Finite Group Databasebeta

← #314836

Subgroups — #314836

Order 512.

611 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27679C211yesmaximal, characteristic
2#27678C211yesmaximal, characteristic
3#22133C211yesmaximal, characteristic
4#21982C211yesmaximal, characteristic, complemented
5#10656C211yesmaximal, characteristic, complemented
6#10652C211yesmaximal, characteristic, complemented
7#10622C211yesmaximal, characteristic, complemented
8#10616C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9615C211yesmaximal, characteristic, complemented
11#8727C211yesmaximal, characteristic, complemented
12#8724C211yesmaximal, characteristic, complemented
13#8648C211yesmaximal, characteristic, complemented
14#8644C211yesmaximal, characteristic, complemented
15#8599C211yesmaximal, characteristic, complemented
16C2×C8×D4C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C4×#121C2211yescharacteristic
19#2285C2211yescharacteristic
20C2×#384C2211yescharacteristic
21#2101—21no—
22#2100—21no—
23#2094—21no—
24#2092—21no—
25#1954—21no—
26#1954C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1947—21no—
31#1947—21no—
32#1947C2211yescharacteristic
33#1939C2211yescharacteristic
34#1923—21no—
35#1875—21no—
36#1874—21no—
37#1869—21no—
38#1854—21no—
39#1853—21no—
40#1852—21no—
41#1852—21no—
42#1850—21no—
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1797C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×#121D411yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C42C2311yescharacteristic
76C8×D4—22no—
77C8×D4—22no—
78#431—22no—
79#431—22no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8D411yescharacteristic
108C2×C22⋊C8D411yescharacteristic
109C2×C4×C8—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#390D411yescharacteristic
113#389D411yescharacteristic
114#388—21no—
115#386—21no—
116#384—21no—
117#384—21no—
118#384—21no—
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#382—21no—
124#380—21no—
125#380—21no—
126#380D411yescharacteristic
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#335—22no—
149#335—22no—
150C22×D4—21no—
151C22×D4C2×D411yescharacteristic
152C23×C4—21no—
153C23×C4—41no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4#4111yescharacteristic
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—42no—
164C22×C8—81no—
165C22×C8—81no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170#121—22no—
171#121Q1614yes—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—81no—
177C4×D4—81no—
178#117—81no—
179#117—81no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4—81no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—41no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—41no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42C2411yescharacteristic
220#107—81no—
221#107—81no—
222#107—81no—
223#107—81no—
224#107—81no—
225#106—81no—
226#106—81no—
227#106—81no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—81no—
243#102—81no—
244#102—81no—
245#102—81no—
246#102—81no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—42no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—81no—
253C22⋊C8—81no—
254C22⋊C8—81no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—81no—
258C4×C8#4112yes—
259C4×C8#4112yes—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—41no—
264#95—41no—
265C24—21no—
266C24#12111yescharacteristic
267C2×D4—24no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—81no—
274C2×D4—81no—
275C2×D4C2×Q1612yes—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4C2×Q1612yes—
310C22×C4#13512yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—84no—
343C2×C8#12312yes—
344C2×C8#12312yes—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4C2×Q1612yes—
375C22⋊C4—24no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—81no—
379C22⋊C4—81no—
380C22⋊C4—81no—
381C22⋊C4—82no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—41no—
389C42—41no—
390C42—82no—
391C42C2×#4111yescharacteristic
392C42C2×#4111yescharacteristic
393C422−1+411yescharacteristic
394C422−1+411yescharacteristic
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—22no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23#46712yes—
409C23C2×#12111yescharacteristic
410D4—22no—
411D4—22no—
412D4—41no—
413D4—41no—
414D4—84no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—164no—
461C2×C4#46712yes—
462C2×C4C2×#12311yescharacteristic
463C2×C4#53611yescharacteristic
464C2×C4#54711yescharacteristic
465C2×C4#54711yescharacteristic
466C2×C4#55211yescharacteristic
467C2×C4#55211yescharacteristic
468C2×C4C22×Q1611yescharacteristic
469C2×C4C2×#13511yescharacteristic
470C2×C4#57611yescharacteristic
471C2×C4#57611yescharacteristic
472C2×C4#57611yescharacteristic
473C2×C4#57711yescharacteristic
474C2×C4#57711yescharacteristic
475C2×C4#57711yescharacteristic
476C8—22no—
477C8—22no—
478C8—41no—
479C8—41no—
480C8—81no—
481C8—81no—
482C8—81no—
483C8—81no—
484C8—81no—
485C8—81no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22#248611yescharacteristic
501C22#304211yescharacteristic
502C22#304511yescharacteristic
503C22C2×#46711yescharacteristic
504C22#305811yescharacteristic
505C22#306811yescharacteristic
506C22#306911yescharacteristic
507C4—22no—
508C4—21no—
509C4—21no—
510C4—41no—
511C4—41no—
512C4—42no—
513C4—81no—
514C4—81no—
515C4—161no—
516C4—161no—
517C4—161no—
518C4—161no—
519C4#309611yescharacteristic
520C4#309911yescharacteristic
521C4#324111yescharacteristic
522C4#324411yescharacteristic
523C4#333811yescharacteristic
524C4#333811yescharacteristic
525C4#333911yescharacteristic
526C4#333911yescharacteristic
527C2—42no—
528C2—82no—
529C2#1939311yescharacteristic
530C2#1941111yescharacteristic
531C2#1951411yescharacteristic
532C2#1953811yescharacteristic
533C2#2765411yescharacteristic
534C2#2766211yescharacteristic
535C2#2767611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21982C2semidirect
5. #10656C2semidirect
6. #10652C2semidirect
7. #10622C2semidirect
8. #10616C2semidirect
9. #9616C2semidirect
10. #9615C2semidirect
11. #8727C2semidirect
12. #8724C2semidirect
13. #8648C2semidirect
14. #8644C2semidirect
15. #8599C2semidirect

Maximal subgroups

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

subgroupindex
#276792
#276782
#221332
#219822
#106562
#106522
#106222
#106162
#96162
#96152
#87272
#87242
#86482
#86442
#85992