Finite Group Databasebeta

← #218983

Subgroups — #218983

Order 512.

670 conjugacy classes of subgroups.

The subgroup lattice has 609 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
4#9545C211yesmaximal, characteristic, complemented
5#9541C211yesmaximal, characteristic, complemented
6#9540C211yesmaximal, characteristic, complemented
7#9536C211yesmaximal, characteristic, complemented
8#9078C211yesmaximal, characteristic, complemented
9#9078C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8847C211yesmaximal, characteristic, complemented
13#8843C211yesmaximal, characteristic, complemented
14#8599C211yesmaximal, characteristic, complemented
15#7850C211yesmaximal, 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
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×D8C2×C411yescharacteristic
68C22×M16—21no—
69C22×M16—21no—
70C23×C8—21no—
71C2×#121C2×C411yescharacteristic
72C2×#121C2×C411yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C4×D8—41no—
79C4×D8—41no—
80#436—41no—
81#436—41no—
82C8×D4—41no—
83C8×D4—41no—
84#431—41no—
85#431—41no—
86#431—41no—
87#431—41no—
88C2×#107C2×C411yescharacteristic
89C2×#107C2×C411yescharacteristic
90C2×#107C2×C411yescharacteristic
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2×C411yescharacteristic
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8D411yescharacteristic
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
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⋊C8C2×C411yescharacteristic
116C2×C22⋊C8C2×C411yescharacteristic
117C2×C22⋊C8D411yescharacteristic
118C2×C22⋊C8D411yescharacteristic
119C4×M16—41no—
120C4×M16—41no—
121C2×#97—21no—
122C2×#97—21no—
123C2×C4×C8—21no—
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130#390C2×C411yescharacteristic
131#390C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#388—21no—
135#388—21no—
136#386—21no—
137#386—21no—
138#380C2×C411yescharacteristic
139#380C2×C411yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#335—41no—
157#335—41no—
158D4⋊C8—41no—
159D4⋊C8—41no—
160D4⋊C8—41no—
161D4⋊C8—41no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C22×C411yescharacteristic
165C22×D4C22×C411yescharacteristic
166C23×C4—21no—
167C23×C4—21no—
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4#4111yescharacteristic
171C2×D8—22no—
172C2×D8—41no—
173C2×D8—41no—
174C2×M16—42no—
175C2×M16—42no—
176C2×M16—42no—
177C2×M16—42no—
178C2×M16—81no—
179C2×M16—81no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—42no—
187C22×C8—81no—
188C22×C8—81no—
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C22×C411yescharacteristic
192C22×C8C22×C411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197#121—22no—
198#121—22no—
199#121—41no—
200#121—41no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—81no—
210C4×D4—81no—
211C4×D4—81no—
212C4×D4—81no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42C22×C411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42C2411yescharacteristic
241#107—22no—
242#107—41no—
243#107—41no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4⋊C8—81no—
257C4⋊C8—81no—
258#102—22no—
259#102—22no—
260#102—41no—
261#102—41no—
262#102—41no—
263#102—41no—
264C22⋊C8—41no—
265C22⋊C8—41no—
266C22⋊C8—41no—
267C22⋊C8—41no—
268C22⋊C8—42no—
269C22⋊C8—42no—
270C22⋊C8—81no—
271C22⋊C8—81no—
272C22⋊C8—81no—
273C22⋊C8—81no—
274C22⋊C8—81no—
275C22⋊C8—81no—
276#97—42no—
277#97—81no—
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—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—81no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297C24—21no—
298C24—21no—
299C24C4×D411yescharacteristic
300C2×D4—24no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—42no—
313C2×D4—81no—
314C2×D4—81no—
315C2×D4—81no—
316C2×D4—81no—
317C2×D4#13112yes—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
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—41no—
339C22×C4—41no—
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×C4C4×D411yescharacteristic
353C22×C4C4×D411yescharacteristic
354C22×C4C4×D411yescharacteristic
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4111yescharacteristic
358D8—44no—
359M16—48no—
360M16—84no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
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—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
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—84no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—81no—
416C4⋊C4—81no—
417C4⋊C4#13112yes—
418C22⋊C4—24no—
419C22⋊C4—42no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—81no—
425C22⋊C4—81no—
426C22⋊C4—81no—
427C22⋊C4—81no—
428C22⋊C4—81no—
429C22⋊C4—81no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42H(5,𝔽2)11yescharacteristic
441C42H(5,𝔽2)11yescharacteristic
442C42H(5,𝔽2)11yescharacteristic
443C422−1+411yescharacteristic
444C23—21no—
445C23—22no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23C2×C4×D411yescharacteristic
466D4—22no—
467D4—22no—
468D4—41no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—84no—
475D4—84no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
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—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4#51811yescharacteristic
524C2×C4#52011yescharacteristic
525C2×C4#53411yescharacteristic
526C2×C4#53711yescharacteristic
527C2×C4#53811yescharacteristic
528C2×C4#53911yescharacteristic
529C2×C4C2×#13111yescharacteristic
530C2×C4C2×#13111yescharacteristic
531C2×C4#56811yescharacteristic
532C2×C4#56811yescharacteristic
533C2×C4#57611yescharacteristic
534C2×C4#57611yescharacteristic
535C2×C4#57611yescharacteristic
536C2×C4#57811yescharacteristic
537C8—42no—
538C8—42no—
539C8—41no—
540C8—41no—
541C8—41no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—81no—
548C8—81no—
549C8—81no—
550C8—81no—
551C22—41no—
552C22—42no—
553C22—41no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—42no—
559C22—41no—
560C22—41no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22#232511yescharacteristic
573C22#292911yescharacteristic
574C22#293011yescharacteristic
575C22#294811yescharacteristic
576C22#295111yescharacteristic
577C22#295611yescharacteristic
578C22#295611yescharacteristic
579C4—21no—
580C4—21no—
581C4—42no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4#297611yescharacteristic
591C4#298111yescharacteristic
592C4#299211yescharacteristic
593C4#299311yescharacteristic
594C4#319911yescharacteristic
595C4#320311yescharacteristic
596C4#325911yescharacteristic
597C4#326211yescharacteristic
598C2—81no—
599C2—82no—
600C2—161no—
601C2#1792211yescharacteristic
602C2#1792611yescharacteristic
603C2#1795011yescharacteristic
604C2#1795411yescharacteristic
605C2#2217311yescharacteristic
606C2#2219611yescharacteristic
607C2#2220711yescharacteristic
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. #9545C2semidirect
5. #9541C2semidirect
6. #9540C2semidirect
7. #9536C2semidirect
8. #9078C2semidirect
9. #9078C2semidirect
10. #8951C2semidirect
11. #8947C2semidirect
12. #8847C2semidirect
13. #8843C2semidirect
14. #8599C2semidirect
15. #7850C2semidirect

Maximal subgroups

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

subgroupindex
#236962
#221392
#221332
#95452
#95412
#95402
#95362
#90782
#89512
#89472
#88472
#88432
#85992
#78502