Finite Group Databasebeta

← #385135

Subgroups — #385135

Order 512.

727 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21032C211yesmaximal, characteristic
2#21024C211yesmaximal, characteristic
3#20755C211yesmaximal, characteristic
4#11376C211yesmaximal, characteristic
5#11326C211yesmaximal, characteristic
6#11317C211yesmaximal, characteristic
7#11317C211yesmaximal, characteristic
8#11258C211yesmaximal, characteristic
9#11209C211yesmaximal, characteristic
10#10910C211yesmaximal, characteristic
11#10910C211yesmaximal, characteristic
12#10877C211yesmaximal, characteristic
13#10860C211yesmaximal, characteristic
14#10819C211yesmaximal, characteristic
15#10808C211yesmaximal, characteristic
16C22×#108C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×#405C2211yescharacteristic
22C22×#97C2211yescharacteristic
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#1917—21no—
34#1917—21no—
35#1916—21no—
36#1916—21no—
37#1832—21no—
38#1832—21no—
39#1831—21no—
40#1831—21no—
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43#1735C2211yescharacteristic
44#1735C2211yescharacteristic
45#1735C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61#1728C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C4×#95C2211yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#108—22no—
83C2×#108—22no—
84C2×#108—22no—
85C2×#108—41no—
86C2×#108—41no—
87C2×#108—41no—
88#427—22no—
89#427—22no—
90C2×#107C2×C411yescharacteristic
91C2×#107C2×C411yescharacteristic
92C2×#106C2×C411yescharacteristic
93C2×#106C2×C411yescharacteristic
94#423—41no—
95#423—41no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—41no—
99C2×C4⋊C8—41no—
100C2×C4⋊C8D411yescharacteristic
101C2×C4⋊C8D411yescharacteristic
102C2×C4⋊C8D411yescharacteristic
103C2×C4⋊C8D411yescharacteristic
104#405—41no—
105#405—41no—
106C2×#97—21no—
107C2×#97—22no—
108C2×#97—22no—
109C2×#97—41no—
110C2×C4×C8—21no—
111#389C2×C411yescharacteristic
112#389C2×C411yescharacteristic
113#389C2×C411yescharacteristic
114#389C2×C411yescharacteristic
115#385—21no—
116#385—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125#376—21no—
126C2×#95—21no—
127C2×#95—21no—
128C43—21no—
129#341—22no—
130#341—41no—
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#339—22no—
144#339—22no—
145#339—41no—
146#339—41no—
147#339—41no—
148#339—41no—
149#339—41no—
150#339—41no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#336—21no—
162#336—21no—
163#336—21no—
164#336—21no—
165#336—22no—
166#336—21no—
167#336—21no—
168#336—22no—
169#336—41no—
170#336C2×C411yescharacteristic
171#336C2×C411yescharacteristic
172#336C2×C411yescharacteristic
173#336C2×C411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4#4111yescharacteristic
177C23×C4#4111yescharacteristic
178C23×C4C2411yescharacteristic
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—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16#4112yes—
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×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8C22×C411yescharacteristic
230C22×C8C22×C411yescharacteristic
231C22×C8C22×C411yescharacteristic
232C22×C8C22×C411yescharacteristic
233C22×C8C2×D411yescharacteristic
234C22×C8C2×D411yescharacteristic
235C22×C8C2×D411yescharacteristic
236C22×C8C2×D411yescharacteristic
237C22×C8#4112yes—
238C22×C8#4112yes—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—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×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42#4112yes—
299#108—44no—
300#108—44no—
301#107—22no—
302#107—22no—
303#106—22no—
304#106—22no—
305C4⋊C8—41no—
306C4⋊C8—41no—
307C4⋊C8—41no—
308C4⋊C8—41no—
309C4⋊C8—42no—
310C4⋊C8—42no—
311#97—44no—
312#97—42no—
313C4×C8—42no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—42no—
323#95—42no—
324C24C2×#4111yescharacteristic
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
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—22no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
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×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4C23×C411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C42−1+411yescharacteristic
369C22×C42−1+411yescharacteristic
370C22×C42−1+411yescharacteristic
371C22×C42−1+411yescharacteristic
372C22×C42−1+411yescharacteristic
373M16—22no—
374M16—22no—
375M16—41no—
376M16—41no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—22no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8#11712yes—
440C2×C8#11712yes—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—22no—
478C42—22no—
479C42—42no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—42no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—81no—
496C42—81no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23C2×C4×D411yescharacteristic
502C23#53911yescharacteristic
503C23#53911yescharacteristic
504C23#55211yescharacteristic
505C23#55511yescharacteristic
506C23#55611yescharacteristic
507C23#55611yescharacteristic
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4#44412yes—
562C2×C4C2×#11711yescharacteristic
563C2×C4C4×#4111yescharacteristic
564C2×C4#51911yescharacteristic
565C2×C4#52011yescharacteristic
566C2×C4#52011yescharacteristic
567C2×C4#52011yescharacteristic
568C2×C4#53411yescharacteristic
569C2×C4#53611yescharacteristic
570C2×C4#53811yescharacteristic
571C2×C4#53911yescharacteristic
572C2×C4#54111yescharacteristic
573C2×C4#54111yescharacteristic
574C2×C4#54111yescharacteristic
575C2×C4#54211yescharacteristic
576C2×C4#54211yescharacteristic
577C2×C4#54211yescharacteristic
578C2×C4#54711yescharacteristic
579C2×C4#54711yescharacteristic
580C2×C4#54811yescharacteristic
581C2×C4#55011yescharacteristic
582C2×C4#55111yescharacteristic
583C2×C4#55111yescharacteristic
584C2×C4#55211yescharacteristic
585C2×C4#55211yescharacteristic
586C2×C4#56211yescharacteristic
587C2×C4#56311yescharacteristic
588C2×C4#56311yescharacteristic
589C2×C4#56411yescharacteristic
590C8—22no—
591C8—22no—
592C8—41no—
593C8—41no—
594C8—41no—
595C8—41no—
596C8—41no—
597C8—41no—
598C8—42no—
599C8—81no—
600C8—81no—
601C8—81no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22#232211yescharacteristic
615C22#235411yescharacteristic
616C22#236211yescharacteristic
617C22#274411yescharacteristic
618C22#276711yescharacteristic
619C22#277511yescharacteristic
620C22#278911yescharacteristic
621C22C2×#44411yescharacteristic
622C22#295511yescharacteristic
623C22#295611yescharacteristic
624C22#295711yescharacteristic
625C4—22no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—82no—
630C4—82no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4—81no—
637C4—81no—
638C4—81no—
639C4#232011yescharacteristic
640C4#233411yescharacteristic
641C4#234311yescharacteristic
642C4#238211yescharacteristic
643C4#271711yescharacteristic
644C4#271911yescharacteristic
645C4#272111yescharacteristic
646C4#273011yescharacteristic
647C2—21no—
648C2—21no—
649C2—21no—
650C2—21no—
651C2#1440011yescharacteristic
652C2#2209611yescharacteristic
653C2#2209711yescharacteristic
654C2#2280311yescharacteristic
655C2#2280411yescharacteristic
656C2#2308111yescharacteristic
657C2#2308211yescharacteristic
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
#210242
#207552
#113762
#113262
#113172
#112582
#112092
#109102
#108772
#108602
#108192
#108082