Finite Group Databasebeta

← #360466

Subgroups — #360466

Order 512.

1382 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1844C211yesmaximal, characteristic, complemented
2C2×#1807C211yesmaximal, characteristic
3C2×#1807C211yesmaximal, characteristic
4C2×C4×#106C211yesmaximal, characteristic, complemented
5C2×C8×C22⋊C4C211yesmaximal, characteristic
6#13876C212yesmaximal
7#11267C212yesmaximal, complemented
8#11095C212yesmaximal
9#10908C212yesmaximal, complemented
10#10861C211yesmaximal, characteristic, complemented
11#10861C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×#106C2212yes—
16C22×#106C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#388—22no—
22C2×#384C2212yes—
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1844—24no—
28#1807—24no—
29#1807—24no—
30C4×#106—24no—
31C8×C22⋊C4C418yescomplemented
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1442—22no—
40#1439—22no—
41#1436—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C24×C4C2311yescharacteristic
45C23×C8—22no—
46C23×C8—22no—
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—22no—
51C23×C8C2×C414yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4D411yescharacteristic
65C22×C22⋊C4D411yescharacteristic
66C22×C42—22no—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×#106—22no—
70C2×#106—22no—
71C2×#106—22no—
72C2×#106—22no—
73C2×#106—22no—
74C2×#106—22no—
75C2×#106—22no—
76C2×#106—24no—
77C2×#106—22no—
78C2×#106—22no—
79C2×#106—22no—
80C2×#106—24no—
81C2×#106—42no—
82C2×#106D414yes—
83C2×#106D414yes—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8C2×C414yes—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8C2×C414yes—
92C2×C4×C8C2×C414yes—
93#388—48no—
94#384—24no—
95#384—42no—
96#380—24no—
97#380—24no—
98#380—42no—
99#380—42no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—42no—
102C4×C22⋊C4C2×C414yes—
103C4×C22⋊C4Q814yes—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95D411yescharacteristic
112C2×#95D411yescharacteristic
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—42no—
118#336C2×C414yes—
119#336C2×C414yes—
120C25C2×D411yescharacteristic
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4C22×C412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×Q812yes—
144C23×C4C2411yescharacteristic
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—28no—
162C22×C8—28no—
163C22×C8—28no—
164C22×C8—44no—
165C22×C8—44no—
166C22×C8—44no—
167C22×C8—44no—
168C22×C8C22×C412yes—
169C22×C8C22×C412yes—
170C22×C8C22×C412yes—
171C22×C8C22×C412yes—
172C22×C8C2×D412yes—
173C22×C8C2×D412yes—
174C22×C8#4112yes—
175C22×C8#4112yes—
176C22×C8#4112yes—
177C22×C8#4112yes—
178C22×C8#4112yes—
179C22×C8#4112yes—
180C2×C4⋊C4—22no—
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—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D414yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—82no—
216C2×C22⋊C4—82no—
217C2×C22⋊C4C4⋊C414yes—
218C2×C22⋊C4C4⋊C414yes—
219C2×C22⋊C4SD1618yes—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42—82no—
231C2×C42C22×C412yes—
232C2×C42C22×C412yes—
233C2×C42C2×Q812yes—
234C2×C42C2×Q812yes—
235#106—28no—
236#106—28no—
237#106—48no—
238#106—44no—
239#106—44no—
240C22⋊C8—28no—
241C4×C8—28no—
242C4×C8—28no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—82no—
250#95C4⋊C414yes—
251#95C4⋊C414yes—
252C24—21no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24C2×C4⋊C412yes—
262C24C2×SD1614yes—
263C24C22×D411yescharacteristic
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
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×C4C2×C4⋊C412yes—
340C22×C4C2×C4⋊C412yes—
341C22×C4C2×C4⋊C412yes—
342C22×C4C2×C4⋊C412yes—
343C22×C4C2×C4⋊C412yes—
344C22×C4C2×SD1614yes—
345C22×C4C2×SD1614yes—
346C22×C4#13514yes—
347C22×C4C23×C411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×Q811yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C42−1+411yescharacteristic
360C22×C42−1+411yescharacteristic
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—416no—
382C2×C8#11714yes—
383C2×C8#11714yes—
384C2×C8C4×D414yes—
385C2×C8C4×D414yes—
386C2×C8#12114yes—
387C2×C8#12314yes—
388C2×C8#12414yes—
389C2×C8#12614yes—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—42no—
395C4⋊C4—42no—
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—88no—
407C4⋊C4—88no—
408C22⋊C4—28no—
409C22⋊C4—88no—
410C22⋊C4—88no—
411C22⋊C4#106116yescomplemented
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—88no—
421C23—24no—
422C23—24no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—24no—
427C23—24no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—22no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—42no—
442C23—42no—
443C23C2×#10614yes—
444C23C22×C4⋊C411yescharacteristic
445C23#53211yescharacteristic
446C23#53211yescharacteristic
447C23D4211yescharacteristic
448C23D4211yescharacteristic
449C23#54711yescharacteristic
450C23#54711yescharacteristic
451C23C22×SD1612yes—
452C23C2×#13512yes—
453C23#57712yes—
454C23#57712yes—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
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×C4C2×#10614yes—
516C2×C4C2×#10614yes—
517C2×C4#42714yes—
518C2×C4C2×#11711yescharacteristic
519C2×C4C2×#11711yescharacteristic
520C2×C4C2×C4×D411yescharacteristic
521C2×C4C2×C4×D411yescharacteristic
522C2×C4#52011yescharacteristic
523C2×C4#52011yescharacteristic
524C2×C4C2×#12111yescharacteristic
525C2×C4C2×#12311yescharacteristic
526C2×C4C2×#12411yescharacteristic
527C2×C4C2×#12611yescharacteristic
528C2×C4#53311yescharacteristic
529C2×C4#53311yescharacteristic
530C2×C4#54011yescharacteristic
531C2×C4#54111yescharacteristic
532C2×C4#54211yescharacteristic
533C2×C4#54211yescharacteristic
534C2×C4#54611yescharacteristic
535C2×C4#54711yescharacteristic
536C2×C4#54811yescharacteristic
537C2×C4D4×Q811yescharacteristic
538C2×C4D4×Q811yescharacteristic
539C2×C4#55011yescharacteristic
540C2×C4#55211yescharacteristic
541C2×C4#55311yescharacteristic
542C2×C4#55411yescharacteristic
543C2×C4#55411yescharacteristic
544C2×C4#55511yescharacteristic
545C2×C4#55611yescharacteristic
546C8—24no—
547C8—42no—
548C8—42no—
549C8#38818yescomplemented
550C22—24no—
551C22—24no—
552C22—24no—
553C22—42no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22#228511yescharacteristic
562C22D4×C4⋊C411yescharacteristic
563C22D4×C4⋊C411yescharacteristic
564C22#242211yescharacteristic
565C22#242811yescharacteristic
566C22#249111yescharacteristic
567C22#249611yescharacteristic
568C22C22×#10611yescharacteristic
569C22C2×#42711yescharacteristic
570C22#291211yescharacteristic
571C22#291211yescharacteristic
572C22#316712yes—
573C22#316712yes—
574C22#317312yes—
575C22#317412yes—
576C22D4×SD1612yes—
577C22D4×SD1612yes—
578C22#328212yes—
579C22#328412yes—
580C22#333512yes—
581C22#333512yes—
582C22#333712yes—
583C22#334012yes—
584C4—24no—
585C4—24no—
586C4—24no—
587C4—42no—
588C4—42no—
589C4—84no—
590C4—84no—
591C4—162no—
592C4—162no—
593C4C2×#38811yescharacteristic
594C4#231411yescharacteristic
595C4#235211yescharacteristic
596C4#235411yescharacteristic
597C4#242311yescharacteristic
598C4#242911yescharacteristic
599C4#249211yescharacteristic
600C4#249711yescharacteristic
601C2—44no—
602C2#1387611yescharacteristic
603C2#2081611yescharacteristic
604C2#2082011yescharacteristic
605C2D4×#10611yescharacteristic
606C2D4×#10611yescharacteristic
607C2#2265411yescharacteristic
608C2#2265711yescharacteristic
609C2#2503012yes—
610C2#2541012yes—
611C2#2778712yes—
612C2#2783512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1844C2semidirect
4. C2×C4×#106C2semidirect
7. #11267C2semidirect
9. #10908C2semidirect
10. #10861C2semidirect
11. #10861C2semidirect
31. C8×C22⋊C4C4semidirect
411. C22⋊C4#106semidirect
549. C8#388semidirect

Maximal subgroups

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

subgroupindex
C2×#18442
C2×#18072
C2×C4×#1062
C2×C8×C22⋊C42
#138762
#112672
#110952
#109082
#108612