Finite Group Databasebeta

← #6458409

Subgroups — #6458409

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#27971C218yesmaximal, complemented
4#27962C212yesmaximal, complemented
5#27962C212yesmaximal, complemented
6#27961C212yesmaximal, complemented
7#27960C212yesmaximal, complemented
8#23748C214yesmaximal, complemented
9#21324C211yesmaximal, characteristic, complemented
10C2×#1990C211yesmaximal, characteristic, complemented
11#20764C211yesmaximal, characteristic, complemented
12#20760C211yesmaximal, characteristic, complemented
13#20703C214yesmaximal
14C4×#408C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C2×#520C2212yes—
17C2×C4×#41C2211yescharacteristic
18C2×#408C2212yes—
19C2×C4×M16C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2857C2212yes—
22#2857C2212yes—
23#2628C2214yes—
24#2568—21no—
25#2567—21no—
26#2563—21no—
27#2542—21no—
28#2487—22no—
29#2409—22no—
30Q8×C4⋊C4—21no—
31Q8×C4⋊C4—21no—
32D4×C4⋊C4—21no—
33D4×C4⋊C4—22no—
34D4×C4⋊C4—22no—
35D4×C4⋊C4—21no—
36#2317C2212yes—
37#2317C2214yes—
38#2317C2212yes—
39#2311C2212yes—
40#2285C2211yescharacteristic
41#2285C2211yescharacteristic
42#2285C2212yes—
43C2×#384C2211yescharacteristic
44C2×#384C2212yes—
45#2101C2218yes—
46#2101C2218yes—
47#2100C2218yes—
48#2099C2218yes—
49#1993C2214yes—
50#1990C2212yes—
51#1990C2212yes—
52#1990C2212yes—
53#1990C2212yes—
54#1990C2214yes—
55#1836C2214yes—
56#1835C2214yes—
57#1834C2211yescharacteristic
58#1834C2211yescharacteristic
59#1834C2211yescharacteristic
60#1834C2211yescharacteristic
61#1834C2212yes—
62#1834C2212yes—
63#1825C2218yes—
64#1825C2218yes—
65#1824C2218yes—
66#1823C2218yes—
67#1750C2214yes—
68C4×C22⋊C8C2212yes—
69C4×C22⋊C8C2212yes—
70#1736C2214yes—
71#1736C2214yes—
72#1735C2214yes—
73C2×#340C2214yes—
74#1729C2212yes—
75C22×#41C2311yescharacteristic
76C23×Q8—21no—
77C22×M16C2312yes—
78C23×C8C2312yes—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—22no—
87#520—216no—
88C4×#41—24no—
89C4×#41—24no—
90C2×C4×Q8—22no—
91C2×C4×Q8D411yescharacteristic
92C2×C4×Q8D411yescharacteristic
93C2×C4×D4—24no—
94C2×C4×D4—22no—
95C2×C4×D4D411yescharacteristic
96C2×C4×D4D412yes—
97C2×C4×D4D412yes—
98C2×C4×D4D411yescharacteristic
99C2×#117C2312yes—
100C2×#117C2314yes—
101C2×#117C2312yes—
102C2×#117C2311yescharacteristic
103C2×#117C2312yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#408—28no—
115C2×C22⋊C8C2314yes—
116C2×C22⋊C8C2314yes—
117C4×M16—24no—
118C2×#97C2312yes—
119C2×C4×C8—24no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2312yes—
123C2×C4×C8C2312yes—
124#400—24no—
125#399—24no—
126#395—24no—
127#393—24no—
128#391—22no—
129#391—22no—
130#391—22no—
131#390—22no—
132#389—22no—
133#389—22no—
134#389—24no—
135#389—24no—
136#389—22no—
137#389—22no—
138#389C2314yes—
139#389C2314yes—
140#386—24no—
141#385—24no—
142#385—24no—
143#385—24no—
144#385—24no—
145#384—22no—
146#384—22no—
147#384—22no—
148#384—21no—
149#384—21no—
150#384C2312yes—
151#384C2314yes—
152#384C2311yescharacteristic
153#384C2311yescharacteristic
154#384C2311yescharacteristic
155#384C2312yes—
156#384C2311yescharacteristic
157#382—22no—
158#382C2312yes—
159#382C2314yes—
160#382C2312yes—
161#380—22no—
162#380—22no—
163#380—22no—
164#380—22no—
165#380—21no—
166#380—21no—
167#380—21no—
168#380—21no—
169#380C2314yes—
170#380C2314yes—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177#340C2314yes—
178#340C2318yes—
179#340C2318yes—
180#340C2318yes—
181#340C2314yes—
182#340C2318yes—
183#340C2318yes—
184#340C2314yes—
185#340C2314yes—
186#340C2318yes—
187#336C2314yes—
188#336C2314yes—
189C2×#41—22no—
190C2×#41—22no—
191C2×#41—24no—
192C2×#41#4118yes—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—44no—
196C22×Q8—82no—
197C22×Q8—82no—
198C22×Q8C2×D411yescharacteristic
199C22×D4C2×D412yes—
200C22×D4C2×D411yescharacteristic
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4C2×D412yes—
205C23×C4C2×D412yes—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C2×M16—24no—
211C2×M16—24no—
212C2×M16—24no—
213C22×C8—24no—
214C22×C8—24no—
215C22×C8—24no—
216C22×C8C2412yes—
217C22×C8C2412yes—
218C22×C8C2414yes—
219C4⋊Q8—24no—
220C4⋊Q8—24no—
221C4⋊Q8—24no—
222C4⋊Q8—24no—
223C4⋊Q8—44no—
224C4⋊Q8—44no—
225C22⋊Q8—84no—
226C22⋊Q8—84no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—48no—
230C4×D4—28no—
231C4×D4—28no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—416no—
235C4×D4—48no—
236#117—22no—
237#117—22no—
238#117—24no—
239#117—24no—
240#117—216no—
241#117—28no—
242#117—28no—
243C2×C4⋊C4—216no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—82no—
270C2×C4⋊C4—84no—
271C2×C4⋊C4—82no—
272C2×C4⋊C4—84no—
273C2×C4⋊C4C2×D411yescharacteristic
274C2×C4⋊C4C2×D411yescharacteristic
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2414yes—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C22⋊C4C2×D412yes—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42#4114yes—
301C2×C42#4114yes—
302C2×C42C2412yes—
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C22⋊C8—28no—
306C22⋊C8—28no—
307#97—24no—
308C4×C8—22no—
309C4×C8—22no—
310C4×C8—24no—
311C4×C8—24no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322C24C22×D412yes—
323C24C22×D411yescharacteristic
324#41—216no—
325#41—216no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—82no—
333C2×Q8—82no—
334C2×Q8—816no—
335C2×Q8#12414yes—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4#12418yes—
341C2×D4#12414yes—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—44no—
362C22×C4—48no—
363C22×C4—44no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—84no—
368C22×C4—82no—
369C22×C4—84no—
370C22×C4#12314yes—
371C22×C4#12314yes—
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D412yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C2×#4118yes—
381C22×C4C2×#4112yes—
382C22×C4C2×#4112yes—
383C22×C4C2×#4114yes—
384C22×C42−1+414yes—
385C22×C4C2511yescharacteristic
386M16—48no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—24no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—24no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—24no—
396C2×C8—48no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—28no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—48no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C4⋊C4—84no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—48no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C42—24no—
438C42—24no—
439C42—22no—
440C42—22no—
441C42—48no—
442C42—44no—
443C42—44no—
444C42#12314yes—
445C42#12314yes—
446C42C2×#4112yes—
447C42C2×#4112yes—
448C42C2×#4112yes—
449C42C2×#4112yes—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23C2×#12312yes—
461C23C2×#12414yes—
462C23C2×#12412yes—
463C23#53612yes—
464C23C23×D411yescharacteristic
465Q8—28no—
466Q8—88no—
467Q8—88no—
468D4—216no—
469D4—28no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—48no—
487C2×C4—48no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—88no—
507C2×C4—164no—
508C2×C4—168no—
509C2×C4C2×#12312yes—
510C2×C4C2×#12312yes—
511C2×C4C2×#12312yes—
512C2×C4C2×#12312yes—
513C2×C4C2×#12312yes—
514C2×C4C2×#12414yes—
515C2×C4C2×#12412yes—
516C2×C4#53612yes—
517C2×C4#55218yes—
518C2×C4#55218yes—
519C2×C4#57612yes—
520C2×C4#57712yes—
521C2×C4#57712yes—
522C2×C4#57812yes—
523C2×C4C22×#4111yescharacteristic
524C2×C4C22×#4111yescharacteristic
525C2×C4C22×#4112yes—
526C2×C4C2×2−1+412yes—
527C8—42no—
528C8—42no—
529C8—44no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—24no—
537C22—24no—
538C22—24no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22#250314yes—
544C22C22×#12311yescharacteristic
545C22C22×#12411yescharacteristic
546C22C2×#53611yescharacteristic
547C22C2×#57611yescharacteristic
548C22C2×#57711yescharacteristic
549C22C2×#57711yescharacteristic
550C22C2×#57811yescharacteristic
551C4—22no—
552C4—22no—
553C4—24no—
554C4—42no—
555C4—42no—
556C4—84no—
557C4—84no—
558C4—82no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4#250314yes—
563C4#309614yes—
564C4#309714yes—
565C4C2×#55212yes—
566C4C2×#55212yes—
567C2—24no—
568C2—44no—
569C2C2×#250311yescharacteristic
570C2C2×#309611yescharacteristic
571C2C2×#309711yescharacteristic
572C2#6062211yescharacteristic
573C2#6062311yescharacteristic
574C2#6088011yescharacteristic
575C2#6088111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27971C2semidirect
4. #27962C2semidirect
5. #27962C2semidirect
6. #27961C2semidirect
7. #27960C2semidirect
8. #23748C2semidirect
9. #21324C2semidirect
10. C2×#1990C2semidirect
11. #20764C2semidirect
12. #20760C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#341832
#279712
#279622
#279612
#279602
#237482
#213242
C2×#19902
#207642
#207602
#207032
C4×#4082