Finite Group Databasebeta

← #356473

Subgroups — #356473

Order 512.

2316 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#440C212yesmaximal
2C22×C4×SD16C211yesmaximal, characteristic
3Q8×#95C211yesmaximal, characteristic, complemented
4D4×#95C211yesmaximal, characteristic
5#10863C212yesmaximal, complemented
6#10862C211yesmaximal, characteristic, complemented
7#10789C212yesmaximal, complemented
8#10789C211yesmaximal, characteristic, complemented
9#10788C212yesmaximal
10#10788C211yesmaximal, characteristic
11#10696C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×C4×D4C2211yescharacteristic
16C22×C4×D4C2212yes—
17C2×#440—28no—
18C2×#440—28no—
19C2×#440C4116yescomplemented
20C2×C4×SD16—28no—
21C2×C4×SD16C418yescomplemented
22C22×#107C2212yes—
23C22×#106C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2212yes—
26C22×#102C2211yescharacteristic
27C22×#102C2212yes—
28C22×#97C2212yes—
29C22×C4×C8C2211yescharacteristic
30C22×#95—21no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36C4⋊#95—21no—
37C4⋊#95—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C22⋊#95—21no—
43C22⋊#95—22no—
44C4×#95—21no—
45C4×#95C2211yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×D4C2311yescharacteristic
48C24×C4—22no—
49C24×C4—21no—
50C22×SD16C2×C414yes—
51C22×SD16C2×C418yes—
52C22×SD16D418yes—
53C22×SD16D414yes—
54C22×SD16Q814yes—
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C23×C8C2311yescharacteristic
58C2×C4×Q8—22no—
59C2×C4×Q8—24no—
60C2×C4×Q8—24no—
61C2×C4×Q8—24no—
62C2×C4×Q8—24no—
63C2×C4×Q8—42no—
64C2×C4×Q8—42no—
65C2×C4×Q8—42no—
66C2×C4×Q8C2×C414yescomplemented
67C2×C4×Q8C2×C418yescomplemented
68C2×C4×D4—24no—
69C2×C4×D4—24no—
70C2×C4×D4—24no—
71C2×C4×D4—22no—
72C2×C4×D4—24no—
73C2×C4×D4—42no—
74C2×C4×D4—42no—
75C2×C4×D4—42no—
76C2×C4×D4C2×C414yes—
77C2×C4×D4C2×C418yes—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C42—22no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2312yes—
90#440—264no—
91C4×SD16—232no—
92C2×#107—24no—
93C2×#107—24no—
94C2×#107C2×C418yescomplemented
95C2×#106—24no—
96C2×#106C2×C414yescomplemented
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103C2×C414yescomplemented
101C2×#103C2×C418yescomplemented
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102C2×C414yes—
106C2×#102C2×C418yes—
107C2×#97—24no—
108C2×#97—24no—
109C2×#97C2×C418yescomplemented
110C2×C4×C8—24no—
111C2×C4×C8C2×C414yescomplemented
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—44no—
121C2×#95D411yescharacteristic
122C2×#95D411yescharacteristic
123C2×#95D412yes—
124#340—42no—
125#340—42no—
126#340—42no—
127#336—24no—
128#336—22no—
129C25—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8C22×C412yes—
136C22×Q8C22×C414yes—
137C22×Q8C2×D414yes—
138C22×Q8C2×D412yes—
139C22×Q8C2×Q812yes—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C22×C412yes—
146C22×D4C22×C414yes—
147C22×D4C2×D414yes—
148C22×D4C2×D412yes—
149C22×D4C2×Q812yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—28no—
156C23×C4—24no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—82no—
165C23×C4—82no—
166C23×C4—82no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C2×SD16C42116yes—
173C2×SD16C22⋊C4116yes—
174C2×SD16C22⋊C4132yes—
175C2×SD16C4⋊C4116yes—
176C2×SD16C4⋊C4132yes—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8C22×C412yes—
188C22×C8C22×C414yes—
189C22×C8C22×C412yes—
190C22×C8C22×C414yes—
191C22×C8C2×D414yes—
192C22×C8C2×D412yes—
193C22×C8C2×Q812yes—
194C4×Q8—28no—
195C4×Q8—216no—
196C4×Q8—44no—
197C4×Q8—48no—
198C4×Q8—48no—
199C4×Q8—48no—
200C4×Q8—48no—
201C4×D4—28no—
202C4×D4—216no—
203C4×D4—44no—
204C4×D4—48no—
205C4×D4—48no—
206C4×D4—48no—
207C4×D4—48no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C414yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C414yes—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—28no—
245C2×C42—24no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42C22×C412yes—
250C2×C42C22×C414yes—
251C2×C42#4114yes—
252C2×C42#4112yes—
253#107—216no—
254#106—28no—
255#103—28no—
256#103—216no—
257#102—28no—
258#102—216no—
259#97—216no—
260C4×C8—28no—
261#95—22no—
262#95—22no—
263#95—42no—
264#95—44no—
265#95—44no—
266#95—42no—
267#95—42no—
268#95—81no—
269#95—81no—
270#95—82no—
271#95—82no—
272C24—22no—
273C24—24no—
274C24—24no—
275C24—22no—
276C24—22no—
277C24—41no—
278C24—41no—
279C24—42no—
280C24—41no—
281C24—42no—
282C24—41no—
283C24C22×D411yescharacteristic
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—22no—
287C2×Q8—24no—
288C2×Q8—22no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8C2×C4214yes—
298C2×Q8C2×C22⋊C414yes—
299C2×Q8C2×C22⋊C418yes—
300C2×Q8C2×C4⋊C414yes—
301C2×Q8C2×C4⋊C418yes—
302C2×D4—22no—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4—24no—
306C2×D4—22no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—22no—
312C2×D4—24no—
313C2×D4—22no—
314C2×D4—24no—
315C2×D4C2×C4214yes—
316C2×D4C2×C22⋊C414yes—
317C2×D4C2×C22⋊C418yes—
318C2×D4C2×C4⋊C414yes—
319C2×D4C2×C4⋊C418yes—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—48no—
348C22×C4—416no—
349C22×C4—416no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—48no—
371C22×C4—84no—
372C22×C4—88no—
373C22×C4—88no—
374C22×C4—88no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4—88no—
379C22×C4C4×D414yes—
380C22×C4C4×D412yes—
381C22×C4C4×D414yes—
382C22×C4C4×D412yes—
383C22×C4AGL(1,ℤ/8)12yes—
384C22×C4#13712yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C23×C412yes—
387C22×C4C22×D412yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C22×Q811yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4112yes—
392C22×C4H(5,𝔽2)12yes—
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C42−1+411yescharacteristic
395SD16#95164yescomplemented
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—22no—
401C2×C8—24no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8C2×C4214yes—
409C2×C8C2×C22⋊C414yes—
410C2×C8C2×C22⋊C418yes—
411C2×C8C2×C4⋊C414yes—
412C2×C8C2×C4⋊C418yes—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—48no—
422C4⋊C4—416no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C22⋊C4—416no—
430C22⋊C4—48no—
431C42—22no—
432C42—24no—
433C42—22no—
434C42—24no—
435C42—24no—
436C42—24no—
437C42—24no—
438C42—48no—
439C42—416no—
440C23—24no—
441C23—28no—
442C23—24no—
443C23—28no—
444C23—24no—
445C23—42no—
446C23—42no—
447C23—44no—
448C23—42no—
449C23—42no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—42no—
455C23—42no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—42no—
461C23—44no—
462C23—42no—
463C23C2×C4×D411yescharacteristic
464C23C2×C4×D412yes—
465C23D4212yes—
466C23D4211yescharacteristic
467C23D4×Q811yescharacteristic
468C23C2×AGL(1,ℤ/8)11yescharacteristic
469C23C2×#13711yescharacteristic
470C23#57512yes—
471C23#57611yescharacteristic
472C23#57712yes—
473C23#57811yescharacteristic
474Q8—24no—
475Q8—28no—
476Q8—24no—
477Q8—28no—
478Q8—24no—
479Q8C2×#9518yescomplemented
480D4—24no—
481D4—28no—
482D4—24no—
483D4—28no—
484D4—24no—
485D4C2×#9518yes—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—816no—
527C2×C4—832no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4#44014yescomplemented
541C2×C4C22×C4211yescharacteristic
542C2×C4C22×C22⋊C411yescharacteristic
543C2×C4C22×C22⋊C412yes—
544C2×C4C22×C4⋊C411yescharacteristic
545C2×C4C22×C4⋊C412yes—
546C2×C4C4×#4111yescharacteristic
547C2×C4C4×#4112yes—
548C2×C4#51811yescharacteristic
549C2×C4#51812yes—
550C2×C4#52011yescharacteristic
551C2×C4#52012yes—
552C2×C4#54612yes—
553C2×C4#54612yes—
554C2×C4#54612yes—
555C2×C4#54711yescharacteristic
556C2×C4#54712yes—
557C2×C4#55411yescharacteristic
558C2×C4#55412yes—
559C8—42no—
560C8—44no—
561C8C2×#9518yescomplemented
562C22—28no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—48no—
568C22—48no—
569C22—44no—
570C22—44no—
571C22—48no—
572C22—48no—
573C22C42×D411yescharacteristic
574C22D4×C22⋊C411yescharacteristic
575C22D4×C22⋊C412yes—
576C22D4×C4⋊C411yescharacteristic
577C22D4×C4⋊C412yes—
578C22C2×#44011yescharacteristic
579C22#294311yescharacteristic
580C22C4×AGL(1,ℤ/8)12yes—
581C22C4×#13712yes—
582C22#294612yes—
583C22#294712yes—
584C22#294911yescharacteristic
585C22#295111yescharacteristic
586C22#326512yes—
587C22#326612yes—
588C22#326812yes—
589C22#326912yes—
590C22#327411yescharacteristic
591C22#327512yes—
592C22#327611yescharacteristic
593C22#338511yescharacteristic
594C22#338611yescharacteristic
595C22#338812yes—
596C4—24no—
597C4—28no—
598C4—42no—
599C4—44no—
600C4—44no—
601C4—82no—
602C4—82no—
603C4—84no—
604C4—84no—
605C4C22×#9511yescharacteristic
606C4#227511yescharacteristic
607C4#233911yescharacteristic
608C4#233912yes—
609C4#234911yescharacteristic
610C4#234912yes—
611C2—48no—
612C2D4×#9511yescharacteristic
613C2C4×#44011yescharacteristic
614C2#2050411yescharacteristic
615C2#2243711yescharacteristic
616C2#2243911yescharacteristic
617C2#2244312yes—
618C2#2244412yes—
619C2#2267111yescharacteristic
620C2#2267311yescharacteristic
621C2#2267712yes—
622C2#2267812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. Q8×#95C2semidirect
5. #10863C2semidirect
6. #10862C2semidirect
7. #10789C2semidirect
8. #10789C2semidirect
11. #10696C2semidirect
19. C2×#440C4semidirect
21. C2×C4×SD16C4semidirect
66. C2×C4×Q8C2×C4semidirect
67. C2×C4×Q8C2×C4semidirect
94. C2×#107C2×C4semidirect
96. C2×#106C2×C4semidirect
100. C2×#103C2×C4semidirect
101. C2×#103C2×C4semidirect
109. C2×#97C2×C4semidirect
111. C2×C4×C8C2×C4semidirect
395. SD16#95semidirect
479. Q8C2×#95semidirect
540. C2×C4#440semidirect
561. C8C2×#95semidirect

Maximal subgroups

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

subgroupindex
C22×#4402
C22×C4×SD162
Q8×#952
D4×#952
#108632
#108622
#107892
#107882
#106962