Finite Group Databasebeta

← #189794

Subgroups — #189794

Order 512.

1039 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34121C211yesmaximal, characteristic
2#21374C211yesmaximal, characteristic
3#20629C211yesmaximal, characteristic
4#8711C212yesmaximal, complemented
5#8710C212yesmaximal, complemented
6#8708C212yesmaximal, complemented
7#8706C212yesmaximal, complemented
8#7903C211yesmaximal, characteristic, complemented
9#7899C211yesmaximal, characteristic, complemented
10#7895C211yesmaximal, characteristic, complemented
11#7891C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C2×#408—21no—
17C2×C4×M16C2211yescharacteristic
18#2357—21no—
19#2356—21no—
20#2355C412yes—
21#2354C412yes—
22#2353—21no—
23#2353—21no—
24#2352C412yes—
25#2352C412yes—
26#2298—21no—
27#2298—21no—
28#2297—21no—
29#2285C2211yescharacteristic
30#2283C2211yescharacteristic
31#2283C2211yescharacteristic
32C4×#117—21no—
33C4×#117C2211yescharacteristic
34C4×#117C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1985—21no—
37#1955—21no—
38#1870—21no—
39#1870—21no—
40#1798—21no—
41#1798—21no—
42#1797C2212yes—
43#1797C2212yes—
44C4×C22⋊C8C2212yes—
45C4×C22⋊C8C2212yes—
46#1729—21no—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55C22×#41D411yescharacteristic
56C22×M16—21no—
57C22×M16—21no—
58C23×C8—21no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41—22no—
62C4×#41—22no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×#117—21no—
72C2×#117C2×C412yes—
73C2×#117C2×C412yes—
74C2×#117D411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117D411yescharacteristic
77C22×C4⋊C4C2×C412yes—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#431—22no—
83#431—22no—
84#423—22no—
85#423—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93#408—44no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C4×M16—22no—
100C4×M16—22no—
101C2×#97—21no—
102C2×C4×C8C2312yes—
103C2×C4×C8C2312yes—
104#391—22no—
105#390—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#386—22no—
115#386—22no—
116#384C2×C412yes—
117#383—21no—
118#383—21no—
119#382—21no—
120#382C2×C412yes—
121#382C2×C412yes—
122#381—21no—
123#381—21no—
124#380—21no—
125#380C2×C412yes—
126#380C2×C412yes—
127#380C2×C412yes—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2×C412yes—
134C4×C4⋊C4C2×C412yes—
135C4×C4⋊C4C2×C412yes—
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4—22no—
148#376—21no—
149#376—21no—
150#376—21no—
151#376—21no—
152C2×#95—21no—
153C43—21no—
154C43—21no—
155#336—22no—
156#336—22no—
157#336—22no—
158#336—22no—
159C2×#41—22no—
160C2×#41—22no—
161C2×#41—22no—
162C2×#41—22no—
163C2×#41—41no—
164C2×#41—41no—
165C22×Q8C22⋊C411yescharacteristic
166C22×D4C22⋊C411yescharacteristic
167C22×D4C22⋊C411yescharacteristic
168C22×D4C22⋊C411yescharacteristic
169C23×C4—22no—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C2×M16—44no—
180C2×M16—42no—
181C2×M16—42no—
182C2×M16—42no—
183C2×M16—42no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—42no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207#117—22no—
208#117—22no—
209#117—22no—
210#117—22no—
211#117—22no—
212#117—22no—
213#117—24no—
214#117—24no—
215#117—42no—
216#117—42no—
217#117—42no—
218#117—42no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4C22⋊C411yescharacteristic
240C2×C22⋊C4C22⋊C411yescharacteristic
241C2×C22⋊C4C22⋊C411yescharacteristic
242C2×C22⋊C4C22⋊C411yescharacteristic
243C2×C22⋊C4C22⋊C411yescharacteristic
244C2×C22⋊C4C22⋊C411yescharacteristic
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42#4112yes—
274C2×C42#4112yes—
275C2×C42#4112yes—
276C2×C42#4112yes—
277C2×C42C2411yescharacteristic
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—42no—
283C4⋊C8—42no—
284C4⋊C8—42no—
285C22⋊C8—44no—
286C22⋊C8—44no—
287C22⋊C8—44no—
288C22⋊C8—42no—
289C22⋊C8—44no—
290#97—42no—
291C4×C8—22no—
292C4×C8—22no—
293C4×C8—22no—
294C4×C8—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309C24C2×C22⋊C411yescharacteristic
310C24C2×C22⋊C411yescharacteristic
311C24C2×C22⋊C411yescharacteristic
312#41—44no—
313#41—44no—
314#41—44no—
315#41—44no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—41no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—82no—
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4#11712yes—
376C22×C4#11712yes—
377C22×C4#12312yes—
378C22×C4#12312yes—
379C22×C4#12312yes—
380C22×C4#12312yes—
381C22×C4AGL(1,ℤ/8)12yes—
382C22×C4#13712yes—
383C22×C4C23×C411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386M16—48no—
387M16—84no—
388C2×C8—24no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—84no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—82no—
417C4⋊C4#13114yes—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—44no—
442C42—44no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—44no—
452C42—82no—
453C42—82no—
454C42#11712yes—
455C42#11712yes—
456C42#12312yes—
457C42#12312yes—
458C42#12312yes—
459C42#12312yes—
460C42C2×#4111yescharacteristic
461C42C2×#4111yescharacteristic
462C42C2×#4111yescharacteristic
463C42C2×#4111yescharacteristic
464C23—22no—
465C23—22no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—22no—
471C23—21no—
472C23—22no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—21no—
477C23#38112yes—
478C23#41712yes—
479C23C22×C22⋊C411yescharacteristic
480Q8—44no—
481D4—44no—
482D4—44no—
483D4—44no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—88no—
542C2×C4#38112yes—
543C2×C4#38112yes—
544C2×C4#38112yes—
545C2×C4#41712yes—
546C2×C4C2×#11711yescharacteristic
547C2×C4C2×#11711yescharacteristic
548C2×C4C2×#12311yescharacteristic
549C2×C4C2×#12311yescharacteristic
550C2×C4C2×#12311yescharacteristic
551C2×C4C2×#12311yescharacteristic
552C2×C4C2×#13112yes—
553C2×C4#56812yes—
554C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
555C2×C4C2×#13711yescharacteristic
556C2×C4#57511yescharacteristic
557C2×C4#57511yescharacteristic
558C8—44no—
559C8—82no—
560C8—82no—
561C8—82no—
562C22—24no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#38111yescharacteristic
578C22#288012yes—
579C22C2×#41711yescharacteristic
580C22#289611yescharacteristic
581C22#290112yes—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—21no—
586C4—21no—
587C4—22no—
588C4—44no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4#292211yescharacteristic
598C4#292311yescharacteristic
599C4#292512yes—
600C4#309411yescharacteristic
601C4#309411yescharacteristic
602C4#309511yescharacteristic
603C4#309511yescharacteristic
604C2—42no—
605C2—42no—
606C2—81no—
607C2#1751212yes—
608C2#1751612yes—
609C2#2068411yescharacteristic
610C2#2070211yescharacteristic
611C2#2070611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8711C2semidirect
5. #8710C2semidirect
6. #8708C2semidirect
7. #8706C2semidirect
8. #7903C2semidirect
9. #7899C2semidirect
10. #7895C2semidirect
11. #7891C2semidirect

Maximal subgroups

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

subgroupindex
#341212
#213742
#206292
#87112
#87102
#87082
#87062
#79032
#78992
#78952
#78912