Finite Group Databasebeta

← #210597

Subgroups — #210597

Order 512.

727 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23680C211yesmaximal, characteristic
2#22208C211yesmaximal, characteristic, complemented
3#20565C211yesmaximal, characteristic, complemented
4C2×#1580C211yesmaximal, characteristic
5C2×C8×SD16C211yesmaximal, characteristic, complemented
6#9676C211yesmaximal, characteristic, complemented
7#9605C211yesmaximal, characteristic, complemented
8#9547C211yesmaximal, characteristic, complemented
9#9546C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic
11#8949C211yesmaximal, characteristic, complemented
12#8567C211yesmaximal, characteristic, complemented
13#8264C211yesmaximal, characteristic, complemented
14#8201C211yesmaximal, characteristic, complemented
15C8×#102C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C4×#124C2211yescharacteristic
23#2293C2211yescharacteristic
24#2085C411yescharacteristic
25#2073C411yescharacteristic
26#2068C411yescharacteristic
27#2065C411yescharacteristic
28#2055C411yescharacteristic
29#2054C411yescharacteristic
30#2042C411yescharacteristic
31#2012C411yescharacteristic
32#1984C2211yescharacteristic
33#1954C2211yescharacteristic
34#1923—21no—
35#1923—21no—
36#1850—21no—
37#1850—21no—
38#1842C2211yescharacteristic
39#1770C2211yescharacteristic
40C8×C4⋊C4C2211yescharacteristic
41#1767C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#102C2211yescharacteristic
46#1753—21no—
47#1752—21no—
48#1750—21no—
49#1744C2211yescharacteristic
50#1729—21no—
51#1580—24no—
52C8×SD16—24no—
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊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×D8C2×C411yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C23×C8—21no—
74C2×#124C2×C411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C2×#117—21no—
79C8×Q8—22no—
80C8×Q8—22no—
81C4×SD16—22no—
82C4×SD16—22no—
83C4×D8—22no—
84C4×D8—22no—
85#436—22no—
86#436—22no—
87C8×D4—22no—
88C8×D4—22no—
89C8×D4—22no—
90C8×D4D414yes—
91C2×#107C2×C411yescharacteristic
92C2×#106C2×C411yescharacteristic
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2×C411yescharacteristic
98C2×C4⋊C8C2×C411yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×#97—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#390C2×C411yescharacteristic
129#390C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#388—21no—
133#388—21no—
134#388—21no—
135#388—21no—
136#382C2×C411yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C43C2311yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336C2×C411yescharacteristic
153#336C2×C411yescharacteristic
154#336D411yescharacteristic
155#336D411yescharacteristic
156#336D411yescharacteristic
157#336D411yescharacteristic
158#334—22no—
159#334—22no—
160Q8⋊C8—22no—
161Q8⋊C8—22no—
162D4⋊C8—22no—
163D4⋊C8—22no—
164D4⋊C8—22no—
165D4⋊C8—22no—
166#322—22no—
167C82—22no—
168C22×Q8C22×C411yescharacteristic
169C22×D4C22×C411yescharacteristic
170C22×D4C22×C411yescharacteristic
171C23×C4—21no—
172C23×C4—21no—
173C2×SD16—22no—
174C2×SD16—22no—
175C2×SD16—22no—
176C2×D8—22no—
177C2×D8—22no—
178C2×D8—22no—
179C2×M16—24no—
180C2×M16—81no—
181C2×M16—81no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—24no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—42no—
190C22×C8—81no—
191C22×C8—81no—
192C22×C8—81no—
193C22×C8—81no—
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C22×C411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202#124—22no—
203#124—22no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4C2×D412yes—
215C4×D4#4112yes—
216#117—81no—
217#117—81no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42C2411yescharacteristic
245#107—22no—
246#106—22no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—42no—
256C4⋊C8—42no—
257C4⋊C8—81no—
258C4⋊C8—81no—
259C4⋊C8C2×D412yes—
260C4⋊C8#4112yes—
261#103—22no—
262#103—22no—
263#103—22no—
264#102—22no—
265#102—22no—
266#102—22no—
267#102—22no—
268#102—22no—
269C22⋊C8—24no—
270C22⋊C8—24no—
271C22⋊C8—42no—
272C22⋊C8—81no—
273C22⋊C8—81no—
274#97—81no—
275#97—81no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—42no—
287C4×C8—81no—
288C4×C8C2×D412yes—
289C4×C8#4112yes—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297C24—21no—
298C24—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8#13112yes—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4C4×D412yes—
316C2×D4#13112yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4C4×D411yescharacteristic
336C22×C4C4×D411yescharacteristic
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4C23×C411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C2×#4111yescharacteristic
342SD16—28no—
343D8—28no—
344M16—84no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—22no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—22no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—22no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
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—41no—
382C2×C8—41no—
383C2×C8—84no—
384C2×C8—84no—
385C2×C8#12112yes—
386C2×C8#12112yes—
387C2×C8#13112yes—
388C2×C8#13112yes—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—81no—
401C4⋊C4—81no—
402C4⋊C4C4×D412yes—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—81no—
410C22⋊C4—81no—
411C22⋊C4—81no—
412C22⋊C4—81no—
413C22⋊C4—81no—
414C22⋊C4—81no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42C22×D411yescharacteristic
430C42C2×#4111yescharacteristic
431C42C2×#4111yescharacteristic
432C42C2×#4111yescharacteristic
433C23—22no—
434C23—22no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23C2×C4×D411yescharacteristic
448Q8—22no—
449Q8—22no—
450Q8—41no—
451Q8—41no—
452D4—22no—
453D4—22no—
454D4—22no—
455D4—22no—
456D4—41no—
457D4—41no—
458D4—41no—
459D4—41no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4C2×C4×D411yescharacteristic
501C2×C4C4×#4111yescharacteristic
502C2×C4C2×#12111yescharacteristic
503C2×C4#52511yescharacteristic
504C2×C4#52911yescharacteristic
505C2×C4#53211yescharacteristic
506C2×C4C2×#13111yescharacteristic
507C2×C4C2×#13111yescharacteristic
508C2×C4C2×#13111yescharacteristic
509C2×C4C2×#13111yescharacteristic
510C2×C4#57611yescharacteristic
511C2×C4#57711yescharacteristic
512C2×C4#57711yescharacteristic
513C2×C4#57711yescharacteristic
514C8—22no—
515C8—22no—
516C8—22no—
517C8—22no—
518C8—42no—
519C8—41no—
520C8—42no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—42no—
525C8—81no—
526C8—81no—
527C8—81no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22C4×#12111yescharacteristic
543C22#292911yescharacteristic
544C22#293111yescharacteristic
545C22#294611yescharacteristic
546C22#295211yescharacteristic
547C22#295611yescharacteristic
548C22#295711yescharacteristic
549C4—21no—
550C4—21no—
551C4—41no—
552C4—42no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4#292911yescharacteristic
562C4#292911yescharacteristic
563C4#296611yescharacteristic
564C4#296611yescharacteristic
565C4#304211yescharacteristic
566C4#304511yescharacteristic
567C4#311811yescharacteristic
568C4#312511yescharacteristic
569C2—82no—
570C2—82no—
571C2#1792011yescharacteristic
572C2#1792711yescharacteristic
573C2#1796811yescharacteristic
574C2#1797411yescharacteristic
575C2#2149111yescharacteristic
576C2#2151011yescharacteristic
577C2#2151911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×C4
LCSG ▷ C22×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
2. #22208C2semidirect
3. #20565C2semidirect
5. C2×C8×SD16C2semidirect
6. #9676C2semidirect
7. #9605C2semidirect
8. #9547C2semidirect
9. #9546C2semidirect
11. #8949C2semidirect
12. #8567C2semidirect
13. #8264C2semidirect
14. #8201C2semidirect
15. C8×#102C2semidirect

Maximal subgroups

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

subgroupindex
#236802
#222082
#205652
C2×#15802
C2×C8×SD162
#96762
#96052
#95472
#95462
#89512
#89492
#85672
#82642
#82012
C8×#1022