Finite Group Databasebeta

← #210944

Subgroups — #210944

Order 512.

740 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23696C211yesmaximal, characteristic
2#22139C211yesmaximal, characteristic, complemented
3#22133C211yesmaximal, characteristic, complemented
4C2×#1586C211yesmaximal, characteristic, complemented
5C2×#1453C211yesmaximal, characteristic, complemented
6#9659C211yesmaximal, characteristic, complemented
7#9595C211yesmaximal, characteristic, complemented
8#9554C211yesmaximal, characteristic, complemented
9#9554C211yesmaximal, characteristic, complemented
10#9078C211yesmaximal, characteristic, complemented
11#9078C211yesmaximal, characteristic, complemented
12#8598C211yesmaximal, characteristic, complemented
13#8288C211yesmaximal, characteristic, complemented
14#8210C211yesmaximal, characteristic, complemented
15#8202C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22C4×#121C2211yescharacteristic
23C4×#121C2211yescharacteristic
24#2086C411yescharacteristic
25#2075C411yescharacteristic
26#2067C411yescharacteristic
27#2067C411yescharacteristic
28#2054C411yescharacteristic
29#2033C411yescharacteristic
30#2033C411yescharacteristic
31#1999C411yescharacteristic
32#1955—21no—
33#1954—21no—
34#1954C2211yescharacteristic
35#1954C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1924—21no—
39#1923—21no—
40#1870—21no—
41#1869—21no—
42#1851—21no—
43#1850—21no—
44#1797C2211yescharacteristic
45#1797C2211yescharacteristic
46C4×#107C2211yescharacteristic
47#1770C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C4×#102C2211yescharacteristic
50C4×#102C2211yescharacteristic
51#1586—24no—
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58#1453—24no—
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×M16—21no—
71C22×M16—21no—
72C23×C8—21no—
73C2×#121C2×C411yescharacteristic
74C2×#121C2×C411yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2311yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C4×D8—22no—
81C4×D8—22no—
82#436—22no—
83#436—22no—
84C8×D4—22no—
85C8×D4—22no—
86#431—22no—
87#431—22no—
88#431—22no—
89#431—22no—
90C2×#107C2×C411yescharacteristic
91C2×#107C2×C411yescharacteristic
92C2×#107C2×C411yescharacteristic
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2×C411yescharacteristic
96C2×C4⋊C8C2×C411yescharacteristic
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8D411yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8C2×C411yescharacteristic
118C2×C22⋊C8C2×C411yescharacteristic
119C2×C22⋊C8D411yescharacteristic
120C2×C22⋊C8D411yescharacteristic
121C4×M16—22no—
122C4×M16—22no—
123C2×#97—21no—
124C2×#97—21no—
125C2×C4×C8—21no—
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132#390C2×C411yescharacteristic
133#390C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#388—21no—
137#388—21no—
138#386—21no—
139#386—21no—
140#380C2×C411yescharacteristic
141#380C2×C411yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#334—22no—
159#334—22no—
160D4⋊C8—22no—
161D4⋊C8—22no—
162D4⋊C8—22no—
163D4⋊C8—22no—
164#322—22no—
165#322—22no—
166#322D414yescomplemented
167C82—22no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4C22×C411yescharacteristic
171C22×D4C22×C411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4C22×C411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4#4111yescharacteristic
177C2×D8—22no—
178C2×D8—22no—
179C2×D8—41no—
180C2×M16—24no—
181C2×M16—42no—
182C2×M16—42no—
183C2×M16—42no—
184C2×M16—81no—
185C2×M16—81no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—42no—
193C22×C8—81no—
194C22×C8—81no—
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C22×C411yescharacteristic
198C22×C8C22×C411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203#121—22no—
204#121—22no—
205#121—41no—
206#121—41no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—81no—
216C4×D4—81no—
217C4×D4—81no—
218C4×D4—81no—
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×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42C2411yescharacteristic
247#107—22no—
248#107—22no—
249#107—41no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—81no—
263C4⋊C8—81no—
264#102—22no—
265#102—22no—
266#102—22no—
267#102—22no—
268#102—41no—
269#102—41no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—81no—
277C22⋊C8—81no—
278C22⋊C8—81no—
279C22⋊C8—81no—
280C22⋊C8—81no—
281C22⋊C8—81no—
282#97—24no—
283#97—81no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—81no—
291C4×C8C2×D412yes—
292C4×C8C2×D412yes—
293C4×C8C2×D412yes—
294C4×C8#4112yes—
295C4×C8#4112yes—
296C4×C8#4112yes—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303C24—21no—
304C24—21no—
305C24C4×D411yescharacteristic
306C2×D4—24no—
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×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—42no—
319C2×D4—81no—
320C2×D4—81no—
321C2×D4—81no—
322C2×D4—81no—
323C2×D4#13112yes—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4C4×D411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4C23×C411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C2×#4111yescharacteristic
364D8—44no—
365M16—48no—
366M16—84no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—84no—
406C2×C8C4×D412yes—
407C2×C8C4×D412yes—
408C2×C8#12112yes—
409C2×C8#12112yes—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—81no—
422C4⋊C4—81no—
423C4⋊C4#13112yes—
424C22⋊C4—24no—
425C22⋊C4—42no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—81no—
431C22⋊C4—81no—
432C22⋊C4—81no—
433C22⋊C4—81no—
434C22⋊C4—81no—
435C22⋊C4—81no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42C22×D411yescharacteristic
447C42C2×#4111yescharacteristic
448C42C2×#4111yescharacteristic
449C42C2×#4111yescharacteristic
450C23—22no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23C2×C4×D411yescharacteristic
472D4—22no—
473D4—22no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—41no—
480D4—84no—
481D4—84no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4C2×C4×D411yescharacteristic
530C2×C4C4×#4111yescharacteristic
531C2×C4C2×#12111yescharacteristic
532C2×C4#52511yescharacteristic
533C2×C4#52911yescharacteristic
534C2×C4#53211yescharacteristic
535C2×C4C2×#13111yescharacteristic
536C2×C4C2×#13111yescharacteristic
537C2×C4#56811yescharacteristic
538C2×C4#56811yescharacteristic
539C2×C4#57611yescharacteristic
540C2×C4#57611yescharacteristic
541C2×C4#57711yescharacteristic
542C2×C4#57711yescharacteristic
543C8—24no—
544C8—22no—
545C8—22no—
546C8—22no—
547C8—22no—
548C8—42no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—41no—
553C8—81no—
554C8—81no—
555C8—81no—
556C8—81no—
557C22—41no—
558C22—42no—
559C22—41no—
560C22—42no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—42no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22C4×#12111yescharacteristic
579C22#292911yescharacteristic
580C22#293011yescharacteristic
581C22#294811yescharacteristic
582C22#295211yescharacteristic
583C22#295611yescharacteristic
584C22#295611yescharacteristic
585C4—21no—
586C4—21no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4#293311yescharacteristic
597C4#293511yescharacteristic
598C4#296811yescharacteristic
599C4#296911yescharacteristic
600C4#306211yescharacteristic
601C4#306211yescharacteristic
602C4#317911yescharacteristic
603C4#317911yescharacteristic
604C2—81no—
605C2—82no—
606C2—161no—
607C2#1792211yescharacteristic
608C2#1792711yescharacteristic
609C2#1795011yescharacteristic
610C2#1795511yescharacteristic
611C2#2149511yescharacteristic
612C2#2152911yescharacteristic
613C2#2154911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radical#1999C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22139C2semidirect
3. #22133C2semidirect
4. C2×#1586C2semidirect
5. C2×#1453C2semidirect
6. #9659C2semidirect
7. #9595C2semidirect
8. #9554C2semidirect
9. #9554C2semidirect
10. #9078C2semidirect
11. #9078C2semidirect
12. #8598C2semidirect
13. #8288C2semidirect
14. #8210C2semidirect
15. #8202C2semidirect
166. #322D4semidirect

Maximal subgroups

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

subgroupindex
#236962
#221392
#221332
C2×#15862
C2×#14532
#96592
#95952
#95542
#90782
#85982
#82882
#82102
#82022