Finite Group Databasebeta

← #358517

Subgroups — #358517

Order 512.

1448 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34060C211yesmaximal, characteristic
2C4×#428C211yesmaximal, characteristic, complemented
3C4×#417C211yesmaximal, characteristic
4C2×C4×#104C211yesmaximal, characteristic
5C4×#384C211yesmaximal, characteristic, complemented
6#11340C212yesmaximal, complemented
7#11276C212yesmaximal, complemented
8#11113C212yesmaximal
9#11078C212yesmaximal
10C4×#343C211yesmaximal, characteristic, complemented
11C4×#339C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#428C2212yes—
14C22×#104C2211yescharacteristic
15C22×#104C2211yescharacteristic
16C2×#417C2212yes—
17C2×C4×M16C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19#2320—21no—
20#2319—21no—
21#2318—22no—
22#2317C2212yes—
23#2314—22no—
24#2311C2212yes—
25#2295—21no—
26#2293—21no—
27#2290—22no—
28#2285—21no—
29C2×#384C2212yes—
30#2283—21no—
31C4×#117C2211yescharacteristic
32C2×C4×C4⋊C4—22no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C43C2211yescharacteristic
36#1776—22no—
37C4×#107—21no—
38C4×#106—21no—
39#1763—21no—
40#1762—21no—
41C4×#103—21no—
42C4×#102—21no—
43C4×#104C418yescomplemented
44C2×#343C2212yes—
45C2×#343C2212yes—
46#1736C2212yes—
47#1736C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C4⋊#95—22no—
53C4⋊#95C2212yes—
54C22×#41C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2312yes—
57C22×M16C2311yescharacteristic
58C4×#41—22no—
59C4×#41—22no—
60C4×#41C2×C414yes—
61C4×#41Q814yes—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×#117—22no—
69C2×#117—21no—
70C2×#117C2312yes—
71C2×#117C2311yescharacteristic
72C22×C4⋊C4—24no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42D412yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#428—24no—
84#428—24no—
85C2×#107—22no—
86C2×#106—22no—
87C2×#104—24no—
88C2×#104—24no—
89C2×#104C2×C414yes—
90C2×#104C2×C414yes—
91#417—24no—
92#417—24no—
93C2×#103—22no—
94C2×#102—22no—
95C4×M16—22no—
96C4×M16—22no—
97C4×M16—22no—
98C4×M16C2×C414yes—
99C2×#97—21no—
100C2×#97—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103#391—22no—
104#390—22no—
105#389—22no—
106#389—22no—
107#388—24no—
108#388—24no—
109#387—24no—
110#387—24no—
111#386—24no—
112#385—24no—
113#384—22no—
114#384—24no—
115#384—24no—
116#383—22no—
117#382—22no—
118#382—22no—
119#381—22no—
120#380—22no—
121#380—22no—
122#380—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—48no—
133C4×C4⋊C4D412yes—
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4D411yescharacteristic
136C4×C22⋊C4—22no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139#376—22no—
140#376—21no—
141C2×#95—21no—
142C2×#95—24no—
143C2×#95D412yes—
144C43—21no—
145C43—22no—
146C43C2×C414yes—
147#343—24no—
148#343—24no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153#339—24no—
154#339—24no—
155#339C2×C414yes—
156#339C2×C414yes—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—22no—
160C2×#41—22no—
161C2×#41—22no—
162C2×#41—22no—
163C2×#41C22×C412yes—
164C2×#41C2×Q812yes—
165C22×Q8C2×D411yescharacteristic
166C22×D4—21no—
167C22×D4C2×D411yescharacteristic
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D414yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2411yescharacteristic
178C2×M16—21no—
179C2×M16—22no—
180C2×M16—21no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16C22×C412yes—
189C2×M16C22×C412yes—
190C2×M16#4114yes—
191C2×M16#4114yes—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8C4⋊C414yescomplemented
198C4×Q8C4⋊C414yescomplemented
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4C4⋊C414yescomplemented
206C4×D4C4⋊C414yescomplemented
207#117—21no—
208#117—21no—
209#117—24no—
210#117—24no—
211#117—22no—
212#117—22no—
213#117—22no—
214#117—22no—
215#117—84no—
216#117C22×C412yes—
217#117C2×Q812yes—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—216no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—48no—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4#4114yes—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—44no—
268C2×C42—44no—
269C2×C42—44no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42C22×C412yes—
273C2×C42C22×C412yes—
274C2×C42C22×C412yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×Q812yes—
280C2×C42#4114yes—
281C2×C42#4112yes—
282C2×C42#4112yes—
283#107—28no—
284#106—28no—
285#104—28no—
286#104—28no—
287#103—28no—
288#102—28no—
289#97—24no—
290#97—42no—
291C4×C8—24no—
292C4×C8—42no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—22no—
298#95—22no—
299#95—21no—
300#95—21no—
301#95—22no—
302#95—48no—
303#95—41no—
304#95—41no—
305C24—21no—
306C24C22×D411yescharacteristic
307#41—22no—
308#41—22no—
309#41—22no—
310#41—22no—
311#41—24no—
312#41—24no—
313#41—42no—
314#41—42no—
315#41—42no—
316#41—42no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8C2×C4⋊C412yes—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4C2×C4⋊C412yes—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—82no—
366C22×C4C4×D412yes—
367C22×C4C4×D412yes—
368C22×C4#12011yescharacteristic
369C22×C4#12011yescharacteristic
370C22×C4#12011yescharacteristic
371C22×C4#12011yescharacteristic
372C22×C4C22⋊Q812yes—
373C22×C4C22⋊Q812yes—
374C22×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×Q811yescharacteristic
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4112yes—
382C22×C4C2×#4112yes—
383C22×C4C2×#4111yescharacteristic
384M16—22no—
385M16—24no—
386M16—22no—
387M16—22no—
388M16—24no—
389M16—24no—
390M16—42no—
391M16—42no—
392M16—42no—
393M16#11714yescomplemented
394C2×C8—22no—
395C2×C8—24no—
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—432no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4C2×C4⋊C412yes—
421C4⋊C4C2×C4⋊C412yes—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—84no—
429C22⋊C4—84no—
430C42—216no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—24no—
435C42—24no—
436C42—24no—
437C42—22no—
438C42—22no—
439C42—48no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—44no—
444C42—42no—
445C42—42no—
446C42—82no—
447C42—82no—
448C42—82no—
449C42—82no—
450C42C2×C4⋊C412yes—
451C42C2×C4⋊C412yes—
452C42#11714yescomplemented
453C42C4×D412yes—
454C42C4×D412yes—
455C42C22⋊Q812yes—
456C42C22⋊Q812yes—
457C42#12312yes—
458C42#12312yes—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—22no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23C2×C4×D411yescharacteristic
468C23C2×#12011yescharacteristic
469C23C2×C22⋊Q811yescharacteristic
470C23#52512yes—
471C23#53212yes—
472Q8—22no—
473Q8—22no—
474Q8—42no—
475D4—22no—
476D4—22no—
477D4—44no—
478D4—42no—
479D4—42no—
480D4—42no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—48no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—84no—
527C2×C4#38512yes—
528C2×C4#38512yes—
529C2×C4#38512yes—
530C2×C4#38512yes—
531C2×C4#45312yes—
532C2×C4#45612yes—
533C2×C4C22×C4⋊C411yescharacteristic
534C2×C4C2×#11711yescharacteristic
535C2×C4C2×#11711yescharacteristic
536C2×C4C2×C4×D411yescharacteristic
537C2×C4C4×#4111yescharacteristic
538C2×C4C4×#4111yescharacteristic
539C2×C4C2×C22⋊Q811yescharacteristic
540C2×C4C2×#12311yescharacteristic
541C2×C4#52512yes—
542C2×C4#52512yes—
543C2×C4#52912yes—
544C2×C4#52912yes—
545C2×C4#52912yes—
546C2×C4#52912yes—
547C2×C4#53212yes—
548C2×C4#53212yes—
549C2×C4#53312yes—
550C2×C4#53312yes—
551C8—24no—
552C8—44no—
553C8—42no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—82no—
566C22—82no—
567C22C2×#38511yescharacteristic
568C22#229011yescharacteristic
569C22#229311yescharacteristic
570C22#238812yes—
571C22#239612yes—
572C22C2×#45311yescharacteristic
573C22C2×#45611yescharacteristic
574C22#301812yes—
575C4—28no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—44no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—82no—
584C4—84no—
585C4—84no—
586C4—82no—
587C4#185914yes—
588C4#228511yescharacteristic
589C4#229011yescharacteristic
590C4#229311yescharacteristic
591C4#229611yescharacteristic
592C4#239512yes—
593C4#239612yes—
594C2—21no—
595C2—21no—
596C2—22no—
597C2—82no—
598C2#1384411yescharacteristic
599C2C2×#185911yescharacteristic
600C2#2086311yescharacteristic
601C2#2358712yes—
602C2#2359012yes—
subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#428C2semidirect
5. C4×#384C2semidirect
6. #11340C2semidirect
7. #11276C2semidirect
10. C4×#343C2semidirect
11. C4×#339C2semidirect
43. C4×#104C4semidirect
197. C4×Q8C4⋊C4semidirect
198. C4×Q8C4⋊C4semidirect
205. C4×D4C4⋊C4semidirect
206. C4×D4C4⋊C4semidirect
393. M16#117semidirect
452. C42#117semidirect

Maximal subgroups

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

subgroupindex
#340602
C4×#4282
C4×#4172
C2×C4×#1042
C4×#3842
#113402
#112762
#111132
#110782
C4×#3432
C4×#3392