Finite Group Databasebeta

← #314835

Subgroups — #314835

Order 512.

631 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27646C211yesmaximal, characteristic
2#27645C211yesmaximal, characteristic
3#22133C211yesmaximal, characteristic
4#21982C211yesmaximal, characteristic, complemented
5#10655C211yesmaximal, characteristic, complemented
6#10651C211yesmaximal, characteristic, complemented
7#10621C211yesmaximal, characteristic, complemented
8#10615C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9615C211yesmaximal, characteristic, complemented
11#8726C211yesmaximal, characteristic, complemented
12#8723C211yesmaximal, characteristic, complemented
13#8648C211yesmaximal, characteristic, complemented
14#8644C211yesmaximal, characteristic, complemented
15#8599C211yesmaximal, characteristic, complemented
16C2×#481—21no—
17C2×#482—21no—
18C2×C8×D4C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C4×#121C2211yescharacteristic
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#2085—21no—
24#2084—21no—
25#2073—21no—
26#2071—21no—
27#1954—21no—
28#1954C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1947—21no—
33#1947—21no—
34#1947C2211yescharacteristic
35#1939C2211yescharacteristic
36#1923—21no—
37#1869—21no—
38#1852—21no—
39#1852—21no—
40#1850—21no—
41#1845C2211yescharacteristic
42#1845C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1825—21no—
46#1824—21no—
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
76#482—44no—
77#481—44no—
78C8×D4—22no—
79C8×D4—22no—
80#431—22no—
81#431—22no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8D411yescharacteristic
92C2×C4⋊C8D411yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8D411yescharacteristic
110C2×C22⋊C8D411yescharacteristic
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#390D411yescharacteristic
115#389D411yescharacteristic
116#388—21no—
117#386—21no—
118#384—21no—
119#384—21no—
120#384—21no—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#382—21no—
126#380—21no—
127#380—21no—
128#380D411yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—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#340—21no—
144#340—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#335—22no—
151#335—22no—
152C22×D4—21no—
153C22×D4C2×D411yescharacteristic
154C23×C4—21no—
155C23×C4—41no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4#4111yescharacteristic
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—42no—
166C22×C8—81no—
167C22×C8—81no—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172#121—22no—
173#121D814yes—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—81no—
179C4×D4—81no—
180#117—42no—
181#117—42no—
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—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—81no—
201C2×C4⋊C4—81no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—41no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—41no—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42C2411yescharacteristic
222#107—42no—
223#107—81no—
224#107—81no—
225#107—81no—
226#107—81no—
227#106—42no—
228#106—42no—
229#106—42no—
230C4⋊C8—22no—
231C4⋊C8—22no—
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—21no—
243C4⋊C8—21no—
244C4⋊C8—81no—
245#102—42no—
246#102—42no—
247#102—42no—
248#102—42no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254C22⋊C8—81no—
255C22⋊C8—81no—
256C22⋊C8—81no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—81no—
260C4×C8#4112yes—
261C4×C8#4112yes—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—41no—
266#95—41no—
267C24—21no—
268C24#12111yescharacteristic
269C2×D4—24no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—81no—
276C2×D4—81no—
277C2×D4C2×D812yes—
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—21no—
292C22×C4—21no—
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—41no—
304C22×C4—41no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4C2×D812yes—
312C22×C4#13512yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4111yescharacteristic
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—21no—
325C2×C8—21no—
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—41no—
343C2×C8—41no—
344C2×C8—84no—
345C2×C8#12312yes—
346C2×C8#12312yes—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
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—41no—
367C4⋊C4—41no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4—81no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4C2×D812yes—
377C22⋊C4—24no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—81no—
381C22⋊C4—81no—
382C22⋊C4—81no—
383C22⋊C4—82no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—41no—
391C42—41no—
392C42—82no—
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C422−1+411yescharacteristic
396C422−1+411yescharacteristic
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—22no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23#46612yes—
411C23C2×#12111yescharacteristic
412D4—22no—
413D4—22no—
414D4—41no—
415D4—41no—
416D4—84no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—42no—
433C2×C4—42no—
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—41no—
445C2×C4—41no—
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—81no—
461C2×C4—81no—
462C2×C4—164no—
463C2×C4#46612yes—
464C2×C4C2×#12311yescharacteristic
465C2×C4#53611yescharacteristic
466C2×C4#54711yescharacteristic
467C2×C4#54711yescharacteristic
468C2×C4#55211yescharacteristic
469C2×C4#55211yescharacteristic
470C2×C4C22×D811yescharacteristic
471C2×C4C2×#13511yescharacteristic
472C2×C4#57611yescharacteristic
473C2×C4#57711yescharacteristic
474C2×C4#57711yescharacteristic
475C2×C4#57711yescharacteristic
476C2×C4#57811yescharacteristic
477C2×C4#57811yescharacteristic
478C8—22no—
479C8—22no—
480C8—41no—
481C8—41no—
482C8—81no—
483C8—81no—
484C8—81no—
485C8—81no—
486C8—81no—
487C8—81no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22#248611yescharacteristic
503C22#304211yescharacteristic
504C22#304511yescharacteristic
505C22C2×#46611yescharacteristic
506C22#305911yescharacteristic
507C22#306811yescharacteristic
508C22#306911yescharacteristic
509C4—22no—
510C4—21no—
511C4—21no—
512C4—41no—
513C4—41no—
514C4—42no—
515C4—81no—
516C4—81no—
517C4—161no—
518C4—161no—
519C4—161no—
520C4—161no—
521C4#309611yescharacteristic
522C4#309911yescharacteristic
523C4#324111yescharacteristic
524C4#324411yescharacteristic
525C4#333411yescharacteristic
526C4#333411yescharacteristic
527C4#333611yescharacteristic
528C4#333611yescharacteristic
529C2—42no—
530C2—82no—
531C2#1939211yescharacteristic
532C2#1941011yescharacteristic
533C2#1951511yescharacteristic
534C2#1953911yescharacteristic
535C2#2765411yescharacteristic
536C2#2766111yescharacteristic
537C2#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. #10655C2semidirect
6. #10651C2semidirect
7. #10621C2semidirect
8. #10615C2semidirect
9. #9616C2semidirect
10. #9615C2semidirect
11. #8726C2semidirect
12. #8723C2semidirect
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
#276462
#276452
#221332
#219822
#106552
#106512
#106212
#106152
#96162
#96152
#87262
#87232
#86482
#86442
#85992