Finite Group Databasebeta

← #204550

Subgroups — #204550

Order 512.

1100 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34979C211yesmaximal, characteristic
2#20809C211yesmaximal, characteristic, complemented
3#20546C211yesmaximal, characteristic, complemented
4C2×#1497C211yesmaximal, characteristic, complemented
5C2×#1481C211yesmaximal, characteristic, complemented
6C2×#1488C211yesmaximal, characteristic
7C2×#1477C211yesmaximal, characteristic
8#9677C212yesmaximal, complemented
9#9676C212yesmaximal, complemented
10#9675C212yesmaximal, complemented
11#9673C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18#2611—21no—
19#2611—21no—
20#2568C411yescharacteristic
21#2547C411yescharacteristic
22#2546C411yescharacteristic
23#2542C411yescharacteristic
24#2499C412yes—
25#2432C412yes—
26#2415—22no—
27#2395—21no—
28#2395—21no—
29C4×C4⋊Q8C2212yes—
30C4×#124C2212yes—
31C4×C22⋊Q8—22no—
32C4×#117C2211yescharacteristic
33#1843—21no—
34#1832—21no—
35#1798—22no—
36#1771—22no—
37#1767C2212yes—
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2212yes—
40C4×C4⋊C8C2212yes—
41#1752—21no—
42C8×C22⋊C4—21no—
43#1497—24no—
44#1488—24no—
45#1481—24no—
46#1477—24no—
47C2×Q8⋊C8C2211yescharacteristic
48C2×Q8⋊C8C2211yescharacteristic
49C2×Q8⋊C8C2212yes—
50C2×Q8⋊C8C2212yes—
51C2×Q8⋊C8C2212yes—
52C2×Q8⋊C8C2212yes—
53C2×Q8⋊C8C2212yes—
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2212yes—
57C22×#41C2×C411yescharacteristic
58C23×C8—21no—
59#533—22no—
60#533—22no—
61#533—22no—
62#533—22no—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C412yes—
65C2×C4⋊Q8C2×C411yescharacteristic
66C2×#125—21no—
67C2×#124C2×C412yes—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8C2×C412yes—
71C4×#41—22no—
72C4×#41—22no—
73C4×#41—22no—
74C4×#41—22no—
75C2×C4×Q8—21no—
76C2×C4×Q8C2×C411yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C2×C4×Q8C2312yes—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C2×#117—21no—
84C2×#117—21no—
85C2×#117C2×C411yescharacteristic
86C2×#117D412yes—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#432—22no—
90#432D414yes—
91#431—22no—
92#431—22no—
93#431—22no—
94#431—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8D414yes—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2312yes—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×#97—21no—
109C2×#97—22no—
110C2×C4×C8—22no—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2312yes—
114#400—21no—
115#400—21no—
116#399—21no—
117#399—21no—
118#398—21no—
119#398—21no—
120#397—22no—
121#397—22no—
122#397—21no—
123#397—21no—
124#394—22no—
125#394—22no—
126#391—22no—
127#391C2×C412yes—
128#391C2×C412yes—
129#391C2×C412yes—
130#390C2×C412yes—
131#389—22no—
132#389—22no—
133#389—22no—
134#388—22no—
135#388—22no—
136#387—22no—
137#387—22no—
138#385—22no—
139#385—22no—
140#384C2×C412yes—
141#382C2×C412yes—
142#380—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4C2×C411yescharacteristic
146C4×C4⋊C4C2×C411yescharacteristic
147C4×C4⋊C4D412yes—
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150C4×C22⋊C4C2×C412yes—
151C4×C22⋊C4D411yescharacteristic
152C4×C22⋊C4D411yescharacteristic
153#376D411yescharacteristic
154#376D411yescharacteristic
155C43C2312yes—
156#336—22no—
157#336—22no—
158#336—22no—
159#336—22no—
160Q8⋊C8—22no—
161Q8⋊C8—22no—
162Q8⋊C8—24no—
163Q8⋊C8—24no—
164Q8⋊C8—24no—
165Q8⋊C8—24no—
166Q8⋊C8—24no—
167D4⋊C8—22no—
168D4⋊C8—22no—
169D4⋊C8—24no—
170C2×#41—22no—
171C2×#41—22no—
172C2×#41—22no—
173C2×#41—22no—
174C2×#41—41no—
175C2×#41—41no—
176C22×Q8—21no—
177C22×Q8C22×C412yes—
178C22×Q8C22×C411yescharacteristic
179C22×D4—21no—
180C22×D4C22×C411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4C22×C411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—24no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—21no—
194C22×C8—82no—
195C4⋊Q8—24no—
196C4⋊Q8—22no—
197C4⋊Q8—22no—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200C4⋊Q8—42no—
201#125—42no—
202#125—42no—
203#124—24no—
204#124—42no—
205C22⋊Q8—24no—
206C22⋊Q8—24no—
207C22⋊Q8—48no—
208C22⋊Q8—42no—
209C22⋊Q8—42no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—42no—
221C4×Q8—42no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—42no—
229C4×D4—42no—
230C4×D4—42no—
231C4×D4—42no—
232#117—22no—
233#117—22no—
234#117—44no—
235#117—42no—
236#117—42no—
237#117—42no—
238#117—42no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C22⋊C412yes—
251C2×C4⋊C4C22⋊C412yes—
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C412yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4C22⋊C412yes—
264C2×C22⋊C4C22⋊C412yes—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—22no—
268C2×C42—21no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42C22×C412yes—
272C2×C42C22×C412yes—
273C2×C42C22×C411yescharacteristic
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42#4112yes—
283C2×C42#4112yes—
284C2×C42C2411yescharacteristic
285C4⋊C8—24no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—22no—
289C4⋊C8—44no—
290C4⋊C8—44no—
291C4⋊C8—42no—
292C4⋊C8—42no—
293C4⋊C8C2×D412yes—
294C4⋊C8C2×D412yes—
295C4⋊C8C2×D414yes—
296C22⋊C8—24no—
297C22⋊C8—42no—
298C22⋊C8—42no—
299#97—24no—
300#97—82no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—22no—
304C4×C8—22no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—82no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95C22⋊C412yes—
317#95C22⋊C412yes—
318#95C22⋊C411yescharacteristic
319#95C22⋊C411yescharacteristic
320#95C22⋊C411yescharacteristic
321#95C22⋊C411yescharacteristic
322C24—21no—
323C24C2×C22⋊C411yescharacteristic
324#41—28no—
325#41—44no—
326#41—44no—
327#41—44no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—21no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—44no—
340C2×Q8—42no—
341C2×Q8#13112yes—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4#13112yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C412yes—
379C22×C4C4×D412yes—
380C22×C4C4×D412yes—
381C22×C4#12112yes—
382C22×C4#12112yes—
383C22×C4#12412yes—
384C22×C4#12712yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C2×C8—24no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—42no—
402C2×C8—84no—
403C4⋊C4—24no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C22⋊C4—24no—
425C22⋊C4—42no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—42no—
434C22⋊C4—44no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—24no—
444C42—42no—
445C42—41no—
446C42—41no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42C4×D412yes—
452C42C4×D412yes—
453C42#12112yes—
454C42#12112yes—
455C42#12412yes—
456C42#12712yes—
457C42C22×D411yescharacteristic
458C42C22×D411yescharacteristic
459C42C2×#4111yescharacteristic
460C42C2×#4111yescharacteristic
461C23—21no—
462C23—21no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—41no—
470C23#39012yes—
471C23C22×C22⋊C411yescharacteristic
472Q8—22no—
473Q8—22no—
474Q8—41no—
475Q8—41no—
476Q8—42no—
477Q8—42no—
478Q8—42no—
479Q8—42no—
480Q8—84no—
481D4—22no—
482D4—22no—
483D4—48no—
484D4—41no—
485D4—41no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—48no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—84no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4#39012yes—
530C2×C4#39012yes—
531C2×C4#39012yes—
532C2×C4#42114yes—
533C2×C4C2×C4×D411yescharacteristic
534C2×C4C2×C4×D411yescharacteristic
535C2×C4C2×#12111yescharacteristic
536C2×C4C2×#12111yescharacteristic
537C2×C4C2×#12411yescharacteristic
538C2×C4C2×#12711yescharacteristic
539C2×C4C2×#13111yescharacteristic
540C2×C4C2×#13111yescharacteristic
541C2×C4#56811yescharacteristic
542C2×C4#56811yescharacteristic
543C2×C4#57712yes—
544C2×C4#57812yes—
545C8—42no—
546C8—42no—
547C8—44no—
548C8—81no—
549C8—81no—
550C8—82no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—42no—
555C22—42no—
556C22—44no—
557C22—42no—
558C22—81no—
559C22—81no—
560C22—82no—
561C22C2×#39011yescharacteristic
562C22#288011yescharacteristic
563C22#288011yescharacteristic
564C22#289812yes—
565C22C2×#42112yes—
566C4—21no—
567C4—22no—
568C4—21no—
569C4—22no—
570C4—24no—
571C4—22no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—81no—
577C4—82no—
578C4—81no—
579C4—82no—
580C4#292911yescharacteristic
581C4#293011yescharacteristic
582C4#293311yescharacteristic
583C4#293311yescharacteristic
584C4#304211yescharacteristic
585C4#304411yescharacteristic
586C4#304612yes—
587C2—42no—
588C2—82no—
589C2#1752312yes—
590C2#1752412yes—
591C2#2119111yescharacteristic
592C2#2125211yescharacteristic
593C2#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. #20809C2semidirect
3. #20546C2semidirect
4. C2×#1497C2semidirect
5. C2×#1481C2semidirect
8. #9677C2semidirect
9. #9676C2semidirect
10. #9675C2semidirect
11. #9673C2semidirect

Maximal subgroups

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

subgroupindex
#349792
#208092
#205462
C2×#14972
C2×#14812
C2×#14882
C2×#14772
#96772
#96762
#96752
#96732