Finite Group Databasebeta

← #193489

Subgroups — #193489

Order 512.

1724 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#423C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3#20714C211yesmaximal, characteristic
4C2×#1487C212yesmaximal, complemented
5#8942C214yesmaximal, complemented
6#8937C214yesmaximal, complemented
7#7901C212yesmaximal, complemented
8C23×M16—21no—
9C22×C22⋊Q8C411yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#431C2212yes—
13C2×#423—21no—
14C2×#423—21no—
15C2×#423—28no—
16C2×#423C2212yes—
17C2×#423C2212yes—
18C22×C4⋊C8—21no—
19C22×C4⋊C8C2212yes—
20C4×C22⋊Q8—21no—
21C4×C22⋊Q8—21no—
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8C412yes—
25C4×C22⋊Q8C412yes—
26C4×C22⋊Q8C2212yes—
27C4×C22⋊Q8C2212yes—
28C2×#391C411yescharacteristic
29C2×#389C411yescharacteristic
30C2×#387—21no—
31C2×#385—21no—
32C2×#380C411yescharacteristic
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4—21no—
36#1817—21no—
37#1797—21no—
38#1797—21no—
39#1779—21no—
40C4×C22⋊C8C2212yes—
41C4×C22⋊C8C2212yes—
42#1487—28no—
43C2×Q8⋊C8C2218yescomplemented
44#1279C2218yescomplemented
45C23×Q8C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C22×M16—28no—
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—44no—
53C22×M16—44no—
54C23×C8—22no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—28no—
58C2×C22⋊Q8C2×C412yes—
59C2×C22⋊Q8C2×C412yes—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—24no—
63C2×C4×Q8—24no—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8C2312yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C42—24no—
75C22×C42—24no—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#431—24no—
91#431—24no—
92#423—28no—
93#423—28no—
94#423—24no—
95#423—28no—
96#423D418yes—
97C2×C4⋊C8—28no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—28no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8C2314yes—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C4×C8C2314yes—
108#391—21no—
109#391—21no—
110#391C2×C412yes—
111#391C2×C412yes—
112#389—21no—
113#389—21no—
114#389C2×C412yes—
115#389C2×C412yes—
116#387—22no—
117#387—22no—
118#387—22no—
119#387—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#385—22no—
124#380—21no—
125#380—21no—
126#380C2×C412yes—
127#380C2×C412yes—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2×C412yes—
135C4×C4⋊C4C2312yes—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145#336—22no—
146#336—22no—
147Q8⋊C8—216no—
148C25C22⋊C411yescharacteristic
149C22×Q8—24no—
150C22×Q8—21no—
151C22×Q8—24no—
152C22×Q8—42no—
153C22×Q8C22×C412yes—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4SD1614yes—
178C23×C4Q1614yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C2×M16—28no—
189C2×M16—28no—
190C2×M16—432no—
191C2×M16—44no—
192C2×M16—44no—
193C2×M16—44no—
194C2×M16—44no—
195C22×C8—28no—
196C22×C8—24no—
197C22×C8—24no—
198C22×C8—44no—
199C22×C8—44no—
200C22⋊Q8—28no—
201C22⋊Q8—28no—
202C22⋊Q8—24no—
203C22⋊Q8—24no—
204C22⋊Q8—24no—
205C22⋊Q8M1618yescomplemented
206C4×Q8—28no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—42no—
210C4×Q8—44no—
211C4×Q8—42no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—44no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D414yes—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42#4114yes—
277C2×C42C2411yescharacteristic
278C4⋊C8—28no—
279C4⋊C8—28no—
280C4⋊C8—28no—
281C4⋊C8—24no—
282C4⋊C8—44no—
283C4⋊C8C2×D418yes—
284C22⋊C8—44no—
285C22⋊C8—44no—
286C22⋊C8—44no—
287C4×C8—24no—
288C4×C8—24no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—41no—
304C24—41no—
305C24#10314yes—
306C24C2×C22⋊C411yescharacteristic
307C24C2×C22⋊C411yescharacteristic
308C24C2×C22⋊C411yescharacteristic
309C2×Q8—24no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—416no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8C2×M1614yescomplemented
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—24no—
343C22×C4—24no—
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—22no—
362C22×C4—22no—
363C22×C4—22no—
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—41no—
374C22×C4—416no—
375C22×C4—416no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4#10314yes—
387C22×C4#10314yes—
388C22×C4#10314yes—
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×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C4×D414yes—
399C22×C4#12012yes—
400C22×C4#12114yes—
401C22×C4#12312yes—
402C22×C4C2×M1614yes—
403C22×C4C2×SD1612yes—
404C22×C4C2×SD1612yes—
405C22×C4C2×SD1612yes—
406C22×C4C2×Q1612yes—
407C22×C4C2×Q1612yes—
408C22×C4C2×Q1612yes—
409C22×C4AGL(1,ℤ/8)12yes—
410C22×C4#13712yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414M16—416no—
415M16—416no—
416C2×C8—28no—
417C2×C8—416no—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8—44no—
421C2×C8—44no—
422C2×C8—44no—
423C4⋊C4—24no—
424C4⋊C4—28no—
425C4⋊C4—22no—
426C4⋊C4—28no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4C2×M1614yescomplemented
436C22⋊C4—28no—
437C22⋊C4—28no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C42—22no—
441C42—22no—
442C42—21no—
443C42—21no—
444C42—22no—
445C42—22no—
446C42—21no—
447C42—21no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—42no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—44no—
459C42—44no—
460C42C4×D414yes—
461C42#12012yes—
462C42#12114yes—
463C42#12312yes—
464C42C22×D412yes—
465C42C2×#4112yes—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—28no—
478C23—21no—
479C23—21no—
480C23—28no—
481C23—21no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23#38612yes—
488C23#40712yes—
489C23C2×#10312yes—
490C23C2×#10312yes—
491C23C2×#10312yes—
492C23#41712yes—
493C23#42112yes—
494C23C22×C22⋊C411yescharacteristic
495Q8—28no—
496Q8—48no—
497Q8—44no—
498C2×C4—28no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—28no—
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—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4#38612yes—
558C2×C4#38612yes—
559C2×C4#38612yes—
560C2×C4#40712yes—
561C2×C4C2×#10312yes—
562C2×C4C2×#10312yes—
563C2×C4C2×#10312yes—
564C2×C4#41712yes—
565C2×C4#42112yes—
566C2×C4#45014yes—
567C2×C4#45114yes—
568C2×C4#48314yes—
569C2×C4#48414yes—
570C2×C4C2×C4×D412yes—
571C2×C4C2×#12011yescharacteristic
572C2×C4C2×#12112yes—
573C2×C4C2×#12311yescharacteristic
574C2×C4C22×M1612yes—
575C2×C4#56812yes—
576C2×C4C22×SD1611yescharacteristic
577C2×C4C22×Q1611yescharacteristic
578C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
579C2×C4C2×#13711yescharacteristic
580C8—48no—
581C8—84no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—416no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22#148714yescomplemented
603C22#187214yes—
604C22C2×#38611yescharacteristic
605C22C2×#40711yescharacteristic
606C22#287911yescharacteristic
607C22C22×#10311yescharacteristic
608C22C2×#41711yescharacteristic
609C22C2×#42111yescharacteristic
610C22C2×#42111yescharacteristic
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—44no—
619C4—41no—
620C4—44no—
621C4—41no—
622C4—82no—
623C4—82no—
624C4#187214yes—
625C4D4×M1614yes—
626C4C2×#45011yescharacteristic
627C4C2×#45111yescharacteristic
628C4C2×#48311yescharacteristic
629C4C2×#48411yescharacteristic
630C2—24no—
631C2—24no—
632C2—42no—
633C2C2×#148711yescharacteristic
634C2#1745111yescharacteristic
635C2#1748611yescharacteristic
636C2#1749211yescharacteristic
637C2#2089811yescharacteristic
638C2#2090211yescharacteristic
639C2C2×#187211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1487C2semidirect
5. #8942C2semidirect
6. #8937C2semidirect
7. #7901C2semidirect
43. C2×Q8⋊C8C22semidirect
44. #1279C22semidirect
205. C22⋊Q8M16semidirect
316. C2×Q8C2×M16semidirect
435. C4⋊C4C2×M16semidirect
602. C22#1487semidirect

Maximal subgroups

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

subgroupindex
C22×#4232
C2×C4×C22⋊Q82
#207142
C2×#14872
#89422
#89372
#79012