Finite Group Databasebeta

← #384928

Subgroups — #384928

Order 512.

834 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1938C211yesmaximal, characteristic
2#20863C211yesmaximal, characteristic
3#20762C211yesmaximal, characteristic, complemented
4#11350C211yesmaximal, characteristic, complemented
5#11305C211yesmaximal, characteristic, complemented
6#11262C211yesmaximal, characteristic, complemented
7#11236C211yesmaximal, characteristic, complemented
8#11176C211yesmaximal, characteristic
9#11081C211yesmaximal, characteristic
10#10956C211yesmaximal, characteristic
11#10900C211yesmaximal, characteristic, complemented
12#10893C211yesmaximal, characteristic, complemented
13#10819C211yesmaximal, characteristic, complemented
14#10779C211yesmaximal, characteristic
15#10752C211yesmaximal, characteristic
16#3428C2211yescharacteristic
17#2911C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C22×#104C2211yescharacteristic
21#2896C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#410C2211yescharacteristic
24C2×#408C2211yescharacteristic
25#2873C2211yescharacteristic
26C22×#97C2211yescharacteristic
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29#1938C418yescomplemented
30#1860—22no—
31#1859—22no—
32#1836—22no—
33#1836—22no—
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36#1742C2211yescharacteristic
37#1740C2211yescharacteristic
38#1740C2211yescharacteristic
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41#1735C2211yescharacteristic
42#1735C2211yescharacteristic
43#1735C2211yescharacteristic
44#1735C2211yescharacteristic
45#1735C2211yescharacteristic
46#1733C2211yescharacteristic
47#1733C2211yescharacteristic
48#1733C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×H(3,ℤ/4)C2211yescharacteristic
53C2×H(3,ℤ/4)C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56C22×#41C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63#520—22no—
64#520—22no—
65#519—21no—
66#518—21no—
67#518—41no—
68C4×#41—41no—
69C4×#41—41no—
70C2×#117—41no—
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#428—22no—
80#428—22no—
81#427—41no—
82#424—41no—
83#424C2×C414yescomplemented
84C2×C4⋊C8—41no—
85#421—22no—
86#421—22no—
87C2×#104—21no—
88C2×#104—21no—
89C2×#104—22no—
90C2×#104—41no—
91C2×#104C2×C414yes—
92#419—21no—
93#418—21no—
94#417—22no—
95#416—22no—
96#416—41no—
97#416C2×C414yes—
98C2×#103—21no—
99C2×#102—21no—
100#410—22no—
101#410—41no—
102C2×#99—41no—
103#408—41no—
104#408C2×C414yes—
105C2×C22⋊C8—41no—
106#405—41no—
107#405—41no—
108C4×M16—41no—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97—22no—
112C2×#97C2×C414yescomplemented
113#384—21no—
114#384D412yes—
115#382—21no—
116#382D412yes—
117#380—41no—
118#380—41no—
119C4×C4⋊C4—41no—
120C4×C4⋊C4—41no—
121#344—22no—
122#344—41no—
123#343—22no—
124#342—41no—
125#342—41no—
126#341—22no—
127#341C2×C414yescomplemented
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#339—21no—
134#339—21no—
135#339—22no—
136#339—41no—
137#339—41no—
138#339—41no—
139#339C2×C414yescomplemented
140H(3,ℤ/4)—41no—
141H(3,ℤ/4)—41no—
142#336—41no—
143#336—41no—
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×#41—21no—
154C2×#41C22×C412yes—
155C2×#41#4112yes—
156C22×Q8#4111yescharacteristic
157C22×D4—21no—
158C22×D4#4111yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C2411yescharacteristic
162C2×M16—21no—
163C2×M16—21no—
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×M16C22×C412yes—
184C2×M16C22×C412yes—
185C2×M16#4112yes—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8C22×C412yes—
203C22×C8C22×C412yes—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—44no—
209C4×D4—44no—
210#117—21no—
211#117—21no—
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—21no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—42no—
238#117—42no—
239#117—42no—
240#117C22×C412yes—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4#4112yes—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—42no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—42no—
270C2×C42C22×C412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42#4112yes—
274#107—42no—
275#106—42no—
276C4⋊C8—44no—
277#104—24no—
278#104—24no—
279#104—44no—
280#103—24no—
281#103—42no—
282#102—24no—
283#102—42no—
284#99—44no—
285C22⋊C8—44no—
286#97—24no—
287#97—24no—
288#97—42no—
289C4×C8—42no—
290#95—42no—
291#95—42no—
292C24—21no—
293C24H(5,𝔽2)11yescharacteristic
294#41—22no—
295#41—22no—
296#41—22no—
297#41—22no—
298#41—22no—
299#41—22no—
300#41—22no—
301#41—22no—
302#41—41no—
303#41—41no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309#41—41no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8#11712yes—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4#11712yes—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—44no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—82no—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C4H(5,𝔽2)11yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C42−1+411yescharacteristic
360C22×C42−1+411yescharacteristic
361C22×C42−1+411yescharacteristic
362M16—22no—
363M16—22no—
364M16—22no—
365M16—22no—
366M16—22no—
367M16—22no—
368M16—41no—
369M16—41no—
370M16—41no—
371M16—41no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—41no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—44no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C22⋊C4—22no—
422C22⋊C4—44no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C42—22no—
437C42—21no—
438C42—21no—
439C42—22no—
440C42—22no—
441C42—44no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42C4×D412yes—
456C42C4×D412yes—
457C23—22no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23—82no—
462C23#52011yescharacteristic
463C23#53411yescharacteristic
464C23#53811yescharacteristic
465C23#53811yescharacteristic
466C23#53811yescharacteristic
467C23#53911yescharacteristic
468C23#54211yescharacteristic
469Q8—22no—
470Q8—22no—
471Q8—41no—
472Q8—41no—
473D4—22no—
474D4—22no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—41no—
480D4—41no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—164no—
509C2×C4C2×#11711yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C4×#4111yescharacteristic
512C2×C4#51811yescharacteristic
513C2×C4#52011yescharacteristic
514C2×C4#52011yescharacteristic
515C2×C4#53611yescharacteristic
516C2×C4#53911yescharacteristic
517C2×C4#53911yescharacteristic
518C2×C4#53911yescharacteristic
519C2×C4#53911yescharacteristic
520C2×C4#54111yescharacteristic
521C2×C4#54611yescharacteristic
522C2×C4#54611yescharacteristic
523C2×C4#54811yescharacteristic
524C2×C4#55011yescharacteristic
525C2×C4#55111yescharacteristic
526C2×C4#55111yescharacteristic
527C2×C4#55211yescharacteristic
528C2×C4#55211yescharacteristic
529C2×C4#55211yescharacteristic
530C2×C4#55311yescharacteristic
531C2×C4#55311yescharacteristic
532C2×C4#55511yescharacteristic
533C2×C4H(3,𝔽4)11yescharacteristic
534C2×C4#56211yescharacteristic
535C2×C4#56211yescharacteristic
536C2×C4#56311yescharacteristic
537C8—42no—
538C8—42no—
539C8—42no—
540C8—42no—
541C8—42no—
542C8—81no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—42no—
551C22—82no—
552C22—82no—
553C22—161no—
554C22#231811yescharacteristic
555C22#232211yescharacteristic
556C22#233411yescharacteristic
557C22#268411yescharacteristic
558C22#273011yescharacteristic
559C22#276511yescharacteristic
560C22#278111yescharacteristic
561C4—42no—
562C4—42no—
563C4—42no—
564C4—82no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—161no—
569C4—161no—
570C4—161no—
571C4#234311yescharacteristic
572C4#235211yescharacteristic
573C4#236311yescharacteristic
574C4#237911yescharacteristic
575C4#269011yescharacteristic
576C4#269411yescharacteristic
577C4#274411yescharacteristic
578C4#278211yescharacteristic
579C2—21no—
580C2—21no—
581C2—21no—
582C2—21no—
583C2—21no—
584C2—21no—
585C2—82no—
586C2#1439311yescharacteristic
587C2#3128012yes—
subgroupG/H
centerC2×C4H(3,𝔽4)
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2684
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
3. #20762C2semidirect
4. #11350C2semidirect
5. #11305C2semidirect
6. #11262C2semidirect
7. #11236C2semidirect
11. #10900C2semidirect
12. #10893C2semidirect
13. #10819C2semidirect
29. #1938C4semidirect
83. #424C2×C4semidirect
112. C2×#97C2×C4semidirect
127. #341C2×C4semidirect
139. #339C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19382
#208632
#207622
#113502
#113052
#112622
#112362
#111762
#110812
#109562
#109002
#108932
#108192
#107792
#107522