Finite Group Databasebeta

← #195944

Subgroups — #195944

Order 512.

1578 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3#20714C211yesmaximal, characteristic
4C2×#1461C212yesmaximal, complemented
5#9086C214yesmaximal, complemented
6#9085C214yesmaximal
7#7908C212yesmaximal, complemented
8C22×C22⋊Q8C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C23×C42C2211yescharacteristic
11#2920—28no—
12C2×#432—21no—
13C2×#432C2212yes—
14C2×#431—21no—
15C2×#431C2212yes—
16C2×#431C2212yes—
17C2×#423—21no—
18C2×#407C2212yes—
19C2×C4×M16—21no—
20C4×C22⋊Q8—21no—
21C4×C22⋊Q8—21no—
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—21no—
27C4×C22⋊Q8—21no—
28C2×#391C411yescharacteristic
29C2×#389C411yescharacteristic
30C2×#387C2212yes—
31C2×#385C2212yes—
32C2×#380C411yescharacteristic
33C2×C4×C4⋊C4C412yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C412yes—
36#1817C2212yes—
37#1797—21no—
38#1797—21no—
39#1779C2212yes—
40C4×C22⋊C8—21no—
41C4×C22⋊C8—21no—
42#1461—28no—
43C2×#324C2218yes—
44#1280C2218yes—
45C23×Q8C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—24no—
56C2×C4×Q8—24no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—24no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C412yes—
70C22×C42—24no—
71C22×C42—24no—
72C22×C42C2×C412yes—
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C412yes—
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#432—28no—
87#432—28no—
88#431—28no—
89#431—24no—
90#431—24no—
91#431—24no—
92#431D418yes—
93#423—28no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97#407—28no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8C2314yes—
102C2×C22⋊C8C2314yes—
103C4×M16—28no—
104C2×#97—22no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107#391—21no—
108#391—21no—
109#391—21no—
110#391—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#387—22no—
116#387—22no—
117#387—22no—
118#387—22no—
119#385—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C2×#95C2×C412yes—
140C2×#95C2×C412yes—
141C2×#95C2×C412yes—
142C2×#95C2×C412yes—
143C2×#95C2312yes—
144#336—22no—
145#336—22no—
146#324—216no—
147C25C22×C411yescharacteristic
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—24no—
151C22×Q8—22no—
152C22×Q8—24no—
153C22×Q8M1614yescomplemented
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C412yes—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C22×C412yes—
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2411yescharacteristic
187C2×M16—28no—
188C2×M16—44no—
189C2×M16—44no—
190C22×C8—24no—
191C22×C8—24no—
192C22⋊Q8—28no—
193C22⋊Q8—28no—
194C22⋊Q8—28no—
195C22⋊Q8—28no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—44no—
199C4×Q8—44no—
200C4×Q8—42no—
201C4×Q8—42no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4M1614yescomplemented
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—44no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—28no—
251C2×C42—44no—
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D414yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42#4114yes—
270C4⋊C8—28no—
271C4⋊C8—28no—
272C4⋊C8—44no—
273C22⋊C8—28no—
274C22⋊C8—24no—
275C22⋊C8—24no—
276C22⋊C8—24no—
277C22⋊C8—44no—
278C22⋊C8—44no—
279C22⋊C8C2×D418yes—
280#97—28no—
281C4×C8—28no—
282C4×C8—44no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293C24—21no—
294C24—21no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—41no—
300C24—41no—
301C24C2×C22⋊C411yescharacteristic
302C24C2×C22⋊C411yescharacteristic
303C24C23×C411yescharacteristic
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—216no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—48no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—416no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4#10114yes—
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C4×D414yes—
396C22×C4C4×D414yes—
397C22×C4C2×M1614yes—
398C22×C4C2×M1614yes—
399C22×C4C2×M1612yes—
400C22×C4C2×M1612yes—
401C22×C4#13114yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C2×#4112yes—
406M16—416no—
407C2×C8—28no—
408C2×C8—44no—
409C2×C8—44no—
410C2×C8—44no—
411C2×C8—44no—
412C2×C8—44no—
413C4⋊C4—28no—
414C4⋊C4—28no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C22⋊C4—28no—
427C22⋊C4—28no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C42—21no—
431C42—21no—
432C42—22no—
433C42—22no—
434C42—21no—
435C42—21no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—44no—
445C42—44no—
446C42—44no—
447C42—44no—
448C42#12012yes—
449C42#12012yes—
450C42#12114yes—
451C42#12114yes—
452C42#12312yes—
453C42#12312yes—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—24no—
459C23—24no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23#41012yes—
481C23C2×#10112yes—
482C23C22×C22⋊C411yescharacteristic
483C23C2×C4×D412yes—
484C23C22×M1611yescharacteristic
485C23C22×M1611yescharacteristic
486C23C2×#13112yes—
487Q8—48no—
488Q8—48no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—28no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4#38612yes—
551C2×C4#38612yes—
552C2×C4#38612yes—
553C2×C4#38612yes—
554C2×C4#40712yes—
555C2×C4#40712yes—
556C2×C4#40812yes—
557C2×C4#40812yes—
558C2×C4#41012yes—
559C2×C4C2×#10112yes—
560C2×C4C2×#10112yes—
561C2×C4#41312yes—
562C2×C4#43518yes—
563C2×C4#43618yes—
564C2×C4C2×#12011yescharacteristic
565C2×C4C2×#12112yes—
566C2×C4C2×#12311yescharacteristic
567C8—48no—
568C8—84no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—48no—
584C22—48no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#38611yescharacteristic
594C22C2×#40711yescharacteristic
595C22C2×#40811yescharacteristic
596C22C2×#41011yescharacteristic
597C22C2×#41011yescharacteristic
598C22C22×#10111yescharacteristic
599C22C2×#41311yescharacteristic
600C22C2×#43512yes—
601C22C2×#43612yes—
602C4—21no—
603C4—22no—
604C4—21no—
605C4—22no—
606C4—22no—
607C4—22no—
608C4—41no—
609C4—41no—
610C4—44no—
611C4—44no—
612C4—41no—
613C4—41no—
614C4—82no—
615C4—82no—
616C4#187014yes—
617C4#187014yes—
618C4#190114yes—
619C2—22no—
620C2—22no—
621C2—42no—
622C2—42no—
623C2#1736511yescharacteristic
624C2#1736611yescharacteristic
625C2#1738311yescharacteristic
626C2#1738511yescharacteristic
627C2C2×#187011yescharacteristic
628C2#2098311yescharacteristic
629C2C2×#190111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1461C2semidirect
5. #9086C2semidirect
7. #7908C2semidirect
153. C22×Q8M16semidirect
222. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×C4×C22⋊Q82
#207142
C2×#14612
#90862
#90852
#79082