Finite Group Databasebeta

← #207238

Subgroups — #207238

Order 512.

1378 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic
3C4×#380C211yesmaximal, characteristic
4#9913C214yesmaximal, complemented
5#9910C214yesmaximal, complemented
6#8023C212yesmaximal
7#8023C212yesmaximal
8C22×#125C411yescharacteristic, complemented
9C23×C42C2211yescharacteristic
10C22×C4⋊C8—21no—
11C22×C22⋊C8C2212yes—
12#2851—21no—
13#2851—21no—
14#2851—21no—
15#2851—21no—
16#2851C412yescomplemented
17#2851C412yescomplemented
18#2851C2212yes—
19#2851C2212yes—
20C2×#389C411yescharacteristic, complemented
21C2×#389C411yescharacteristic, complemented
22C2×#387—21no—
23C2×#387—21no—
24C2×#385—21no—
25C2×#385—21no—
26C2×#382C411yescharacteristic, complemented
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C22⋊C4—21no—
32#1797—21no—
33#1797—21no—
34#1797C2212yes—
35#1797C2212yes—
36C2×#336—21no—
37C4⋊#95—21no—
38#1434C2212yes—
39C22⋊#95—21no—
40C4×#95C2212yes—
41#1294C2218yes—
42#1294C2218yes—
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8—22no—
47C2×#125—21no—
48C2×#125—21no—
49C2×#125—44no—
50C2×#125C2×C412yes—
51C2×#125C2×C412yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C42—22no—
65C22×C42—44no—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—44no—
81C2×C22⋊C8—28no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8C2314yes—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389C2×C412yes—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#382—21no—
109#382—21no—
110#382C2×C412yes—
111#382C2×C412yes—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—24no—
117#380C2×C412yes—
118#380C2×C412yes—
119#380Q818yes—
120#380C2312yes—
121#380C2312yes—
122C4×C4⋊C4—44no—
123C4×C22⋊C4—44no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95C2314yes—
134#336—22no—
135#336—22no—
136C25C2×D411yescharacteristic
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×Q814yes—
169C23×C4#4114yes—
170C23×C4C2411yescharacteristic
171C22×C8—24no—
172C22×C8—44no—
173C22×C8—44no—
174C22×C8—44no—
175#125—44no—
176#125—44no—
177#125—44no—
178#125—44no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—42no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42—44no—
231C2×C42C22⋊C411yescharacteristic
232C2×C42C22⋊C411yescharacteristic
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C4⋊C8—44no—
248C4⋊C8—44no—
249C22⋊C8—28no—
250C22⋊C8—44no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—24no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95C2×Q818yes—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24C2×C22⋊C411yescharacteristic
273C24C2×C22⋊C411yescharacteristic
274C24C22⋊Q814yes—
275C24#13514yes—
276C24AGL(1,ℤ/8)12yes—
277C24#13712yes—
278C24C22×D411yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—28no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—48no—
337C22×C4—416no—
338C22×C4—48no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C4×Q814yes—
368C22×C4C4×Q814yes—
369C22×C4C22⋊Q814yes—
370C22×C4#12412yes—
371C22×C4#12412yes—
372C22×C4C4⋊Q812yes—
373C22×C4C4⋊Q812yes—
374C22×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×Q812yes—
377C22×C4C2×#4112yes—
378C2×C8—44no—
379C2×C8—44no—
380C2×C8—44no—
381C2×C8—88no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—416no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—416no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—88no—
421C42#9914yes—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—44no—
435C23—44no—
436C23—42no—
437C23—42no—
438C23—44no—
439C23—44no—
440C23—42no—
441C23—42no—
442C23#39112yes—
443C23#39112yes—
444C23#41612yes—
445C23#41612yes—
446C23#41712yes—
447C23#41712yes—
448C23#47818yes—
449C23#47918yes—
450C23C22×C22⋊C411yescharacteristic
451C23C2×C22⋊Q812yes—
452C23C2×#13512yes—
453C23C2×AGL(1,ℤ/8)11yescharacteristic
454C23C2×#13711yescharacteristic
455C2×C4—24no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—28no—
473C2×C4—24no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—44no—
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—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—48no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4—84no—
526C2×C4—84no—
527C2×C4#39112yes—
528C2×C4#39112yes—
529C2×C4C2×#9912yes—
530C2×C4C2×#9912yes—
531C2×C4C2×#9912yes—
532C2×C4#41012yes—
533C2×C4#42112yes—
534C2×C4#42112yes—
535C2×C4C2×C4×Q812yes—
536C2×C4C2×#12411yescharacteristic
537C2×C4C2×C4⋊Q811yescharacteristic
538C8—84no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—21no—
552C22—22no—
553C22—21no—
554C22—22no—
555C22—21no—
556C22—21no—
557C22—22no—
558C22—22no—
559C22—48no—
560C22—48no—
561C22#198914yes—
562C22#198914yes—
563C22C2×#39111yescharacteristic
564C22C22×#9911yescharacteristic
565C22C2×#41011yescharacteristic
566C22C2×#41611yescharacteristic
567C22C2×#41711yescharacteristic
568C22C2×#42111yescharacteristic
569C22C2×#42111yescharacteristic
570C22C2×#47812yes—
571C22C2×#47912yes—
572C4—21no—
573C4—22no—
574C4—21no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—22no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—88no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—82no—
589C4#199614yes—
590C2—22no—
591C2—22no—
592C2—24no—
593C2—24no—
594C2#1756711yescharacteristic
595C2#1757211yescharacteristic
596C2#1758111yescharacteristic
597C2#1759211yescharacteristic
598C2C2×#198911yescharacteristic
599C2#2132911yescharacteristic
600C2C2×#199611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9913C2semidirect
5. #9910C2semidirect
8. C22×#125C4semidirect
16. #2851C4semidirect
17. #2851C4semidirect
20. C2×#389C4semidirect
21. C2×#389C4semidirect
26. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#17972
C4×#3802
#99132
#99102
#80232