Finite Group Databasebeta

← #497439

Subgroups — #497439

Order 512.

1100 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#509C211yesmaximal, characteristic
2C2×#2175C211yesmaximal, characteristic, complemented
3C2×C4×SD32C211yesmaximal, characteristic, complemented
4#22451C211yesmaximal, characteristic, complemented
5C22⋊C4×D8C211yesmaximal, characteristic, complemented
6#13239C211yesmaximal, characteristic, complemented
7#13235C211yesmaximal, characteristic, complemented
8#13013C211yesmaximal, characteristic, complemented
9#13013C211yesmaximal, characteristic, complemented
10#13012C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic, complemented
12#12836C211yesmaximal, characteristic, complemented
13#12733C211yesmaximal, characteristic, complemented
14#12731C211yesmaximal, characteristic, complemented
15#12612C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C23×D8C2211yescharacteristic
18C2×#509—24no—
19C2×#509—24no—
20C2×#509C418yescomplemented
21C22×SD32C2211yescharacteristic
22C22×SD32C2211yescharacteristic
23C22×D16C2211yescharacteristic
24C22×D16C2211yescharacteristic
25C22×M32C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic, complemented
27C2×C4×Q16C2211yescharacteristic, complemented
28C2×C4×D8C2211yescharacteristic
29C2×C4×D8C2211yescharacteristic
30#2341—21no—
31D4×C22⋊C4—21no—
32#2175—24no—
33C4×SD32—24no—
34C2×#367C2211yescharacteristic, complemented
35C2×#366C2211yescharacteristic, complemented
36C2×#358C2211yescharacteristic, complemented
37C2×#358C2211yescharacteristic, complemented
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×#357C2211yescharacteristic
41C2×#357C2211yescharacteristic
42C2×#357C2211yescharacteristic, complemented
43C2×#357C2211yescharacteristic, complemented
44C2×C22⋊C16C2211yescharacteristic
45C2×#346C2211yescharacteristic, complemented
46C2×C4×C16C2211yescharacteristic, complemented
47#1970C2211yescharacteristic
48#1970C2211yescharacteristic
49#1970C2211yescharacteristic, complemented
50#1970C2211yescharacteristic, complemented
51#1880—21no—
52#1880—21no—
53#1879—21no—
54#1879—21no—
55#1808C2211yescharacteristic
56#1808C2211yescharacteristic
57#1788—21no—
58#1786—21no—
59C8×C22⋊C4C2211yescharacteristic
60#1380C2211yescharacteristic, complemented
61#1380C2211yescharacteristic, complemented
62#1374C2211yescharacteristic, complemented
63#1374C2211yescharacteristic, complemented
64C22×#41—21no—
65C23×D4—21no—
66C2×#135—22no—
67C2×#135—22no—
68C2×#135—22no—
69C2×#135—41no—
70C2×#135—41no—
71C2×#135C2×C414yescomplemented
72C22×Q16C2311yescharacteristic
73C22×SD16—21no—
74C22×D8—22no—
75C22×D8—22no—
76C22×D8—21no—
77C22×D8—22no—
78C22×D8—41no—
79C22×D8—41no—
80C22×D8C2×C414yes—
81C22×D8C2311yescharacteristic
82C22×D8C2311yescharacteristic
83C22×D8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×C4×Q8—21no—
86C2×C4×Q8—21no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C22×C22⋊C4—41no—
90#509—232no—
91C2×SD32—22no—
92C2×SD32—22no—
93C2×SD32C2×C414yes—
94C2×SD32C2×C414yescomplemented
95C2×SD32D414yes—
96C2×SD32D414yes—
97C2×D16—22no—
98C2×D16—22no—
99C2×D16—22no—
100C2×D16—22no—
101C2×D16C2×C414yes—
102C2×D16C2×C414yescomplemented
103C2×M32—22no—
104C2×M32—22no—
105C2×M32C2×C414yescomplemented
106C22×C16C2311yescharacteristic
107C22×C16C2311yescharacteristic
108C22×C16C2311yescharacteristic
109C22×C16C2311yescharacteristic
110C4×Q16—22no—
111C4×Q16—22no—
112C4×Q16—41no—
113C4×Q16—41no—
114C4×D8—22no—
115C4×D8—22no—
116C4×D8—41no—
117C4×D8—41no—
118C2×#107C2311yescharacteristic
119C2×#107C2311yescharacteristic
120C2×#107C2311yescharacteristic
121C2×#107C2311yescharacteristic
122C2×#103—21no—
123C2×#103—21no—
124C2×#102—21no—
125C2×#102—21no—
126C2×#102—21no—
127C2×#102—21no—
128C2×#102—21no—
129C2×#102—21no—
130C2×C22⋊C8D411yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#390—21no—
134#390—21no—
135#390—21no—
136#390—21no—
137#390—41no—
138#390—41no—
139#386—41no—
140#386—41no—
141#380—21no—
142#380—21no—
143#380—41no—
144#379—41no—
145C4×C22⋊C4—41no—
146C4×C22⋊C4D411yescharacteristic
147#367—22no—
148#366—22no—
149#358—22no—
150#358—22no—
151#357—22no—
152#357—22no—
153#357—41no—
154#357—41no—
155#357—41no—
156#357—41no—
157C22⋊C16—22no—
158#346—22no—
159C4×C16—22no—
160#340—21no—
161#340—21no—
162#336D411yescharacteristic
163#336D411yescharacteristic
164C25—41no—
165C2×#41—24no—
166C2×#41—42no—
167C2×#41—42no—
168C2×#41—42no—
169C2×#41—81no—
170C2×#41—81no—
171C22×Q8—21no—
172C22×D4—24no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—41no—
179C22×D4—41no—
180C22×D4—42no—
181C22×D4—81no—
182C22×D4—81no—
183C23×C4—41no—
184C23×C4—41no—
185C23×C4—41no—
186C23×C4C2×D411yescharacteristic
187#135—28no—
188#135—44no—
189#135—44no—
190#135—44no—
191C2×Q16—21no—
192C2×Q16—21no—
193C2×Q16—21no—
194C2×Q16C22×C412yes—
195C2×Q16C2×D412yes—
196C2×Q16C2×D412yes—
197C2×SD16—24no—
198C2×SD16—42no—
199C2×SD16—42no—
200C2×D8—28no—
201C2×D8—24no—
202C2×D8—21no—
203C2×D8—21no—
204C2×D8—21no—
205C2×D8—21no—
206C2×D8—21no—
207C2×D8—21no—
208C2×D8—21no—
209C2×D8—21no—
210C2×D8—21no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—44no—
216C2×D8—44no—
217C2×D8—44no—
218C2×D8—42no—
219C2×D8—42no—
220C2×D8C22×C412yes—
221C2×D8C22×C412yes—
222C2×D8C22×C412yes—
223C2×D8C2×D412yes—
224C2×D8C2×D412yes—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8C22×C412yes—
230C22×C8C2×D411yescharacteristic
231C22×C8C2×D411yescharacteristic
232C22×C8C2×D411yescharacteristic
233C22×C8#4112yes—
234C22×C8C2411yescharacteristic
235C4×Q8—42no—
236C4×Q8—42no—
237C4×Q8—81no—
238C4×Q8—81no—
239C4×D4—42no—
240C4×D4—42no—
241C4×D4—81no—
242C4×D4—81no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—84no—
262C2×C22⋊C4D811yescharacteristic
263C2×C22⋊C4D811yescharacteristic
264C2×C42—41no—
265C2×C42—41no—
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268SD32—28no—
269SD32C22⋊C4116yescomplemented
270D16—28no—
271D16—28no—
272M32—28no—
273C2×C16—21no—
274C2×C16—21no—
275C2×C16—21no—
276C2×C16—21no—
277C2×C16—21no—
278C2×C16—21no—
279C2×C16—21no—
280C2×C16—21no—
281C2×C16C22×C412yes—
282C2×C16C22×C412yes—
283C2×C16C2×D412yes—
284C2×C16C2×D412yes—
285#107—21no—
286#107—21no—
287#107—21no—
288#107—21no—
289#107—21no—
290#107—21no—
291#107—21no—
292#107—21no—
293#103—42no—
294#103—42no—
295#102—42no—
296#102—42no—
297#102—81no—
298#102—81no—
299#102—81no—
300#102—81no—
301C22⋊C8—22no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306#95—41no—
307#95—41no—
308#95D811yescharacteristic
309#95D811yescharacteristic
310C24—41no—
311C24—41no—
312C24—41no—
313C24—44no—
314C24—82no—
315C24—82no—
316C24—82no—
317C24—82no—
318C24—161no—
319C24—161no—
320C24C2×D811yescharacteristic
321#41—48no—
322#41—84no—
323#41—84no—
324#41—84no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—48no—
337C2×D4—44no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—44no—
352C2×D4—84no—
353C2×D4—84no—
354C2×D4—84no—
355C2×D4—82no—
356C2×D4—82no—
357C2×D4—82no—
358C2×D4—82no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—41no—
364C22×C4—44no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—161no—
376C22×C4—161no—
377C22×C4C4×D412yes—
378C22×C4C2×D811yescharacteristic
379C22×C4C2×D811yescharacteristic
380C22×C4C2×D811yescharacteristic
381C22×C4C2×D811yescharacteristic
382C22×C4C2×D811yescharacteristic
383C22×C4#13512yes—
384C22×C4C22×D411yescharacteristic
385Q16—22no—
386Q16—22no—
387Q16—22no—
388Q16C2×C22⋊C414yescomplemented
389SD16—48no—
390D8—22no—
391D8—22no—
392D8—22no—
393D8—22no—
394D8—22no—
395D8—22no—
396D8—22no—
397D8—48no—
398D8—41no—
399D8—41no—
400D8—41no—
401D8—41no—
402D8C2×C22⋊C414yes—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8C4×D412yes—
416C2×C8C23×C411yescharacteristic
417C2×C8C22×D411yescharacteristic
418C2×C8C22×D411yescharacteristic
419C2×C8C2×#4111yescharacteristic
420C2×C8H(5,𝔽2)11yescharacteristic
421C2×C8H(5,𝔽2)11yescharacteristic
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C22⋊C4—22no—
433C22⋊C4—41no—
434C22⋊C4—82no—
435C22⋊C4—82no—
436C22⋊C4—82no—
437C22⋊C4—82no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C22⋊C4—161no—
442C22⋊C4—161no—
443C22⋊C4—161no—
444C22⋊C4—161no—
445C22⋊C4—161no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—82no—
451C42—82no—
452C16—22no—
453C16—41no—
454C16—41no—
455C16C2×C22⋊C414yescomplemented
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—88no—
468C23—88no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—164no—
486C23—164no—
487C23—164no—
488C23—164no—
489C23—164no—
490C23C4×D812yes—
491C23C22×D811yescharacteristic
492Q8—24no—
493Q8—42no—
494Q8—42no—
495Q8—42no—
496D4—24no—
497D4—42no—
498D4—42no—
499D4—42no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—42no—
504D4—88no—
505D4—88no—
506D4—81no—
507D4—81no—
508D4—81no—
509D4—81no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—88no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—164no—
544C2×C4—164no—
545C2×C4C4×D812yes—
546C2×C4#50912yescomplemented
547C2×C4C2×C4×D411yescharacteristic
548C2×C4D4211yescharacteristic
549C2×C4D4211yescharacteristic
550C2×C4C2×#13511yescharacteristic
551C2×C4#57611yescharacteristic
552C2×C4#57611yescharacteristic
553C8—22no—
554C8—41no—
555C8—41no—
556C8C22×C22⋊C411yescharacteristic
557C8#51811yescharacteristic
558C8#54611yescharacteristic
559C8#54611yescharacteristic
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—44no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—1616no—
581C22—161no—
582C22—161no—
583C22—161no—
584C22—161no—
585C22—161no—
586C22—161no—
587C22—161no—
588C22—161no—
589C22C2×C4×D811yescharacteristic
590C22D4×D811yescharacteristic
591C22D4×D811yescharacteristic
592C22C2×#50911yescharacteristic
593C22#341411yescharacteristic
594C22#341511yescharacteristic
595C22#341611yescharacteristic
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—44no—
602C4—161no—
603C4—161no—
604C4—161no—
605C4—161no—
606C4D4×C22⋊C411yescharacteristic
607C4#294811yescharacteristic
608C4#328011yescharacteristic
609C4#328011yescharacteristic
610C2—42no—
611C2—84no—
612C2—162no—
613C2—162no—
614C2C22⋊C4×D811yescharacteristic
615C2C4×#50911yescharacteristic
616C2#3345411yescharacteristic
617C2#3383111yescharacteristic
618C2#3383411yescharacteristic
619C2#3384011yescharacteristic
620C2#3384111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×#509C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2175C2semidirect
3. C2×C4×SD32C2semidirect
4. #22451C2semidirect
5. C22⋊C4×D8C2semidirect
6. #13239C2semidirect
7. #13235C2semidirect
8. #13013C2semidirect
9. #13013C2semidirect
10. #13012C2semidirect
11. #13012C2semidirect
12. #12836C2semidirect
13. #12733C2semidirect
14. #12731C2semidirect
15. #12612C2semidirect
20. C2×#509C4semidirect
26. C2×C4×Q16C22semidirect
27. C2×C4×Q16C22semidirect
34. C2×#367C22semidirect
35. C2×#366C22semidirect
36. C2×#358C22semidirect
37. C2×#358C22semidirect
42. C2×#357C22semidirect
43. C2×#357C22semidirect
45. C2×#346C22semidirect
46. C2×C4×C16C22semidirect
49. #1970C22semidirect
50. #1970C22semidirect
60. #1380C22semidirect
61. #1380C22semidirect
62. #1374C22semidirect
63. #1374C22semidirect
71. C2×#135C2×C4semidirect
94. C2×SD32C2×C4semidirect
102. C2×D16C2×C4semidirect
105. C2×M32C2×C4semidirect
269. SD32C22⋊C4semidirect
388. Q16C2×C22⋊C4semidirect
455. C16C2×C22⋊C4semidirect
546. C2×C4#509semidirect

Maximal subgroups

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

subgroupindex
C22×#5092
C2×#21752
C2×C4×SD322
#224512
C22⋊C4×D82
#132392
#132352
#130132
#130122
#128362
#127332
#127312
#126122