Finite Group Databasebeta

← #204544

Subgroups — #204544

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#536C211yesmaximal, characteristic
2#21350C211yesmaximal, characteristic, complemented
3#20549C211yesmaximal, characteristic, complemented
4C2×#1489C211yesmaximal, characteristic
5C2×#1491C211yesmaximal, characteristic, complemented
6C2×#1488C211yesmaximal, characteristic
7C2×#1487C211yesmaximal, characteristic, complemented
8#9677C212yesmaximal, complemented
9#9676C212yesmaximal, complemented
10#9675C212yesmaximal, complemented
11#9673C212yesmaximal, complemented
12C2×#536C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19#2327C411yescharacteristic
20#2327C411yescharacteristic
21#2326C411yescharacteristic
22#2326C411yescharacteristic
23#2323—22no—
24#2322—22no—
25#2317C411yescharacteristic
26#2311C411yescharacteristic
27#2310C411yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×C4⋊Q8C2211yescharacteristic
30C4×#124C2211yescharacteristic
31C4×#124C2211yescharacteristic
32C4×C22⋊Q8—22no—
33C4×#123—22no—
34C4×#117C2211yescharacteristic
35#1940—22no—
36#1851—22no—
37#1771—22no—
38#1767C2212yes—
39C4×C4⋊C8C2212yes—
40C4×C4⋊C8C2212yes—
41C4×C4⋊C8C2212yes—
42#1753—22no—
43#1491—24no—
44#1489—24no—
45#1488—24no—
46#1487—24no—
47C2×Q8⋊C8C2212yes—
48C2×Q8⋊C8C2212yes—
49C2×Q8⋊C8C2212yes—
50C2×Q8⋊C8C2212yes—
51C2×Q8⋊C8C2212yes—
52C2×Q8⋊C8C2212yes—
53C2×D4⋊C8C2212yes—
54C2×D4⋊C8C2212yes—
55C22×#41C2×C411yescharacteristic
56C23×Q8C2×C411yescharacteristic
57C22×M16—21no—
58C22×M16—21no—
59#536—24no—
60#536—24no—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×#124C2×C411yescharacteristic
64C2×#124C2×C411yescharacteristic
65C2×#123—22no—
66C2×C22⋊Q8—22no—
67C4×#41—22no—
68C4×#41—22no—
69C4×#41—22no—
70C4×#41—22no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×Q8C2×C411yescharacteristic
79C2×C4×Q8C2×C411yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82C2×C4×Q8C2311yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2311yescharacteristic
87C2×#117—21no—
88C2×#117C2×C411yescharacteristic
89C2×#117C2×C411yescharacteristic
90C2×#117C2×C411yescharacteristic
91C22×C4⋊C4—21no—
92C22×C4⋊C4C2×C411yescharacteristic
93C22×C42—21no—
94C22×C42—21no—
95C22×C42C2311yescharacteristic
96#432—22no—
97#432D414yes—
98#431—22no—
99#431—22no—
100#423—22no—
101#423D414yes—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2312yes—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C4×M16—22no—
111C4×M16—22no—
112C2×#97—21no—
113C2×#97—21no—
114C2×#97—22no—
115C2×C4×C8—22no—
116C2×C4×C8C2312yes—
117C2×C4×C8C2312yes—
118#391—22no—
119#391C2×C411yescharacteristic
120#391C2×C411yescharacteristic
121#391C2×C411yescharacteristic
122#391C2×C411yescharacteristic
123#391C2×C411yescharacteristic
124#391C2×C411yescharacteristic
125#390C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#389—22no—
128#389—22no—
129#389—22no—
130#388—22no—
131#388—22no—
132#388—22no—
133#388—22no—
134#387—22no—
135#387—22no—
136#387—22no—
137#387—22no—
138#386—22no—
139#385—22no—
140#385—22no—
141#385—22no—
142#384C2×C411yescharacteristic
143#384C2×C411yescharacteristic
144#382C2×C411yescharacteristic
145#382C2×C411yescharacteristic
146#381—22no—
147#380—22no—
148C4×C4⋊C4—22no—
149C4×C4⋊C4—22no—
150C4×C4⋊C4D412yes—
151C4×C4⋊C4D412yes—
152C4×C22⋊C4—22no—
153C4×C22⋊C4—22no—
154C4×C22⋊C4D412yes—
155C4×C22⋊C4D412yes—
156#376C2×C411yescharacteristic
157#376C2×C411yescharacteristic
158C43C2311yescharacteristic
159C43C2311yescharacteristic
160#336—22no—
161#336—22no—
162#336—22no—
163#336—22no—
164Q8⋊C8—24no—
165Q8⋊C8—24no—
166Q8⋊C8—24no—
167Q8⋊C8—24no—
168Q8⋊C8—24no—
169Q8⋊C8—24no—
170D4⋊C8—24no—
171D4⋊C8—24no—
172C2×#41—22no—
173C2×#41—22no—
174C2×#41—22no—
175C2×#41—22no—
176C2×#41—41no—
177C2×#41—41no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—21no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—41no—
184C22×Q8—41no—
185C22×Q8C22×C411yescharacteristic
186C22×Q8C22×C411yescharacteristic
187C22×Q8C22×C411yescharacteristic
188C22×D4—21no—
189C22×D4C22×C411yescharacteristic
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4C22×C411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C2×M16—24no—
199C2×M16—44no—
200C2×M16—42no—
201C2×M16—82no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—22no—
208C4⋊Q8—22no—
209C4⋊Q8M1614yescomplemented
210#124—22no—
211#124—22no—
212#124—22no—
213#124M1614yescomplemented
214#123—216no—
215C22⋊Q8—216no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8—42no—
235C4×Q8—42no—
236C4×Q8—42no—
237C4×Q8—42no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—42no—
245C4×D4—42no—
246C4×D4—42no—
247C4×D4—42no—
248#117—22no—
249#117—22no—
250#117—22no—
251#117—22no—
252#117—41no—
253#117—41no—
254#117—42no—
255#117—42no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4C22⋊C412yes—
272C2×C4⋊C4C22⋊C412yes—
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C411yescharacteristic
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4C22⋊C412yes—
285C2×C22⋊C4C22⋊C412yes—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C22×C411yescharacteristic
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42#4112yes—
311C2×C42#4112yes—
312C2×C42C2411yescharacteristic
313C4⋊C8—24no—
314C4⋊C8—22no—
315C4⋊C8—22no—
316C4⋊C8—44no—
317C4⋊C8—44no—
318C4⋊C8—42no—
319C4⋊C8C2×D414yes—
320C4⋊C8C2×D414yes—
321C22⋊C8—24no—
322C22⋊C8—42no—
323#97—24no—
324#97—42no—
325#97—82no—
326C4×C8—22no—
327C4×C8—22no—
328C4×C8—22no—
329C4×C8—22no—
330C4×C8—82no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95C22⋊C412yes—
340#95C22⋊C412yes—
341#95C22⋊C412yes—
342#95C22⋊C412yes—
343C24—21no—
344C24C2×C22⋊C411yescharacteristic
345#41—44no—
346#41—44no—
347#41—44no—
348#41—44no—
349C2×Q8—24no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8C2×M1612yes—
372C2×Q8C2×M1612yes—
373C2×Q8C2×M1612yes—
374C2×D4—24no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4C2×M1612yes—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C22⋊C412yes—
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C412yes—
418C22×C4C4×D412yes—
419C22×C4C4×D412yes—
420C22×C4#12112yes—
421C22×C4#12112yes—
422C22×C4#12412yes—
423C22×C4#12712yes—
424C22×C4C23×C411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427M16—48no—
428M16—84no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C22⋊C4—28no—
466C22⋊C4—28no—
467C22⋊C4—24no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—42no—
473C22⋊C4—82no—
474C22⋊C4—82no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—42no—
486C42—42no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—42no—
494C42—42no—
495C42—42no—
496C42—42no—
497C42C4×D412yes—
498C42C4×D412yes—
499C42#12112yes—
500C42#12112yes—
501C42#12412yes—
502C42#12712yes—
503C42C2×M1612yes—
504C42C2×M1612yes—
505C42C22×D411yescharacteristic
506C42C22×D411yescharacteristic
507C42C2×#4111yescharacteristic
508C42C2×#4111yescharacteristic
509C23—21no—
510C23—21no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23#39012yes—
519C23C22×C22⋊C411yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8—22no—
524Q8—22no—
525Q8—22no—
526Q8—48no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—41no—
531Q8—41no—
532Q8—41no—
533D4—22no—
534D4—22no—
535D4—48no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—82no—
587C2×C4#39012yes—
588C2×C4#39012yes—
589C2×C4#39012yes—
590C2×C4#40714yes—
591C2×C4C2×C4×D411yescharacteristic
592C2×C4C2×C4×D411yescharacteristic
593C2×C4C2×#12111yescharacteristic
594C2×C4C2×#12111yescharacteristic
595C2×C4C2×#12411yescharacteristic
596C2×C4C2×#12711yescharacteristic
597C2×C4C22×M1611yescharacteristic
598C2×C4C22×M1611yescharacteristic
599C2×C4#56811yescharacteristic
600C2×C4#56811yescharacteristic
601C2×C4#57712yes—
602C2×C4#57812yes—
603C8—44no—
604C8—44no—
605C8—44no—
606C8—82no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—82no—
617C22C2×#39011yescharacteristic
618C22C2×#40711yescharacteristic
619C22#287911yescharacteristic
620C22#289812yes—
621C22#290112yes—
622C4—21no—
623C4—22no—
624C4—21no—
625C4—22no—
626C4—24no—
627C4—22no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—81no—
635C4—81no—
636C4—81no—
637C4—81no—
638C4D4×M1611yescharacteristic
639C4D4×M1611yescharacteristic
640C4D4×M1611yescharacteristic
641C4D4×M1611yescharacteristic
642C4#304211yescharacteristic
643C4#304411yescharacteristic
644C4#304612yes—
645C2—42no—
646C2—82no—
647C2#1745412yes—
648C2#1750612yes—
649C2#2119011yescharacteristic
650C2#2125311yescharacteristic
651C2#2126211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21350C2semidirect
3. #20549C2semidirect
5. C2×#1491C2semidirect
7. C2×#1487C2semidirect
8. #9677C2semidirect
9. #9676C2semidirect
10. #9675C2semidirect
11. #9673C2semidirect
209. C4⋊Q8M16semidirect
213. #124M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5362
#213502
#205492
C2×#14892
C2×#14912
C2×#14882
C2×#14872
#96772
#96762
#96752
#96732