Finite Group Databasebeta

← #366269

Subgroups — #366269

Order 512.

957 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1932C211yesmaximal, characteristic
2C2×#1841C211yesmaximal, characteristic, complemented
3C2×C4×#104C211yesmaximal, characteristic
4#11367C211yesmaximal, characteristic, complemented
5#11263C211yesmaximal, characteristic, complemented
6#11151C211yesmaximal, characteristic
7#11085C211yesmaximal, characteristic
8#10945C211yesmaximal, characteristic
9#10897C211yesmaximal, characteristic, complemented
10#10878C211yesmaximal, characteristic, complemented
11#10824C211yesmaximal, characteristic, complemented
12#10805C211yesmaximal, characteristic, complemented
13#10772C211yesmaximal, characteristic
14#10758C211yesmaximal, characteristic
15C4×#339C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#421C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C22×#104C2211yescharacteristic
21#2896C2211yescharacteristic
22C2×#413C2211yescharacteristic
23C2×#410C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26#2296C2211yescharacteristic
27C4×#117C2211yescharacteristic
28C2×C43C2211yescharacteristic
29#1932C418yescomplemented
30#1841—24no—
31C4×#104—24no—
32C2×#344C2211yescharacteristic
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36#1742C2211yescharacteristic
37#1740C2211yescharacteristic
38#1733C2211yescharacteristic
39C2×#339C2211yescharacteristic
40C2×#339C2211yescharacteristic
41C2×#339C2211yescharacteristic
42C2×#339C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#338C2211yescharacteristic
50C2×H(3,ℤ/4)C2211yescharacteristic
51C2×H(3,ℤ/4)C2211yescharacteristic
52C2×H(3,ℤ/4)C2211yescharacteristic
53#1728C2211yescharacteristic
54C22×#41C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C4×#41—22no—
62C4×#41—22no—
63C4×#41—22no—
64C4×#41—41no—
65C2×C4×Q8—21no—
66C2×C4×D4—21no—
67C2×C4×D4—41no—
68C2×#117—21no—
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C22×C42—21no—
75C22×C42—41no—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#430—41no—
80#430C2×C414yescomplemented
81C2×#108—41no—
82#421—41no—
83#421C2×C414yes—
84C2×#104—22no—
85C2×#104—22no—
86C2×#104—22no—
87C2×#104—22no—
88C2×#104—21no—
89C2×#104—21no—
90C2×#104—41no—
91C2×#104C2×C414yes—
92#419—21no—
93#418—21no—
94#417—22no—
95#416—41no—
96#413—22no—
97#413C2×C414yes—
98C2×#101—21no—
99C2×#100—21no—
100#410—22no—
101#410—22no—
102C2×#99—41no—
103C4×M16—22no—
104C4×M16—22no—
105C4×M16—41no—
106C4×M16C2×C414yescomplemented
107C2×#97—21no—
108C2×#97—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111#389—22no—
112#385—41no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—41no—
115C4×C22⋊C4—41no—
116#376—21no—
117C43—22no—
118C43—22no—
119C43—21no—
120#344—22no—
121#344—22no—
122#344—22no—
123#344C2×C414yescomplemented
124#343—22no—
125#342—41no—
126#339—22no—
127#339—22no—
128#339—22no—
129#339—22no—
130#339—21no—
131#339—22no—
132#339—22no—
133#339—22no—
134#339—21no—
135#339—41no—
136#339C2×C414yescomplemented
137#339D414yes—
138#338—22no—
139H(3,ℤ/4)—22no—
140H(3,ℤ/4)—41no—
141H(3,ℤ/4)—41no—
142#336—41no—
143C2×#41—21no—
144C2×#41—21no—
145C2×#41—21no—
146C2×#41—21no—
147C2×#41—21no—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41C22×C412yes—
154C2×#41#4112yes—
155C22×Q8#4111yescharacteristic
156C22×D4—21no—
157C22×D4#4111yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—41no—
163C23×C4C2411yescharacteristic
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
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—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16C22×C412yes—
196C2×M16C22×C412yes—
197C2×M16C22×C412yes—
198C2×M16C22×C412yes—
199C2×M16C2×D412yes—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C4×Q8—24no—
207C4×Q8—42no—
208C4×D4—24no—
209C4×D4—42no—
210C4×D4—44no—
211C4×D4—44no—
212#117—21no—
213#117—21no—
214#117—21no—
215#117—21no—
216#117—21no—
217#117—21no—
218#117—21no—
219#117—21no—
220#117—21no—
221#117—21no—
222#117—21no—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—21no—
227#117—21no—
228#117—21no—
229#117—42no—
230#117—42no—
231#117C22×C412yes—
232#117C2×D412yes—
233#117#4112yes—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4#4112yes—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—41no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—42no—
264C2×C42—41no—
265C2×C42—44no—
266C2×C42—48no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42C22×C412yes—
270C2×C42C2×D412yes—
271C2×C42#4112yes—
272C2×C42#4111yescharacteristic
273C2×C42#4111yescharacteristic
274C2×C42#4111yescharacteristic
275C2×C42#4112yes—
276C2×C42#4112yes—
277C2×C42#4111yescharacteristic
278#108—44no—
279#104—28no—
280#104—24no—
281#104—24no—
282#104—44no—
283#103—42no—
284#102—42no—
285#101—24no—
286#100—24no—
287#99—44no—
288#97—24no—
289#97—42no—
290C4×C8—24no—
291C4×C8—42no—
292#95—42no—
293#95—42no—
294C24—21no—
295C24H(5,𝔽2)11yescharacteristic
296#41—22no—
297#41—22no—
298#41—22no—
299#41—22no—
300#41—22no—
301#41—22no—
302#41—22no—
303#41—22no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309#41—41no—
310#41—41no—
311#41—41no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8#11712yes—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4#11712yes—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—84no—
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C42−1+411yescharacteristic
372M16—22no—
373M16—22no—
374M16—22no—
375M16—22no—
376M16—22no—
377M16—22no—
378M16—22no—
379M16—22no—
380M16—22no—
381M16—22no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—41no—
394M16—41no—
395M16—41no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—44no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—84no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—44no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C42—22no—
438C42—22no—
439C42—21no—
440C42—21no—
441C42—22no—
442C42—22no—
443C42—21no—
444C42—21no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—44no—
451C42—416no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—84no—
462C42C4×D412yes—
463C42C4×D412yes—
464C42#12612yes—
465C42#12612yes—
466C23—22no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—82no—
471C23#52011yescharacteristic
472C23#53811yescharacteristic
473C23#53811yescharacteristic
474C23#54611yescharacteristic
475C23#55111yescharacteristic
476C23#55111yescharacteristic
477C23#55111yescharacteristic
478Q8—22no—
479Q8—22no—
480Q8—41no—
481Q8—41no—
482D4—22no—
483D4—22no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—41no—
489D4—41no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—88no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4C2×#11711yescharacteristic
526C2×C4C2×C4×D411yescharacteristic
527C2×C4C4×#4111yescharacteristic
528C2×C4C4×#4111yescharacteristic
529C2×C4C4×#4111yescharacteristic
530C2×C4#51811yescharacteristic
531C2×C4#52511yescharacteristic
532C2×C4#52511yescharacteristic
533C2×C4C2×#12611yescharacteristic
534C2×C4#52911yescharacteristic
535C2×C4#52911yescharacteristic
536C2×C4#52911yescharacteristic
537C2×C4#52911yescharacteristic
538C2×C4#52911yescharacteristic
539C2×C4#54211yescharacteristic
540C2×C4#54211yescharacteristic
541C2×C4#54611yescharacteristic
542C2×C4#54711yescharacteristic
543C2×C4#55011yescharacteristic
544C2×C4#55111yescharacteristic
545C2×C4#55111yescharacteristic
546C2×C4#55211yescharacteristic
547C2×C4#55211yescharacteristic
548C2×C4#55211yescharacteristic
549C2×C4#55311yescharacteristic
550C2×C4#55311yescharacteristic
551C2×C4#55311yescharacteristic
552C2×C4#55511yescharacteristic
553C8—42no—
554C8—42no—
555C8—42no—
556C8—42no—
557C8—42no—
558C8—81no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—42no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22#233711yescharacteristic
571C22#234311yescharacteristic
572C22#235911yescharacteristic
573C22#246711yescharacteristic
574C22#248011yescharacteristic
575C22#257211yescharacteristic
576C22#257811yescharacteristic
577C4—24no—
578C4—24no—
579C4—42no—
580C4—42no—
581C4—44no—
582C4—42no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4—161no—
588C4#229011yescharacteristic
589C4C4×#12611yescharacteristic
590C4#235311yescharacteristic
591C4#236311yescharacteristic
592C4#246811yescharacteristic
593C4#247911yescharacteristic
594C4#249511yescharacteristic
595C4#251511yescharacteristic
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—82no—
603C2#1399011yescharacteristic
604C2#2923212yes—
subgroupG/H
centerC2×C4#555
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2572
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1841C2semidirect
4. #11367C2semidirect
5. #11263C2semidirect
9. #10897C2semidirect
10. #10878C2semidirect
11. #10824C2semidirect
12. #10805C2semidirect
15. C4×#339C2semidirect
29. #1932C4semidirect
80. #430C2×C4semidirect
106. C4×M16C2×C4semidirect
123. #344C2×C4semidirect
136. #339C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19322
C2×#18412
C2×C4×#1042
#113672
#112632
#111512
#110852
#109452
#108972
#108782
#108242
#108052
#107722
#107582
C4×#3392