Finite Group Databasebeta

← #188485

Subgroups — #188485

Order 512.

1574 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic
3C2×#1520C212yesmaximal
4#14686C211yesmaximal, characteristic
5#8578C214yesmaximal, complemented
6#8572C214yesmaximal, complemented
7#8023C212yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10C23×C42C2211yescharacteristic
11C22×C4⋊C8—21no—
12C22×C22⋊C8C2212yes—
13C4×C22⋊Q8—21no—
14C4×C22⋊Q8—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8C412yescomplemented
18C4×C22⋊Q8C412yescomplemented
19C4×C22⋊Q8C2212yes—
20C4×C22⋊Q8C2212yes—
21C2×#391C411yescharacteristic, complemented
22C2×#389C411yescharacteristic, complemented
23C2×#387—21no—
24C2×#385—21no—
25C2×#384—21no—
26C2×#380—21no—
27C2×#380—21no—
28C2×#380C2212yes—
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C411yescharacteristic, complemented
32C2×C4×C22⋊C4—21no—
33#1797—21no—
34#1797—21no—
35#1797C2212yes—
36#1797C2212yes—
37C2×#336—21no—
38#1520—28no—
39C2×#328C2218yes—
40C4⋊#95—21no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43#1294C2218yes—
44C23×Q8C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8—22no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8C2×C412yes—
54C2×C22⋊Q8D414yes—
55C2×C22⋊Q8Q814yes—
56C2×C22⋊Q8C2312yes—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—44no—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C412yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C42—21no—
75C22×C42—21no—
76C22×C42—44no—
77C22×C42C2×C411yescharacteristic
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
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—44no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8C2314yes—
96#391—21no—
97#391—21no—
98#391C2×C412yes—
99#391C2×C412yes—
100#389—21no—
101#389—21no—
102#389C2×C412yes—
103#389C2×C412yes—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#385—22no—
109#385—22no—
110#385—22no—
111#385—22no—
112#384—44no—
113#380—21no—
114#380—21no—
115#380—24no—
116#380—24no—
117#380—44no—
118#380—44no—
119#380C2×C412yes—
120#380C2312yes—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C412yes—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95C2314yes—
141#336—22no—
142#336—22no—
143#328—216no—
144C25C2×D411yescharacteristic
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8C4⋊C414yescomplemented
150C22×Q8C22×C412yes—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—24no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4SD1614yes—
173C23×C4Q1614yes—
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2×Q812yes—
185C23×C4#4112yes—
186C23×C4#4112yes—
187C23×C4C2411yescharacteristic
188C22×C8—24no—
189C22×C8—44no—
190C22×C8—44no—
191C22×C8—44no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C22⋊Q8—24no—
195C22⋊Q8—24no—
196C22⋊Q8—24no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8C4⋊C418yescomplemented
200C22⋊Q8C2×D414yes—
201C22⋊Q8C2×Q814yes—
202C4×Q8—44no—
203C4×Q8—44no—
204C4×Q8—44no—
205C4×Q8—44no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C4⋊C414yescomplemented
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C412yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C4⋊C8—44no—
281C4⋊C8—44no—
282C22⋊C8—24no—
283C22⋊C8—24no—
284C22⋊C8—44no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—24no—
297#95—44no—
298#95—44no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24C2×C22⋊C411yescharacteristic
304C24C2×C22⋊C411yescharacteristic
305C24#12112yes—
306C24C22⋊Q812yes—
307C24C2×SD1612yes—
308C24C2×Q1612yes—
309C24AGL(1,ℤ/8)12yes—
310C24#13712yes—
311C24C22×D411yescharacteristic
312C2×Q8—22no—
313C2×Q8—24no—
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8C2×C4⋊C414yes—
321C22×C4—21no—
322C22×C4—21no—
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—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—24no—
342C22×C4—24no—
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—48no—
354C22×C4—48no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
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—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4#10314yes—
397C22×C4#10314yes—
398C22×C4#10314yes—
399C22×C4#10314yes—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C4⋊C412yes—
411C22×C4C2×C4⋊C412yes—
412C22×C4#11712yes—
413C22×C4#11712yes—
414C22×C4#12112yes—
415C22×C4#12112yes—
416C22×C4#12112yes—
417C22×C4C22⋊Q812yes—
418C22×C4C22⋊Q812yes—
419C22×C4C22⋊Q812yes—
420C22×C4C2×SD1612yes—
421C22×C4C2×SD1612yes—
422C22×C4C2×Q1612yes—
423C22×C4C2×Q1612yes—
424C22×C4C23×C411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×Q811yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C2×C8—44no—
431C2×C8—44no—
432C2×C8—44no—
433C2×C8—88no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—24no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—88no—
448C4⋊C4—88no—
449C4⋊C4—88no—
450C4⋊C4C2×C4⋊C414yes—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—28no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—44no—
457C22⋊C4—88no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—41no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—41no—
472C42—41no—
473C42—44no—
474C42—44no—
475C42—44no—
476C42—44no—
477C42#9914yes—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23#38012yes—
503C23#38012yes—
504C23C2×#10312yes—
505C23C2×#10312yes—
506C23#41712yes—
507C23#41712yes—
508C23#46014yes—
509C23#46214yes—
510C23#47514yes—
511C23#47714yes—
512C23C22×C22⋊C411yescharacteristic
513C23C2×#12111yescharacteristic
514C23C2×C22⋊Q811yescharacteristic
515C23C22×SD1611yescharacteristic
516C23C22×Q1611yescharacteristic
517C23C2×AGL(1,ℤ/8)11yescharacteristic
518C23C2×#13711yescharacteristic
519Q8—44no—
520Q8—44no—
521Q8—48no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4#38012yes—
598C2×C4#38012yes—
599C2×C4C2×#9912yes—
600C2×C4C2×#9912yes—
601C2×C4C2×#9912yes—
602C2×C4#41012yes—
603C2×C4C2×#10312yes—
604C2×C4C2×#10312yes—
605C2×C4C2×#10312yes—
606C2×C4C2×#10312yes—
607C2×C4#42112yes—
608C2×C4#42112yes—
609C2×C4C22×C4⋊C411yescharacteristic
610C2×C4C2×#11711yescharacteristic
611C2×C4C2×#12111yescharacteristic
612C2×C4C2×C22⋊Q811yescharacteristic
613C8—84no—
614C22—22no—
615C22—21no—
616C22—21no—
617C22—22no—
618C22—21no—
619C22—21no—
620C22—22no—
621C22—22no—
622C22—21no—
623C22—21no—
624C22—22no—
625C22—22no—
626C22—21no—
627C22—22no—
628C22—21no—
629C22—22no—
630C22—22no—
631C22—21no—
632C22—21no—
633C22—22no—
634C22—21no—
635C22—21no—
636C22—22no—
637C22—22no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22#180314yes—
643C22#180314yes—
644C22C2×#38011yescharacteristic
645C22C22×#9911yescharacteristic
646C22C2×#41011yescharacteristic
647C22C22×#10311yescharacteristic
648C22C2×#41711yescharacteristic
649C22C2×#42111yescharacteristic
650C22C2×#42111yescharacteristic
651C22C2×#46011yescharacteristic
652C22C2×#46211yescharacteristic
653C22C2×#47511yescharacteristic
654C22C2×#47711yescharacteristic
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—44no—
666C4—44no—
667C4—41no—
668C4—41no—
669C4—82no—
670C4—82no—
671C4—164no—
672C4#152014yescomplemented
673C4#179914yes—
674C2—21no—
675C2—21no—
676C2—21no—
677C2—24no—
678C2—24no—
679C2—21no—
680C2C2×#152011yescharacteristic
681C2#1757511yescharacteristic
682C2#1758111yescharacteristic
683C2#1759211yescharacteristic
684C2C2×#179911yescharacteristic
685C2C2×#180311yescharacteristic
686C2#2065411yescharacteristic
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
5. #8578C2semidirect
6. #8572C2semidirect
9. C22×C4×Q8C4semidirect
17. C4×C22⋊Q8C4semidirect
18. C4×C22⋊Q8C4semidirect
21. C2×#391C4semidirect
22. C2×#389C4semidirect
31. C2×C4×C4⋊C4C4semidirect
149. C22×Q8C4⋊C4semidirect
199. C22⋊Q8C4⋊C4semidirect
231. C2×C4⋊C4C4⋊C4semidirect
672. C4#1520semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C2×#17972
C2×#15202
#146862
#85782
#85722
#80232