Finite Group Databasebeta

← #358516

Subgroups — #358516

Order 512.

1360 conjugacy classes of subgroups.

The subgroup lattice has 600 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×#382C211yesmaximal, characteristic, complemented
6#11339C212yesmaximal, complemented
7#11276C212yesmaximal, complemented
8#11113C212yesmaximal
9#11079C212yesmaximal
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—
29#2283—21no—
30C2×#382C2212yes—
31C4×#117C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C43C2211yescharacteristic
35#1776—22no—
36C4×#107—21no—
37C4×#106—21no—
38#1763—21no—
39#1762—21no—
40C4×#103—21no—
41C4×#102—21no—
42C4×#104C418yescomplemented
43C2×#343C2212yes—
44C2×#343C2212yes—
45#1736C2212yes—
46#1736C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C4⋊#95C2212yes—
52#1434—22no—
53C4×#95—22no—
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⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C42—21no—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42D412yes—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#428—24no—
83#428—24no—
84C2×#107—22no—
85C2×#106—22no—
86C2×#104—24no—
87C2×#104—24no—
88C2×#104C2×C414yes—
89C2×#104C2×C414yes—
90#417—24no—
91#417—24no—
92C2×#103—22no—
93C2×#102—22no—
94C4×M16—22no—
95C4×M16—22no—
96C4×M16—22no—
97C4×M16C2×C414yes—
98C2×#97—21no—
99C2×#97—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102#391—22no—
103#390—22no—
104#389—22no—
105#389—22no—
106#388—24no—
107#388—24no—
108#387—24no—
109#387—24no—
110#386—24no—
111#385—24no—
112#384—22no—
113#383—22no—
114#382—22no—
115#382—22no—
116#382—24no—
117#382—24no—
118#381—22no—
119#380—22no—
120#380—22no—
121#380—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
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⋊C4D412yes—
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C22⋊C4—22no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137#376—22no—
138#376—21no—
139C2×#95—24no—
140C2×#95—24no—
141C2×#95—21no—
142C2×#95D412yes—
143C43—21no—
144C43—22no—
145C43C2×C414yes—
146#343—24no—
147#343—24no—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#339—24no—
153#339—24no—
154#339C2×C414yes—
155#339C2×C414yes—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41—22no—
161C2×#41—22no—
162C2×#41C22×C412yes—
163C2×#41C2×Q812yes—
164C22×Q8C2×D411yescharacteristic
165C22×D4—21no—
166C22×D4C2×D411yescharacteristic
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D414yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2411yescharacteristic
177C2×M16—21no—
178C2×M16—22no—
179C2×M16—21no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16C22×C412yes—
188C2×M16C22×C412yes—
189C2×M16#4114yes—
190C2×M16#4114yes—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8C4⋊C414yes—
197C4×Q8C4⋊C414yes—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4C4⋊C414yes—
205C4×D4C4⋊C414yes—
206#117—21no—
207#117—21no—
208#117—24no—
209#117—24no—
210#117—22no—
211#117—22no—
212#117—22no—
213#117—22no—
214#117—84no—
215#117C22×C412yes—
216#117C2×Q812yes—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4#4114yes—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—41no—
267C2×C42—41no—
268C2×C42C22×C412yes—
269C2×C42C22×C412yes—
270C2×C42C22×C412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×Q812yes—
276C2×C42#4114yes—
277C2×C42#4112yes—
278C2×C42#4112yes—
279#107—28no—
280#106—28no—
281#104—28no—
282#104—28no—
283#103—28no—
284#102—28no—
285#97—24no—
286#97—42no—
287C4×C8—24no—
288C4×C8—42no—
289#95—216no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—21no—
297#95—21no—
298#95—22no—
299#95—48no—
300#95—41no—
301#95—41no—
302C24—21no—
303C24C22×D411yescharacteristic
304#41—22no—
305#41—22no—
306#41—22no—
307#41—22no—
308#41—24no—
309#41—24no—
310#41—42no—
311#41—42no—
312#41—42no—
313#41—42no—
314C2×Q8—21no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8C2×C4⋊C412yes—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—21no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4C2×C4⋊C412yes—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—82no—
363C22×C4C4×D412yes—
364C22×C4C4×D412yes—
365C22×C4#12011yescharacteristic
366C22×C4#12011yescharacteristic
367C22×C4#12011yescharacteristic
368C22×C4#12011yescharacteristic
369C22×C4C22⋊Q812yes—
370C22×C4C22⋊Q812yes—
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×Q811yescharacteristic
375C22×C4C2×#4112yes—
376C22×C4C2×#4112yes—
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4C2×#4111yescharacteristic
381M16—22no—
382M16—24no—
383M16—22no—
384M16—22no—
385M16—24no—
386M16—24no—
387M16—42no—
388M16—42no—
389M16—42no—
390M16#11714yescomplemented
391C2×C8—22no—
392C2×C8—24no—
393C2×C8—22no—
394C2×C8—24no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C4⋊C4C2×C4⋊C412yes—
417C4⋊C4C2×C4⋊C412yes—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—84no—
425C22⋊C4—84no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—24no—
430C42—24no—
431C42—24no—
432C42—22no—
433C42—22no—
434C42—48no—
435C42—48no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—44no—
441C42—42no—
442C42—82no—
443C42—82no—
444C42—82no—
445C42—82no—
446C42C2×C4⋊C412yes—
447C42C2×C4⋊C412yes—
448C42#11714yescomplemented
449C42C4×D412yes—
450C42C4×D412yes—
451C42C22⋊Q812yes—
452C42C22⋊Q812yes—
453C42#12312yes—
454C42#12312yes—
455C23—21no—
456C23—21no—
457C23—22no—
458C23—22no—
459C23—42no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23C2×C4×D411yescharacteristic
464C23C2×#12011yescharacteristic
465C23C2×C22⋊Q811yescharacteristic
466C23#52512yes—
467C23#53212yes—
468Q8—22no—
469Q8—22no—
470Q8—42no—
471D4—22no—
472D4—22no—
473D4—44no—
474D4—42no—
475D4—42no—
476D4—42no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—48no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—84no—
523C2×C4#38512yes—
524C2×C4#38512yes—
525C2×C4#38512yes—
526C2×C4#38512yes—
527C2×C4#45412yes—
528C2×C4#45512yes—
529C2×C4C22×C4⋊C411yescharacteristic
530C2×C4C2×#11711yescharacteristic
531C2×C4C2×#11711yescharacteristic
532C2×C4C2×C4×D411yescharacteristic
533C2×C4C4×#4111yescharacteristic
534C2×C4C4×#4111yescharacteristic
535C2×C4C2×C22⋊Q811yescharacteristic
536C2×C4C2×#12311yescharacteristic
537C2×C4#52512yes—
538C2×C4#52512yes—
539C2×C4#52912yes—
540C2×C4#52912yes—
541C2×C4#52912yes—
542C2×C4#52912yes—
543C2×C4#53212yes—
544C2×C4#53212yes—
545C2×C4#53312yes—
546C2×C4#53312yes—
547C8—24no—
548C8—44no—
549C8—42no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—82no—
562C22—82no—
563C22C2×#38511yescharacteristic
564C22#229011yescharacteristic
565C22#229311yescharacteristic
566C22#238812yes—
567C22#239612yes—
568C22C2×#45511yescharacteristic
569C22C2×#45411yescharacteristic
570C22#301812yes—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—44no—
575C4—44no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—82no—
580C4—84no—
581C4—82no—
582C4—84no—
583C4#186014yes—
584C4#228511yescharacteristic
585C4#229011yescharacteristic
586C4#229311yescharacteristic
587C4#229611yescharacteristic
588C4#239512yes—
589C4#239612yes—
590C2—21no—
591C2—21no—
592C2—22no—
593C2—82no—
594C2#1384411yescharacteristic
595C2C2×#186011yescharacteristic
596C2#2086311yescharacteristic
597C2#2358812yes—
598C2#2358912yes—
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×#382C2semidirect
6. #11339C2semidirect
7. #11276C2semidirect
10. C4×#343C2semidirect
11. C4×#339C2semidirect
42. C4×#104C4semidirect
390. M16#117semidirect
448. 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×#3822
#113392
#112762
#111132
#110792
C4×#3432
C4×#3392