Finite Group Databasebeta

← #241308

Subgroups — #241308

Order 512.

1012 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34938C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21171C211yesmaximal, characteristic
4C2×#1947C211yesmaximal, characteristic
5#10581C212yesmaximal, complemented
6#10579C212yesmaximal, complemented
7#10575C212yesmaximal, complemented
8#10573C212yesmaximal, complemented
9#8045C211yesmaximal, characteristic
10#7901C211yesmaximal, characteristic
11#7891C211yesmaximal, characteristic
12C2×#533—21no—
13C2×#431C2211yescharacteristic
14C2×#428—21no—
15C22×#107—21no—
16C2×#423C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18#2552C2211yescharacteristic
19#2552C2211yescharacteristic
20#2551C2211yescharacteristic
21#2551C2211yescharacteristic
22#2550—21no—
23#2550—21no—
24#2549—22no—
25#2440—21no—
26#2440—21no—
27#2399—22no—
28C4×C4⋊Q8—21no—
29C4×C4⋊Q8—21no—
30#2296—21no—
31#2285C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C4×#117C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1948—21no—
36#1948C2212yes—
37#1947—21no—
38#1947—21no—
39#1947—21no—
40#1947—22no—
41#1947C2212yes—
42#1947C2212yes—
43#1947C2212yes—
44#1852—22no—
45#1845C2212yes—
46#1834C2212yes—
47#1736—21no—
48C2×#340—21no—
49#1296C2212yes—
50#1296C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55C22×M16—21no—
56C23×C8—21no—
57#533—44no—
58#533—44no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×#125—21no—
64C2×#125—21no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×#117—21no—
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79#431—22no—
80#431Q814yescomplemented
81#428—44no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#107—24no—
86C2×#107—44no—
87C2×#106—22no—
88#423—22no—
89#423Q814yescomplemented
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8Q814yescomplemented
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2312yes—
97C2×C22⋊C8—21no—
98C2×C4×C8C2312yes—
99#398—22no—
100#398—22no—
101#398—22no—
102#398—22no—
103#395—22no—
104#395—22no—
105#395—22no—
106#395—22no—
107#391—22no—
108#391—22no—
109#389—22no—
110#389—22no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#385—22no—
120#385—22no—
121#384—21no—
122#384—21no—
123#384—21no—
124#384—21no—
125#384—21no—
126#384C2312yes—
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#382—21no—
130#380—21no—
131#380D412yes—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4D412yes—
140C4×C4⋊C4D411yescharacteristic
141C4×C4⋊C4D411yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145C4×C4⋊C4C2311yescharacteristic
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148#376—21no—
149C2×#95—21no—
150C43—21no—
151#340—24no—
152#340—22no—
153#340—22no—
154#340—22no—
155#340—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—41no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C2×M16—24no—
165C2×M16—42no—
166C2×M16—42no—
167C22×C8—24no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—42no—
172C22×C8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4⋊Q8—42no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181#125—42no—
182#125—42no—
183#125—42no—
184#125—42no—
185C22⋊Q8—48no—
186C22⋊Q8—48no—
187C4×Q8—48no—
188C4×Q8—48no—
189#117—22no—
190#117—22no—
191#117—42no—
192#117—42no—
193#117D814yes—
194#117Q1614yes—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×Q812yes—
256C2×C42C2×Q812yes—
257C2×C42C2×Q812yes—
258C2×C42#4112yes—
259C2×C42C2411yescharacteristic
260#107—44no—
261#107—44no—
262#107—44no—
263#107—84no—
264#106—44no—
265C4⋊C8—24no—
266C4⋊C8—24no—
267C4⋊C8—24no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8C2×Q814yes—
271C4⋊C8C2×Q814yes—
272C22⋊C8—24no—
273C4×C8—22no—
274C4×C8C2×Q814yes—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—22no—
284#95—22no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289C24#12011yescharacteristic
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—44no—
294C2×Q8—82no—
295C2×Q8—82no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4#12011yescharacteristic
323C22×C4#12011yescharacteristic
324C22×C4#12011yescharacteristic
325C22×C4C22⋊Q812yes—
326C22×C4C22⋊Q812yes—
327C22×C4C22⋊Q812yes—
328C22×C4C4⋊Q812yes—
329C22×C4C4⋊Q812yes—
330C22×C4C4⋊Q812yes—
331C22×C4C2×D812yes—
332C22×C4C2×Q1612yes—
333C22×C4AGL(1,ℤ/8)12yes—
334C22×C4#13712yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338M16—48no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—48no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4C2×D812yes—
390C4⋊C4C2×D812yes—
391C4⋊C4C2×Q1612yes—
392C4⋊C4C2×Q1612yes—
393C4⋊C4AGL(1,ℤ/8)12yes—
394C4⋊C4#13712yes—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—44no—
398C42—24no—
399C42—24no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—42no—
409C42—42no—
410C42—44no—
411C42—81no—
412C42—81no—
413C42C22⋊Q812yes—
414C42C22⋊Q812yes—
415C42C22⋊Q812yes—
416C42C4⋊Q812yes—
417C42C4⋊Q812yes—
418C42C4⋊Q812yes—
419C42C22×Q811yescharacteristic
420C42C22×Q811yescharacteristic
421C42C22×Q811yescharacteristic
422C42C2×#4111yescharacteristic
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23#39512yes—
428C23#44712yes—
429C23#45112yes—
430C23C2×#12011yescharacteristic
431Q8—88no—
432Q8—88no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—84no—
476C2×C4—84no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—164no—
481C2×C4#39512yes—
482C2×C4#39512yes—
483C2×C4#39512yes—
484C2×C4#44712yes—
485C2×C4#45112yes—
486C2×C4#47414yes—
487C2×C4#47714yes—
488C2×C4#50014yes—
489C2×C4#50114yes—
490C2×C4C2×C22⋊Q811yescharacteristic
491C2×C4C2×C22⋊Q811yescharacteristic
492C2×C4C2×C22⋊Q811yescharacteristic
493C2×C4C2×C4⋊Q811yescharacteristic
494C2×C4C2×C4⋊Q811yescharacteristic
495C2×C4C2×C4⋊Q811yescharacteristic
496C2×C4C22×D811yescharacteristic
497C2×C4C22×Q1611yescharacteristic
498C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
499C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
500C2×C4C2×#13711yescharacteristic
501C2×C4C2×#13711yescharacteristic
502C2×C4#57511yescharacteristic
503C2×C4#57511yescharacteristic
504C8—44no—
505C8—44no—
506C8—44no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22#205714yes—
514C22C2×#39511yescharacteristic
515C22C2×#44711yescharacteristic
516C22C2×#45111yescharacteristic
517C22#300611yescharacteristic
518C22#300711yescharacteristic
519C22#302111yescharacteristic
520C22#302211yescharacteristic
521C4—22no—
522C4—21no—
523C4—21no—
524C4—22no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—81no—
532C4—82no—
533C4—81no—
534C4—82no—
535C4—162no—
536C4#205714yes—
537C4C2×#47411yescharacteristic
538C4C2×#47711yescharacteristic
539C4#307711yescharacteristic
540C4#307811yescharacteristic
541C4C2×#50011yescharacteristic
542C4C2×#50111yescharacteristic
543C4#316511yescharacteristic
544C4#316611yescharacteristic
545C2—24no—
546C2#1878611yescharacteristic
547C2#1878911yescharacteristic
548C2#1879811yescharacteristic
549C2#1880111yescharacteristic
550C2C2×#205711yescharacteristic
551C2#2355411yescharacteristic
552C2#2356211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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. #10581C2semidirect
6. #10579C2semidirect
7. #10575C2semidirect
8. #10573C2semidirect
80. #431Q8semidirect
89. #423Q8semidirect
94. C2×C4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#349382
#219822
#211712
C2×#19472
#105812
#105792
#105752
#105732
#80452
#79012
#78912