Finite Group Databasebeta

← #239745

Subgroups — #239745

Order 512.

1210 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4#20648C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic, complemented
6C2×#1476C211yesmaximal, characteristic
7#10419C214yesmaximal, complemented
8#10416C214yesmaximal, complemented
9#7897C211yesmaximal, characteristic, complemented
10C2×#536—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×C4×#41C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#417—21no—
16C22×C4×C8C2211yescharacteristic
17#2857—21no—
18#2857—21no—
19#2621—22no—
20#2547—21no—
21#2547C2212yes—
22#2546—21no—
23#2546C2212yes—
24#2486—22no—
25#2486—22no—
26#2310—21no—
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29#1990—22no—
30#1990C2214yes—
31#1947C2214yes—
32#1852—22no—
33#1845—21no—
34#1834—21no—
35#1803—21no—
36#1803C2212yes—
37#1802—21no—
38#1802C2212yes—
39#1736—21no—
40C2×#340—22no—
41#1546—24no—
42C2×#332C2214yes—
43#1476—24no—
44C2×Q8⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46#1279C2214yes—
47C22×#41D411yescharacteristic
48C23×Q8—21no—
49C22×M16—21no—
50C23×C8—21no—
51#536—44no—
52#536—44no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8C2312yes—
62C2×#124—22no—
63C2×#123—24no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C4×#41—22no—
67C4×#41—22no—
68C4×#41—22no—
69C4×#41—22no—
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4D411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117—21no—
77C2×#117—21no—
78C2×#117D411yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D412yes—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#431—22no—
87#431D414yes—
88C2×#107—24no—
89#423—22no—
90#423—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94#417—44no—
95C2×#103—22no—
96C2×#102—22no—
97C2×C22⋊C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101C2×C4×C8—21no—
102C2×C4×C8C2312yes—
103#398—24no—
104#398—24no—
105#397—24no—
106#397—24no—
107#391—22no—
108#391—22no—
109#391—22no—
110#391—22no—
111#389—22no—
112#389—22no—
113#389—22no—
114#389—22no—
115#384—21no—
116#384—21no—
117#384—21no—
118#384D411yescharacteristic
119#384D411yescharacteristic
120#384C2312yes—
121#382—21no—
122#381—24no—
123#380—22no—
124#380—22no—
125#380—21no—
126#380—22no—
127#380—22no—
128#380D412yes—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2312yes—
131#340—22no—
132#340—22no—
133#340—24no—
134#340—28no—
135#332—28no—
136Q8⋊C8—24no—
137D4⋊C8—24no—
138C2×#41—22no—
139C2×#41—22no—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41—22no—
143C2×#41D814yes—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8C2×D411yescharacteristic
153C22×D4—21no—
154C22×D4C2×D411yescharacteristic
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—41no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×M16—42no—
164C2×M16—42no—
165C2×M16—42no—
166C22×C8—24no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—42no—
171C22×C8—42no—
172C4⋊Q8—24no—
173C4⋊Q8—24no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—44no—
179C4⋊Q8—44no—
180C4⋊Q8—42no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184#124—44no—
185#124—44no—
186#123—416no—
187C22⋊Q8—48no—
188C22⋊Q8—48no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—42no—
204C4×D4—42no—
205#117—22no—
206#117—22no—
207#117—42no—
208#117—42no—
209#117—42no—
210#117—42no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42#4112yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260C2×C42C2411yescharacteristic
261#107—84no—
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—42no—
266C4⋊C8C2×D414yes—
267#103—44no—
268#102—44no—
269C22⋊C8—24no—
270C4×C8—24no—
271C4×C8—22no—
272C4×C8—22no—
273C4×C8—42no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—22no—
279#95—22no—
280C24—21no—
281C24#12011yescharacteristic
282#41—28no—
283#41—28no—
284#41—44no—
285#41—44no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—44no—
298C2×Q8—44no—
299C2×Q8—44no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×Q8C2×D812yes—
305C2×D4—24no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4C2×D812yes—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4#12112yes—
345C22×C4#12112yes—
346C22×C4#12112yes—
347C22×C4#12312yes—
348C22×C4#12312yes—
349C22×C4#12412yes—
350C22×C4C2×D812yes—
351C22×C4#13512yes—
352C22×C4AGL(1,ℤ/8)12yes—
353C22×C4#13712yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357M16—84no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—42no—
361C2×C8—48no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C4⋊C4—24no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—24no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—48no—
377C4⋊C4—48no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C22⋊C4—48no—
397C22⋊C4—48no—
398C22⋊C4—42no—
399C22⋊C4—44no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—42no—
403C42—41no—
404C42—41no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42#12112yes—
410C42#12112yes—
411C42#12112yes—
412C42#12312yes—
413C42#12312yes—
414C42#12412yes—
415C42C22×D411yescharacteristic
416C42C2×#4111yescharacteristic
417C42C2×#4111yescharacteristic
418C42C2×#4111yescharacteristic
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—22no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23#39412yes—
431C23#44712yes—
432C23#45212yes—
433C23C2×#12011yescharacteristic
434Q8—22no—
435Q8—22no—
436Q8—41no—
437Q8—41no—
438Q8—82no—
439Q8—82no—
440Q8—82no—
441Q8—82no—
442Q8—88no—
443D4—22no—
444D4—22no—
445D4—48no—
446D4—41no—
447D4—41no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—48no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—88no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—84no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—164no—
483C2×C4#39412yes—
484C2×C4#39412yes—
485C2×C4#39412yes—
486C2×C4#44712yes—
487C2×C4#45212yes—
488C2×C4#45412yes—
489C2×C4#45612yes—
490C2×C4#46614yes—
491C2×C4#46814yes—
492C2×C4#48014yes—
493C2×C4#48214yes—
494C2×C4C2×#12111yescharacteristic
495C2×C4C2×#12111yescharacteristic
496C2×C4C2×#12111yescharacteristic
497C2×C4C2×#12311yescharacteristic
498C2×C4C2×#12311yescharacteristic
499C2×C4C2×#12411yescharacteristic
500C2×C4C22×D811yescharacteristic
501C2×C4C2×#13511yescharacteristic
502C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
503C2×C4C2×#13711yescharacteristic
504C2×C4#57712yes—
505C2×C4#57812yes—
506C8—44no—
507C8—44no—
508C8—82no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22#203314yes—
523C22C2×#39411yescharacteristic
524C22C2×#44711yescharacteristic
525C22C2×#45211yescharacteristic
526C22#301011yescharacteristic
527C22#301111yescharacteristic
528C22C2×#45411yescharacteristic
529C22C2×#45611yescharacteristic
530C4—21no—
531C4—21no—
532C4—22no—
533C4—42no—
534C4—84no—
535C4—81no—
536C4—82no—
537C4—81no—
538C4—82no—
539C4—162no—
540C4#203314yes—
541C4#203814yes—
542C4#304612yes—
543C4C2×#46611yescharacteristic
544C4C2×#46811yescharacteristic
545C4C2×#48011yescharacteristic
546C4C2×#48211yescharacteristic
547C4#313712yes—
548C2—24no—
549C2—82no—
550C2#1864011yescharacteristic
551C2#1864511yescharacteristic
552C2#1865011yescharacteristic
553C2#1865511yescharacteristic
554C2C2×#203311yescharacteristic
555C2C2×#203811yescharacteristic
556C2#2351011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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
2. #21982C2semidirect
3. C2×#1990C2semidirect
5. C2×#1546C2semidirect
7. #10419C2semidirect
8. #10416C2semidirect
9. #7897C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#219822
C2×#19902
#206482
C2×#15462
C2×#14762
#104192
#104162
#78972