Finite Group Databasebeta

← #190365

Subgroups — #190365

Order 512.

1159 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
1#34186C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3C2×#1465C211yesmaximal, characteristic
4#8779C214yesmaximal, complemented
5#8778C214yesmaximal, complemented
6#8051C212yesmaximal, complemented
7#7942C212yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9C22×#99—21no—
10C2×#407C2212yes—
11#2357C412yescomplemented
12#2357C412yescomplemented
13#2357C2212yes—
14#2357C2212yes—
15#2354—21no—
16#2354—21no—
17#2354—21no—
18#2354—21no—
19#2312—21no—
20#2312—21no—
21#2309—21no—
22#2285C411yescharacteristic, complemented
23#2285C411yescharacteristic, complemented
24#2280C411yescharacteristic, complemented
25#2280C411yescharacteristic, complemented
26#2280C2211yescharacteristic
27#2280C2211yescharacteristic
28#2274—21no—
29C2×C4×C22⋊C4C2212yes—
30C4×#99—21no—
31C4×#99—21no—
32C4×#99C2212yes—
33C4×#99C2212yes—
34C2×#342—21no—
35#1465—21no—
36#1465—21no—
37#1465C2212yes—
38#1465C2212yes—
39C2×#324—21no—
40C2×#323—21no—
41#1297C2218yescomplemented
42#1284C2218yescomplemented
43C23×D4D411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D411yescharacteristic
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C2×C4×D4D411yescharacteristic
57C2×C4×D4C2311yescharacteristic
58C2×C4×D4C2311yescharacteristic
59C2×#117—21no—
60C2×#117—21no—
61C2×#117—21no—
62C2×#117—21no—
63C2×#117—21no—
64C2×#117—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C22⋊C4—22no—
67C22×C22⋊C4D411yescharacteristic
68C22×C22⋊C4D411yescharacteristic
69C22×C42—22no—
70C22×C42C2×C411yescharacteristic
71C2×#99—28no—
72C2×#99—22no—
73C2×#99—22no—
74#407—22no—
75#407—22no—
76#407—22no—
77#407C2314yes—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80#390—22no—
81#390—22no—
82#390—22no—
83#390—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#388—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#384—21no—
93#384—21no—
94#382C2×C412yescomplemented
95#382C2×C412yescomplemented
96#381C2×C412yes—
97#381C2×C412yescomplemented
98#381C2×C412yescomplemented
99#381C2312yes—
100#380—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105#380—21no—
106#380C2×C411yescharacteristic
107#380C2×C411yescharacteristic
108#380C2×C411yescharacteristic
109#380C2×C411yescharacteristic
110#379—21no—
111#379—21no—
112#379—21no—
113#379—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yescomplemented
121C4×C4⋊C4C2×C412yescomplemented
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4C2×C412yes—
130C4×C22⋊C4C2×C412yescomplemented
131C4×C22⋊C4C2×C412yescomplemented
132C4×C22⋊C4C2314yes—
133C4×C22⋊C4C2312yes—
134C2×#95—22no—
135#342—22no—
136#342—22no—
137#324—22no—
138#324—22no—
139#323—22no—
140#323—22no—
141C25C22⋊C411yescharacteristic
142C25C2×D411yescharacteristic
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—24no—
146C22×D4—42no—
147C22×D4C22⋊C411yescharacteristic
148C22×D4C22⋊C411yescharacteristic
149C22×D4C2×D411yescharacteristic
150C22×D4C2×D411yescharacteristic
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—42no—
156C23×C4—82no—
157C23×C4—82no—
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4#4112yes—
171C23×C4#4112yes—
172C23×C4#4112yes—
173C23×C4#4112yes—
174C23×C4C2411yescharacteristic
175C2×M16—24no—
176C2×M16—42no—
177C2×M16—84no—
178C22×C8—42no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—22no—
195#117—44no—
196#117—44no—
197#117—82no—
198#117—82no—
199#117—82no—
200#117—82no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4C22⋊C411yescharacteristic
236C2×C22⋊C4C22⋊C411yescharacteristic
237C2×C22⋊C4C22⋊C411yescharacteristic
238C2×C22⋊C4C22⋊C411yescharacteristic
239C2×C22⋊C4C2×D411yescharacteristic
240C2×C22⋊C4C2×D411yescharacteristic
241C2×C22⋊C4C2×D411yescharacteristic
242C2×C22⋊C4C2×D411yescharacteristic
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—42no—
255C2×C42—84no—
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260#99—28no—
261#99—28no—
262C22⋊C8—24no—
263C22⋊C8—24no—
264C22⋊C8—42no—
265C22⋊C8—42no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—24no—
273#95—42no—
274#95C22×C412yes—
275#95C22×C412yes—
276#95C22×C412yes—
277#95C22×C412yes—
278C24—22no—
279C24—22no—
280C24—21no—
281C24—21no—
282C24—22no—
283C24—22no—
284C24—81no—
285C24—81no—
286C24—81no—
287C24—81no—
288C24C2×C22⋊C411yescharacteristic
289C24C2×C22⋊C411yescharacteristic
290C24C2×C22⋊C411yescharacteristic
291C24C2×C22⋊C411yescharacteristic
292C24#12312yes—
293C24#12312yes—
294C24C22×D411yescharacteristic
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—48no—
308C2×D4—48no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—44no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—48no—
336C22×C4—44no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—81no—
340C22×C4—81no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—81no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4—81no—
349C22×C4—81no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4#10414yes—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4#11711yescharacteristic
362C22×C4#11711yescharacteristic
363C22×C4#11711yescharacteristic
364C22×C4#11711yescharacteristic
365C22×C4#11712yes—
366C22×C4#11712yes—
367C22×C4#12312yes—
368C22×C4#12312yes—
369C22×C4#12312yes—
370C22×C4#12312yes—
371C22×C4#12312yes—
372C22×C4#12312yes—
373C22×C4C23×C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C2×#4112yes—
376C22×C4C2×#4112yes—
377M16—84no—
378M16—84no—
379C2×C8—44no—
380C2×C8—84no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—28no—
402C22⋊C4—48no—
403C22⋊C4—44no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—44no—
425C42—44no—
426C42—84no—
427C42—82no—
428C42—82no—
429C42—82no—
430C42—82no—
431C42—84no—
432C23—24no—
433C23—24no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—42no—
445C23—41no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—81no—
464C23—84no—
465C23—84no—
466C23—84no—
467C23—84no—
468C23#38112yes—
469C23#38111yescharacteristic
470C23#38111yescharacteristic
471C23C2×#10412yes—
472C23#42112yes—
473C23C22×C22⋊C411yescharacteristic
474C23C2×#12312yes—
475D4—44no—
476D4—44no—
477D4—44no—
478D4—44no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—816no—
539C2×C4#38111yescharacteristic
540C2×C4#38112yes—
541C2×C4#38111yescharacteristic
542C2×C4C2×#10412yes—
543C2×C4C2×#10412yes—
544C2×C4C2×#11711yescharacteristic
545C2×C4C2×#11711yescharacteristic
546C2×C4C2×#12312yes—
547C8—84no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—42no—
561C22—42no—
562C22—48no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—84no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—84no—
579C22—84no—
580C22#182612yes—
581C22#182612yes—
582C22C2×#38111yescharacteristic
583C22C22×#10411yescharacteristic
584C22C2×#42111yescharacteristic
585C4—22no—
586C4—22no—
587C4—22no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—88no—
601C4#182812yes—
602C2—21no—
603C2—21no—
604C2—22no—
605C2—22no—
606C2—42no—
607C2—42no—
608C2—81no—
609C2—81no—
610C2C2×#182611yescharacteristic
611C2#2070811yescharacteristic
612C2C2×#182811yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8779C2semidirect
5. #8778C2semidirect
6. #8051C2semidirect
7. #7942C2semidirect
11. #2357C4semidirect
12. #2357C4semidirect
22. #2285C4semidirect
23. #2285C4semidirect
24. #2280C4semidirect
25. #2280C4semidirect
41. #1297C22semidirect
42. #1284C22semidirect
94. #382C2×C4semidirect
95. #382C2×C4semidirect
97. #381C2×C4semidirect
98. #381C2×C4semidirect
120. C4×C4⋊C4C2×C4semidirect
121. C4×C4⋊C4C2×C4semidirect
130. C4×C22⋊C4C2×C4semidirect
131. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341862
C2×C4×#992
C2×#14652
#87792
#87782
#80512
#79422