Finite Group Databasebeta

← #364841

Subgroups — #364841

Order 512.

1578 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3C2×#1997C211yesmaximal, characteristic, complemented
4#20646C211yesmaximal, characteristic
5C4×#418C211yesmaximal, characteristic
6#11211C212yesmaximal, complemented
7#11054C212yesmaximal
8#10958C212yesmaximal
9#10892C212yesmaximal, complemented
10#10852C211yesmaximal, characteristic, complemented
11#10851C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#418C2211yescharacteristic
16C2×#418C2212yes—
17C2×#418C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19#2563—24no—
20#2563C418yescomplemented
21C2×#393—22no—
22#2369—22no—
23#2365—22no—
24#2345—22no—
25Q8×C22⋊C4—22no—
26#2331—22no—
27#2316C2212yes—
28C4×C4⋊Q8—22no—
29C2×#390C2212yes—
30#2285C2211yescharacteristic
31C2×#382C2212yes—
32C2×#380—22no—
33#2274C2212yes—
34C4×#117C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1997—24no—
37#1804—22no—
38#1802—22no—
39#1762—22no—
40C4×#102—22no—
41C2×#343C2212yes—
42C2×#343C2212yes—
43#1736C2211yescharacteristic
44#1736C2212yes—
45#1736C2211yescharacteristic
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50C23×Q8D412yes—
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C22×M16C2312yes—
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8C2×C414yes—
63C2×C4⋊Q8D414yes—
64C2×C4×Q8—24no—
65C2×C4×D4—22no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4C2×C414yes—
73C2×#117C2311yescharacteristic
74C2×#117C2312yes—
75C2×#117C2311yescharacteristic
76C22×C4⋊C4—21no—
77C22×C4⋊C4D412yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—24no—
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84#423—22no—
85#423—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88#418—22no—
89#418—22no—
90#418—22no—
91#418—24no—
92#418—42no—
93#418Q814yescomplemented
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—24no—
97C4×M16—22no—
98C4×M16—22no—
99C2×#97—21no—
100C2×C4×C8—21no—
101#393—216no—
102#391—24no—
103#391—28no—
104#390—24no—
105#390C2×C418yes—
106#389—24no—
107#389—24no—
108#389—24no—
109#387—28no—
110#384D412yes—
111#382—24no—
112#382C2×C418yes—
113#382D412yes—
114#380—24no—
115#380—21no—
116#380—21no—
117#380—216no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—24no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2×C414yes—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D412yes—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—24no—
129C4×C22⋊C4—21no—
130#376—21no—
131C2×#95—21no—
132C2×#95—24no—
133C43—21no—
134#343—24no—
135#343—24no—
136#340—24no—
137#340—22no—
138#340—22no—
139#339—24no—
140#339—24no—
141#339—42no—
142#339—42no—
143C25—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—24no—
147C22×Q8—84no—
148C22×Q8C22⋊C418yes—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4C22×C412yes—
161C22×D4C2×Q812yes—
162C22×D4#4112yes—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—24no—
166C23×C4—48no—
167C23×C4—41no—
168C23×C4—41no—
169C23×C4—42no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D414yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2411yescharacteristic
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16C2×Q812yes—
191C2×M16#4112yes—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—22no—
195C4⋊Q8—24no—
196C4⋊Q8—24no—
197C4⋊Q8—24no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—24no—
201C4⋊Q8—24no—
202C4⋊Q8—24no—
203C4×Q8—88no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212#117—21no—
213#117—21no—
214#117—22no—
215#117—22no—
216#117—21no—
217#117—21no—
218#117—21no—
219#117—22no—
220#117—22no—
221#117—21no—
222#117C2×Q812yes—
223#117#4112yes—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4C22⋊C418yes—
242C2×C4⋊C4C22×C412yes—
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4C2×D412yes—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—216no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—88no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—28no—
268C2×C42C22×C414yes—
269C2×C42C22×C412yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D412yes—
275C2×C42#4112yes—
276C2×C42#4112yes—
277C4⋊C8—42no—
278C4⋊C8—42no—
279#102—28no—
280#102—28no—
281#102—48no—
282#97—42no—
283C4×C8—42no—
284#95—28no—
285#95—28no—
286#95—24no—
287#95—216no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—44no—
296C24—44no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24C22×D411yescharacteristic
302C2×Q8—28no—
303C2×Q8—28no—
304C2×Q8—48no—
305C2×Q8—84no—
306C2×Q8—88no—
307C2×Q8—88no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4C4×Q814yescomplemented
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—42no—
345C22×C4—416no—
346C22×C4—48no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—84no—
357C22×C4—88no—
358C22×C4—84no—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C418yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4#12011yescharacteristic
363C22×C4#12011yescharacteristic
364C22×C4#12011yescharacteristic
365C22×C4#12011yescharacteristic
366C22×C4#12312yes—
367C22×C4#12312yes—
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×Q811yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C42−1+411yescharacteristic
380C22×C42−1+411yescharacteristic
381C22×C42−1+411yescharacteristic
382M16—22no—
383M16—22no—
384M16—22no—
385M16—22no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390M16—42no—
391M16—42no—
392M16—42no—
393M16—42no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—28no—
409C4⋊C4—28no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—48no—
417C4⋊C4—44no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—44no—
421C4⋊C4—88no—
422C4⋊C4C22⋊Q814yes—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—42no—
426C22⋊C4—41no—
427C22⋊C4—42no—
428C22⋊C4—41no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—816no—
432C42—22no—
433C42—22no—
434C42—24no—
435C42—24no—
436C42—44no—
437C42—42no—
438C42—42no—
439C42—41no—
440C42—41no—
441C42—42no—
442C42—42no—
443C42—44no—
444C42—88no—
445C42C4×D412yes—
446C42C4×D412yes—
447C42C4×D412yes—
448C42C4×D412yes—
449C42#12112yes—
450C42#12112yes—
451C42#12112yes—
452C42#12112yes—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—44no—
457C23—44no—
458C23—48no—
459C23—48no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—82no—
467C23—84no—
468C23—84no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23C22×C22⋊C411yescharacteristic
473C23C2×#12011yescharacteristic
474C23C2×#12311yescharacteristic
475C23#54811yescharacteristic
476C23#54811yescharacteristic
477C23D4×Q811yescharacteristic
478C23#55011yescharacteristic
479Q8—416no—
480Q8—88no—
481D4—24no—
482D4—24no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—42no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—88no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—88no—
520C2×C4#38612yes—
521C2×C4#38612yes—
522C2×C4#38612yes—
523C2×C4#38612yes—
524C2×C4#45414yes—
525C2×C4C2×C4×D411yescharacteristic
526C2×C4C2×C4×D411yescharacteristic
527C2×C4C2×C4×Q811yescharacteristic
528C2×C4C4×#4111yescharacteristic
529C2×C4#51911yescharacteristic
530C2×C4#52011yescharacteristic
531C2×C4C2×#12111yescharacteristic
532C2×C4C2×#12111yescharacteristic
533C2×C4C2×C22⋊Q811yescharacteristic
534C2×C4#52511yescharacteristic
535C2×C4#52911yescharacteristic
536C2×C4#53311yescharacteristic
537C2×C4#53611yescharacteristic
538C2×C4#53711yescharacteristic
539C2×C4#54111yescharacteristic
540C2×C4#54211yescharacteristic
541C2×C4#54611yescharacteristic
542C2×C4#54711yescharacteristic
543C2×C4#54811yescharacteristic
544C2×C4#54811yescharacteristic
545C2×C4#54811yescharacteristic
546C2×C4D4×Q811yescharacteristic
547C2×C4D4×Q811yescharacteristic
548C2×C4#55011yescharacteristic
549C2×C4#55211yescharacteristic
550C2×C4#55211yescharacteristic
551C2×C4#55411yescharacteristic
552C2×C4#55611yescharacteristic
553C8—42no—
554C8—42no—
555C8—82no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—48no—
565C22—44no—
566C22—84no—
567C22—84no—
568C22—84no—
569C22—84no—
570C22—84no—
571C22—82no—
572C22—82no—
573C22C2×#38611yescharacteristic
574C22Q8×C22⋊C411yescharacteristic
575C22#234111yescharacteristic
576C22#240711yescharacteristic
577C22#240911yescharacteristic
578C22#242111yescharacteristic
579C22#242311yescharacteristic
580C22C2×#45412yes—
581C22#301912yes—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—44no—
586C4—48no—
587C4—84no—
588C4—82no—
589C4—84no—
590C4—82no—
591C4#188814yes—
592C4C4×C22⋊Q811yescharacteristic
593C4#232311yescharacteristic
594C4#236011yescharacteristic
595C4#236211yescharacteristic
596C4#242611yescharacteristic
597C4#242911yescharacteristic
598C4#253611yescharacteristic
599C4#253811yescharacteristic
600C2—21no—
601C2—21no—
602C2—21no—
603C2—21no—
604C2—84no—
605C2#1396911yescharacteristic
606C2C2×#188811yescharacteristic
607C2#2095311yescharacteristic
608C2#2417112yes—
609C2#2432112yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
3. C2×#1997C2semidirect
6. #11211C2semidirect
9. #10892C2semidirect
10. #10852C2semidirect
11. #10851C2semidirect
20. #2563C4semidirect
93. #418Q8semidirect
335. C2×D4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
#342162
C2×#19972
#206462
C4×#4182
#112112
#110542
#109582
#108922
#108522
#108512