Finite Group Databasebeta

← #238189

Subgroups — #238189

Order 512.

1122 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic, complemented
2#21884C211yesmaximal, characteristic
3#21401C211yesmaximal, characteristic
4#21258C211yesmaximal, characteristic
5#10356C212yesmaximal, complemented
6#10354C212yesmaximal, complemented
7#10350C212yesmaximal, complemented
8#10348C212yesmaximal, complemented
9#8045C211yesmaximal, characteristic, complemented
10#7897C211yesmaximal, characteristic, complemented
11C2×#1279C211yesmaximal, characteristic, complemented
12C2×#544—21no—
13C2×#532C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C2×#417—21no—
17C2×#417—21no—
18C22×C4×C8C2211yescharacteristic
19#2857—22no—
20#2659—22no—
21#2552—21no—
22#2552C2212yes—
23#2551—21no—
24#2551C2212yes—
25#2550—22no—
26#2550—22no—
27#2316—21no—
28#2285C2211yescharacteristic
29#2285C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1988C2212yes—
34#1983—22no—
35#1982C2212yes—
36#1982C2212yes—
37#1981C2212yes—
38#1957C2211yescharacteristic
39#1957C2211yescharacteristic
40#1893—22no—
41#1893—22no—
42#1802—21no—
43#1802—21no—
44C4×#102—22no—
45#1296C2212yes—
46#1296C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51C22×#41—21no—
52C22×M16—21no—
53C23×C8—21no—
54C23×C8—21no—
55#544—44no—
56#544—44no—
57#532—22no—
58#532—22no—
59#532—22no—
60#532Q814yescomplemented
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8C2311yescharacteristic
65C2×#127C2311yescharacteristic
66C2×#125—22no—
67C2×#125—22no—
68C2×#124—21no—
69C2×C22⋊Q8—24no—
70C2×#121—21no—
71C2×#121—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×#117—21no—
75C2×#117—21no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#423—22no—
81#423—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88#417—44no—
89#417—44no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—22no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#398—24no—
104#398—24no—
105#395—24no—
106#395—24no—
107#391—24no—
108#389—22no—
109#389—22no—
110#389—24no—
111#389—22no—
112#389—22no—
113#389—22no—
114#389—22no—
115#384—22no—
116#384D411yescharacteristic
117#384D411yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#382—21no—
121#382—21no—
122#380—24no—
123#380—21no—
124#380—21no—
125#380D412yes—
126#380D412yes—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4D412yes—
132C4×C4⋊C4D412yes—
133C4×C4⋊C4C2312yes—
134C4×C4⋊C4C2312yes—
135C2×#95—21no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140C2×#41—24no—
141C2×#41—42no—
142C2×#41—42no—
143C2×#41—42no—
144C2×#41—42no—
145C2×#41—42no—
146C22×Q8—24no—
147C22×Q8—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—41no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C2×M16—42no—
159C2×M16—42no—
160C2×M16—42no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—42no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8—42no—
169C22×C8—81no—
170C22×C8—81no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—44no—
175C4⋊Q8—48no—
176C4⋊Q8—44no—
177C4⋊Q8—82no—
178C4⋊Q8—82no—
179C4⋊Q8C2×Q812yes—
180#127—21no—
181#127—21no—
182#127—21no—
183#127C2×Q812yes—
184#125—44no—
185#125—48no—
186#125—44no—
187#124—24no—
188#124—42no—
189C22⋊Q8—416no—
190#121—24no—
191#121—24no—
192#121—42no—
193#121—42no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—42no—
197C4×D4—42no—
198#117—42no—
199#117—42no—
200#117—42no—
201#117—42no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4D814yes—
222C2×C4⋊C4SD1614yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—81no—
238C2×C42—81no—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×Q812yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246C2×C42C2411yescharacteristic
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—42no—
252C4⋊C8—42no—
253#103—44no—
254#103—44no—
255#102—44no—
256#102—44no—
257#102—44no—
258#102—44no—
259#102—82no—
260#102—82no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—42no—
266#95—24no—
267#95—24no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274C24—41no—
275C24#12011yescharacteristic
276#41—48no—
277#41—48no—
278#41—84no—
279#41—84no—
280C2×Q8—22no—
281C2×Q8—48no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—84no—
288C2×Q8—84no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—82no—
307C2×D4—82no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—44no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4#12011yescharacteristic
331C22×C4#12011yescharacteristic
332C22×C4#12011yescharacteristic
333C22×C4C22⋊Q812yes—
334C22×C4C22⋊Q812yes—
335C22×C4C22⋊Q812yes—
336C22×C4#12412yes—
337C22×C4#12412yes—
338C22×C4#12412yes—
339C22×C4C2×D812yes—
340C22×C4C2×SD1612yes—
341C22×C4AGL(1,ℤ/8)12yes—
342C22×C4#13712yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346M16—84no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—48no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—84no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—48no—
374C4⋊C4—48no—
375C4⋊C4—48no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—81no—
389C4⋊C4—81no—
390C4⋊C4—84no—
391C4⋊C4—84no—
392C4⋊C4C2×D814yes—
393C4⋊C4C2×SD1614yes—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42—42no—
402C42—44no—
403C42—82no—
404C42C22⋊Q812yes—
405C42C22⋊Q812yes—
406C42C22⋊Q812yes—
407C42#12412yes—
408C42#12412yes—
409C42#12412yes—
410C42C22×Q811yescharacteristic
411C42C2×#4111yescharacteristic
412C42C2×#4111yescharacteristic
413C42C2×#4111yescharacteristic
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—42no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23#39312yes—
426C23#44712yes—
427C23#44812yes—
428C23C2×#12011yescharacteristic
429Q8—42no—
430Q8—42no—
431Q8—81no—
432Q8—81no—
433Q8—168no—
434D4—44no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—42no—
439D4—82no—
440D4—88no—
441D4—81no—
442D4—81no—
443D4—81no—
444D4—81no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—84no—
477C2×C4—84no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—88no—
481C2×C4#39312yes—
482C2×C4#39312yes—
483C2×C4#39312yes—
484C2×C4#44712yes—
485C2×C4#44812yes—
486C2×C4#47414yes—
487C2×C4#47614yes—
488C2×C4#48614yes—
489C2×C4#48914yes—
490C2×C4C2×C22⋊Q811yescharacteristic
491C2×C4C2×C22⋊Q811yescharacteristic
492C2×C4C2×C22⋊Q811yescharacteristic
493C2×C4C2×#12411yescharacteristic
494C2×C4C2×#12411yescharacteristic
495C2×C4C2×#12411yescharacteristic
496C2×C4C22×D811yescharacteristic
497C2×C4C22×SD1611yescharacteristic
498C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
499C2×C4C2×#13711yescharacteristic
500C2×C4#57611yescharacteristic
501C2×C4#57711yescharacteristic
502C2×C4#57711yescharacteristic
503C2×C4#57811yescharacteristic
504C8—42no—
505C8—42no—
506C8—82no—
507C8—82no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—161no—
518C22—161no—
519C22—161no—
520C22—161no—
521C22#202314yes—
522C22C2×#39311yescharacteristic
523C22C2×#44711yescharacteristic
524C22C2×#44811yescharacteristic
525C22#301211yescharacteristic
526C22#301311yescharacteristic
527C22#301811yescharacteristic
528C22#301811yescharacteristic
529C4—22no—
530C4—21no—
531C4—21no—
532C4—22no—
533C4—44no—
534C4—44no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4#202314yes—
541C4C2×#47411yescharacteristic
542C4C2×#47611yescharacteristic
543C4#308311yescharacteristic
544C4#308411yescharacteristic
545C4C2×#48611yescharacteristic
546C4C2×#48911yescharacteristic
547C4#313611yescharacteristic
548C4#313711yescharacteristic
549C2—24no—
550C2—162no—
551C2#1866211yescharacteristic
552C2#1866411yescharacteristic
553C2#1867411yescharacteristic
554C2#1867611yescharacteristic
555C2C2×#202311yescharacteristic
556C2#2345811yescharacteristic
557C2#2346111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
1. #35167C2semidirect
5. #10356C2semidirect
6. #10354C2semidirect
7. #10350C2semidirect
8. #10348C2semidirect
9. #8045C2semidirect
10. #7897C2semidirect
11. C2×#1279C2semidirect
60. #532Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#218842
#214012
#212582
#103562
#103542
#103502
#103482
#80452
#78972
C2×#12792