039211yescharacteristic
609C2#2247712yes—
610C2#2253512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13802C2semidirect
6. #10854C2semidirect
7. #10801C2semidirect
8. C4×#342C2semidirect
15. C2×#388C4semidirect
16. C2×#387C4semidirect
18. C2×#383C4semidirect
19. C2×#381C4semidirect
22. C2×C4×C22⋊C4C4semidirect
26. C2×#376C4semidirect
28. #1799C4semidirect
29. C4×#99C4semidirect
143. C23×C4C42semidirect
170. C2×C4⋊C4C42semidirect
197. C2×C22⋊C4C42semidirect
219. C2×C42C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23532
C2×#17992
C2×C4×#992
#138022
#109602
#108542
#108012
C4×#3422