Finite Group Databasebeta

← #218933

Subgroups — #218933

Order 512.

696 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21597C211yesmaximal, characteristic
2C8×C22⋊Q8C211yesmaximal, characteristic
3C4×#467C211yesmaximal, characteristic
4#9547C211yesmaximal, characteristic, complemented
5#9543C211yesmaximal, characteristic, complemented
6#9537C211yesmaximal, characteristic, complemented
7#9533C211yesmaximal, characteristic, complemented
8#9071C211yesmaximal, characteristic
9#9071C211yesmaximal, characteristic
10#8950C211yesmaximal, characteristic, complemented
11#8946C211yesmaximal, characteristic, complemented
12#8850C211yesmaximal, characteristic, complemented
13#8846C211yesmaximal, characteristic, complemented
14#8595C211yesmaximal, characteristic
15C2×#1273C211yesmaximal, characteristic
16C2×#467C411yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24C4×C22⋊Q8C2211yescharacteristic
25#1984C2211yescharacteristic
26#1984C2211yescharacteristic
27#1969C411yescharacteristic
28#1942C411yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1933C411yescharacteristic
32#1933C411yescharacteristic
33#1916—21no—
34#1916—21no—
35#1892C411yescharacteristic
36#1892C411yescharacteristic
37#1850—21no—
38#1850—21no—
39#1808C411yescharacteristic
40#1797C2211yescharacteristic
41#1797C2211yescharacteristic
42C4×#107C2211yescharacteristic
43#1771—21no—
44#1770C2211yescharacteristic
45C8×C4⋊C4—21no—
46C8×C4⋊C4C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#103C2211yescharacteristic
49#1752—21no—
50C8×C22⋊C4—21no—
51C2×#334C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×Q16C2×C411yescharacteristic
68C23×C8—21no—
69C23×C8C2×C411yescharacteristic
70C23×C8C2×C411yescharacteristic
71C23×C8D411yescharacteristic
72C23×C8D411yescharacteristic
73C2×C22⋊Q8C2×C411yescharacteristic
74C2×C22⋊Q8C2×C411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#467—24no—
79#446—41no—
80#446—41no—
81C8×Q8—41no—
82C8×Q8—41no—
83C4×Q16—41no—
84C4×Q16—41no—
85C2×#107C2×C411yescharacteristic
86C2×#107C2×C411yescharacteristic
87C2×#107C2×C411yescharacteristic
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#103C2×C411yescharacteristic
97C2×#103C2×C411yescharacteristic
98C2×#103C2×C411yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—41no—
110C2×C4×C8—41no—
111C2×C4×C8—41no—
112C2×C4×C8—41no—
113C2×C4×C8—41no—
114C2×C4×C8—41no—
115C2×C4×C8C2×C411yescharacteristic
116C2×C4×C8C2×C411yescharacteristic
117C2×C4×C8D411yescharacteristic
118C2×C4×C8D411yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#391C2×C411yescharacteristic
126#391C2×C411yescharacteristic
127#389C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#387—21no—
130#387—21no—
131#385—21no—
132#385—21no—
133#380C2×C411yescharacteristic
134#380C2×C411yescharacteristic
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2311yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#336—21no—
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#336—21no—
157#334—41no—
158#334—41no—
159Q8⋊C8—41no—
160Q8⋊C8—41no—
161Q8⋊C8—41no—
162Q8⋊C8—41no—
163C22×Q8C22×C411yescharacteristic
164C22×Q8C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4#4111yescharacteristic
168C2×Q16—22no—
169C2×Q16—22no—
170C2×Q16—41no—
171C22×C8—24no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—41no—
187C22×C8—41no—
188C22×C8—41no—
189C22×C8—41no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8C22×C411yescharacteristic
193C22×C8C22×C411yescharacteristic
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8M1614yes—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42C2411yescharacteristic
240#107—22no—
241#107—22no—
242#107—41no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—42no—
252C4⋊C8—42no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255#103—22no—
256#103—22no—
257#103—22no—
258#103—22no—
259#103—41no—
260#103—41no—
261C22⋊C8—42no—
262C22⋊C8—42no—
263#97—42no—
264#97—42no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—41no—
278C4×C8—41no—
279C4×C8—41no—
280C4×C8—41no—
281C4×C8—42no—
282C4×C8—42no—
283C4×C8—42no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294C24C4×D411yescharacteristic
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8C2×M1612yes—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
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×C4C4×D411yescharacteristic
328C22×C4C4×D411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C2×M1612yes—
331C22×C4#13112yes—
332C22×C4#13712yes—
333C22×C4#13712yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C2×#4111yescharacteristic
337Q16—44no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—44no—
361C2×C8—44no—
362C2×C8—44no—
363C2×C8—44no—
364C2×C8—48no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8#13112yes—
382C2×C8#13112yes—
383C2×C8#13512yes—
384C2×C8#13512yes—
385C4⋊C4—21no—
386C4⋊C4—24no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4C2×M1612yes—
405C22⋊C4—24no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42H(5,𝔽2)11yescharacteristic
424C42H(5,𝔽2)11yescharacteristic
425C42H(5,𝔽2)11yescharacteristic
426C422−1+411yescharacteristic
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23#43512yes—
432C23#44112yes—
433C23#44312yes—
434C23C2×C4×D411yescharacteristic
435Q8—22no—
436Q8—22no—
437Q8—41no—
438Q8—41no—
439Q8—41no—
440Q8—41no—
441Q8—41no—
442Q8—41no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4#43512yes—
486C2×C4#44112yes—
487C2×C4#44412yes—
488C2×C4#51811yescharacteristic
489C2×C4#52011yescharacteristic
490C2×C4#53411yescharacteristic
491C2×C4#53711yescharacteristic
492C2×C4#53811yescharacteristic
493C2×C4#53911yescharacteristic
494C2×C4C22×M1611yescharacteristic
495C2×C4C2×#13111yescharacteristic
496C2×C4C2×#13111yescharacteristic
497C2×C4C2×#13111yescharacteristic
498C2×C4C2×#13511yescharacteristic
499C2×C4C2×#13511yescharacteristic
500C2×C4C2×#13711yescharacteristic
501C2×C4C2×#13711yescharacteristic
502C8—22no—
503C8—22no—
504C8—22no—
505C8—22no—
506C8—42no—
507C8—42no—
508C8—41no—
509C8—41no—
510C8—41no—
511C8—41no—
512C8—81no—
513C8—81no—
514C8—81no—
515C8—81no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22#158714yes—
523C22#232511yescharacteristic
524C22C2×#43511yescharacteristic
525C22#293111yescharacteristic
526C22C4×#13511yescharacteristic
527C22C2×#44111yescharacteristic
528C22C2×#44311yescharacteristic
529C22C2×#44411yescharacteristic
530C4—22no—
531C4—22no—
532C4—21no—
533C4—21no—
534C4—22no—
535C4—42no—
536C4—42no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4—81no—
543C4#297411yescharacteristic
544C4#297711yescharacteristic
545C4#299011yescharacteristic
546C4#299411yescharacteristic
547C4#319011yescharacteristic
548C4#319011yescharacteristic
549C4#324811yescharacteristic
550C4#324811yescharacteristic
551C2—24no—
552C2#1786311yescharacteristic
553C2C2×#158711yescharacteristic
554C2#1795911yescharacteristic
555C2#1796111yescharacteristic
556C2#2217211yescharacteristic
557C2#2218311yescharacteristic
558C2#2220411yescharacteristic
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 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9547C2semidirect
5. #9543C2semidirect
6. #9537C2semidirect
7. #9533C2semidirect
10. #8950C2semidirect
11. #8946C2semidirect
12. #8850C2semidirect
13. #8846C2semidirect

Maximal subgroups

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

subgroupindex
#215972
C8×C22⋊Q82
C4×#4672
#95472
#95432
#95372
#95332
#90712
#89502
#89462
#88502
#88462
#85952
C2×#12732