Finite Group Databasebeta

← #210457

Subgroups — #210457

Order 512.

777 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22238C211yesmaximal, characteristic, complemented
2C4×#461C211yesmaximal, characteristic
3#20555C211yesmaximal, characteristic
4C2×SD16⋊C8C211yesmaximal, characteristic
5C2×C8×SD16C211yesmaximal, characteristic
6#9673C211yesmaximal, characteristic, complemented
7#9603C211yesmaximal, characteristic, complemented
8#9544C211yesmaximal, characteristic, complemented
9#9543C211yesmaximal, characteristic, complemented
10#8945C211yesmaximal, characteristic
11#8945C211yesmaximal, characteristic
12#8565C211yesmaximal, characteristic
13#8264C211yesmaximal, characteristic, complemented
14#8201C211yesmaximal, characteristic, complemented
15C8×#103C211yesmaximal, characteristic, complemented
16C2×#461C411yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#1984C2211yescharacteristic
26#1984C2211yescharacteristic
27#1981C411yescharacteristic
28#1948C411yescharacteristic
29#1927C411yescharacteristic
30#1927C411yescharacteristic
31#1916—21no—
32#1916—21no—
33#1916—21no—
34#1916—21no—
35#1877C411yescharacteristic
36#1877C411yescharacteristic
37#1831C2211yescharacteristic
38#1802C411yescharacteristic
39C8×C4⋊C4C2211yescharacteristic
40C8×C4⋊C4C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C8×C22⋊C4—21no—
47C8×C22⋊C4—21no—
48C4×C22⋊C8—21no—
49#1744C2211yescharacteristic
50C2×#336—21no—
51SD16⋊C8—24no—
52C8×SD16—24no—
53C2×#335C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C23×C8—21no—
72C23×C8—21no—
73C2×C4⋊Q8C2×C411yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C22×C42—21no—
80#461C818yescomplemented
81C8×Q8—22no—
82C8×Q8—22no—
83C8×Q8—22no—
84C8×Q8—22no—
85C4×SD16—22no—
86C4×SD16—22no—
87C4×SD16—22no—
88C4×SD16—22no—
89C8×D4—22no—
90C8×D4—22no—
91C8×D4—22no—
92C8×D4D414yes—
93C2×#106C2×C411yescharacteristic
94C2×#106C2×C411yescharacteristic
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2×C411yescharacteristic
99C2×C4⋊C8C2×C411yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×#103C2×C411yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#391C2×C411yescharacteristic
126#391C2×C411yescharacteristic
127#389—21no—
128#389—21no—
129#384C2×C411yescharacteristic
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C43C2311yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336C2×C411yescharacteristic
154#336C2×C411yescharacteristic
155#336D411yescharacteristic
156#336D411yescharacteristic
157#336D411yescharacteristic
158#336D411yescharacteristic
159#335—22no—
160#334—22no—
161Q8⋊C8—22no—
162Q8⋊C8—22no—
163Q8⋊C8—22no—
164Q8⋊C8—22no—
165D4⋊C8—22no—
166D4⋊C8—22no—
167#322—22no—
168C82—22no—
169C22×Q8C22×C411yescharacteristic
170C22×Q8C22×C411yescharacteristic
171C22×D4C22×C411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C2×SD16—22no—
176C2×SD16—22no—
177C2×SD16—22no—
178C2×SD16—22no—
179C2×SD16C2×C814yes—
180C2×SD16C2×C814yes—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—24no—
185C22×C8—24no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—81no—
190C22×C8—81no—
191C22×C8—81no—
192C22×C8—81no—
193C22×C8C22×C411yescharacteristic
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C4⋊Q8—22no—
202C4⋊Q8C2×C814yescomplemented
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4C2×C814yes—
217C4×D4C2×D412yes—
218C4×D4#4112yes—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—81no—
239C2×C42—81no—
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42C2411yescharacteristic
247#106—22no—
248#106—22no—
249C4⋊C8—22no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—42no—
257C4⋊C8—42no—
258C4⋊C8—42no—
259C4⋊C8C2×C814yescomplemented
260C4⋊C8C2×D412yes—
261C4⋊C8#4112yes—
262#103—22no—
263#103—22no—
264#103—22no—
265#103—22no—
266#103C2×C814yescomplemented
267#103C2×C814yescomplemented
268#102—22no—
269#102—22no—
270C22⋊C8—24no—
271C22⋊C8—24no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—42no—
283C4×C8—42no—
284C4×C8—81no—
285C4×C8—81no—
286C4×C8—81no—
287C4×C8C2×D412yes—
288C4×C8#4112yes—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296C24—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8C22×C812yes—
308C2×Q8C22×C812yes—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4C4×D412yes—
313C2×D4C22×C812yes—
314C2×D4#13112yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—81no—
331C22×C4—81no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4C4×D411yescharacteristic
335C22×C4C4×D411yescharacteristic
336C22×C4C4×D411yescharacteristic
337C22×C4C4×D411yescharacteristic
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C2×#4111yescharacteristic
341SD16—28no—
342SD16—28no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—22no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—84no—
382C2×C8—84no—
383C2×C8#12112yes—
384C2×C8#12112yes—
385C2×C8C22×C812yes—
386C2×C8C22×C812yes—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—81no—
401C4⋊C4—81no—
402C4⋊C4C4×D412yes—
403C4⋊C4C22×C812yes—
404C4⋊C4#13112yes—
405C22⋊C4—24no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—24no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—81no—
424C42—81no—
425C42C22×C812yes—
426C42#13112yes—
427C42C22×D411yescharacteristic
428C42C2×#4111yescharacteristic
429C42C2×#4111yescharacteristic
430C42C2×#4111yescharacteristic
431C23—22no—
432C23—22no—
433C23—22no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23C2×C4×D411yescharacteristic
439Q8—22no—
440Q8—22no—
441Q8—22no—
442Q8—22no—
443Q8—41no—
444Q8—41no—
445Q8—41no—
446Q8—41no—
447D4—22no—
448D4—22no—
449D4—22no—
450D4C8×D414yes—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4C8×D412yes—
491C2×C4C8×D412yes—
492C2×C4C2×C4×D411yescharacteristic
493C2×C4C4×#4111yescharacteristic
494C2×C4C2×#12111yescharacteristic
495C2×C4#52511yescharacteristic
496C2×C4#52911yescharacteristic
497C2×C4#53211yescharacteristic
498C2×C4C23×C811yescharacteristic
499C2×C4C2×#13111yescharacteristic
500C2×C4C2×#13111yescharacteristic
501C2×C4C2×#13111yescharacteristic
502C2×C4#57711yescharacteristic
503C2×C4#57711yescharacteristic
504C2×C4#57711yescharacteristic
505C2×C4#57811yescharacteristic
506C8—22no—
507C8—22no—
508C8—22no—
509C8—22no—
510C8—42no—
511C8—41no—
512C8—41no—
513C8—42no—
514C8—41no—
515C8—41no—
516C8—42no—
517C8—81no—
518C8—81no—
519C8—81no—
520C22—24no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22C4×#12111yescharacteristic
528C22C2×C8×D411yescharacteristic
529C22#292911yescharacteristic
530C22#294711yescharacteristic
531C22#294911yescharacteristic
532C22#295611yescharacteristic
533C22#295711yescharacteristic
534C4—21no—
535C4—21no—
536C4—22no—
537C4—22no—
538C4—41no—
539C4—41no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—81no—
546C4—81no—
547C4C2×C8×D411yescharacteristic
548C4#293111yescharacteristic
549C4C8×#4111yescharacteristic
550C4#296611yescharacteristic
551C4#304211yescharacteristic
552C4#304611yescharacteristic
553C4#312011yescharacteristic
554C4#312311yescharacteristic
555C2—44no—
556C2#1785311yescharacteristic
557C2#1785511yescharacteristic
558C2#1792111yescharacteristic
559C2#1792411yescharacteristic
560C2C8×#12111yescharacteristic
561C2#2151411yescharacteristic
562C2#2151911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22238C2semidirect
6. #9673C2semidirect
7. #9603C2semidirect
8. #9544C2semidirect
9. #9543C2semidirect
13. #8264C2semidirect
14. #8201C2semidirect
15. C8×#103C2semidirect
80. #461C8semidirect
202. C4⋊Q8C2×C8semidirect
259. C4⋊C8C2×C8semidirect
266. #103C2×C8semidirect
267. #103C2×C8semidirect

Maximal subgroups

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

subgroupindex
#222382
C4×#4612
#205552
C2×SD16⋊C82
C2×C8×SD162
#96732
#96032
#95442
#95432
#89452
#85652
#82642
#82012
C8×#1032