Finite Group Databasebeta

← #189798

Subgroups — #189798

Order 512.

1078 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34129C211yesmaximal, characteristic
2#21849C211yesmaximal, characteristic
3C4×#408C211yesmaximal, characteristic
4#8710C212yesmaximal, complemented
5#8709C212yesmaximal, complemented
6#8708C212yesmaximal, complemented
7#8708C212yesmaximal, complemented
8#7903C211yesmaximal, characteristic, complemented
9#7897C211yesmaximal, characteristic, complemented
10#7896C211yesmaximal, characteristic, complemented
11#7890C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#408—21no—
16C2×C4×M16C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2354C411yescharacteristic
19#2354C411yescharacteristic
20#2354C412yes—
21#2354C412yes—
22Q8×C4⋊C4C411yescharacteristic
23Q8×C4⋊C4C411yescharacteristic
24#2346—21no—
25#2346—22no—
26#2341—21no—
27#2293—22no—
28#2293—21no—
29#2285C2212yes—
30#2285C2211yescharacteristic
31C42×Q8—21no—
32C4×#117C2212yes—
33C4×#117C2211yescharacteristic
34#1984—21no—
35#1954—21no—
36#1870—22no—
37#1797C2212yes—
38#1797C2212yes—
39#1750—22no—
40C4×C22⋊C8C2212yes—
41C4×C22⋊C8C2212yes—
42#1729—21no—
43#1279C2211yescharacteristic
44#1279C2211yescharacteristic
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2211yescharacteristic
51#1279C2212yes—
52#1279C2211yescharacteristic
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C22×M16—21no—
56C23×C8—21no—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×D4C2×C412yes—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D412yes—
68C2×C4×D4D411yescharacteristic
69C2×#117—22no—
70C2×#117—21no—
71C2×#117C2×C412yes—
72C2×#117C2×C412yes—
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117D412yes—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79#431—22no—
80#431—22no—
81#423—22no—
82#423—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87#408—28no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C4×M16—22no—
94C4×M16—22no—
95C2×#97—21no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2312yes—
103#390—21no—
104#390—21no—
105#390—22no—
106#390—22no—
107#389—21no—
108#389—21no—
109#389—22no—
110#389—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#384C2×C412yes—
118#384C2×C412yes—
119#384C2×C411yescharacteristic
120#384C2×C411yescharacteristic
121#382—22no—
122#382—21no—
123#380—22no—
124#380—21no—
125#380C2×C412yes—
126#380C2×C412yes—
127#380C2×C412yes—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2×C412yes—
135C4×C4⋊C4C2×C412yes—
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2312yes—
140C4×C4⋊C4C2311yescharacteristic
141C4×C22⋊C4—22no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—21no—
145#376—22no—
146#376—21no—
147C43—22no—
148C43—21no—
149#336—22no—
150#336—22no—
151#336—22no—
152#336—22no—
153C2×#41—24no—
154C2×#41—22no—
155C2×#41—22no—
156C2×#41—42no—
157C22×Q8C22⋊C411yescharacteristic
158C22×D4C22⋊C411yescharacteristic
159C22×D4C22⋊C412yes—
160C23×C4C22×C412yes—
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—24no—
167C2×M16—42no—
168C2×M16—42no—
169C2×M16—42no—
170C2×M16—81no—
171C2×M16—81no—
172C22×C8—24no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—42no—
180C22×C8—42no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—44no—
186C4×Q8—44no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—22no—
195#117—22no—
196#117—22no—
197#117—24no—
198#117—24no—
199#117—22no—
200#117—22no—
201#117—24no—
202#117—24no—
203#117—41no—
204#117—41no—
205#117—42no—
206#117—42no—
207#117—81no—
208#117—82no—
209#117—81no—
210#117—82no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4C22⋊C412yes—
237C2×C22⋊C4C22⋊C412yes—
238C2×C22⋊C4C22⋊C412yes—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—21no—
254C2×C42C22×C412yes—
255C2×C42C22×C411yescharacteristic
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4112yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264C2×C42C2411yescharacteristic
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—42no—
272C22⋊C8—28no—
273C22⋊C8—28no—
274C22⋊C8—42no—
275C22⋊C8—82no—
276C22⋊C8—82no—
277#97—42no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—22no—
283C4×C8—22no—
284C4×C8—42no—
285C4×C8—42no—
286#95—22no—
287#95—22no—
288#95—21no—
289#95—21no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—21no—
294#95—22no—
295#95—21no—
296C24C2×C22⋊C412yes—
297C24C2×C22⋊C411yescharacteristic
298#41—48no—
299#41—48no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8#10414yes—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4—24no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—42no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—42no—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4#11712yes—
348C22×C4#11712yes—
349C22×C4#12312yes—
350C22×C4#12312yes—
351C22×C4#12312yes—
352C22×C4#12312yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356M16—48no—
357M16—84no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—24no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—48no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—41no—
374C2×C8—42no—
375C2×C8—41no—
376C2×C8—41no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—81no—
396C4⋊C4—81no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4#13112yes—
400C4⋊C4#13112yes—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—42no—
418C42—42no—
419C42—44no—
420C42—44no—
421C42—44no—
422C42—42no—
423C42—41no—
424C42—41no—
425C42—42no—
426C42—41no—
427C42—41no—
428C42—44no—
429C42—42no—
430C42—42no—
431C42#11712yes—
432C42#11712yes—
433C42#12312yes—
434C42#12312yes—
435C42#12312yes—
436C42#12312yes—
437C42C2×#4111yescharacteristic
438C42C2×#4111yescharacteristic
439C42C2×#4111yescharacteristic
440C42C2×#4111yescharacteristic
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23#38112yes—
455C23C22×C22⋊C411yescharacteristic
456Q8—28no—
457D4—48no—
458D4—44no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4#38112yes—
512C2×C4#38112yes—
513C2×C4#38112yes—
514C2×C4C2×#10414yes—
515C2×C4C2×#10412yes—
516C2×C4#42112yes—
517C2×C4C2×#11711yescharacteristic
518C2×C4C2×#11711yescharacteristic
519C2×C4C2×#12311yescharacteristic
520C2×C4C2×#12311yescharacteristic
521C2×C4C2×#12311yescharacteristic
522C2×C4C2×#12311yescharacteristic
523C2×C4C2×#13111yescharacteristic
524C2×C4C2×#13111yescharacteristic
525C2×C4#56811yescharacteristic
526C2×C4#56811yescharacteristic
527C2×C4#57611yescharacteristic
528C2×C4#57711yescharacteristic
529C2×C4#57711yescharacteristic
530C2×C4#57811yescharacteristic
531C8—42no—
532C8—42no—
533C8—44no—
534C8—82no—
535C8—81no—
536C8—81no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22C2×#38111yescharacteristic
552C22#288011yescharacteristic
553C22#288011yescharacteristic
554C22#289711yescharacteristic
555C22#289811yescharacteristic
556C22C22×#10411yescharacteristic
557C22C2×#42111yescharacteristic
558C4—22no—
559C4—22no—
560C4—21no—
561C4—21no—
562C4—24no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—81no—
568C4—82no—
569C4—82no—
570C4—81no—
571C4#182614yes—
572C4#292211yescharacteristic
573C4#292311yescharacteristic
574C4#292511yescharacteristic
575C4#292511yescharacteristic
576C4#309611yescharacteristic
577C4#309711yescharacteristic
578C4#309811yescharacteristic
579C4#309911yescharacteristic
580C2—42no—
581C2—82no—
582C2#1751811yescharacteristic
583C2#1752011yescharacteristic
584C2#1752311yescharacteristic
585C2#1752611yescharacteristic
586C2#2068411yescharacteristic
587C2#2070311yescharacteristic
588C2C2×#182611yescharacteristic
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. #8710C2semidirect
5. #8709C2semidirect
6. #8708C2semidirect
7. #8708C2semidirect
8. #7903C2semidirect
9. #7897C2semidirect
10. #7896C2semidirect
11. #7890C2semidirect

Maximal subgroups

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

subgroupindex
#341292
#218492
C4×#4082
#87102
#87092
#87082
#79032
#78972
#78962
#78902