Finite Group Databasebeta

← #203594

Subgroups — #203594

Order 512.

658 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21972C211yesmaximal, characteristic
2#21954C211yesmaximal, characteristic
3#21362C211yesmaximal, characteristic
4C2×#1573C211yesmaximal, characteristic
5C2×#1454C211yesmaximal, characteristic
6#9600C211yesmaximal, characteristic, complemented
7#9599C211yesmaximal, characteristic, complemented
8#9598C211yesmaximal, characteristic, complemented
9#9597C211yesmaximal, characteristic, complemented
10#9596C211yesmaximal, characteristic, complemented
11#9595C211yesmaximal, characteristic, complemented
12#9594C211yesmaximal, characteristic, complemented
13#9593C211yesmaximal, characteristic, complemented
14#7902C211yesmaximal, characteristic
15#7893C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2285C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1948C411yescharacteristic
24#1948C411yescharacteristic
25#1947C411yescharacteristic
26#1947C411yescharacteristic
27#1940—21no—
28#1940—21no—
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1939C2211yescharacteristic
36#1939C2211yescharacteristic
37#1854C411yescharacteristic
38#1854C411yescharacteristic
39#1854C411yescharacteristic
40#1854C411yescharacteristic
41#1850—21no—
42#1850—21no—
43#1842—21no—
44#1831—21no—
45#1798—21no—
46#1798—21no—
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C4×#106C2211yescharacteristic
50C4×#106C2211yescharacteristic
51#1573—24no—
52C2×#335C2211yescharacteristic
53C2×#335C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×#334C2211yescharacteristic
56#1454—24no—
57C2×#322C2211yescharacteristic
58C2×#322C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69C23×C8—21no—
70C2×#117—21no—
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C22×C42C2311yescharacteristic
74#432—22no—
75#432—22no—
76#431—22no—
77#431—22no—
78#431—22no—
79#431D414yes—
80#431Q814yes—
81#431Q814yes—
82C2×#107C2×C411yescharacteristic
83C2×#107C2×C411yescharacteristic
84C2×#107C2×C411yescharacteristic
85C2×#107C2×C411yescharacteristic
86C2×#106C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88C2×#106C2×C411yescharacteristic
89C2×#106C2×C411yescharacteristic
90C2×C4⋊C8—21no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8C2×C411yescharacteristic
103C2×C4⋊C8C2×C411yescharacteristic
104C2×C4⋊C8D411yescharacteristic
105C2×C4⋊C8D411yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8C2×C411yescharacteristic
113C2×C22⋊C8C2×C411yescharacteristic
114C2×C22⋊C8D411yescharacteristic
115C2×C22⋊C8D411yescharacteristic
116C2×#97—21no—
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#384C2×C411yescharacteristic
124#384C2×C411yescharacteristic
125#382—21no—
126#380—21no—
127#380C2×C411yescharacteristic
128#380C2×C411yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138#376—21no—
139C43—21no—
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#335—22no—
157#335—22no—
158#334—22no—
159#334—22no—
160#322—22no—
161#322—22no—
162#322—22no—
163C82—22no—
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4#4111yescharacteristic
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—42no—
174C22×C8—81no—
175C22×C8—81no—
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184#117—22no—
185#117—22no—
186#117—41no—
187#117—41no—
188#117—81no—
189#117—81no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×Q812yes—
218C2×C42C2×Q812yes—
219C2×C42#4112yes—
220C2×C42#4111yescharacteristic
221C2×C42C2411yescharacteristic
222#107—22no—
223#107—22no—
224#107—41no—
225#107—41no—
226#106—22no—
227#106—22no—
228#106—41no—
229#106—41no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—24no—
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—42no—
242C4⋊C8—81no—
243C4⋊C8—81no—
244C4⋊C8—81no—
245C4⋊C8—81no—
246C4⋊C8—81no—
247C4⋊C8—81no—
248C4⋊C8—81no—
249C4⋊C8—81no—
250C4⋊C8C4⋊C414yes—
251C4⋊C8C4⋊C414yes—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—24no—
255C22⋊C8—42no—
256C22⋊C8C4⋊C414yes—
257C22⋊C8C4⋊C414yes—
258#97—42no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8C2×D412yes—
266C4×C8C2×D412yes—
267C4×C8C2×Q812yes—
268C4×C8C2×Q812yes—
269C4×C8C2×Q812yes—
270C4×C8C2×Q812yes—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279C24C4×D411yescharacteristic
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—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4C2×C4⋊C412yes—
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4C4×Q812yes—
303C22×C4#12112yes—
304C22×C4C22⋊Q812yes—
305C22×C4#12512yes—
306C22×C4C4⋊Q812yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
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—22no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
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—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—84no—
349C2×C8C2×C4⋊C412yes—
350C2×C8C2×C4⋊C412yes—
351C2×C8C2×C4⋊C412yes—
352C2×C8C2×C4⋊C412yes—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C4⋊C4#13112yes—
372C4⋊C4#13112yes—
373C22⋊C4—24no—
374C22⋊C4—42no—
375C22⋊C4—81no—
376C22⋊C4—81no—
377C42—24no—
378C42—42no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—81no—
388C42—81no—
389C42C2×C4⋊C412yes—
390C42C4×Q812yes—
391C42#12112yes—
392C42C22⋊Q812yes—
393C42#12512yes—
394C42C4⋊Q812yes—
395C42C22×D411yescharacteristic
396C42C22×Q811yescharacteristic
397C42C22×Q811yescharacteristic
398C42C2×#4111yescharacteristic
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23#38912yes—
406C23C2×C4×D411yescharacteristic
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
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—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4#38912yes—
450C2×C4#38912yes—
451C2×C4#38912yes—
452C2×C4C22×C4⋊C411yescharacteristic
453C2×C4C2×C4×Q811yescharacteristic
454C2×C4C2×#12111yescharacteristic
455C2×C4C2×C22⋊Q811yescharacteristic
456C2×C4C2×#12511yescharacteristic
457C2×C4C2×C4⋊Q811yescharacteristic
458C2×C4C2×#13111yescharacteristic
459C2×C4C2×#13111yescharacteristic
460C2×C4#56811yescharacteristic
461C2×C4#56811yescharacteristic
462C2×C4#57611yescharacteristic
463C2×C4#57711yescharacteristic
464C2×C4#57711yescharacteristic
465C2×C4#57811yescharacteristic
466C8—22no—
467C8—22no—
468C8—22no—
469C8—22no—
470C8—42no—
471C8—42no—
472C8—41no—
473C8—41no—
474C8—41no—
475C8—41no—
476C8—81no—
477C8—81no—
478C8—81no—
479C8—81no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22C2×#38911yescharacteristic
488C22#292911yescharacteristic
489C22#293011yescharacteristic
490C22#294611yescharacteristic
491C22#294711yescharacteristic
492C22#295711yescharacteristic
493C22#295711yescharacteristic
494C4—22no—
495C4—22no—
496C4—21no—
497C4—21no—
498C4—22no—
499C4—42no—
500C4—42no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4#290611yescharacteristic
508C4#290611yescharacteristic
509C4#296211yescharacteristic
510C4#296211yescharacteristic
511C4#306211yescharacteristic
512C4#306311yescharacteristic
513C4#316311yescharacteristic
514C4#316411yescharacteristic
515C2—42no—
516C2#1792011yescharacteristic
517C2#1792111yescharacteristic
518C2#1794811yescharacteristic
519C2#1794911yescharacteristic
520C2#2114711yescharacteristic
521C2#2116011yescharacteristic
522C2#2116311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
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
6. #9600C2semidirect
7. #9599C2semidirect
8. #9598C2semidirect
9. #9597C2semidirect
10. #9596C2semidirect
11. #9595C2semidirect
12. #9594C2semidirect
13. #9593C2semidirect

Maximal subgroups

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

subgroupindex
#219722
#219542
#213622
C2×#15732
C2×#14542
#96002
#95992
#95982
#95972
#95962
#95952
#95942
#95932
#79022
#78932