Finite Group Databasebeta

← #389682

Subgroups — #389682

Order 512.

876 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23601C211yesmaximal, characteristic
2#21178C211yesmaximal, characteristic, complemented
3#21042C211yesmaximal, characteristic, complemented
4#20886C211yesmaximal, characteristic
5#20885C211yesmaximal, characteristic
6#11343C211yesmaximal, characteristic, complemented
7#11304C211yesmaximal, characteristic, complemented
8#11250C211yesmaximal, characteristic, complemented
9#11218C211yesmaximal, characteristic, complemented
10#11146C211yesmaximal, characteristic
11#10956C211yesmaximal, characteristic
12#10930C211yesmaximal, characteristic
13#10883C211yesmaximal, characteristic, complemented
14#10865C211yesmaximal, characteristic, complemented
15#10753C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C22×#108C2211yescharacteristic
21#2911C2211yescharacteristic
22#2905C2211yescharacteristic
23#2905C2211yescharacteristic
24C2×#424C2211yescharacteristic
25C2×#421C2211yescharacteristic
26C22×#104C2211yescharacteristic
27#2896C2211yescharacteristic
28C2×#416C2211yescharacteristic
29C2×#413C2211yescharacteristic
30C2×#408C2211yescharacteristic
31#2873C2211yescharacteristic
32C22×#97C2211yescharacteristic
33#2064C414yescomplemented
34#2063C414yescomplemented
35#1950—24no—
36#1922—24no—
37#1866—24no—
38#1866—24no—
39C2×#344C2211yescharacteristic
40C2×#344C2211yescharacteristic
41#1742C2211yescharacteristic
42#1738C2211yescharacteristic
43#1738C2211yescharacteristic
44C2×#341C2211yescharacteristic
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47C2×#341C2211yescharacteristic
48#1735C2211yescharacteristic
49#1733C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1728C2211yescharacteristic
56#1728C2211yescharacteristic
57C22×#41C2311yescharacteristic
58#568—22no—
59#568—41no—
60#568—41no—
61#568—41no—
62C2×#131—22no—
63C2×#131—22no—
64C22×M16—41no—
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#430—41no—
80#430—41no—
81#430—41no—
82#430—41no—
83C2×#108—22no—
84C2×#108—22no—
85C2×#108—41no—
86C2×#108—41no—
87C2×#108—41no—
88C2×#108—41no—
89C2×#108C2×C414yes—
90C2×#108D414yes—
91#428—22no—
92#428C2×C414yes—
93#427—22no—
94#424—22no—
95#424—22no—
96#424—22no—
97#424—22no—
98#424—41no—
99#424C2×C414yes—
100#423—22no—
101#423—41no—
102C2×C4⋊C8—22no—
103#421—22no—
104#421—41no—
105C2×#104—22no—
106C2×#104—22no—
107C2×#104—22no—
108C2×#104—41no—
109#419C2×C412yes—
110#418C2×C412yes—
111#417—22no—
112#416—22no—
113#416—22no—
114#416—22no—
115C2×#103C2×C412yes—
116C2×#102C2×C412yes—
117#413—41no—
118#413—41no—
119C2×#101C2×C412yes—
120C2×#100C2×C412yes—
121#408—41no—
122#408Q814yes—
123C2×C22⋊C8—41no—
124#405—22no—
125#405—41no—
126C4×M16—22no—
127C2×#97—22no—
128C2×#97—41no—
129C2×#97—41no—
130#344—41no—
131#344—41no—
132#343C2×C414yes—
133#341—22no—
134#341—22no—
135#341—22no—
136#341—41no—
137#341—41no—
138#341—41no—
139#340—22no—
140#339—22no—
141#339—22no—
142#339—41no—
143#339D414yes—
144#336—22no—
145#336—22no—
146#336—22no—
147#336—41no—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41C2×Q812yes—
159C2×#41#4112yes—
160C22×Q8C22×C411yescharacteristic
161C22×D4—21no—
162C22×D4C22×C411yescharacteristic
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C2411yescharacteristic
166#131—44no—
167#131—44no—
168#131—44no—
169#131—44no—
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×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—42no—
212C2×M16—44no—
213C2×M16—44no—
214C2×M16C22×C412yes—
215C2×M16C22×C412yes—
216C2×M16C22×C412yes—
217C2×M16C2×D412yes—
218C2×M16C2×D412yes—
219C2×M16C2×D412yes—
220C2×M16C2×Q812yes—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—42no—
240C22×C8C22×C412yes—
241C22×C8C2×D412yes—
242C22×C8C2×Q812yes—
243C22×C8#4112yes—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117C22×C412yes—
255#117C2×D412yes—
256C2×C4⋊C4—21no—
257C2×C4⋊C4C22×C412yes—
258C2×C4⋊C4#4112yes—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42C2×D412yes—
271#108—44no—
272#108—44no—
273#108—44no—
274#108—44no—
275#107—24no—
276#106—24no—
277C4⋊C8—28no—
278C4⋊C8—44no—
279C4⋊C8—44no—
280#104—44no—
281#104—44no—
282#103—24no—
283#103—24no—
284#102—24no—
285#102—24no—
286#101—42no—
287#100—42no—
288C22⋊C8—44no—
289#97—42no—
290#97—44no—
291C4×C8—42no—
292C24—21no—
293C242−1+411yescharacteristic
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×Q8C4×Q812yes—
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×D4C4×Q812yes—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—82no—
342C22×C4C23×C411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×Q811yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4H(5,𝔽2)11yescharacteristic
350C22×C4H(5,𝔽2)11yescharacteristic
351C22×C4H(5,𝔽2)11yescharacteristic
352C22×C4H(5,𝔽2)11yescharacteristic
353C22×C4H(5,𝔽2)11yescharacteristic
354C22×C42−1+411yescharacteristic
355C22×C42−1+411yescharacteristic
356M16—22no—
357M16—22no—
358M16—22no—
359M16—22no—
360M16—22no—
361M16—22no—
362M16—22no—
363M16—22no—
364M16—22no—
365M16—22no—
366M16—22no—
367M16—22no—
368M16—22no—
369M16—22no—
370M16—48no—
371M16—41no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—48no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8C4×D414yes—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4C4×D414yes—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C23—22no—
451C23—42no—
452C23—41no—
453C23—41no—
454C23—82no—
455C23#52011yescharacteristic
456C23#53911yescharacteristic
457C23#54411yescharacteristic
458C23#54711yescharacteristic
459C23#54811yescharacteristic
460C23#56311yescharacteristic
461C23#56311yescharacteristic
462Q8—22no—
463Q8—22no—
464Q8—41no—
465Q8—41no—
466D4—22no—
467D4—22no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—41no—
473D4—41no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4C2×C4×D411yescharacteristic
492C2×C4C2×C4×D411yescharacteristic
493C2×C4C2×C4×Q811yescharacteristic
494C2×C4#51811yescharacteristic
495C2×C4#52011yescharacteristic
496C2×C4#52011yescharacteristic
497C2×C4#53411yescharacteristic
498C2×C4#53511yescharacteristic
499C2×C4#53711yescharacteristic
500C2×C4#53711yescharacteristic
501C2×C4#53811yescharacteristic
502C2×C4#53811yescharacteristic
503C2×C4#53911yescharacteristic
504C2×C4#54011yescharacteristic
505C2×C4#54211yescharacteristic
506C2×C4#54311yescharacteristic
507C2×C4D4211yescharacteristic
508C2×C4#54611yescharacteristic
509C2×C4#54611yescharacteristic
510C2×C4D4×Q811yescharacteristic
511C2×C4D4×Q811yescharacteristic
512C2×C4#55211yescharacteristic
513C2×C4#55211yescharacteristic
514C2×C4#55311yescharacteristic
515C2×C4#55411yescharacteristic
516C2×C4#55411yescharacteristic
517C2×C4#56011yescharacteristic
518C2×C4#56311yescharacteristic
519C8—24no—
520C8—42no—
521C8—42no—
522C8—42no—
523C8—42no—
524C8—42no—
525C8—42no—
526C8—42no—
527C8—42no—
528C8—81no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—42no—
537C22—82no—
538C22—82no—
539C22—161no—
540C22#232911yescharacteristic
541C22#236111yescharacteristic
542C22#238311yescharacteristic
543C22#269511yescharacteristic
544C22#270711yescharacteristic
545C22#272011yescharacteristic
546C22#272911yescharacteristic
547C4—44no—
548C4—42no—
549C4—82no—
550C4—82no—
551C4—161no—
552C4#232411yescharacteristic
553C4#232511yescharacteristic
554C4#236011yescharacteristic
555C4#236011yescharacteristic
556C4#267811yescharacteristic
557C4#268311yescharacteristic
558C4#276511yescharacteristic
559C4#278011yescharacteristic
560C2—21no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—21no—
565C2—21no—
566C2—82no—
567C2#1454911yescharacteristic
568C2#2304312yes—
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2361
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. #21178C2semidirect
3. #21042C2semidirect
6. #11343C2semidirect
7. #11304C2semidirect
8. #11250C2semidirect
9. #11218C2semidirect
13. #10883C2semidirect
14. #10865C2semidirect
33. #2064C4semidirect
34. #2063C4semidirect

Maximal subgroups

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

subgroupindex
#236012
#211782
#210422
#208862
#208852
#113432
#113042
#112502
#112182
#111462
#109562
#109302
#108832
#108652
#107532