Finite Group Databasebeta

← #204552

Subgroups — #204552

Order 512.

1097 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34972C211yesmaximal, characteristic
2#20809C211yesmaximal, characteristic, complemented
3#20550C211yesmaximal, characteristic, complemented
4C2×#1495C211yesmaximal, characteristic
5C2×#1482C211yesmaximal, characteristic
6C2×#1487C211yesmaximal, characteristic, complemented
7C2×#1479C211yesmaximal, characteristic, complemented
8#9677C212yesmaximal, complemented
9#9676C212yesmaximal, complemented
10#9675C212yesmaximal, complemented
11#9673C212yesmaximal, complemented
12C2×#529C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#423C2211yescharacteristic
18#2604—21no—
19#2604—21no—
20#2565C412yes—
21#2545C412yes—
22#2503C411yescharacteristic
23#2501C411yescharacteristic
24#2436C411yescharacteristic
25#2434C411yescharacteristic
26#2413—22no—
27#2396—21no—
28#2396—21no—
29C4×C4⋊Q8C2212yes—
30C4×#124C2212yes—
31C4×#123—22no—
32C4×#117C2211yescharacteristic
33#1843—21no—
34#1832—21no—
35#1798—22no—
36#1770—21no—
37C8×C4⋊C4—21no—
38#1767C2212yes—
39C4×C4⋊C8C2212yes—
40C4×C4⋊C8C2212yes—
41C4×C4⋊C8C2212yes—
42#1753—22no—
43#1495—24no—
44#1487—24no—
45#1482—24no—
46#1479—24no—
47C2×Q8⋊C8C2212yes—
48C2×Q8⋊C8C2212yes—
49C2×Q8⋊C8C2212yes—
50C2×Q8⋊C8C2212yes—
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×Q8⋊C8C2212yes—
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2212yes—
57C23×Q8C2×C411yescharacteristic
58C22×M16—21no—
59C2×C4⋊Q8C2×C412yes—
60#529—22no—
61#529—22no—
62#529—22no—
63#529—22no—
64C2×#126—21no—
65C2×#125C2×C411yescharacteristic
66C2×#124C2×C411yescharacteristic
67C2×#124C2×C412yes—
68C2×#123—21no—
69C2×#123—22no—
70C2×C22⋊Q8C2×C411yescharacteristic
71C2×#121C2×C411yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8C2×C412yes—
79C2×C4×Q8C2311yescharacteristic
80C2×C4×Q8C2312yes—
81C2×C4×D4—21no—
82C2×C4×D4C2311yescharacteristic
83C2×#117—21no—
84C2×#117D412yes—
85C22×C4⋊C4—21no—
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#431—22no—
90#431—22no—
91#431—22no—
92#431—22no—
93#431—22no—
94#431D414yes—
95#423—22no—
96#423D414yes—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8C2312yes—
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×#97—22no—
109C2×C4×C8—21no—
110C2×C4×C8—22no—
111C2×C4×C8C2312yes—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#400—21no—
115#400—21no—
116#399—21no—
117#399—21no—
118#398—22no—
119#398—22no—
120#396—22no—
121#396—22no—
122#395—21no—
123#395—21no—
124#393—21no—
125#393—21no—
126#391C2×C412yes—
127#391C2×C412yes—
128#391C2×C412yes—
129#390C2×C412yes—
130#389—22no—
131#389—22no—
132#388—22no—
133#388—22no—
134#388—22no—
135#388—22no—
136#387—22no—
137#387—22no—
138#386—22no—
139#385—22no—
140#384C2×C412yes—
141#382C2×C412yes—
142#381—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4C2×C412yes—
146C4×C4⋊C4D411yescharacteristic
147C4×C4⋊C4D411yescharacteristic
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150C4×C22⋊C4C2×C411yescharacteristic
151C4×C22⋊C4C2×C411yescharacteristic
152C4×C22⋊C4D412yes—
153#376D411yescharacteristic
154#376D411yescharacteristic
155C43C2312yes—
156#336—22no—
157#336—22no—
158#336—22no—
159#336—22no—
160Q8⋊C8—24no—
161Q8⋊C8—24no—
162Q8⋊C8—22no—
163Q8⋊C8—22no—
164Q8⋊C8—24no—
165Q8⋊C8—24no—
166Q8⋊C8—24no—
167D4⋊C8—22no—
168D4⋊C8—22no—
169D4⋊C8—24no—
170C22×Q8—21no—
171C22×Q8—24no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—42no—
175C22×Q8C22×C411yescharacteristic
176C22×Q8C22×C412yes—
177C22×D4—21no—
178C22×D4C22×C411yescharacteristic
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C2×M16—24no—
186C2×M16—81no—
187C2×M16—81no—
188C22×C8—22no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—21no—
194C4⋊Q8—24no—
195C4⋊Q8—42no—
196#126—42no—
197#126—42no—
198#125—22no—
199#125—22no—
200#124—24no—
201#124—22no—
202#124—22no—
203#124—42no—
204#123—48no—
205#123—42no—
206#123—42no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209#121—22no—
210#121—22no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—42no—
222C4×Q8—42no—
223C4×Q8—42no—
224C4×Q8—42no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—42no—
230C4×D4—42no—
231#117—44no—
232#117—42no—
233#117—42no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C22⋊C412yes—
250C2×C4⋊C4C22⋊C412yes—
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
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—22no—
266C2×C42—21no—
267C2×C42—22no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42C22×C412yes—
271C2×C42C22×C412yes—
272C2×C42C22×C411yescharacteristic
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42#4112yes—
282C2×C42#4112yes—
283C2×C42C2411yescharacteristic
284C4⋊C8—24no—
285C4⋊C8—22no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—44no—
289C4⋊C8—44no—
290C4⋊C8—42no—
291C4⋊C8—42no—
292C4⋊C8C2×D414yes—
293C4⋊C8C2×D412yes—
294C4⋊C8C2×D412yes—
295C22⋊C8—24no—
296C22⋊C8—42no—
297C22⋊C8—42no—
298#97—82no—
299C4×C8—24no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—22no—
303C4×C8—22no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—82no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95C22⋊C411yescharacteristic
316#95C22⋊C411yescharacteristic
317#95C22⋊C411yescharacteristic
318#95C22⋊C411yescharacteristic
319#95C22⋊C412yes—
320#95C22⋊C412yes—
321C24—21no—
322C24C2×C22⋊C411yescharacteristic
323C2×Q8—22no—
324C2×Q8—28no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—44no—
334C2×Q8—48no—
335C2×Q8—44no—
336C2×Q8—42no—
337C2×Q8#13114yes—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—44no—
343C2×D4—42no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C412yes—
373C22×C4C4×D412yes—
374C22×C4C4×D412yes—
375C22×C4#12112yes—
376C22×C4#12112yes—
377C22×C4#12412yes—
378C22×C4#12712yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382M16—84no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—42no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—42no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C22⋊C4—24no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—42no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—44no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—24no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—41no—
445C42—41no—
446C42—44no—
447C42C4×D412yes—
448C42C4×D412yes—
449C42#12112yes—
450C42#12112yes—
451C42#12412yes—
452C42#12712yes—
453C42C22×D411yescharacteristic
454C42C22×D411yescharacteristic
455C42C2×#4111yescharacteristic
456C42C2×#4111yescharacteristic
457C23—22no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—41no—
466C23#39012yes—
467C23C22×C22⋊C411yescharacteristic
468Q8—24no—
469Q8—24no—
470Q8—48no—
471Q8—42no—
472Q8—42no—
473Q8—41no—
474Q8—41no—
475Q8—42no—
476D4—41no—
477D4—41no—
478D4—42no—
479D4—84no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—84no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4#39012yes—
525C2×C4#39012yes—
526C2×C4#39012yes—
527C2×C4#42114yes—
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×C4×D411yescharacteristic
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×#12111yescharacteristic
532C2×C4C2×#12411yescharacteristic
533C2×C4C2×#12711yescharacteristic
534C2×C4C2×#13112yes—
535C2×C4#56812yes—
536C2×C4#57711yescharacteristic
537C2×C4#57711yescharacteristic
538C2×C4#57811yescharacteristic
539C2×C4#57811yescharacteristic
540C8—44no—
541C8—42no—
542C8—42no—
543C8—81no—
544C8—81no—
545C8—82no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—82no—
554C22—82no—
555C22—81no—
556C22C2×#39011yescharacteristic
557C22#288012yes—
558C22#289811yescharacteristic
559C22#289811yescharacteristic
560C22C2×#42112yes—
561C4—21no—
562C4—21no—
563C4—22no—
564C4—22no—
565C4—24no—
566C4—22no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—44no—
571C4—82no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4#292911yescharacteristic
576C4#293011yescharacteristic
577C4#293312yes—
578C4#304211yescharacteristic
579C4#304411yescharacteristic
580C4#304611yescharacteristic
581C4#304611yescharacteristic
582C2—42no—
583C2—161no—
584C2#1752412yes—
585C2#1752512yes—
586C2#2119111yescharacteristic
587C2#2125211yescharacteristic
588C2#2126211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#121C2×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. #20550C2semidirect
6. C2×#1487C2semidirect
7. C2×#1479C2semidirect
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
#349722
#208092
#205502
C2×#14952
C2×#14822
C2×#14872
C2×#14792
#96772
#96762
#96752
#96732