Finite Group Databasebeta

← #380508

Subgroups — #380508

Order 512.

750 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21032C211yesmaximal, characteristic
2#21020C211yesmaximal, characteristic
3C4×#424C211yesmaximal, characteristic
4#11329C211yesmaximal, characteristic
5#11317C211yesmaximal, characteristic
6#11317C211yesmaximal, characteristic
7#11291C211yesmaximal, characteristic
8#11291C211yesmaximal, characteristic
9#11258C211yesmaximal, characteristic
10#11221C211yesmaximal, characteristic
11#11221C211yesmaximal, characteristic
12#10911C211yesmaximal, characteristic
13#10886C211yesmaximal, characteristic
14#10808C211yesmaximal, characteristic
15#10797C211yesmaximal, characteristic
16C22×#108C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2296C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1921C411yescharacteristic, complemented
26#1921C411yescharacteristic, complemented
27#1920C411yescharacteristic, complemented
28#1920C411yescharacteristic, complemented
29#1919C411yescharacteristic, complemented
30#1919C411yescharacteristic, complemented
31#1918C411yescharacteristic, complemented
32#1918C411yescharacteristic, complemented
33#1916—22no—
34#1916—22no—
35#1768—22no—
36C4×C4⋊C8—22no—
37C2×#341C2211yescharacteristic
38C2×#341C2211yescharacteristic
39C2×#341C2211yescharacteristic
40#1735C2211yescharacteristic
41#1735C2211yescharacteristic
42#1735C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51#1729C2211yescharacteristic
52#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1728C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C4×#95C2211yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#108—22no—
79C2×#108—41no—
80C2×#108—41no—
81#427—22no—
82#427—22no—
83C2×#107C2×C411yescharacteristic
84C2×#107C2×C411yescharacteristic
85C2×#106C2×C411yescharacteristic
86C2×#106C2×C411yescharacteristic
87#424—22no—
88#424—22no—
89#424—22no—
90#424—22no—
91#424—41no—
92#424—41no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C4×M16—41no—
97C4×M16—41no—
98C2×#97—21no—
99C2×C4×C8—21no—
100C2×C4×C8—41no—
101C2×C4×C8—41no—
102C2×C4×C8D412yes—
103C2×C4×C8D412yes—
104#389C2×C411yescharacteristic
105#389C2×C411yescharacteristic
106#389C2×C411yescharacteristic
107#389C2×C411yescharacteristic
108#385—21no—
109#385—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4C2×C411yescharacteristic
113C4×C4⋊C4C2×C411yescharacteristic
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118#376—21no—
119C2×#95—21no—
120C2×#95—21no—
121C43—21no—
122#341—22no—
123#341—41no—
124#341—41no—
125#340C2×C411yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#339—22no—
138#339—22no—
139#339—22no—
140#339—22no—
141#339—41no—
142#339—41no—
143#339—41no—
144#339—41no—
145#336—22no—
146#336—22no—
147#336—22no—
148#336—22no—
149#336—22no—
150#336—21no—
151#336—21no—
152#336—41no—
153#336—41no—
154#336—41no—
155#336—41no—
156#336C2×C411yescharacteristic
157#336C2×C411yescharacteristic
158#336C2×C411yescharacteristic
159#336C2×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4#4111yescharacteristic
163C23×C4#4111yescharacteristic
164C23×C4C2411yescharacteristic
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16#4112yes—
188C2×M16#4112yes—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C22×C411yescharacteristic
221C22×C8C2×D412yes—
222C22×C8C2×D412yes—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—21no—
227#117—21no—
228#117—21no—
229#117—21no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117#4112yes—
238#117#4112yes—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283#108—44no—
284#107—22no—
285#107—22no—
286#106—22no—
287#106—22no—
288C4⋊C8—28no—
289C4⋊C8—28no—
290C4⋊C8—44no—
291#97—42no—
292C4×C8—42no—
293C4×C8—42no—
294C4×C8—42no—
295#95—24no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—42no—
305C24C2×#4111yescharacteristic
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
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×C4—41no—
336C22×C4C4×D411yescharacteristic
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4C4×D411yescharacteristic
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C4H(5,𝔽2)11yescharacteristic
348C22×C4H(5,𝔽2)11yescharacteristic
349C22×C42−1+411yescharacteristic
350C22×C42−1+411yescharacteristic
351C22×C42−1+411yescharacteristic
352C22×C42−1+411yescharacteristic
353C22×C42−1+411yescharacteristic
354M16—22no—
355M16—22no—
356M16—22no—
357M16—22no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
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—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—41no—
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—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4#11712yes—
435C4⋊C4#11712yes—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—22no—
449C42—22no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—42no—
455C42—42no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—42no—
463C42—42no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—41no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23C2×C4×D411yescharacteristic
477C23#52911yescharacteristic
478C23#52911yescharacteristic
479C23#53911yescharacteristic
480C23#53911yescharacteristic
481C23#54111yescharacteristic
482C23#54211yescharacteristic
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
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—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4#44312yes—
536C2×C4#44412yes—
537C2×C4C2×#11711yescharacteristic
538C2×C4C4×#4111yescharacteristic
539C2×C4#51911yescharacteristic
540C2×C4#52011yescharacteristic
541C2×C4#52011yescharacteristic
542C2×C4#52011yescharacteristic
543C2×C4#52511yescharacteristic
544C2×C4#52511yescharacteristic
545C2×C4#53411yescharacteristic
546C2×C4#53611yescharacteristic
547C2×C4#53611yescharacteristic
548C2×C4#53711yescharacteristic
549C2×C4#53811yescharacteristic
550C2×C4#53911yescharacteristic
551C2×C4#53911yescharacteristic
552C2×C4#53911yescharacteristic
553C2×C4#54011yescharacteristic
554C2×C4#54111yescharacteristic
555C2×C4#54111yescharacteristic
556C2×C4#54111yescharacteristic
557C2×C4#54111yescharacteristic
558C2×C4#54111yescharacteristic
559C2×C4#54111yescharacteristic
560C2×C4#54211yescharacteristic
561C2×C4#54211yescharacteristic
562C2×C4#54211yescharacteristic
563C2×C4#54211yescharacteristic
564C2×C4#54211yescharacteristic
565C8—41no—
566C8—41no—
567C8—41no—
568C8—41no—
569C8—42no—
570C8—42no—
571C8—42no—
572C8—42no—
573C8—81no—
574C8—81no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22#229011yescharacteristic
588C22#232211yescharacteristic
589C22#232311yescharacteristic
590C22#265011yescharacteristic
591C22#265411yescharacteristic
592C22#265511yescharacteristic
593C22#265511yescharacteristic
594C22C2×#44311yescharacteristic
595C22C2×#44411yescharacteristic
596C22#295511yescharacteristic
597C22#295511yescharacteristic
598C4—22no—
599C4—22no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—42no—
604C4—82no—
605C4—82no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4#231411yescharacteristic
613C4#232011yescharacteristic
614C4#233411yescharacteristic
615C4#233611yescharacteristic
616C4#263011yescharacteristic
617C4#263411yescharacteristic
618C4#263711yescharacteristic
619C4#263811yescharacteristic
620C2—21no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2#1427011yescharacteristic
625C2#2113711yescharacteristic
626C2#2113811yescharacteristic
627C2#2209511yescharacteristic
628C2#2209611yescharacteristic
629C2#2213011yescharacteristic
630C2#2213111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1921C4semidirect
26. #1921C4semidirect
27. #1920C4semidirect
28. #1920C4semidirect
29. #1919C4semidirect
30. #1919C4semidirect
31. #1918C4semidirect
32. #1918C4semidirect

Maximal subgroups

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

subgroupindex
#210322
#210202
C4×#4242
#113292
#113172
#112912
#112582
#112212
#109112
#108862
#108082
#107972