Finite Group Databasebeta

← #6611545

Subgroups — #6611545

Order 512.

1968 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36179C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic, complemented
3#33289C211yesmaximal, characteristic, complemented
4#33048C212yesmaximal, complemented
5#29219C211yesmaximal, characteristic, complemented
6#28716C211yesmaximal, characteristic, complemented
7#27961C212yesmaximal, complemented
8#27822C214yesmaximal, complemented
9#27709C218yesmaximal, complemented
10#25905C212yesmaximal, complemented
11#22500C214yesmaximal, complemented
12C4×#486C211yesmaximal, characteristic, complemented
13#21322C212yesmaximal, complemented
14C4×#436C211yesmaximal, characteristic, complemented
15C2×#547—21no—
16C2×#547—21no—
17C2×#547C2212yes—
18C2×#533C2212yes—
19C2×#486C2211yescharacteristic
20C2×#486C2212yes—
21C2×#436C2212yes—
22C2×C4×M16C2212yes—
23#2868C2211yescharacteristic
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2856C2212yes—
27#2552C2212yes—
28#2542—21no—
29#2542—21no—
30#2542—21no—
31#2542C2212yes—
32#2538—21no—
33#2501—22no—
34#2501—22no—
35#2501C2214yes—
36#2441C2214yes—
37#2429—22no—
38#2423—22no—
39#2365—21no—
40#2354C2212yes—
41Q8×C4⋊C4—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C2×#384C2212yes—
45C42×D4C2211yescharacteristic
46#2100C2218yes—
47#2099C2214yes—
48#2071C2214yes—
49#2025C2218yes—
50#1992C2218yes—
51#1990C2212yes—
52#1990C2212yes—
53#1990C2214yes—
54#1990C2212yes—
55#1990C2212yes—
56#1990C2211yescharacteristic
57#1988C2214yes—
58#1957C2211yescharacteristic
59#1936C2218yes—
60#1934C2214yes—
61#1895C2218yes—
62#1875C2218yes—
63#1834C2214yes—
64#1834C2212yes—
65#1834C2211yescharacteristic
66#1824C2218yes—
67#1802C2214yes—
68#1770C2212yes—
69C4×C4⋊C8C2211yescharacteristic
70C4×#102C2214yes—
71#1753C2214yes—
72C4×C22⋊C8C2212yes—
73#1736C2218yes—
74C42×C8C2211yescharacteristic
75C22×#41—22no—
76C22×M16C2314yes—
77#547—216no—
78#547—48no—
79#547—48no—
80#533—28no—
81#533—44no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2311yescharacteristic
89C2×#127C2311yescharacteristic
90C2×#125C2314yes—
91C2×#123—24no—
92C2×#123—24no—
93C2×#123—24no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—22no—
100C2×#121—22no—
101C2×#121—22no—
102C2×#121C2314yes—
103C2×C4×Q8—21no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4D411yescharacteristic
109C2×C4×D4D411yescharacteristic
110C2×C4×D4C2312yes—
111C2×C4×D4C2311yescharacteristic
112C2×#117C2314yes—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4C2314yes—
116C22×C42C2312yes—
117#486—24no—
118#486—44no—
119#436—28no—
120C2×C4⋊C8C2312yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2318yes—
124C2×C22⋊C8C2314yes—
125C4×M16—44no—
126C2×#97C2314yes—
127C2×C4×C8C2312yes—
128C2×C4×C8C2312yes—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2312yes—
132#399—24no—
133#399—24no—
134#399—24no—
135#399—24no—
136#398—24no—
137#396—24no—
138#391—21no—
139#391—22no—
140#390—22no—
141#390—21no—
142#390—21no—
143#390—22no—
144#390—22no—
145#390C2312yes—
146#390C2314yes—
147#389—22no—
148#389—21no—
149#389—22no—
150#389—24no—
151#389C2312yes—
152#389C2314yes—
153#388—24no—
154#387—24no—
155#384—22no—
156#384—21no—
157#384—21no—
158#384—21no—
159#384—21no—
160#384—22no—
161#384—22no—
162#384C2311yescharacteristic
163#384C2311yescharacteristic
164#384C2312yes—
165#384C2312yes—
166#384C2312yes—
167#384C2312yes—
168#384C2314yes—
169#380—22no—
170#380C2314yes—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4D411yescharacteristic
174C4×C4⋊C4D411yescharacteristic
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2311yescharacteristic
177C4×C22⋊C4D412yes—
178C4×C22⋊C4D412yes—
179C4×C22⋊C4C2312yes—
180C43C2311yescharacteristic
181#340C2318yes—
182#340C2314yes—
183#340C2318yes—
184#340C2318yes—
185#340C2318yes—
186#340C2318yes—
187#340C2314yes—
188#336C2312yes—
189#336C2314yes—
190C2×#41—48no—
191C2×#41—48no—
192C2×#41—48no—
193C22×Q8—21no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×D4—24no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4C2×D411yescharacteristic
200C22×D4C2412yes—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4C2412yes—
206C2×M16—28no—
207C2×M16—28no—
208C2×M16—44no—
209C22×C8C2414yes—
210C22×C8C2412yes—
211C22×C8C2412yes—
212C4⋊Q8—24no—
213C4⋊Q8—28no—
214C4⋊Q8—22no—
215C4⋊Q8—24no—
216C4⋊Q8—44no—
217C4⋊Q8—44no—
218C4⋊Q8—44no—
219C4⋊Q8—82no—
220C4⋊Q8#4114yescomplemented
221#127—22no—
222#127—22no—
223#125—28no—
224#125—44no—
225#123—416no—
226#123—416no—
227#123—416no—
228C22⋊Q8—48no—
229C22⋊Q8—48no—
230C22⋊Q8—48no—
231C22⋊Q8—48no—
232C22⋊Q8—48no—
233C22⋊Q8—84no—
234#121—216no—
235#121—48no—
236#121—48no—
237C4×Q8—48no—
238C4×Q8—48no—
239C4×Q8—44no—
240C4×D4—24no—
241C4×D4—24no—
242C4×D4—28no—
243C4×D4—44no—
244C4×D4—44no—
245C4×D4#4118yes—
246#117—28no—
247#117—44no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—216no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—48no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4C2×D411yescharacteristic
274C2×C4⋊C4C2×D411yescharacteristic
275C2×C4⋊C4C2×D411yescharacteristic
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2414yes—
278C2×C4⋊C4C2414yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4C2×D412yes—
290C2×C22⋊C4C2×D412yes—
291C2×C22⋊C4C2×D412yes—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2412yes—
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306C4⋊C8—24no—
307#102—28no—
308#102—28no—
309#102—48no—
310C22⋊C8—28no—
311#97—44no—
312C4×C8—24no—
313C4×C8—24no—
314C4×C8—22no—
315C4×C8—22no—
316C4×C8—24no—
317#95—24no—
318#95—22no—
319#95—22no—
320#95—24no—
321#95C2×D412yes—
322#95C2×D412yes—
323#95C2×D412yes—
324C24—22no—
325C24C22×D412yes—
326#41—832no—
327C2×Q8—24no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—42no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—42no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4#12414yes—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4118yes—
382C22×C4C2511yescharacteristic
383M16—416no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—22no—
391C2×C8—24no—
392C2×C82−1+414yes—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—28no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—48no—
403C4⋊C4—48no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—84no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4#12414yes—
424C4⋊C4C2×#4114yes—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—44no—
431C22⋊C4—48no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—84no—
435C22⋊C4#12418yes—
436C42—24no—
437C42—24no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—48no—
444C42—44no—
445C42AGL(1,ℤ/8)14yes—
446C42AGL(1,ℤ/8)14yes—
447C42C2×#4114yes—
448C42C2×#4112yes—
449C42H(5,𝔽2)12yes—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23C2×#12414yes—
464C23C23×D411yescharacteristic
465Q8—48no—
466Q8—88no—
467Q8—88no—
468D4—28no—
469D4—44no—
470D4—44no—
471D4—816no—
472D4—84no—
473D4—84no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—48no—
494C2×C4—48no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—816no—
508C2×C4—88no—
509C2×C4C2×#12412yes—
510C2×C4C2×#12414yes—
511C2×C4C2×#12412yes—
512C2×C4#54612yes—
513C2×C4#54612yes—
514C2×C4C2×AGL(1,ℤ/8)12yes—
515C2×C4C2×AGL(1,ℤ/8)12yes—
516C2×C4C2×AGL(1,ℤ/8)12yes—
517C2×C4C2×AGL(1,ℤ/8)12yes—
518C2×C4C2×AGL(1,ℤ/8)12yes—
519C2×C4C2×AGL(1,ℤ/8)12yes—
520C2×C4#57712yes—
521C2×C4#57812yes—
522C2×C4C22×#4111yescharacteristic
523C2×C4C22×#4112yes—
524C2×C4C2×H(5,𝔽2)11yescharacteristic
525C2×C4C2×2−1+412yes—
526C8—24no—
527C8—42no—
528C8—44no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22C22×#12411yescharacteristic
544C22C2×#54611yescharacteristic
545C22#349911yescharacteristic
546C22C22×AGL(1,ℤ/8)11yescharacteristic
547C22C22×AGL(1,ℤ/8)11yescharacteristic
548C22C2×#57711yescharacteristic
549C22C2×#57811yescharacteristic
550C4—22no—
551C4—22no—
552C4—24no—
553C4—22no—
554C4—42no—
555C4—44no—
556C4—42no—
557C4—82no—
558C4—82no—
559C4—84no—
560C4—84no—
561C4#313814yes—
562C4#333014yes—
563C4#333214yes—
564C4#348212yes—
565C4#352012yes—
566C2—44no—
567C2—162no—
568C2#3630311yescharacteristic
569C2C2×#313811yescharacteristic
570C2#6062311yescharacteristic
571C2C2×#333011yescharacteristic
572C2C2×#333211yescharacteristic
573C2#6247711yescharacteristic
574C2#6247911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36134C2semidirect
3. #33289C2semidirect
4. #33048C2semidirect
5. #29219C2semidirect
6. #28716C2semidirect
7. #27961C2semidirect
8. #27822C2semidirect
9. #27709C2semidirect
10. #25905C2semidirect
11. #22500C2semidirect
12. C4×#486C2semidirect
13. #21322C2semidirect
14. C4×#436C2semidirect
220. C4⋊Q8#41semidirect

Maximal subgroups

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

subgroupindex
#361792
#361342
#332892
#330482
#292192
#287162
#279612
#278222
#277092
#259052
#225002
C4×#4862
#213222
C4×#4362