Finite Group Databasebeta

← #274747

Subgroups — #274747

Order 512.

749 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32906C211yesmaximal, characteristic
2#27877C211yesmaximal, characteristic
3Q8×C22⋊C8C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5#10623C211yesmaximal, characteristic, complemented
6#10617C211yesmaximal, characteristic, complemented
7#10478C211yesmaximal, characteristic, complemented
8#10478C211yesmaximal, characteristic, complemented
9#9218C211yesmaximal, characteristic, complemented
10#9218C211yesmaximal, characteristic, complemented
11#9139C211yesmaximal, characteristic
12#8832C211yesmaximal, characteristic
13#8726C211yesmaximal, characteristic
14#8649C211yesmaximal, characteristic, complemented
15#8645C211yesmaximal, characteristic, complemented
16C22×C4×Q8C2211yescharacteristic
17C2×#462C2211yescharacteristic
18C2×#462C2211yescharacteristic
19C2×#451—21no—
20C2×C8×Q8C2211yescharacteristic
21C2×C8×Q8C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#419—21no—
24#2857C2211yescharacteristic
25#2285C2211yescharacteristic
26#2075—21no—
27#2075—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1984—21no—
31#1984—21no—
32#1984C2211yescharacteristic
33#1984C2211yescharacteristic
34#1918—21no—
35#1918—21no—
36#1878—21no—
37#1878—21no—
38#1872—21no—
39#1845C2211yescharacteristic
40#1834C2211yescharacteristic
41#1805—21no—
42#1805—21no—
43#1803C2211yescharacteristic
44#1803C2211yescharacteristic
45#1803C2211yescharacteristic
46#1803C2211yescharacteristic
47#1803C2211yescharacteristic
48#1803C2211yescharacteristic
49#1797—21no—
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52C4×C22⋊C8—21no—
53C4×C22⋊C8C2211yescharacteristic
54#1736—21no—
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×Q8D411yescharacteristic
68C22×Q16—21no—
69C22×Q16—21no—
70C22×M16—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×C22⋊Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—41no—
78C2×C4×Q8—41no—
79C2×C4×Q8D411yescharacteristic
80C2×C4×Q8D411yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82C2×C4×Q8C2311yescharacteristic
83C2×#117—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D411yescharacteristic
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#462—22no—
89#462—22no—
90#462—41no—
91#462—41no—
92#451—44no—
93C8×Q8—22no—
94C8×Q8—22no—
95C8×Q8—41no—
96C8×Q8—41no—
97C2×#107—21no—
98C2×#107—21no—
99#423—41no—
100#423—41no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110#419—44no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×#103—21no—
115C2×#103—21no—
116C2×#103—21no—
117C2×#103—21no—
118C2×#103—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8D411yescharacteristic
122C2×C22⋊C8D411yescharacteristic
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#391—21no—
128#389—21no—
129#389—21no—
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#382—21no—
134#380—21no—
135#380—21no—
136#380—21no—
137#380—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336D411yescharacteristic
159#336D411yescharacteristic
160#335—41no—
161#335—41no—
162#332—41no—
163#332—41no—
164C22×Q8—21no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—21no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8D814yes—
172C22×Q8C2×D411yescharacteristic
173C22×Q8C2×D411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4#4111yescharacteristic
179C2×Q16—42no—
180C2×Q16—42no—
181C2×Q16—42no—
182C2×Q16—42no—
183C2×Q16—42no—
184C2×Q16—42no—
185C2×M16—42no—
186C2×M16—42no—
187C2×M16—42no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8—42no—
213C4×Q8—42no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—42no—
217C4×Q8—42no—
218C4×Q8#4112yes—
219C4×Q8#4112yes—
220#117—42no—
221#117—42no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—41no—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42C2411yescharacteristic
262#107—42no—
263#107—42no—
264C4⋊C8—21no—
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—42no—
277C4⋊C8—42no—
278C4⋊C8—42no—
279#103—42no—
280#103—42no—
281#103—42no—
282#103—42no—
283#103—42no—
284#103—42no—
285#103—42no—
286#103—42no—
287C22⋊C8—22no—
288C22⋊C8—41no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—42no—
296C4×C8—42no—
297#95—41no—
298#95—41no—
299#95—41no—
300#95—41no—
301C24#12111yescharacteristic
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—24no—
307C2×Q8—28no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8#12312yes—
324C2×Q8C2×D812yes—
325C2×Q8C2×D812yes—
326C2×Q8#13712yes—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—81no—
355C22×C4#12111yescharacteristic
356C22×C4#12111yescharacteristic
357C22×C4#12111yescharacteristic
358C22×C4C2×D812yes—
359C22×C4#13512yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4111yescharacteristic
363Q16—84no—
364Q16—84no—
365M16—84no—
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—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
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#13712yes—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—81no—
423C4⋊C4—81no—
424C4⋊C4—81no—
425C4⋊C4—82no—
426C4⋊C4#12312yes—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C42—21no—
430C42—21no—
431C42—22no—
432C42—22no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—42no—
438C42—42no—
439C42—41no—
440C42—41no—
441C42—82no—
442C42C2×#4111yescharacteristic
443C42C2×#4111yescharacteristic
444C42H(5,𝔽2)11yescharacteristic
445C42H(5,𝔽2)11yescharacteristic
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23#46612yes—
453C23C2×#12111yescharacteristic
454Q8—22no—
455Q8—22no—
456Q8—22no—
457Q8—22no—
458Q8—22no—
459Q8—48no—
460Q8—41no—
461Q8—41no—
462Q8—81no—
463Q8—81no—
464Q8—81no—
465Q8—81no—
466Q8#48014yes—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—48no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
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—81no—
499C2×C4—81no—
500C2×C4—84no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4#46612yes—
514C2×C4C2×#12311yescharacteristic
515C2×C4#53411yescharacteristic
516C2×C4#54611yescharacteristic
517C2×C4#54611yescharacteristic
518C2×C4#55111yescharacteristic
519C2×C4#55111yescharacteristic
520C2×C4C22×D811yescharacteristic
521C2×C4C2×#13511yescharacteristic
522C2×C4C2×#13711yescharacteristic
523C2×C4C2×#13711yescharacteristic
524C2×C4#57511yescharacteristic
525C2×C4#57511yescharacteristic
526C2×C4#57811yescharacteristic
527C2×C4#57811yescharacteristic
528C8—22no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—81no—
533C8—81no—
534C8—81no—
535C8—81no—
536C8—81no—
537C8—81no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22#241211yescharacteristic
546C22#303711yescharacteristic
547C22#304411yescharacteristic
548C22C2×#46611yescharacteristic
549C22#305611yescharacteristic
550C22#306711yescharacteristic
551C22#306911yescharacteristic
552C4—22no—
553C4—24no—
554C4—21no—
555C4—21no—
556C4—41no—
557C4—42no—
558C4—41no—
559C4—42no—
560C4—81no—
561C4—161no—
562C4—161no—
563C4—161no—
564C4—161no—
565C4C2×#48011yescharacteristic
566C4#309111yescharacteristic
567C4#318611yescharacteristic
568C4#319011yescharacteristic
569C4#331211yescharacteristic
570C4#331311yescharacteristic
571C4#332711yescharacteristic
572C4#332811yescharacteristic
573C2—42no—
574C2#1926111yescharacteristic
575C2#1931111yescharacteristic
576C2#1940411yescharacteristic
577C2#1947811yescharacteristic
578C2#2454011yescharacteristic
579C2#2455011yescharacteristic
580C2#2457411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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
5. #10623C2semidirect
6. #10617C2semidirect
7. #10478C2semidirect
8. #10478C2semidirect
9. #9218C2semidirect
10. #9218C2semidirect
14. #8649C2semidirect
15. #8645C2semidirect

Maximal subgroups

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

subgroupindex
#329062
#278772
Q8×C22⋊C82
#206472
#106232
#106172
#104782
#92182
#91392
#88322
#87262
#86492
#86452