Finite Group Databasebeta

← #6458407

Subgroups — #6458407

Order 512.

1786 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#27970C218yesmaximal, complemented
4#27962C212yesmaximal, complemented
5#27962C212yesmaximal, complemented
6#27961C212yesmaximal, complemented
7#27960C212yesmaximal, complemented
8#23749C214yesmaximal, complemented
9#21322C211yesmaximal, characteristic, complemented
10#21318C211yesmaximal, characteristic, complemented
11#20764C211yesmaximal, characteristic, complemented
12C2×#1834C211yesmaximal, characteristic, complemented
13#20703C214yesmaximal
14C4×#408C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C2×#520C2212yes—
17C2×C4×#41C2211yescharacteristic
18C2×#408C2212yes—
19C2×C4×M16C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2857C2212yes—
22#2857C2212yes—
23#2629C2214yes—
24#2582—21no—
25#2582—21no—
26#2538—21no—
27#2538—21no—
28#2417—22no—
29#2417—22no—
30Q8×C4⋊C4—21no—
31Q8×C4⋊C4—21no—
32D4×C4⋊C4—22no—
33D4×C4⋊C4—22no—
34D4×C4⋊C4—21no—
35D4×C4⋊C4—21no—
36#2317C2214yes—
37#2317C2212yes—
38#2316C2212yes—
39#2311C2212yes—
40#2285C2211yescharacteristic
41#2285C2212yes—
42C2×#384C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×#384C2212yes—
45#2101C2218yes—
46#2101C2218yes—
47#2100C2218yes—
48#2099C2218yes—
49#1992C2214yes—
50#1991C2214yes—
51#1990C2211yescharacteristic
52#1990C2212yes—
53#1990C2211yescharacteristic
54#1990C2212yes—
55#1990C2211yescharacteristic
56#1990C2211yescharacteristic
57#1836C2214yes—
58#1834C2212yes—
59#1834C2212yes—
60#1834C2214yes—
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
76C22×M16C2312yes—
77C23×C8C2312yes—
78#533—24no—
79#533—24no—
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×#125C2312yes—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—22no—
87#520—216no—
88C4×#41—24no—
89C4×#41—24no—
90C2×C4×Q8—22no—
91C2×C4×Q8—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8D411yescharacteristic
94C2×C4×Q8D411yescharacteristic
95C2×C4×D4—24no—
96C2×C4×D4—22no—
97C2×C4×D4D412yes—
98C2×C4×D4D412yes—
99C2×C4×D4D411yescharacteristic
100C2×C4×D4D411yescharacteristic
101C2×#117C2312yes—
102C2×#117C2314yes—
103C2×#117C2312yes—
104C2×#117C2311yescharacteristic
105C2×#117C2312yes—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#408—28no—
118C2×C22⋊C8C2314yes—
119C2×C22⋊C8C2314yes—
120C4×M16—24no—
121C2×#97C2312yes—
122C2×C4×C8—24no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2312yes—
126C2×C4×C8C2312yes—
127#398—24no—
128#398—24no—
129#396—24no—
130#396—24no—
131#391—22no—
132#391—22no—
133#391—22no—
134#390—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—24no—
140#389—24no—
141#389—22no—
142#389—22no—
143#389C2314yes—
144#389C2314yes—
145#386—24no—
146#385—24no—
147#385—24no—
148#385—24no—
149#384—21no—
150#384—21no—
151#384—22no—
152#384—22no—
153#384—21no—
154#384—21no—
155#384C2312yes—
156#384C2314yes—
157#384C2312yes—
158#384C2312yes—
159#384C2312yes—
160#384C2311yescharacteristic
161#384C2311yescharacteristic
162#382C2312yes—
163#382C2314yes—
164#380—22no—
165#380—22no—
166#380—22no—
167#380—22no—
168#380—21no—
169#380—21no—
170#380—24no—
171#380C2314yes—
172#380C2314yes—
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#340C2318yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2314yes—
189C2×#41—22no—
190C2×#41—22no—
191C2×#41—24no—
192C2×#41#4118yes—
193C22×Q8—21no—
194C22×Q8—21no—
195C22×Q8C2×D411yescharacteristic
196C22×D4C2×D412yes—
197C22×D4C2×D411yescharacteristic
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4C2×D412yes—
202C23×C4C2×D412yes—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C2×M16—24no—
208C2×M16—24no—
209C2×M16—24no—
210C22×C8—24no—
211C22×C8—24no—
212C22×C8—24no—
213C22×C8C2412yes—
214C22×C8C2412yes—
215C22×C8C2414yes—
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220#125—24no—
221#125—24no—
222C22⋊Q8—44no—
223C22⋊Q8—44no—
224C22⋊Q8—84no—
225C22⋊Q8—84no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—48no—
229C4×Q8—44no—
230C4×Q8—44no—
231C4×D4—28no—
232C4×D4—28no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—416no—
236C4×D4—48no—
237#117—22no—
238#117—22no—
239#117—24no—
240#117—24no—
241#117—216no—
242#117—28no—
243#117—24no—
244#117—24no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—82no—
272C2×C4⋊C4—84no—
273C2×C4⋊C4—82no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4—84no—
276C2×C4⋊C4—82no—
277C2×C4⋊C4C2×D411yescharacteristic
278C2×C4⋊C4C2×D411yescharacteristic
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2414yes—
283C2×C4⋊C4C2414yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4C2×D412yes—
293C2×C22⋊C4C2×D412yes—
294C2×C22⋊C4C2×D412yes—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42#4114yes—
305C2×C42#4114yes—
306C2×C42C2412yes—
307C2×C42C2411yescharacteristic
308C2×C42C2411yescharacteristic
309C22⋊C8—28no—
310C22⋊C8—28no—
311#97—24no—
312C4×C8—22no—
313C4×C8—22no—
314C4×C8—24no—
315C4×C8—24no—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324C24C22×D412yes—
325C24C22×D411yescharacteristic
326#41—216no—
327#41—216no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8#12414yes—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4#12418yes—
342C2×D4#12414yes—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—48no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—84no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4—84no—
371C22×C4#12314yes—
372C22×C4#12314yes—
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D412yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4118yes—
382C22×C4C2×#4112yes—
383C22×C4C2×#4112yes—
384C22×C4C2×#4114yes—
385C22×C42−1+414yes—
386C22×C4C2511yescharacteristic
387M16—48no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—24no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—24no—
397C2×C8—48no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—28no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—28no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—48no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—82no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4—84no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—44no—
437C22⋊C4—48no—
438C22⋊C4—44no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C42—24no—
442C42—24no—
443C42—22no—
444C42—22no—
445C42—44no—
446C42—48no—
447C42—44no—
448C42#12314yes—
449C42#12314yes—
450C42C2×#4112yes—
451C42C2×#4112yes—
452C42C2×#4112yes—
453C42C2×#4112yes—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23C2×#12312yes—
465C23C2×#12414yes—
466C23C2×#12412yes—
467C23#53612yes—
468C23C23×D411yescharacteristic
469Q8—28no—
470Q8—84no—
471Q8—84no—
472D4—216no—
473D4—28no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—164no—
511C2×C4—164no—
512C2×C4—168no—
513C2×C4C2×#12312yes—
514C2×C4C2×#12312yes—
515C2×C4C2×#12312yes—
516C2×C4C2×#12312yes—
517C2×C4C2×#12312yes—
518C2×C4C2×#12414yes—
519C2×C4C2×#12412yes—
520C2×C4#53612yes—
521C2×C4#55218yes—
522C2×C4#55218yes—
523C2×C4#57612yes—
524C2×C4#57712yes—
525C2×C4#57712yes—
526C2×C4#57812yes—
527C2×C4C22×#4111yescharacteristic
528C2×C4C22×#4111yescharacteristic
529C2×C4C22×#4112yes—
530C2×C4C2×2−1+412yes—
531C8—42no—
532C8—42no—
533C8—44no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22#250314yes—
548C22C22×#12311yescharacteristic
549C22C22×#12411yescharacteristic
550C22C2×#53611yescharacteristic
551C22C2×#57611yescharacteristic
552C22C2×#57711yescharacteristic
553C22C2×#57711yescharacteristic
554C22C2×#57811yescharacteristic
555C4—22no—
556C4—22no—
557C4—24no—
558C4—42no—
559C4—42no—
560C4—82no—
561C4—82no—
562C4—84no—
563C4—84no—
564C4—82no—
565C4—82no—
566C4#250314yes—
567C4#309814yes—
568C4#309914yes—
569C4C2×#55212yes—
570C4C2×#55212yes—
571C2—24no—
572C2—44no—
573C2C2×#250311yescharacteristic
574C2C2×#309811yescharacteristic
575C2C2×#309911yescharacteristic
576C2#6062211yescharacteristic
577C2#6062311yescharacteristic
578C2#6087711yescharacteristic
579C2#6087811yescharacteristic
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. #27970C2semidirect
4. #27962C2semidirect
5. #27962C2semidirect
6. #27961C2semidirect
7. #27960C2semidirect
8. #23749C2semidirect
9. #21322C2semidirect
10. #21318C2semidirect
11. #20764C2semidirect
12. C2×#1834C2semidirect

Maximal subgroups

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

subgroupindex
#352802
#341832
#279702
#279622
#279612
#279602
#237492
#213222
#213182
#207642
C2×#18342
#207032
C4×#4082