Finite Group Databasebeta

← #6484975

Subgroups — #6484975

Order 512.

1704 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#33096C214yesmaximal, complemented
4#29912C214yesmaximal, complemented
5#26321C214yesmaximal, complemented
6#26281C214yesmaximal, complemented
7#25057C214yesmaximal, complemented
8#24991C214yesmaximal, complemented
9#21982C211yesmaximal, characteristic, complemented
10#21859C211yesmaximal, characteristic
11#21818C211yesmaximal, characteristic, complemented
12#21386C211yesmaximal, characteristic
13#21359C211yesmaximal, characteristic, complemented
14C2×#536—21no—
15C2×#532C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2854C2214yes—
20#2618—22no—
21#2548C2212yes—
22#2546C2212yes—
23#2545—22no—
24#2538C2212yes—
25#2538C2212yes—
26#2536—22no—
27#2355—22no—
28#2355—22no—
29#2355C2214yes—
30#2355C2214yes—
31#2311—21no—
32C2×#391—21no—
33C2×#387—22no—
34#2285C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×#382C2212yes—
37C2×C4×C4⋊C4—21no—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40#2101C2218yes—
41#2092C2218yes—
42#2085C2218yes—
43#2073C2218yes—
44#1990C2212yes—
45#1988C2212yes—
46#1984C2214yes—
47#1984C2214yes—
48#1982C2212yes—
49#1957C2212yes—
50#1948C2212yes—
51#1948C2212yes—
52#1947C2214yes—
53#1926C2218yes—
54#1925C2218yes—
55#1921C2218yes—
56#1920C2218yes—
57#1874C2212yes—
58#1874C2214yes—
59#1874C2212yes—
60#1853C2212yes—
61#1853C2212yes—
62#1852C2214yes—
63#1845C2212yes—
64#1834C2212yes—
65#1817C2214yes—
66#1802C2214yes—
67C4×#106C2214yes—
68C4×#102C2214yes—
69C22×#41—21no—
70C23×Q8D411yescharacteristic
71#536—48no—
72#532—24no—
73#532—24no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8C2311yescharacteristic
76C2×#127C2311yescharacteristic
77C2×#124—22no—
78C2×#124—21no—
79C2×#123—24no—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—22no—
82C2×#121—21no—
83C2×#121C2312yes—
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8D412yes—
87C2×C4×Q8D412yes—
88C2×C4×Q8C2312yes—
89C2×C4×Q8C2312yes—
90C2×C4×D4—21no—
91C2×C4×D4C2312yes—
92C2×#117—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4D411yescharacteristic
96C22×C4⋊C4D412yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C42—21no—
101C22×C42—22no—
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104#431—24no—
105#431—24no—
106C2×#107C2318yes—
107C2×#106C2314yes—
108C2×#106C2314yes—
109C2×C4⋊C8C2312yes—
110C2×C4⋊C8C2312yes—
111C2×#102C2314yes—
112C2×#102C2318yes—
113C2×#102C2314yes—
114C2×C22⋊C8C2312yes—
115C2×C22⋊C8C2312yes—
116C2×C4×C8C2312yes—
117C2×C4×C8C2312yes—
118#398—24no—
119#396—24no—
120#396—24no—
121#394—24no—
122#391—24no—
123#391—24no—
124#391—22no—
125#390—22no—
126#390C2314yes—
127#390C2314yes—
128#389C2314yes—
129#389C2314yes—
130#388—24no—
131#388—24no—
132#387—28no—
133#387—28no—
134#386—24no—
135#384—21no—
136#384—21no—
137#384C2312yes—
138#384C2311yescharacteristic
139#384C2312yes—
140#384C2311yescharacteristic
141#382—24no—
142#382—21no—
143#382C2318yes—
144#380—21no—
145#380—22no—
146#380—22no—
147#380C2312yes—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4C2318yes—
155C4×C4⋊C4C2312yes—
156C4×C4⋊C4C2312yes—
157C4×C4⋊C4C2312yes—
158C2×#95—22no—
159C2×#95—21no—
160C2×#95—22no—
161C2×#95—22no—
162#340C2318yes—
163#340C2314yes—
164#340C2318yes—
165#340C2314yes—
166#340C2314yes—
167#340C2314yes—
168#336C2318yes—
169#336C2318yes—
170C2×#41—44no—
171C2×#41—44no—
172C2×#41—44no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—21no—
177C22×Q8C2×D412yes—
178C22×Q8C2×D412yes—
179C22×D4—22no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4C2412yes—
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—21no—
186C23×C4—22no—
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2411yescharacteristic
192C22×C8C2414yes—
193C22×C8C2414yes—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8—48no—
197#127—22no—
198#127—22no—
199#124—48no—
200#124—44no—
201#123—416no—
202C22⋊Q8—48no—
203C22⋊Q8—48no—
204#121—24no—
205#121—24no—
206#121—44no—
207C4×Q8—28no—
208C4×Q8—28no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—44no—
216#117—44no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—82no—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2414yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—28no—
261C2×C42—44no—
262C2×C42—84no—
263C2×C42C2×D414yes—
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42#4114yes—
267C2×C42#4114yes—
268C2×C42C2414yes—
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271#107—48no—
272#106—28no—
273#106—44no—
274C4⋊C8—24no—
275C4⋊C8—24no—
276#102—28no—
277#102—44no—
278#102—48no—
279C22⋊C8—24no—
280C22⋊C8—24no—
281C4×C8—24no—
282C4×C8—24no—
283#95—28no—
284#95—22no—
285#95—22no—
286#95—28no—
287#95—28no—
288#95—22no—
289#95—22no—
290#95—24no—
291#95—24no—
292C24—22no—
293C24C22×D411yescharacteristic
294#41—816no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—416no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—81no—
317C2×D4—81no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4#12314yes—
347C22×C4#12314yes—
348C22×C4#13514yes—
349C22×C4AGL(1,ℤ/8)14yes—
350C22×C4C22×D414yes—
351C22×C4C22×D414yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4112yes—
358C22×C4C2×#4112yes—
359C22×C4C2511yescharacteristic
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—28no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—81no—
389C4⋊C4—81no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C42−1+414yes—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—84no—
399C22⋊C4—84no—
400C42—28no—
401C42—24no—
402C42—24no—
403C42—44no—
404C42—44no—
405C42—44no—
406C42—88no—
407C42#12314yes—
408C42#12314yes—
409C42C2×#4112yes—
410C42C2×#4112yes—
411C42C2×#4112yes—
412C42C2×#4112yes—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23C2×#12312yes—
425C23C2×#13512yes—
426C23C2×AGL(1,ℤ/8)12yes—
427C23C23×D411yescharacteristic
428Q8—48no—
429Q8—48no—
430Q8—84no—
431D4—44no—
432D4—44no—
433D4—84no—
434D4—88no—
435D4—84no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—44no—
452C2×C4—48no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—48no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—84no—
473C2×C4—88no—
474C2×C4—88no—
475C2×C4#482116yes—
476C2×C4C2×#12312yes—
477C2×C4C2×#12312yes—
478C2×C4C2×#12312yes—
479C2×C4C2×#12312yes—
480C2×C4C2×#12312yes—
481C2×C4C2×#13512yes—
482C2×C4C2×#13512yes—
483C2×C4C2×AGL(1,ℤ/8)12yes—
484C2×C4C2×AGL(1,ℤ/8)12yes—
485C2×C4C22×#4111yescharacteristic
486C2×C4C22×#4111yescharacteristic
487C2×C4C2×2−1+412yes—
488C2×C4#58512yes—
489C8—44no—
490C8—44no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22C2×#48214yes—
505C22C22×#12311yescharacteristic
506C22#348611yescharacteristic
507C22#348711yescharacteristic
508C22C22×#13511yescharacteristic
509C22C22×AGL(1,ℤ/8)11yescharacteristic
510C22#358511yescharacteristic
511C22#358611yescharacteristic
512C4—22no—
513C4—22no—
514C4—22no—
515C4—44no—
516C4—44no—
517C4—44no—
518C4—82no—
519C4—82no—
520C4—82no—
521C4—82no—
522C4—84no—
523C4C2×#48214yes—
524C4C2×#48214yes—
525C4#346112yes—
526C4#350712yes—
527C2—24no—
528C2—162no—
529C2#3518211yescharacteristic
530C2C22×#48211yescharacteristic
531C2#6051711yescharacteristic
532C2#6181911yescharacteristic
533C2#6182811yescharacteristic
534C2#6188211yescharacteristic
535C2#6189111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2355C2semidirect
3. #33096C2semidirect
4. #29912C2semidirect
5. #26321C2semidirect
6. #26281C2semidirect
7. #25057C2semidirect
8. #24991C2semidirect
9. #21982C2semidirect
11. #21818C2semidirect
13. #21359C2semidirect

Maximal subgroups

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

subgroupindex
#351782
C2×#23552
#330962
#299122
#263212
#262812
#250572
#249912
#219822
#218592
#218182
#213862
#213592